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https://mathoverflow.net/users/13625/sylvain-julien?tab=answers | Sylvain JULIEN
• Member for 10 years, 10 months
• Last seen this week
• France
Hoping it is not too famous an open problem, I would suggest trying to (dis)prove that Euler's constant $\gamma$, defined as $\displaystyle{\lim_{n\to\infty}H_{n}-\log n}$ where $H_{n}$ is the $n$-th ...
I just read on Google+ that the paper will be published in 2018 in a Japanese journal whose editor-in-chief is Mochizuki himself. See https://plus.google.com/+johncbaez999/posts/DWtbKSG9BWD
In France again, IHES (Institut des Hautes Etudes Scientifiques), maybe a kind of Princeton's IAS "à la française".
Taniyama, Shimura and later Weil conjectured around 60 years ago that the L-function of an elliptic curve arises from a modular form. This conjecture was known to entail the last Fermat theorem after ...
Woodin's omega conjecture. See this pdf
Fermat's conjecture that all numbers of the form $F_{n} : =2^{2^{n}}+1$ are prime. Euler proved that $641\mid F_{5}$ .
What matters here is not how to help him/her to fulfill his/her potential, but how to help him/her be happy in life. Being different is not easy to deal with, and often leads to loneliness or social ...
Marek Wolf, a Polish physicist, is the author of several articles about the distribution of prime numbers. He studied jumping champions and provided a heuristic formula refining Cramer's conjecture, ...
Maybe not exactly what you're looking for, but in his book "Merveilleux nombres premiers" (in French), Jean-Paul Delahaye mentions the so-called Mills' constant $A=1.30637788386...$ which fulfills, ...
Please notice a few differences between French and English. "Un nombre positif" is a non negative number, "supérieur à" is "greater than or equal to". In English 0 is not a natural number, while in ...
Only a partial answer as it is too long for a mere comment. There are known connections between non vanishing of L-functions and some versions of the Goldbach conjecture. Gautami Bhowmik showed that ...
A good principle would be to apply this piece of advice from Hermann Weyl : to understand well a mathematical object, determine and study the structure of its group of automorphisms.
In the French edition, it is said that the considered constant does not exceed $4(1+9\lambda_{1}+\lambda_{1}\lambda_{2}/(2-\lambda_{2})^2)$ where $\lambda_{1}>0$, $0\leq \lambda_{2}<2$ are such ...
This recent preprint may be of interest for you, as the authors first consider L-functions and then find back the algebraic variety they come from.
In number theory, the notation $\log$ is commonly used, especially when asymptotics are considered. One also frequently uses the notation $\log_{k}$ for the $k$ -th iterate of this function. ...
This is related to the fact that the circle has constant curvature. Indeed the curvature of a curve obtained plotting $y=f(x)$ is $\mathcal{C}_{f}(x)=f''(x)(1+(f'(x))^{2})^{-3/2}$, which comes ...
I'd like to have on a usb key a user friendly software that could parse a math article to check the proofs in it without having to learn how to use stuff like Coq and highlight the possible gaps. But ...
Definitely not a pure interplay between two subfields of math, but the omnipresence of quaternions in physics is stunning. I even figured out a few years ago that writing down the equivalent of Cauchy-...
I finally managed to find back the article I was talking about. Just click on the green link in the first message of the following link: link text
Characterizing a class of integers sharing some property $P$ by defining an arithmetic function taking a single value $k_{P}$ at those integers and then give an equivalent of this arithmetic function. ...
Any $n$ which is of the form $4P$ where $P$ is a perfect number is good: just consider as a set $\{a_{i}\}$ the set of its divisors. As the sum of the reciprocals thereof equals $2$, this guarantees ...
I'm not really sure this is a suitable answer to your question, but I'd like to have my recently published novel "Sahelios", in which a Japanese highschool student named Satori (Japanese female given ...
Only a partial answer for now, as it is too long for a comment. From my comment and Carlo Beenakker's answer, it suffices to consider the case where $n$ and $m$ have different radicals but the same ...
Maybe not exactly what you're looking for, but you may be interested in this preprint by Kaczorowski and Perelli: arXiv:1506.07630 where the authors study the links between several kinds of ...
The integral does not converge. See https://academic.oup.com/blms/article-abstract/31/4/424/277640?redirectedFrom=fulltext. On the other hand, a proof that this integral is less than $K.t^{\alpha}$ ...
See https://www.researchgate.net/publication/321187136_Pair_Correlation_of_Zeros_of_the_Real_and_Imaginary_Parts_of_the_Riemann_Zeta-Function where the authors investigate the behavior of the real ...
Grothendieck wrote a text entitled "Structure de la psyché" that might help shed a light on the issues you consider. As a former member of Bourbaki, it seems highly plausible that he tried doing so to ...
The French publisher Assimil is very good (and famous) to get conversational skills in many foreign languages. As for the mathematical words, you can use Wikipedia, that's how I got to know that the ...
RH holds if and only if the group of isometries of the complex plane that preserve globally the multiset of non-trivial zeroes of the Riemann Zeta function is isomorphic to $Z/2Z$ (otherwise, it would ... | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.840569257736206, "perplexity": 465.08967157665523}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320303884.44/warc/CC-MAIN-20220122194730-20220122224730-00619.warc.gz"} |
https://kr.mathworks.com/help/econ/fit-multiplicative-seasonal-model-to-airline-passenger-data.html | # Estimate Multiplicative ARIMA Model
This example shows how to estimate a multiplicative seasonal ARIMA model using `estimate`. The time series is monthly international airline passenger numbers from 1949 to 1960.
### Load Data and Specify Model.
```load Data_Airline y = log(Data); T = length(y); Mdl = arima('Constant',0,'D',1,'Seasonality',12,... 'MALags',1,'SMALags',12);```
### Estimate Model.
Use the first 13 observations as presample data, and the remaining 131 observations for estimation.
```y0 = y(1:13); [EstMdl,EstParamCov] = estimate(Mdl,y(14:end),'Y0',y0)```
``` ARIMA(0,1,1) Model Seasonally Integrated with Seasonal MA(12) (Gaussian Distribution): Value StandardError TStatistic PValue _________ _____________ __________ __________ Constant 0 0 NaN NaN MA{1} -0.37716 0.073426 -5.1366 2.7972e-07 SMA{12} -0.57238 0.093933 -6.0935 1.1047e-09 Variance 0.0013887 0.00015242 9.1115 8.1249e-20 ```
```EstMdl = arima with properties: Description: "ARIMA(0,1,1) Model Seasonally Integrated with Seasonal MA(12) (Gaussian Distribution)" Distribution: Name = "Gaussian" P: 13 D: 1 Q: 13 Constant: 0 AR: {} SAR: {} MA: {-0.377161} at lag [1] SMA: {-0.572379} at lag [12] Seasonality: 12 Beta: [1×0] Variance: 0.00138874 ```
```EstParamCov = 4×4 0 0 0 0 0 0.0054 -0.0015 -0.0000 0 -0.0015 0.0088 0.0000 0 -0.0000 0.0000 0.0000 ```
The fitted model is
`$\Delta {\Delta }_{12}{y}_{t}=\left(1-0.38L\right)\left(1-0.57{L}^{12}\right){\epsilon }_{t},$`
with innovation variance 0.0014.
Notice that the model constant is not estimated, but remains fixed at zero. There is no corresponding standard error or t statistic for the constant term. The row (and column) in the variance-covariance matrix corresponding to the constant term has all zeros.
### Infer Residuals.
Infer the residuals from the fitted model.
```res = infer(EstMdl,y(14:end),'Y0',y0); figure plot(14:T,res) xlim([0,T]) title('Residuals') axis tight```
When you use the first 13 observations as presample data, residuals are available from time 14 onward.
References:
Box, G. E. P., G. M. Jenkins, and G. C. Reinsel. Time Series Analysis: Forecasting and Control. 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1994. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 1, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9689241647720337, "perplexity": 1912.5613489057619}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103205617.12/warc/CC-MAIN-20220626101442-20220626131442-00256.warc.gz"} |
https://www.physicsforums.com/threads/would-gravity-waves-show-doppler-shift.244524/ | # Would Gravity Waves Show Doppler Shift?
1. Jul 11, 2008
### peter0302
Random question that I thought of while trying to fall asleep last night. :)
We all know that galaxies moving very fast away from us exhibit visual redshift.
If gravity waves / gravitons are real, they must then have a frequency of some kind. Does the frequency of a gravity wave affect its strength? (i.e. does a more massive object emit higher frequency gravity waves?) If so, would an object moving away from us be "red shifted" to have a weaker gravitational effect than if it were the same distance away, but moving towad us?
2. Jul 11, 2008
### Redbelly98
Staff Emeritus
The frequency of the wave indicates the rotation rate of the object emitting it.
And yes, they would exhibit a Doppler shift. Doppler shift happens for any periodic emission with a finite propagation speed. Light, sound, and gravity waves all have this property.
3. Jul 11, 2008
### peter0302
Ah, I see. So the frequency of the gravity wave would not affect its strength?
I was wondering if gravity waves would behave like photons, i.e., the frequency - not amplitude - would determine the force it exerted on a particle.
4. Jul 11, 2008
### Redbelly98
Staff Emeritus
Frequency would determine the energy of individual gravitons.
However, in practice the field of a gravitational wave would contain many gravitons. The amplitude of the field would determine the force exerted on a mass.
Similarly, it is the electric field amplitude that determines the force that a laser beam exerts on, say, an electron.
5. Jul 11, 2008
### Antenna Guy
By "rotation", do you mean about an axis through the object's center of gravity, something akin to an orbit, or both?
Regards,
Bill
6. Jul 11, 2008
### MeJennifer
I think in GR it is incorrect to think that gravitational waves are emitted by objects. Gravitational waves exist due to the changing relationships between multiple objects.
7. Jul 11, 2008
### Antenna Guy
I think that is sufficiently vague to cover what I was alluding to.
Regards,
Bill
8. Jul 11, 2008
### Redbelly98
Staff Emeritus
Antenna and MeJen,
I should probably have used the word "system" rather than object. At any rate, I was making a general comment to answer the op's question about the meaning of frequency.
As to the source of gravity waves that people are trying to detect, I'll defer to somebody more knowledgeable than I:
https://www.physicsforums.com/showpost.php?p=1696992&postcount=5
Regards,
Mark
9. Jul 11, 2008
### peter0302
Well, what I (the op was getting at was would an object moving very fast either toward or away from us exert a different instantaneous force than if it were stationary, due to doppler shift of the gravity waves?
If I understand Redbelly correctly, the answer is no.
10. Jul 11, 2008
### Antenna Guy
The "mountain range" on a spinning object is a pretty good example of what I was thinking of when I said "akin to an orbit". I might even wager that such an oscillation would yield the highest frequency waves.
Regards,
Bill
11. Jul 11, 2008
### Antenna Guy
I think E=hf still holds (which would imply the opposite), but I'll defer to someone more knowledgeable than myself.
Regards,
Bill
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http://mathoverflow.net/users/18030/anon?tab=activity | # anon
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Sep23 awarded Yearling Jun21 awarded Nice Answer Apr22 comment Analytic isomorphisms above two etale maps For proofs, see: Grauert and Remmert, Math. Ann. (1958), 245--318; SGA 4, XI.4.3; and SGA 1, XII. If you are willing to use resolution of singularities, the proof is not too hard (see SGA 1). Mar25 comment Any local algebraic group is birationally equivalent to an algebraic group It's not a textbook, but the theorem is proved in the setting of schemes in an expose of Artin in SGA 3. Artin also gives a very sketchy proof of the theorem in his first article in Arithmetic Geometry 1986 (Cornell/Silverman) --- I think I would look there. Mar23 comment Derived Functors Versus Spectral Sequences "This is usually proved using the Leray spectral sequence" I hope not (Example 1). The hypotheses imply that $f_*$ takes an injective resolution of $F$ to an acyclic resolution of $f_*F$, which can be used to compute its cohomology. This gives the result. Mar18 comment Why do twists of an algebraic group over k correspond to k-torsors over G Look at Hom from G to the twist (as a functor). This is a G-torsor. Mar10 comment Soft(?) algebraic groups question Well first you need to do it over Q. If H and G are semisimple and G is split over Q, you can take the split form of H. If G is not split, it's a twist of the split form, so you need the cocycle to come from one on H. Looks dubious to me. Mar7 comment Decomposition of semisimple Lie group into almost simple factors A semisimple Lie group is a covering of a semisimple algebraic group .... Alternatively, use that its Lie algebra is a product of simple Lie algebras. Feb18 comment Lie algebras with abelian Cartan subalgebras. Even simpler, an abelian Lie algebra is a Cartan subalgebra of itself. Feb15 comment Role of fiber functor monoidal structure in Tannakian bialgebra reconstruction My guess is the answer is no. Specifically, I'd guess that there exist really different fibre functors that become isomorphic when you forget their monoidal structures. For example, two fibre functors send an object to vector spaces of the same dimension, so they become equal on objects when you replace the category of vector spaces with its skeleton. Perhaps this can be pushed further to show that sometimes (often? always?) two fibre functors will become isomorphic when you forget their monoidal structures. Jan19 comment Should I write to the referee? It shouldn't "restart the review procedure" --- the editor will just send the new version of the article to his current referee. Jan13 comment The historical development of automorphic geometry "Langlands did quite a good job of suggesting that the Jugendtraum was some sort of wrong turning." Where did he do that? Jan5 revised Permission to use Online Notes added 351 characters in body Jan5 answered Permission to use Online Notes Jan5 awarded Commentator Jan5 comment finite non-commutative local group schemes Over a field of characteristic $p$, there is an obvious action of $\mathbb{G}_m$ on $\alpha_p$, and hence an action of $\mu_p$ on $\alpha_p$. The semi-direct product is a noncommutative connected group scheme of order $p^2$. Jan3 comment Topological examples of profinite groups "I would like to exclude Galois groups". Actually, all profinite groups are Galois groups, so you may be in trouble. More seriously, I agree with KConrad: for most us of a profinite group is a projective limit of finite groups, so it's better to start with that as the definition. Dec20 comment understanding Milne's article “Duality in the flat cohomology of a surface” Have you tried looking at the exposition of the theorem in: Berthelot, P. Le théorème de dualité plate pour les surfaces (d'après J. S. Milne). (French) [The flat duality theorem for surfaces (according to J. S. Milne)] Algebraic surfaces (Orsay, 1976--78), pp. 203--237, Lecture Notes in Math., 868, Springer, Berlin-New York, 1981. MR0638601? Dec16 comment Why is the definition of l-adic sheaves so complicated? With the naive definition, $H^{1}(X,\mathbb{Z}_{\ell))=\Hom_{\text{continuous}}(\pi_{1}(X),\mathbb{Z}_{\ell})$ with the discrete topology on $\mathbb{Z}_{\ell}$. This is generally zero, because (for nice schemes) $\pi_{1}(X)$ is profinite. With the nonnaive definition, it is Hom with the natural $\ell$-adic topology on $\mathbb{Z}_{\ell}$, which is what you want. Nov22 comment Constructive proof of algebraic elements forming a subfield Take the product of all polynomials $X-a'b'$ where $a'$ and $b'$ range over the conjugates of a and b, and use the symmetric function theorem to show that its coefficient lie in $E$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8299926519393921, "perplexity": 618.4849170187418}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1409535922871.14/warc/CC-MAIN-20140901014522-00167-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/determine-the-magnitude-of-the-minimum-acceleration.711570/ | # Determine the magnitude of the minimum acceleration
• #1
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## Homework Statement
A 76-kg petty thief wants to escape from a third-story jail window. Unfortunately, a makeshift rope made of sheets tied together can support a mass of only 58 kg.Determine the magnitude of the minimum acceleration at which the thief can descend using the rope.
i think T-mg=ma?
## The Attempt at a Solution
i know tension would be the upward force and mg (weight) would be the upward force correct?
• #2
haruspex
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The signs are going to depend on how you define the positive direction. Your equation is fine if the aceleration a is defined upward (so expect a negative result).
• #3
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0
The signs are going to depend on how you define the positive direction. Your equation is fine if the aceleration a is defined upward (so expect a negative result).
would i use the mass of the man or the mass that the sheet can with stand?
• #4
haruspex
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would i use the mass of the man or the mass that the sheet can with stand?
The question is not quite right. It should say that the sheets can withstand a weight of 58g N. Mass is not force.
• #5
arildno
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What would the tension be if a man of 58 kg chose to rest from it?
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https://www.aimsciences.org/article/doi/10.3934/jcd.2019005?viewType=html | # American Institute of Mathematical Sciences
June 2019, 6(1): 111-130. doi: 10.3934/jcd.2019005
## Symplectic integration of PDEs using Clebsch variables
1 School of Fundamental Sciences, Massey University, Private Bag 11 222, Palmerston North, 4442, New Zealand 2 Department of Mathematical Sciences, Norwegian University of Science and Technology, Sentralbygg 2, Gløshaugen, Norway
Published July 2019
Fund Project: This research was supported by the Marsden Fund of the Royal Society Te Apārangi.
Many PDEs (Burgers' equation, KdV, Camassa-Holm, Euler's fluid equations, …) can be formulated as infinite-dimensional Lie-Poisson systems. These are Hamiltonian systems on manifolds equipped with Poisson brackets. The Poisson structure is connected to conservation properties and other geometric features of solutions to the PDE and, therefore, of great interest for numerical integration. For the example of Burgers' equations and related PDEs we use Clebsch variables to lift the original system to a collective Hamiltonian system on a symplectic manifold whose structure is related to the original Lie-Poisson structure. On the collective Hamiltonian system a symplectic integrator can be applied. Our numerical examples show excellent conservation properties and indicate that the disadvantage of an increased phase-space dimension can be outweighed by the advantage of symplectic integration.
Citation: Robert I McLachlan, Christian Offen, Benjamin K Tapley. Symplectic integration of PDEs using Clebsch variables. Journal of Computational Dynamics, 2019, 6 (1) : 111-130. doi: 10.3934/jcd.2019005
##### References:
show all references
##### References:
Uniform periodic grids on $S^1 \cong \mathbb{R}/L\mathbb{Z}$, $L>0$
Order-two convergence for the travelling wave solution of the extended Burgers' equation outlined in section 6.2. The plots correspond to the conventional solution (○) and the collective solution (△) and an order-two reference line (). The error is calculated after 512 timesteps, with $L = 8$, $\Delta t = 2^{-14}$ and $\Delta x = L/2^{k}$ for $k = 1,2,3$ and $4$
Inviscid Burgers' equation solutions of the conventional method () and collective method (). The grid parameters are $n_x = 64$, $\Delta x = 0.125$, $L = 8$ and $\Delta t = 2^{-12}$. A shock forms at about $t = 0.4$
The errors corresponding to the conventional () and collective () methods for the inviscid Burgers' equation and $\mathcal{O}(t^2)$ reference lines ()
Travelling wave solutions of the perturbed Burgers' equation (top row) and the positive Fourier modes (bottom row) at $t = 109$ (left column), $t = 218$ (middle column) and $t = 437$ (right column). The plots correspond to the conventional method (), collective method () and the exact travelling wave solution (). The grid parameters are $n_x = 16$, $\Delta x = 0.5$, $L = 8$ and $\Delta t = 2^{-6}$
The errors corresponding to the conventional () and collective () methods for the travelling wave experiment. The reference lines () are $\mathcal{O}(t)$ in figures (a) and (b) and exponential in figure (c)
Periodic bump solutions of the extended Burgers' equation (top row) and the positive Fourier modes (bottom row) at $t = 10$ (left column), $t = 100$ (middle column) and $t = 1000$ (right column). The plots correspond to the conventional method () and the collective method (). The grid parameters are $n_x = 32$, $\Delta x = 0.25$, $L = 8$ and $\Delta t = 2^{-8}$
The errors corresponding to the conventional () and collective () methods for the periodic bump example. The reference line () in figure (a) is $\mathcal{O}(t)$
Overview of the setting
Continuous system Spatially discretised system Collective Hamiltonian system on an infinite-dimensional symplectic vector space in Clebsch variables $q_t = \frac{\delta \bar H}{\delta p}, \quad p_t = -\frac{\delta \bar H}{\delta q}.$ Exact solutions preserve the symplectic structure, the Hamiltonian $\bar H=H\circ J$, all quantities related to the Casimirs of the original PDE and the fibres of the Clebsch map $J(q,p)=u$. Canonical Hamiltonian ODEs in $2N$ variables $\hat q_t = \nabla_{\hat p} \hat {\bar H}, \quad \hat p_t = - \nabla_{\hat q} \hat {\bar H}.$ The exact flow preserves the symplectic structure and the Hamiltonian $\hat {\bar H}$. Time-integration with the midpoint rule is symplectic. Original PDE, interpreted as a Lie-Poisson equation $u_t = \mathrm{ad}^\ast_{\frac {\delta H}{\delta u}}u.$ Exact solutions preserve the Poisson structure, the Hamiltonian $H$ and all Casimirs. Non-Hamiltonian ODEs in $N$ variables $\hat u_t = K(\hat u) \nabla_{\hat u} \hat H, \qquad K^T=-K.$ Exact solutions conserve $\hat H$. Time-integration with the midpoint rule is not symplectic.
Continuous system Spatially discretised system Collective Hamiltonian system on an infinite-dimensional symplectic vector space in Clebsch variables $q_t = \frac{\delta \bar H}{\delta p}, \quad p_t = -\frac{\delta \bar H}{\delta q}.$ Exact solutions preserve the symplectic structure, the Hamiltonian $\bar H=H\circ J$, all quantities related to the Casimirs of the original PDE and the fibres of the Clebsch map $J(q,p)=u$. Canonical Hamiltonian ODEs in $2N$ variables $\hat q_t = \nabla_{\hat p} \hat {\bar H}, \quad \hat p_t = - \nabla_{\hat q} \hat {\bar H}.$ The exact flow preserves the symplectic structure and the Hamiltonian $\hat {\bar H}$. Time-integration with the midpoint rule is symplectic. Original PDE, interpreted as a Lie-Poisson equation $u_t = \mathrm{ad}^\ast_{\frac {\delta H}{\delta u}}u.$ Exact solutions preserve the Poisson structure, the Hamiltonian $H$ and all Casimirs. Non-Hamiltonian ODEs in $N$ variables $\hat u_t = K(\hat u) \nabla_{\hat u} \hat H, \qquad K^T=-K.$ Exact solutions conserve $\hat H$. Time-integration with the midpoint rule is not symplectic.
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Figures and Tables | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8126962184906006, "perplexity": 1864.2446868475258}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347398233.32/warc/CC-MAIN-20200528061845-20200528091845-00095.warc.gz"} |
http://mathoverflow.net/questions/122154/asymptotic-behavior-of-entire-functions | # Asymptotic behavior of entire functions
Which entire function $f\left(x\right)$ goes asymptotically to $\dfrac{e^{-x}}{x}$ as $x$ goes to infinity with $x$ positive? That is, $\left(e^{-x}/x \right)/f \left(x \right) \rightarrow 1$.
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Plenty of them. $(e^x-e^{-2x})/x$ is the simplest example. As a matter of fact, every continuous function $g(x)$ on $\mathbb R$ can be approximated by an entire function with arbitrary continuous precision $\varepsilon(x)>0$. Voting to close. – fedja Feb 18 '13 at 12:39
You cannot say a lot about $f$ without additional assumption on the uniform growth at infinity. For instance if such a $f$ exists then any other entire function $f+g$, where $g$ is a tower of exponential of length greater than $1$ $g(x)=\exp(-\exp(\exp(\ldots(\exp x)\ldots)))$, will also satisfy your assumption. – Loïc Teyssier Feb 18 '13 at 12:40
Dear fedja, Do you mean $\left(e^{-x} - e^{-2x} \right)/x$? Best, davwood83 – davwood83 Feb 18 '13 at 13:48
As mentioned in the comments, the asymptotic behavior of $f$ along the real axis doesn't really tell you anything about the function globally.
For example, your function $f$ can behave in any way you like as $x\to-\infty$.
Indeed, the following is implied by Arakeljan's (or Nersesjan's) approximation theorem: Let $A$ be any finite union of disjoint curves tending to infinity, and let $g:A\to\mathbb{C}$ and $\varepsilon:A\to(0,\infty)$ be continuous.
Then there is an entire function $f$ such that $|f(z)-g(z)|<\varepsilon(z)$ for every $z\in A$.
So, for example, let $A=(-\infty,0] \cup [1,\infty)$, let $g$ be the function $e^{-x}/x$ on $[1,\infty)$ and let $g$ be arbitrary on $(-\infty,0]$. Then you can find an entire function that approximates this function arbitrarily closely. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9889994263648987, "perplexity": 174.3175630362751}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802777418.140/warc/CC-MAIN-20141217075257-00165-ip-10-231-17-201.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/physical-meaning-of-directrix.336922/ | # Physical meaning of directrix?
1. Sep 13, 2009
### homology
I suddenly realized that the very notion of a directrix seems odd to me.
(1) I don't know any physical significance of such a thing
(2) How anyone came to define the parabola as such
Any help? I read some older posts about the history of the directrix/focus definitions of the conic sections and I can see the idea of unifying the conics through the ratio of distances from a point to the focus and from the point to the directrix but perhaps there is something more, something deeper, something physical?
Cheers,
Kevin
2. Sep 13, 2009
### tiny-tim
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https://math.stackexchange.com/questions/923173/andrei-flips-a-coin-over-and-over-again-until-he-gets-a-tail-followed-by-a-head | # Andrei flips a coin over and over again until he gets a tail followed by a head, then he quits. What is the expected number of coin flips?
(a) Andrei flips a coin over and over again until he gets a tail followed by a head, then he quits. What is the expected number of coin flips?
(b) Bela flips a fair coin over and over again until she gets two tails in a row, then she quits. What is the expected number of coin flips?
Hello, I have been having some trouble with this problem. I have tried to use state diagrams, but bonked my head on the table, because obviously, that wouldn't work. I have not yet been able to find any method in doing this. Any help is appreciated
• It may help to work out what some of the admissible sequences are in each case. The first one has a fairly simple characterization, and I imagine the second one as well. – Semiclassical Sep 8 '14 at 1:28
• Conway's algorithm gives $2^2$ for TH, as it does for HT, while it gives $2^1+2^2$ for TT and HH. The curiosity is that if they combine their tosses to see which comes first (i.e. Penney's game), TH and TT are equally likely to come before the other, but if they were looking for TH and HH then TH would be three times as likely to come before HH as the other way round. – Henry Sep 8 '14 at 7:14
In the first case (a) the sequence of the outcomes is something like $H^j T^k H$ with $j\geq 0$ and $k\geq 1$. Such a sequence has length $j+k+1$, hence the expected number of coin flips is given by: $$\sum_{j\geq 0}\sum_{k\geq 1}\frac{j+k+1}{2^{j+k+1}}=\sum_{h,k\geq 1}\frac{h+k}{2^{h+k}}=4.$$ In the second case (b), the sequence of outcomes is a string over $\{H,TH\}$ plus a $TT$ suffix. The number of strings of length $N$ over $\Sigma=\{H,TH\}$ is given by the $(N+1)$-th Fibonacci number $F_{N+1}$, hence the expected number of coin flips is given by: $$2+\sum_{N=0}^{+\infty}\frac{N\cdot F_{N+1}}{2^{N+2}}=6.$$
Hint for the case of TH (tails-heads).
First let $y$ be the expected number of flips until a T is obtained. You can prove that $y = (1/2)\cdot 1 + (1/2)\cdot( 1 + y)$ by considering what happens in the first flip: you have a 50% chance of needing one flip, and a 50% chance of having to start over.
Now, can you explain why the required expected value is $2y$? (Hint: Define two random variables $X_1$ and $X_2$, where $X_1$ is the number of flips you need to obtain your first T, and $X_2$ the number to obtain the first $H$ after that. There's a formula for $E(X_1 + X_2)$.)
Hint for $TT$.
You have a 25% chance of getting TT immediately. You have a 50% chance of starting with H, in which case you start over, etc.
Now do a similar calculation to the first case.
• In case b), how do you deal with the case in which we start with TH ? – Jack D'Aurizio Sep 8 '14 at 2:17
• You start over. So if $x$ is the expectation you're looking for, that counts for $2 + x$. – Dave Sep 8 '14 at 2:25
There are two states:
A: Just flipped a Head; and
B: Just flipped a Tail.
They both might as well start in State A because they need to start with a Tail.
Let $M$ be the average number of flips needed to go from State A to State B, and $N$ be the average number of flips needed to go from State B to finish.
Going from state A to B might take 1 flip, with a tail, or might stay in state A with a head, and take M+1 flips. They are equally likely, so $M=\frac12 1+\frac12(M+1)$. Solve for $M$.
Do the same thing for $N$.
Andrei's allowable sequences are $H^i T^j TH$, where $i \ge 0, j\ge 0$.
The expected length is $\sum_{i=0}^\infty \sum_{j=0}^\infty (i+j+2) {1 \over 2^i} {1 \over 2^j} {1 \over 2^2}$.
Bela's allowable sequences are $S^i TT$, where $S \in \{ H, TH\}$, where $i \ge 0$.
The expected length is $\sum_{i=0}^\infty (i+2) \sum_{k=0}^i \binom{i}{k}({1 \over 2^k} {1 \over 4^{i-k}} ){1 \over 2^2} = \sum_{i=0}^\infty (i+2) ({1 \over 2}+{1 \over 4})^i{1 \over 2^2}$.
(a) $E[X=T] = p(X=T) + 0.5 \cdot (E[X=T] + 1) = 0.5 + 0.5 \cdot E[X=T] + 0.5 <=> E[X=T] = 2$ $E[X=TH] = E[X=T] + p(X=H) + 0.5 \cdot (E[X=T] + 1) = 2 + 0.5 + 0.5 \cdot (2 + 1) = 4$
(b) from above $E[X=T] = 2$ $E[X=TT] = E[X=T] + p(X=T) + 0.5 \cdot (E[X=TT] + 1) = 2 + 0.5 + 0.5 \cdot E[X=TT] + 0.5 <=> 0.5 \cdot E[X=TT] = 3 <=> E[X=TT] = 6$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9253180027008057, "perplexity": 213.90055100713778}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496670255.18/warc/CC-MAIN-20191119195450-20191119223450-00338.warc.gz"} |
http://math.stackexchange.com/questions/78055/decomposition-of-a-manifold/78166 | # Decomposition of a manifold
As a kind of aside to this question, where one of the answers assumed that if $S^n=X \times Y$ then we can assume that $X$ and $Y$ are manifolds.
If we have a manifold $M$, such that $M$ is homeomorphic to $X \times Y$, then must $X$ and $Y$ be manifolds? The converse ($X,Y$ manifolds implies $X \times Y$ is a manifold) is certainly true. I'd like to think it is true, but I have seen enough strange topological behaviour to suggest this may not be true.
For this question take 'manifold' to mean a second countable Hausdorff space that is locally homeomorphic to $\mathbb{R}^n$, for some finite $n$.
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+1 for questioning your intuition! – Grumpy Parsnip Nov 2 '11 at 18:41
Thanks for following up on this! My intuition told me it was false, but maybe I'm just a cynic when it comes to spaces that aren't at least CW complexes. Or does that make me an optimist? I guess it all depends on your perspective :o) – Aaron Mazel-Gee Nov 4 '11 at 5:59
Juan, @Aaron: In case you missed it: there's a new related thread on MO. – t.b. Dec 17 '11 at 3:13
Disclaimer: I'm by no means knowledgeable in this field and I haven't read the papers or books I mention below. I found these by digging in the literature and hope these pointers are useful.
The answer to your question is no.
1. The first example was given by R.H. Bing, The Cartesian Product of a Certain Nonmanifold and a Line is $E^4$, Ann. of Math. (2) 70 (1959) 399–412. MR107228.
Bing describes a topological space $B$ — in fact a quotient space of $\mathbb{R}^3$, sometimes called the Dogbone space — which is not a manifold and has the property that $B \times \mathbb{R}$ is homeomorphic to $\mathbb{R}^4$.
2. Modifying this construction and relying heavily on work of Andrews and Curtis, K.W. Kwun, Product of Euclidean Spaces Modulo an Arc, Ann. of Math. (2) 79 (1964) 104–108, MR159312, produced product decompositions of $\mathbb{R}^n \cong X \times Y$ for $n \geq 6$ where neither $X$ nor $Y$ is a manifold.
3. Here are two freely available papers by A.J. Boals:
4. Quoting C.D. Bass, Some products of topological spaces wich are manifolds, Proc. Amer. Math. Soc. 81 (1981), 641–646, MR601746, Corollary 3 on page 645 “gives abundant examples of factorizations of certain manifolds into nonmanifold factors.”
5. A textbook covering these and many more topics:
Daverman, Robert J., Decompositions of manifolds, Pure and Applied Mathematics, 124, Academic Press, 1986, MR872468. (Reprinted by the AMS, 2007).
Here are two related MO-threads:
Added: A further MO-thread was posted and answered a couple of hours ago:
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Thanks Theo! Nice detective work...I thought about searching through all of Bing's paper's. I had a feeling if it was not true he would have provided a counter example – Juan S Nov 2 '11 at 21:44
@Juan (Qwirk): you're welcome. I wanted to look up this stuff for quite a while, so now I had the opportunity... Interesting stuff, but the details seem somewhat messy. – t.b. Nov 2 '11 at 22:10
Great answer t.b.! You write "I'm by no means knowledgeable in this field" What would it be if you were:-) – Georges Elencwajg Nov 2 '11 at 23:22
Dear @Georges: Thank you, but yes, I meant it. I know little about geometric topology and the question at hand. I knew the Dogbone space and the two MO-questions, but not more. – t.b. Nov 2 '11 at 23:24 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9077019095420837, "perplexity": 776.0289462235723}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246660724.78/warc/CC-MAIN-20150417045740-00029-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://mathematics.huji.ac.il/eventss/events-seminars?page=3 | 2019 May 01
2:00pm to 3:30pm
Ross 63
2019 Mar 13
# Set Theory Seminar - Tom Benhamou (TAU), "Projections of Tree-Prikry forcing"
2:00pm to 3:30pm
## Location:
Ross 63
Title: Projections of Tree-Prikry forcing. Abstract: Gitik, Kanovei and Koepke proved that if U is a normal measure over \kappa then the projections of Prikry forcing with U is essentially Prikry forcing with U. The questions remains regarding to the Tree-Prikry forcing. Gitik and B. showed that without normality, it is possible that a Tree-Prikry generic sequence adds a Add(\kappa,1) generic function. In this talk we wish to examine which forcing notions can be projections of Tree-Prikry forcing under different large cardinals assumptions.
2019 Mar 27
# Set Theory Seminar - Ralf Schindler (Munster), "Paradoxical" sets with no well-ordering of the reals
2:00pm to 3:30pm
## Location:
Ross 63
Title: "Paradoxical" sets with no well-ordering of the reals Abstract: By a Hamel basis we mean a basis for the reals, R, construed as a vecor space over the field of rationals. In 1905, G. Hamel constructed such a basis from a well-ordering of R. In 1975, D. Pincus and K. Prikry asked "whether a Hamel basis exists in any model in which R cannot be well ordered." About two years ago, we answered this positively in a joint paper with M. Beriashvili, L. Wu, and L. Yu. In more recent joint work, additionally with J. Brendle and F. Castiblanco we constructed a model of
2019 Mar 20
# Set Theory Seminar - Tom Benhamou (TAU) (part II)
2:00pm to 3:30pm
## Location:
Ross 63
Title: Projections of Tree-Prikry forcing. Abstract: Gitik, Kanovei and Koepke proved that if U is a normal measure over \kappa then the projections of Prikry forcing with U is essentially Prikry forcing with U. The questions remains regarding to the Tree-Prikry forcing. Gitik and B. showed that without normality, it is possible that a Tree-Prikry generic sequence adds a Add(\kappa,1) generic function. In this talk we wish to examine which forcing notions can be projections of Tree-Prikry forcing under different large cardinals assumptions.
2019 Mar 19
# Dynamics lunch: Jing Zhou "“escaping orbit of some piecewise smooth Fermi acceleration model”
12:00pm to 1:00pm
## Location:
Manchester faculty club
Following "Dynamics of some piecewise smooth Fermi-Ulam Models” by De Simoi and Dolgopyat.
2019 Mar 12
# Dynamics Seminar: Terry Soo (KU) Finitary isomorphism of Bernoulli flows
2:15pm to 3:15pm
## Location:
Ross 70
A powerful theory due to Ornstein and his collaborators has been successfully applied to many random systems to show that they are isomorphic to independent and identically distributed systems. The isomorphisms provided by Ornstein's theory may not be finitary, that is, effectively realizable in practice. Despite the large number of systems known to be Bernoulli, there are only a handful of cases where explicit finitary isomorphisms have been constructed. In this talk, we will discuss classical and recent constructions, and some long standing open problems.
2019 Apr 10
2:30pm to 3:30pm
Sprinzak 24
2019 Jun 19
# Analysis Seminar: Daniel Ofner
12:00pm to 1:00pm
Ross 70
2019 May 22
# Analysis seminar: Yoel Grinshpon
12:00pm to 1:00pm
2019 Mar 13
# Analysis Seminar: Yehuda Pinchover (Technion) "How large can Hardy-weight be?"
12:00pm to 1:00pm
## Location:
Ross 70
Title: How large can Hardy-weight be? Abstract: In the first part of the talk we will discuss the existence of optimal Hardy-type inequalities with 'as large as possible' Hardy-weight for a general second-order elliptic operator defined on a noncompact Riemannian manifold, while the second part of the talk will be devoted to a sharp answer to the question: "How large can Hardy-weight be?"
2019 Feb 07
# Special group actions seminar. On Tame Subgroups of Finitely Presented Groups: Prof. Rita Gitik, University of Michigan, Ann Arbor.
10:00am to 11:00am
## Location:
Ross 70
We describe several examples of tame subgroups of finitely presented groups and prove that the fundamental groups of certain finite graphs of groups are locally tame.
2019 Jan 23
# T&G: Sylvain Cappell (NYU), Atiyah-Bott classes and extending representations of fundamental groups of 3-manifolds from part of the boundary
1:00pm to 2:00pm
## Location:
Room 70, Ross Building, Jerusalem, Israel
We consider the problem of extending a representation of the fundamental group of 3-manifolds from part of the boundary surfaces. Applications to links will be discussed. Combining this with some cohomology classes of Atiyah and Bott leads to new multivariable polynomial invariants of 3-manifolds with boundary. This is joint work with Edward Miller. No background in 3-dimensional topology will be assumed in this survey and research talk.
2019 Mar 19
# Dynamics Seminar: Elon Lindenstrauss (HUJI) - Double variational principle for mean dimension
2:15pm to 3:15pm
Mean dimension is a topological invariant of dynamical systems introduced by Gromov that measures the number of parameters per iteration needed to describe a trajectory in the system. We characterize this invariant (at least for dynamical systems with the marker property, such as infinite minimal systems) using a min-max principle, where choices of both a metric on the topological space and an invariant probability measure on the system are varied. The work I will report on is joint work with M. Tsukamoto.
2019 Jan 15
# T&G: Michael Khanevsky (Technion), Geometry of sets of Hamiltonian isotopic curves in a symplectic surface
2:00pm to 3:30pm
## Location:
Room 209, Manchester Building, Jerusalem, Israel
Given two Hamiltonian isotopic curves in a surface, one would like to tell whether they are "close" or "far apart". A natural way to do that is to consider Hofer's metric which computes mechanical energy needed to deform one curve into the other. However due to lack of tools the large-scale Hofer geometry is only partially understood. On some surfaces (e.g. S^2) literally nothing is known.
2019 May 15
# Analysis Seminar: Matthias Keller (Potsdam)
12:00pm to 1:00pm
Ross 70 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8273458480834961, "perplexity": 2708.7973486253904}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912204885.27/warc/CC-MAIN-20190326075019-20190326101019-00429.warc.gz"} |
https://www.physicsforums.com/threads/a-wagon-wheel-and-inertia.50057/ | # A wagon wheel and inertia
1. Oct 27, 2004
### envscigrl
A wagon wheel 1.00m in diameter consists of a thin rim having a mass of 7.25kg and six spokes each having a mass of 1.30kg. Determine the moment of inertia of the wagon wheel for rotation about its axis.
I thought that I could simply sum the masses (7 of them) and the multiply them by them by the radius squred. Didnt work!
I am having some problems with this chaper on rotational dynamics and torque. If anyone knows of any really good books or websites that could be helpful please let me know. Thanks!
2. Oct 27, 2004
### Tide
You cannot simply multiply the masses by the square of the radius to find their moment of inertia. Calculate (by integration) or look up the moment of inertia of a thin rod then apply the parallel axis theorem for moments. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9510002732276917, "perplexity": 569.2249934914206}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257647567.36/warc/CC-MAIN-20180321023951-20180321043951-00065.warc.gz"} |
http://proceedings.mlr.press/v28/lattimore13.html | # The Sample-Complexity of General Reinforcement Learning
Tor Lattimore, Marcus Hutter, Peter Sunehag ;
Proceedings of the 30th International Conference on Machine Learning, PMLR 28(3):28-36, 2013.
#### Abstract
We study the sample-complexity of reinforcement learning in a general setting without assuming ergodicity or finiteness of the environment. Instead, we define a topology on the space of environments and show that if an environment class is compact with respect to this topology then finite sample-complexity bounds are possible and give an algorithm achieving these bounds. We also show the existence of environment classes that are non-compact where finite sample-complexity bounds are not achievable. A lower bound is presented that matches the upper bound except for logarithmic factors. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9964617490768433, "perplexity": 877.980736984756}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128320226.61/warc/CC-MAIN-20170624050312-20170624070312-00440.warc.gz"} |
https://pressbooks.bccampus.ca/collegephysics/chapter/work-the-scientific-definition/ | Chapter 7 Work, Energy, and Energy Resources
# 42 7.1 Work: The Scientific Definition
### Summary
• Explain how an object must be displaced for a force on it to do work.
• Explain how relative directions of force and displacement determine whether the work done is positive, negative, or zero.
# What It Means to Do Work
The scientific definition of work differs in some ways from its everyday meaning. Certain things we think of as hard work, such as writing an exam or carrying a heavy load on level ground, are not work as defined by a scientist. The scientific definition of work reveals its relationship to energy—whenever work is done, energy is transferred.
For work, in the scientific sense, to be done, a force must be exerted and there must be displacement in the direction of the force.
Formally, the work done on a system by a constant force is defined to be the product of the component of the force in the direction of motion times the distance through which the force acts. For one-way motion in one dimension, this is expressed in equation form as
$\boldsymbol{W=|{F}|(\textbf{cos}\theta)|\textbf{d}|},$
where$\boldsymbol{W}$is work,$\textbf{d}$is the displacement of the system, and$\boldsymbol{\theta}$is the angle between the force vector$\boldsymbol{F}$and the displacement vector$\textbf{d},$as in Figure 1. We can also write this as
$\boldsymbol{W=Fd\textbf{cos}\theta}.$
To find the work done on a system that undergoes motion that is not one-way or that is in two or three dimensions, we divide the motion into one-way one-dimensional segments and add up the work done over each segment.
### WHAT IS WORK?
The work done on a system by a constant force is the product of the component of the force in the direction of motion times the distance through which the force acts. For one-way motion in one dimension, this is expressed in equation form as
$\boldsymbol{W=Fd\:\textbf{cos}\theta},$
where$\boldsymbol{W}$is work,$\boldsymbol{F}$is the magnitude of the force on the system,$\boldsymbol{d}$is the magnitude of the displacement of the system, and$\boldsymbol{\theta}$is the angle between the force vector$\boldsymbol{F}$and the displacement vector$\boldsymbol{d}.$
To examine what the definition of work means, let us consider the other situations shown in Figure 1. The person holding the briefcase in Figure 1(b) does no work, for example. Here$\boldsymbol{d=0},$so$\boldsymbol{W=0}.$Why is it you get tired just holding a load? The answer is that your muscles are doing work against one another, but they are doing no work on the system of interest (the “briefcase-Earth system”—see Chapter 7.3 Gravitational Potential Energy for more details). There must be displacement for work to be done, and there must be a component of the force in the direction of the motion. For example, the person carrying the briefcase on level ground in Figure 1(c) does no work on it, because the force is perpendicular to the motion. That is,$\boldsymbol{\textbf{cos}\:90^0 =0},$and so$\boldsymbol{W=0}.$
In contrast, when a force exerted on the system has a component in the direction of motion, such as in Figure 1(d), work is done—energy is transferred to the briefcase. Finally, in Figure 1(e), energy is transferred from the briefcase to a generator. There are two good ways to interpret this energy transfer. One interpretation is that the briefcase’s weight does work on the generator, giving it energy. The other interpretation is that the generator does negative work on the briefcase, thus removing energy from it. The drawing shows the latter, with the force from the generator upward on the briefcase, and the displacement downward. This makes$\boldsymbol{\theta=180^0},$and$\boldsymbol{\textbf{cos}\:180^0=-1};$therefore,$\boldsymbol{W}$is negative.
# Calculating Work
Work and energy have the same units. From the definition of work, we see that those units are force times distance. Thus, in SI units, work and energy are measured in newton-meters. A newton-meter is given the special name joule (J), and$\boldsymbol{1\textbf{ J}=1\textbf{ N}\cdotp\textbf{ m}=1\textbf{ kg}\cdotp\textbf{m}^2/\textbf{s}^2}.$One joule is not a large amount of energy; it would lift a small 100-gram apple a distance of about 1 meter.
### Example 1: Calculating the Work You Do to Push a Lawn Mower Across a Large Lawn
How much work is done on the lawn mower by the person in Figure 1(a) if he exerts a constant force of$\boldsymbol{75.0\textbf{ N}}$at an angle$\boldsymbol{35^0}$below the horizontal and pushes the mower$\boldsymbol{25.0\textbf{ m}}$on level ground? Convert the amount of work from joules to kilocalories and compare it with this person’s average daily intake of$\boldsymbol{10,000\textbf{ kJ}}$(about$\boldsymbol{2400\textbf{ kcal}}$) of food energy. One calorie (1 cal) of heat is the amount required to warm 1 g of water by$\boldsymbol{1^0C},$and is equivalent to$\boldsymbol{4.184\textbf{ J}},$while one food calorie (1 kcal) is equivalent to$\boldsymbol{4184\textbf{ J}}.$
Strategy
We can solve this problem by substituting the given values into the definition of work done on a system, stated in the equation$\boldsymbol{W=Fd\:\textbf{cos}\theta}.$The force, angle, and displacement are given, so that only the work$\boldsymbol{W}$is unknown.
Solution
The equation for the work is
$\boldsymbol{W=Fd\:\textbf{cos}\theta}.$
Substituting the known values gives
$\begin{array}{lcl} \boldsymbol{W} & = & \boldsymbol{(75.0\textbf{ N})(25.0\textbf{ m}) \;\textbf{cos}\; (35.0^0)} \\ {} & = & \boldsymbol{1536\textbf{ J}=1.54\times10^3\textbf{ J}} \end{array}.$
Converting the work in joules to kilocalories yields$\boldsymbol{W=(1536\textbf{ J})(1\textbf{ kcal}/4184\textbf{ J})=0.367\textbf{ kcal}}.$The ratio of the work done to the daily consumption is
$\boldsymbol{\frac{W}{2400\textbf{ kcal}}}$$\boldsymbol{=\:1.53\times10^{-4}}.$
Discussion
This ratio is a tiny fraction of what the person consumes, but it is typical. Very little of the energy released in the consumption of food is used to do work. Even when we “work” all day long, less than 10% of our food energy intake is used to do work and more than 90% is converted to thermal energy or stored as chemical energy in fat.
# Section Summary
• Work is the transfer of energy by a force acting on an object as it is displaced.
• The work$\boldsymbol{W}$that a force$\boldsymbol{F}$does on an object is the product of the magnitude$\boldsymbol{F}$of the force, times the magnitude$\boldsymbol{d}$of the displacement, times the cosine of the angle$\boldsymbol{\theta}$between them. In symbols,
$\boldsymbol{W=Fd\:\textbf{cos}\theta}.$
• The SI unit for work and energy is the joule (J), where$\boldsymbol{1\textbf{ J}=1\textbf{ N}\cdotp\textbf{ m}=1\textbf{ kg}\cdotp\textbf{ m}^2/\textbf{s }^2}.$
• The work done by a force is zero if the displacement is either zero or perpendicular to the force.
• The work done is positive if the force and displacement have the same direction, and negative if they have opposite direction.
### Conceptual Questions
1: Give an example of something we think of as work in everyday circumstances that is not work in the scientific sense. Is energy transferred or changed in form in your example? If so, explain how this is accomplished without doing work.
2: Give an example of a situation in which there is a force and a displacement, but the force does no work. Explain why it does no work.
3: Describe a situation in which a force is exerted for a long time but does no work. Explain.
### Problems & Exercises
1: How much work does a supermarket checkout attendant do on a can of soup he pushes 0.600 m horizontally with a force of 5.00 N? Express your answer in joules and kilocalories.
2: A 75.0-kg person climbs stairs, gaining 2.50 meters in height. Find the work done to accomplish this task.
3: (a) Calculate the work done on a 1500-kg elevator car by its cable to lift it 40.0 m at constant speed, assuming friction averages 100 N. (b) What is the work done on the lift by the gravitational force in this process? (c) What is the total work done on the lift?
4: Suppose a car travels 108 km at a speed of 30.0 m/s, and uses 2.0 gal of gasoline. Only 30% of the gasoline goes into useful work by the force that keeps the car moving at constant speed despite friction. (See Chapter 7.6 Table 1 for the energy content of gasoline.) (a) What is the magnitude of the force exerted to keep the car moving at constant speed? (b) If the required force is directly proportional to speed, how many gallons will be used to drive 108 km at a speed of 28.0 m/s?
5: Calculate the work done by an 85.0-kg man who pushes a crate 4.00 m up along a ramp that makes an angle of$\boldsymbol{20.0^0}$with the horizontal. (See Figure 2.) He exerts a force of 500 N on the crate parallel to the ramp and moves at a constant speed. Be certain to include the work he does on the crate and on his body to get up the ramp.
6: How much work is done by the boy pulling his sister 30.0 m in a wagon as shown in Figure 3? Assume no friction acts on the wagon.
7: A shopper pushes a grocery cart 20.0 m at constant speed on level ground, against a 35.0 N frictional force. He pushes in a direction$\boldsymbol{25.0^0}$below the horizontal. (a) What is the work done on the cart by friction? (b) What is the work done on the cart by the gravitational force? (c) What is the work done on the cart by the shopper? (d) Find the force the shopper exerts, using energy considerations. (e) What is the total work done on the cart?
8: Suppose the ski patrol lowers a rescue sled and victim, having a total mass of 90.0 kg, down a$\boldsymbol{60.0^0}$slope at constant speed, as shown in Figure 4. The coefficient of friction between the sled and the snow is 0.100. (a) How much work is done by friction as the sled moves 30.0 m along the hill? (b) How much work is done by the rope on the sled in this distance? (c) What is the work done by the gravitational force on the sled? (d) What is the total work done?
## Glossary
energy
the ability to do work
work
the transfer of energy by a force that causes an object to be displaced; the product of the component of the force in the direction of the displacement and the magnitude of the displacement
joule
SI unit of work and energy, equal to one newton-meter
### Solutions
Problems & Exercises
1:
$\boldsymbol{3.00\textbf{ J}=7.17\times10^{-4}\textbf{ kcal}}$
3:
(a)$\boldsymbol{5.92\times10^5\textbf{ J}}$
(b)$\boldsymbol{-5.88\times10^5\textbf{ J}}$
(c) The net force is zero.
5:
$\boldsymbol{3.14\times10^3\textbf{ J}}$
7:
(a)$\boldsymbol{-700\textbf{ J}}$
(b) 0
(c) 700 J
(d) 38.6 N
(e) 0 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9409652948379517, "perplexity": 512.4816228743283}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296943747.51/warc/CC-MAIN-20230321225117-20230322015117-00297.warc.gz"} |
http://math.stackexchange.com/users/29334/indiosmo?tab=activity | # indiosmo
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# 13 Actions
Apr17 awarded Scholar Apr17 accepted Possible ways to numerically solve this equation Apr17 comment Possible ways to numerically solve this equation I'll study the method to understand it better but I tested with several values and it works, so I'm accepting your answer. Apr17 revised Possible ways to numerically solve this equation added 68 characters in body Apr17 revised Possible ways to numerically solve this equation added 61 characters in body Apr17 comment Possible ways to numerically solve this equation Yes, a,b, c are non-zero positive integers. Apr17 awarded Editor Apr17 comment Possible ways to numerically solve this equation Ok, thanks, I updated the question to reflect that. Apr17 revised Possible ways to numerically solve this equation added 182 characters in body Apr17 awarded Student Apr17 comment Possible ways to numerically solve this equation To be fair, my question does ask specifically for a numerical method to solve this, as I realize it can't be solved analytically. Apr17 comment Possible ways to numerically solve this equation Well, WA did solve it though. See eq3 where I have all terms but y. It resulted in "Numerical Solution: y $\approx$ 1.10150000054474..." Apr17 asked Possible ways to numerically solve this equation | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8053232431411743, "perplexity": 1927.3293885312273}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500822560.65/warc/CC-MAIN-20140820021342-00081-ip-10-180-136-8.ec2.internal.warc.gz"} |
http://papers.nips.cc/paper/5972-a-fast-universal-algorithm-to-learn-parametric-nonlinear-embeddings | # NIPS Proceedingsβ
## A fast, universal algorithm to learn parametric nonlinear embeddings
A note about reviews: "heavy" review comments were provided by reviewers in the program committee as part of the evaluation process for NIPS 2015, along with posted responses during the author feedback period. Numerical scores from both "heavy" and "light" reviewers are not provided in the review link below.
[PDF] [BibTeX] [Supplemental] [Reviews]
### Abstract
Nonlinear embedding algorithms such as stochastic neighbor embedding do dimensionality reduction by optimizing an objective function involving similarities between pairs of input patterns. The result is a low-dimensional projection of each input pattern. A common way to define an out-of-sample mapping is to optimize the objective directly over a parametric mapping of the inputs, such as a neural net. This can be done using the chain rule and a nonlinear optimizer, but is very slow, because the objective involves a quadratic number of terms each dependent on the entire mapping's parameters. Using the method of auxiliary coordinates, we derive a training algorithm that works by alternating steps that train an auxiliary embedding with steps that train the mapping. This has two advantages: 1) The algorithm is universal in that a specific learning algorithm for any choice of embedding and mapping can be constructed by simply reusing existing algorithms for the embedding and for the mapping. A user can then try possible mappings and embeddings with less effort. 2) The algorithm is fast, and it can reuse N-body methods developed for nonlinear embeddings, yielding linear-time iterations. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8330363035202026, "perplexity": 876.2726536009152}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218188924.7/warc/CC-MAIN-20170322212948-00493-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://cosmologyquestionoftheweek.blogspot.com/2011/12/ | ## Friday, December 30, 2011
### nucleosynthesis
there is a gap in element production: big bang nucleosynthesis produce everything up to lithium, and stars start at producing carbon, by triple-alpha fusion. in what way are beryllium ($Z=4$) and boron ($Z=5$) produced?
## Friday, December 23, 2011
### coldest place
is there a place in the Universe, which is colder than the CMB? (Planck's cryostats don't count as an answer)
for bonus points: which is the hottest place in the Universe?
## Friday, December 16, 2011
### distances
when defining cosmological distances, on has a choice of 4 definitions: proper, comoving, angular diameter and luminosity distance. normally, distances out to a redshift z appear in this order: angular diameter distance $<$ proper distance $<$ comoving distance $<$ luminosity distance. can you think of a cosmological model, where the order is inversed?
## Friday, December 9, 2011
### change of redshift
it is planned to observe spectral lines of distant quasars with very high accuracy and see if their redshift changes with time. why does it change with time and would there be cosmological models where the redshift stays constant? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8898946642875671, "perplexity": 2323.106393869801}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128320338.89/warc/CC-MAIN-20170624203022-20170624223022-00529.warc.gz"} |
https://stackoverflow.com/questions/26925392/how-to-insert-infinity-symbol-to-x-axis-of-matlab-bar-graph | How to insert infinity symbol to X axis of Matlab Bar graph?
How can I add the infinity symbol to X axis of Matlab Bar graph?
Naturally it is possible to insert the infinity symbol i.e. '\infty' for xlabel, as seen in the last line of the inserted code.
But, I want to add the infinity sign in the x axis bar not in the x axis label.
How can I do that? For the sake of detailed clarification, the following script is added bellow:
data=[1 2 3; 1 3 4; 3 1 2];
bar(data)
set(gca,'YLim',[0 3])
set(gca,'YTick',[0:0.5:3])
set(gca, 'YTickLabel',num2str(get(gca,'YTick')','%02.1f%%'))
set(gca,'Xtick',1:3,'XTickLabel',{'\infty' ; '20 dB'; '15 dB'})
xlabel('\infty dB') % x-axis label
• Have you seen this related question? It might be helpful. – Arpi Nov 14 '14 at 8:19
• Sorry, That doesn't work for the case above. @Arpi – mohsen Nov 14 '14 at 8:32
How about this solution, using format_tick function from File Exchange?:
data=[1 2 3; 1 3 4; 3 1 2];
bar(data)
set(gca,'YLim',[0 3])
set(gca,'YTick',[0:0.5:3])
set(gca, 'YTickLabel',num2str(get(gca,'YTick')','%02.1f%%'))
set(gca,'Xtick',1:3)
format_ticks(gca, {'$\infty$' ; '20 dB'; '15 dB'})
I left out the xlabel because it interfers with the Xtick, but probably that can be easily moved to lower position.
EDIT: To fix the overlap of Xtick and xlabel add this to the end of the code:
xlabh = get(gca,'XLabel');
set(xlabh,'Position',get(xlabh,'Position') - [0 .1 0])
• Thanks for your solution @arpi – mohsen Nov 14 '14 at 8:39
• Nice, thank you. – Rashid Nov 14 '14 at 8:41
• @mohsen You're welcome. I extended the answer with the code which fixes the overlap problem. – Arpi Nov 14 '14 at 8:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8679937720298767, "perplexity": 2162.580058552234}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496667262.54/warc/CC-MAIN-20191113140725-20191113164725-00511.warc.gz"} |
http://mathhelpforum.com/advanced-statistics/54667-normal-distribution.html | # Math Help - Normal Distribution
1. ## Normal Distribution
Y is a rv of 'normal distribution' with mean -2 and variance 25.
a) Find the probability that Y>1, given that Y>0.
b) Find the EY, given that Y>0.
Hint: for b), first derive the cdf of the rv given by X=(Y l Y>0)
The hint did not help me much..
:S
2. Originally Posted by zangbangapda
Y is a rv with mean -2 and variance 25.
a) Find the probability that Y>1, given that Y>0.
b) Find the EY, given that Y>0.
Hint: for b), first derive the cdf of the rv given by X=(Y l Y>0)
The hint did not help me much.. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9608817100524902, "perplexity": 2993.988608510748}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440644064517.22/warc/CC-MAIN-20150827025424-00118-ip-10-171-96-226.ec2.internal.warc.gz"} |
http://www.nag.com/numeric/fl/nagdoc_fl24/html/D05/d05intro.html | D05 Chapter Contents
NAG Library Manual
# NAG Library Chapter IntroductionD05 – Integral Equations
## 1 Scope of the Chapter
This chapter is concerned with the numerical solution of integral equations. Provision will be made for most of the standard types of equation (see below). The following are, however, specifically excluded:
(a) Equations arising in the solution of partial differential equations by integral equation methods. In cases where the prime purpose of an algorithm is the solution of a partial differential equation it will normally be included in Chapter D03. (b) Calculation of inverse integral transforms. This problem falls within the scope of Chapter C06.
## 2 Background to the Problems
### 2.1 Introduction
Any functional equation in which the unknown function appears under the sign of integration is called an integral equation. Integral equations arise in a great many branches of science; for example, in potential theory, acoustics, elasticity, fluid mechanics, radiative transfer, theory of population, etc. In many instances the integral equation originates from the conversion of a boundary value problem or an initial value problem associated with a partial or an ordinary differential equation, but many problems lead directly to integral equations and cannot be formulated in terms of differential equations.
Integral equations are of many types; here we attempt to indicate some of the main distinguishing features with particular regard to the use and construction of algorithms.
### 2.2 Classification of Integral Equations
In the classical theory of integral equations one distinguishes between Fredholm equations and Volterra equations. In a Fredholm equation the region of integration is fixed, whereas in a Volterra equation the region is variable. Thus, the equation
$cyt=ft+λ∫abKt,s,ysds, a≤t≤b$ (1)
is an example of Fredholm equation, and the equation
$cyt=ft+λ∫atKt,s,ysds, a≤t$ (2)
is an example of a Volterra equation.
Here the forcing function $f\left(t\right)$ and the kernel function $K\left(t,s,y\left(s\right)\right)$ are prescribed, while $y\left(t\right)$ is the unknown function to be determined. (More generally the integration and the domain of definition of the functions may extend to more than one dimension.) The parameter $\lambda$ is often omitted; it is, however, of importance in certain theoretical investigations (e.g., stability) and in the eigenvalue problem discussed below.
If in (1) or (2), $c=0$, the integral equation is said to be of the first kind. If $c=1$, the equation is said to be of the second kind.
Equations (1) and (2) are linear if the kernel $K\left(t,s,y\left(s\right)\right)=k\left(t,s\right)y\left(s\right)$, otherwise they are nonlinear.
Note: in a linear integral equation, $k\left(t,s\right)$ is usually referred to as the kernel. We adopt this convention throughout.
These two types of equations are broadly analogous to problems of initial- and boundary value type for an ordinary differential equation (ODE); thus the Volterra equation, characterised by a variable upper limit of integration, is amenable to solution by methods of marching type whilst most methods for treating Fredholm equations lead ultimately to the solution of an approximating system of simultaneous algebraic equations. For comprehensive discussion of numerical methods see Atkinson (1976), Baker (1977), Brunner and van der Houwen (1986) and Delves and Walsh (1974). In what follows, the term ‘integral equation’ is used in its general sense, and the type is distinguished when appropriate.
### 2.3 Structure of Kernel
When considering numerical methods for integral equations, particular attention should be paid to the character of the kernel, which is usually the main factor governing the choice of an appropriate quadrature formula or system of approximating functions. Various commonly occurring types of singularity call for individual treatment.
Likewise provision can be made for cases of symmetry, periodicity or other special structure, where the solution may have special properties and/or economies may be effected in the solution process. We note in particular the following cases to which we shall often have occasion to refer in the description of individual algorithms.
(a) A linear integral equation with a kernel $k\left(t,s\right)=k\left(s,t\right)$ is said to be symmetric. This property plays a key role in the theory of Fredholm integral equations.
(b) If $k\left(t,s\right)=k\left(a+b-t,a+b-s\right)$ in a linear integral equation, the kernel is called centro-symmetric.
(c) If in Equations (1) or (2) the kernel has the form $K\left(t,s,y\left(s\right)\right)=k\left(t-s\right)g\left(s,y\left(s\right)\right)$, the equation is called a convolution integral equation; in the linear case $g\left(s,y\left(s\right)\right)=y\left(s\right)$.
(d) If the kernel in (1) has the form
$Kt,s,ys=K1t,s,ys, a≤s≤t, Kt,s,ys=K2t,s,ys, t
where the functions ${K}_{1}$ and ${K}_{2}$ are well behaved, whilst $K$ or its $s$-derivative is possibly discontinuous, may be described as discontinuous or of ‘split’ type; in the linear case $K\left(t,s,y\left(s\right)\right)=k\left(t,s\right)y\left(s\right)$ and consequently ${K}_{1}={k}_{1}y$ and ${K}_{2}={k}_{2}y$. Examples are the commonly occurring kernels of the type $k\left(\left|t-s\right|\right)$ and the Green's functions (influence functions) which arise in the conversion of ODE boundary value problems to integral equations. It is also of interest to note that the Volterra equation (2) may be conceived as a Fredholm equation with kernel of split type, with ${K}_{2}\left(t,s,y\left(s\right)\right)\equiv 0$; consequently methods designed for the solution of Fredholm equations with split kernels are also applicable to Volterra equations.
### 2.4 Singular and Weakly Singular Equations
An integral equation may be called singular if either
(a) its kernel contains a singularity, or (b) the range of integration is infinite,
and it is said to be weakly singular if the kernel becomes infinite at $s=t$.
Sometimes a solution can be effected by a simple adaptation of a method applicable to a nonsingular equation: for example, an infinite range may be truncated at a suitably chosen point. In other cases, however, theoretical considerations will dictate the need for special methods and algorithms. Examples are:
(i) Integral equations with singular kernels of Cauchy type; (ii) Equations of Wiener–Hopf type; (iii) Various dual integral equations arising in the solution of boundary value problems of mathematical physics; (iv) The well-known Abel integral equation, an equation of Volterra type, whose kernel contains an inverse square root singularity at $s=t$.
Problems of inversion of integral transforms also fall under this heading but, as already remarked, they lie outside the scope of this chapter.
### 2.5 Fredholm Integral Equations
#### 2.5.1 Eigenvalue problem
Closely connected with the linear Fredholm integral equation of the second kind is the eigenvalue problem represented by the homogeneous equation
$yt-λ∫abkt,sysds=0, a≤t≤b.$ (3)
If $\lambda$ is chosen arbitrarily this equation in general possesses only the trivial solution $y\left(t\right)=0$. However, for a certain critical set of values of $\lambda$, the characteristic values or eigenvalues (the latter term is sometimes reserved for the reciprocals $\mu =1/\lambda$), there exist nontrivial solutions $y\left(t\right)$, termed characteristic functions or eigenfunctions, which are of fundamental importance in many investigations. The analogy with the eigenproblem of linear algebra is readily apparent, and indeed most methods of solution of equation (3) entail reduction to an approximately equivalent algebraic problem
$K-μIy=0.$ (4)
#### 2.5.2 Equations of the first kind
The Fredholm integral equation of the first kind
$∫abkt,sysds=ft, a≤t≤b,$ (5)
belong to the class of ‘ill-posed’ problems; even supposing that a solution corresponding to the prescribed $f\left(t\right)$ exists, a slight perturbation of $f\left(t\right)$ may give rise to an arbitrarily large variation in the solution $y\left(t\right)$. Hence the equation may be closely satisfied by a function bearing little resemblance to the ‘true’ solution. The difficulty associated with this instability is aggravated by the fact that in practice the specification of $f\left(t\right)$ is usually inexact.
Nevertheless a great many physical problems (e.g., in radiography, spectroscopy, stereology, chemical analysis) are appropriately formulated in terms of integral equations of the first kind, and useful and meaningful ‘solutions’ can be obtained with the aid of suitable stabilizing procedures. See Chapters 12 and 13 of Delves and Walsh (1974) for further discussion and references.
#### 2.5.3 Equations of the second kind
Consider the nonlinear Fredholm equation of the second kind
$yt=ft+∫abKt,s,ysds, a≤t≤b.$ (6)
The numerical solution of equation (6) is usually accomplished either by simple iteration or by a more sophisticated iterative scheme based on Newton's method; in the latter case it is necessary to solve a sequence of linear integral equations. Convergence may be demonstrated subject to suitable conditions of Lipschitz continuity of the functions $K$ with respect to the argument $y$.
Examples of Fredholm type (for which the provision of algorithms is contemplated) are:
(a) the Uryson equation
$ut-∫01Ft,s,usds=0, 0≤t≤1,$ (7)
(b) the Hammerstein equation
$ut-∫01kt,sgs,usds=0, 0≤t≤1,$ (8)
where $F$ and $g$ are arbitrary functions.
### 2.6 Volterra Integral Equations
#### 2.6.1 Equations of the first kind
Consider the Volterra integral equation of the first kind
$∫atkt,sysds=ft, a≤t.$ (9)
Clearly it is necessary that $f\left(a\right)=0$; otherwise no solution to (9) can exist. The following types of Volterra integral equations of the first kind occur in real life problems:
• equations with unbounded kernel at $s=t$,
• equations with sufficiently smooth kernel.
These types belong also to the class of ‘ill-posed’ problems. However, the instability is appreciably less severe in the equations with unbounded kernel. In general, a nonsingular Volterra equation of the first kind presents less computational difficulty than the Fredholm equation (5) with a smooth kernel.
A Volterra equation of the first kind may, under suitable conditions, be converted by differentiation to one of the second kind or by integration by parts to an equation of the second kind for the integral of the wanted function.
#### 2.6.2 Equations of the second kind
A very general Volterra equation of the second kind is given by
$yt=ft+∫atKt,s,ysds, a≤t.$ (10)
The resemblance of Volterra equations to ODEs suggests that the underlying methods for ODE problems can be applied to Volterra equations. Indeed this turns out to be the case. The main advantages of implementing these methods are their well-developed theoretical background, i.e., convergence and stability; see Brunner and van der Houwen (1986) and Wolkenfelt (1982).
Many Volterra integral equations arising in real life problems have a convolution kernel (see Section 2.3(c)); see Brunner and van der Houwen (1986) for references. However, a subclass of these equations which have kernels of the form
$kt-s=∑j=0Mλjt-sj,$ (11)
where $\left\{{\lambda }_{j}\right\}$ are real, can be converted into a system of linear or nonlinear ODEs; see Brunner and van der Houwen (1986).
For more information on the theoretical and the numerical treatment of integral equations we refer you to Atkinson (1976), Baker (1977), Brunner and van der Houwen (1986), Cochran (1972) and Delves and Walsh (1974).
## 3 Recommendations on Choice and Use of Available Routines
The choice of routine will depend first of all upon the type of integral equation to be solved.
### 3.1 Fredholm Equations of the Second Kind
(a) Linear equations D05AAF is applicable to an equation with a discontinuous or ‘split’ kernel as defined in Section 2.3(d). Here, however, both the functions ${k}_{1}$ and ${k}_{2}$ are required to be defined (and well-behaved) throughout the square $a\le s$, $t\le b$. D05ABF is applicable to an equation with a smooth kernel. Note that D05AAF may also be applied to this case, by setting ${k}_{1}={k}_{2}=k$, but D05ABF is more efficient.
### 3.2 Volterra Equations of the Second Kind
(a) Linear equations
D05AAF may be used to solve a Volterra equation by defining ${k}_{2}$ (or ${k}_{1}$) to be identically zero. (See also (b).)
(b) Nonlinear equations
D05BAF is applicable to a nonlinear convolution Volterra integral equation of the second kind. The kernel function has the form
$Kt,s,ys=kt-sgs,ys.$
The underlying methods used in the routine are the reducible linear multistep methods. You have a choice of variety of these methods. This routine can also be used for linear $g$.
D05BDF is applicable to a nonlinear convolution equation having a weakly-singular kernel (Abel). The kernel function has the form
$Kt,s,ys=kt-s t-s gs,ys.$
The underlying methods used in the routine are the fractional linear multistep methods based on Backward Difference Formula (BDF, see Section 3.1 in the D02 Chapter Introduction) methods. This routine can also be used for linear $g$.
### 3.3 Volterra Equations of the First Kind
(a) Linear equations
See (b).
(b) Nonlinear equations
D05BEF is applicable to a nonlinear equation having a weakly-singular kernel (Abel). The kernel function has the form
$Kt,s,ys=kt-s t-s gs,ys.$
The underlying methods used in the routine are the fractional linear multistep methods based on BDF methods. This routine can also be used for linear $g$.
### 3.4 Utility Routines
D05BWF generates the weights associated with Adams and BDF linear multistep methods. These weights can be used for the solution of nonsingular Volterra integral and integro-differential equations of general type.
D05BYF generates the weights associated with BDF linear multistep methods. These weights can be used for the solution of weakly-singular Volterra (Abel) integral equations of general type.
### 3.5 User-supplied Routines
Many of the routines in this chapter require you to supply procedures defining the kernels and other given functions in the equations. It is important to test these independently before using them in conjunction with NAG Library routines.
## 4 Functionality Index
Fredholm equation of second kind,
linear,
nonsingular discontinuous or ‘split’ kernel D05AAF
nonsingular smooth kernel D05ABF
Volterra equation of first kind,
nonlinear,
weakly-singular,
convolution equation (Abel): D05BEF
Volterra equation of second kind,
nonlinear,
nonsingular,
convolution equation D05BAF
weakly-singular,
convolution equation (Abel): D05BDF
Weight generating routines,
weights for general solution of Volterra equations D05BWF
weights for general solution of Volterra equations with weakly-singular kernel D05BYF
None.
None.
## 7 References
Atkinson K E (1976) A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the Second Kind SIAM, Philadelphia
Baker C T H (1977) The Numerical Treatment of Integral Equations Oxford University Press
Brunner H and van der Houwen P J (1986) The Numerical Solution of Volterra Equations CWI Monographs, North-Holland, Amsterdam
Cochran J A (1972) The Analysis of Linear Integral Equations McGraw–Hill
Delves L M and Walsh J (1974) Numerical Solution of Integral Equations Clarendon Press, Oxford
Wolkenfelt P H M (1982) The construction of reducible quadrature rules for Volterra integral and integro-differential equations IMA J. Numer. Anal. 2 131–152
D05 Chapter Contents | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 64, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9857094883918762, "perplexity": 697.5775150547227}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988718284.75/warc/CC-MAIN-20161020183838-00518-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://byjus.com/sphere-formula/ | Sphere formula
A perfectly symmetrical 3 – Dimensional circular shaped object is a Sphere. The line that connects from the center to the boundary is called radius of the square. You will find a point equidistant from any point on the surface of a sphere. The longest straight line that passes through the center of the sphere is called the diameter of the sphere. It is twice the length of the radius of the sphere.
Sphere Formula
Formulas of a Sphere
There are four main formulas for a sphere which include sphere diameter formula, sphere circumference formula, sphere surface area, and sphere volume area. All these formulas are mentioned in the table given below and an example is also prodided here.
Sphere Formulas
Diameter of a Sphere D = 2 r
Circumference of a Sphere C = 2 π r
Surface Area of a Sphere A = 4 π r2
Volume of a Sphere V = (4 ⁄ 3) π r3
Solved Examples Using Formulas of a Sphere
Question: Calculate the diameter, circumference, surface area and volume of a sphere of radius 9 cm ?
Solution:
Given,
r = 7 cm
Diameter of a sphere
=2r
= 2 × 9
=18 cm
Circumference of a sphere
= 2πr
= 2 × π × 9
= 56.54 cm
Surface area of a sphere
4πr2 = 4 × π × 92
= 4 × π × 81 = 1017.87 cm
Volume of a sphere
4/3 πr3
= 4/3 π93 = 338.2722 cm
= 338.2722 cm
More topics in Sphere Formula Volume of a Sphere Formula Surface Area of a Sphere Formula | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9422557353973389, "perplexity": 1128.953715684934}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514572980.56/warc/CC-MAIN-20190917000820-20190917022820-00446.warc.gz"} |
https://albannalab.science/category/publications/ | Categories
## New paper in eLife: “Spike-timing-dependent ensemble encoding…”
Spike-timing-dependent ensemble encoding by non-classically responsive cortical neurons
## Abstract
Neurons recorded in behaving animals often do not discernibly respond to sensory input and are not overtly task-modulated. These non-classically responsive neurons are difficult to interpret and are typically neglected from analysis, confounding attempts to connect neural activity to perception and behavior. Here we describe a trial-by-trial, spike-timing-based algorithm to reveal the coding capacities of these neurons in auditory and frontal cortex of behaving rats. Classically responsive and non-classically responsive cells contained significant information about sensory stimuli and behavioral decisions. Stimulus category was more accurately represented in frontal cortex than auditory cortex, via ensembles of non-classically responsive cells coordinating the behavioral meaning of spike timings on correct but not error trials. This unbiased approach allows the contribution of all recorded neurons – particularly those without obvious task-related, trial-averaged firing rate modulation – to be assessed for behavioral relevance on single trials.
Categories
## New Paper in Entropy: “Minimum and Maximum Entropy…”
The first paper from the Albanna Lab is out in Entropy!
Link to Paper Online
Link to PDF
Minimum and Maximum Entropy Distributions for Binary Systems with Known Means and Pairwise Correlations
Badr F. Albanna, Christopher Hillar, Jascha Sohl-Dickstein and Michael R. DeWeese
Abstract
Maximum entropy models are increasingly being used to describe the collective activity of neural populations with measured mean neural activities and pairwise correlations, but the full space of probability distributions consistent with these constraints has not been explored. We provide upper and lower bounds on the entropy for the minimum entropy distribution over arbitrarily large collections of binary units with any fixed set of mean values and pairwise correlations. We also construct specific low-entropy distributions for several relevant cases. Surprisingly, the minimum entropy solution has entropy scaling logarithmically with system size for any set of first- and second-order statistics consistent with arbitrarily large systems. We further demonstrate that some sets of these low-order statistics can only be realized by small systems. Our results show how only small amounts of randomness are needed to mimic low-order statistical properties of highly entropic distributions, and we discuss some applications for engineered and biological information transmission systems. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9003475904464722, "perplexity": 2900.2961373436347}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600402132335.99/warc/CC-MAIN-20201001210429-20201002000429-00586.warc.gz"} |
https://arxiv.org/abs/1411.0046 | gr-qc
(what is this?)
# Title: The Dispersion Relation for Matter Waves in a Two-Phase Vacuum
Abstract: The cosmological constant (lambda) of general relativity is a natural consequence of embedding Einstein's theory in a five-dimensional theory of the type needed for unification. The exact 5D solution for lambda less than 0 shows waves in ordinary 3D space with properties similar to those of de Broglie or matter waves. Here the dispersion relation is derived for matter waves in a toy two-phase model, where regions with lambda less than 0 and lambda greater than 0 average on the large scale to lambda = 0, thus providing in principle a resolution of the cosmological-constant problem. A striking result of the analysis is that the dispersion relation is bimodal, with a well-defined window of high-frequency transmission which effectively defines the speed of light.
Subjects: General Relativity and Quantum Cosmology (gr-qc) Journal reference: Mod. Phys. Lett. A v.29, No. 31 (2014), 9 pp DOI: 10.1142/S0217732314501685 Cite as: arXiv:1411.0046 [gr-qc] (or arXiv:1411.0046v1 [gr-qc] for this version)
## Submission history
From: Paul Wesson [view email]
[v1] Sat, 1 Nov 2014 00:22:23 GMT (349kb) | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8982346653938293, "perplexity": 1457.707283735904}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267156314.26/warc/CC-MAIN-20180919235858-20180920015858-00475.warc.gz"} |
https://forum.allaboutcircuits.com/threads/voltage-gain-of-a-transistor-amplifier.74105/ | # voltage gain of a transistor amplifier
#### dumindu89
Joined Oct 28, 2010
113
1. How to find the small signal AC voltage gain of a transistor (NPN) amplifier in common emitter configuration which has a emitter resistor (Re) with a capacitor (Ce) parallel with emitter resistor around 100 MHz?(transistor is BFR520, gainbandwidth product is 9GHz)
2. What will be the case if the Ce capacitor is not there?
3. Is it possible to get a small signal voltage gain of 25 at 100 MHz from BFR520 with the configuration stated above or what configuration will be the most suitable for this? | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8294902443885803, "perplexity": 2512.036836008177}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400189264.5/warc/CC-MAIN-20200918221856-20200919011856-00297.warc.gz"} |
http://mathhelpforum.com/calculus/122918-determine-whether-following-series-convergent-divergent.html | # Math Help - Determine whether the following series is convergent or divergent.
1. ## Determine whether the following series is convergent or divergent.
Hi !
I have this series
Sigma from 1 to infinity ( 1 - n*sin(1/n) )
nth term test fails
and rewrtting it as
Sigma(1) - Sigma( n*sin(1/n) )
will not solve it.
since both series diverges .. and the convergence of the difference of two divergent series is unknown .. it maybe converges or diverges
i want only a hint
I dont want full solution .. because its my favorite chapter =D
2. OK Let me thinking more
N.T.T Failed
I cant apply A.S.T here since there is no (-1)^n or (-1)^(n-1) .. etc
Root and Ratio Failed since its algebric function
its not telescoping << am not sure about this .. maybe it needs some algebric operations to make it telescoping.
i have the comparison tests
but the problem i must prove that ( 1 - n*sin(1/n) ) is positive for all positive integers n > 1
Clearly n*sin(1/n) is positive
since n is positive and (1/n) are angles in the first quadrant for all n > 1
and sine is positive in the first quadrant
but the problem here i have ( 1 - n*sin(1/n) ) !!
if i proved that n*sin(1/n) < 1 for all n then i can use the comparison tests ..
but i cant prove it !
n*sin(1/n) ---> 1 n-->infinity
n*sin(1/n) = sin(1) as n=1
ohhh .. did i prove n*sin(1/n) belongs to [sin(1) , 1) for all n > 1
or this is wrong ?
Any mistakes here?
3. Originally Posted by TWiX
n*sin(1/n) ---> 1 n-->infinity
n*sin(1/n) = sin(1) as n=1
ohhh .. did i prove n*sin(1/n) belongs to [sin(1) , 1) for all n > 1
or this is wrong ?
Any mistakes here?
Although your assertion that $n \sin{\frac{1}{n}} \in [\sin(1), 1)$ for $n\ge 1$ is correct, your proof is not, as you did not prove it for $n \in (1,\infty)$. To show that from $1 \sin{\frac{1}{1}} = \sin(1)$ and $\lim_{n\rightarrow \infty} n \sin{\frac{1}{n}} = 1$, $n \sin{\frac{1}{n}} \in [\sin(1), 1)$ follows, you may try to show that the function $f(x) = x \sin{\frac{1}{x}}$ is increasing in the interval $(1,\infty)$. Another way is to use the inequality that holds for all $x \in \mathbb{R}\setminus\{0\}$: $|\sin{x}|<|x|$
4. Originally Posted by TWiX
Hi !
I have this series
Sigma from 1 to infinity ( 1 - n*sin(1/n) )
nth term test fails
and rewrtting it as
Sigma(1) - Sigma( n*sin(1/n) )
will not solve it.
since both series diverges .. and the convergence of the difference of two divergent series is unknown .. it maybe converges or diverges
i want only a hint
I dont want full solution .. because its my favorite chapter =D
I just did this one for someone recently.
Hint!:
Spoiler:
$\sin\left(\frac{1}{n}\right)=\sum_{\jmath=0}^{\inf ty}\frac{(-1)^{\jmath}}{n^{2\jmath+1}\left(2\jmath+1\right)!}$
Bigger Hint!
Spoiler:
Write our just the first two terms of this series, multiply by $n$ and subtract one. It should be obvious from there.
P.S. As anyone here that has posted for a while can attest, I too love infinite series! PM me if you have any questions or just want some suggestions of areas of study!
5. It just occurred to me that you may not understand the hint I gave you (if you are currently studying the convergence of infinite series). So, here is another hint:
Spoiler:
Conjecture that your series shares convergence/divergence with some other series $\sum_{n\in\mathbb{N}}\frac{1}{n^\lambda}$. Find this lambda.
Here is how:
Spoiler:
We need $\lim_{n\to\infty}\frac{1-n\sin\left(\tfrac{1}{n}\right)}{\frac{1}{n^{\lambd a}}}=C$. Make the substitution $\frac{1}{n}=z$ to get the limit $\lim_{z\to0}\frac{1-\frac{\sin(z)}{z}}{z^{\lambda}}$. Or, equivalently $\lim_{z\to0}\frac{z-\sin(z)}{z^{\lambda+1}}$. Applying L'hopital's (since I assume that is your most familiar means of computing limits) we see that $\frac{1-\cos(z)}{\left(\lambda+1\right)z^{\lambda}}=C$. Now, your experience with limits should bring the limit $\lim_{x\to0}\frac{1-\cos(x)}{x^2}=\frac{1}{2}$ to let you conclude that $\lambda=2$. Lo and behold $\lim_{n\to\infty}\frac{1-n\sin\left(\tfrac{1}{n}\right)}{\frac{1}{n^2}}=\fr ac{1}{6}$. Now, just apply the limit comparison test
6. Alternatively, alternatively.
Spoiler:
Show that you may apply the integral test. And then,
$\int_1^{\infty}\left\{1-n\sin\left(\tfrac{1}{n}\right)\right\}$, from where the substitution $\frac{1}{n}=z$ leads to $\int_0^1\left\{\left(1-\frac{\sin(z)}{z}\right)\cdot\frac{1}{z^2}\right\} =\int_0^1\sum_{\jmath=1}^{\infty}\frac{(-1)^{\jmath}z^{2\jmath-2}}{(2\jmath+1)!}=$ $\sum_{\jmath=1}^{\infty}\int_0^{1}\frac{(-1)^{\jmath+1}z^{2\jmath-2}}{(2\jmath+1)!}=\sum_{\jmath=1}^{\infty}\frac{(-1)^{\jmath+1}z^{2\jmath-1}}{(2\jmath-1)(2\jmath+1)!}$ the last of which I know you know converges. Thus, we may draw our conclusion.
7. Originally Posted by TWiX
Hi !
I have this series
Sigma from 1 to infinity ( 1 - n*sin(1/n) )
nth term test fails
and rewrtting it as
Sigma(1) - Sigma( n*sin(1/n) )
will not solve it.
since both series diverges .. and the convergence of the difference of two divergent series is unknown .. it maybe converges or diverges
i want only a hint
I dont want full solution .. because its my favorite chapter =D
Big-O or Landau notation and operations always makes these things seem simple:
$a_n=1-n\sin(1/n) =$ $1-n\left(\frac{1}{n} + \frac{1}{6n^3}+O(n^{-5}) \right)=\frac{1}{6n^2}+O(n^{-4})=O(n^{-2})$
Which means that there exists a constants $C>0$ and $n_0\in \mathbb{N}$ such that:
$|a_n|
for all $n>n_0$.
(you can put in explicit remainder terms if you wish and get the same result)
CB
8. Originally Posted by Drexel28
It just occurred to me that you may not understand the hint I gave you (if you are currently studying the convergence of infinite series). So, here is another hint:
Spoiler:
Conjecture that your series shares convergence/divergence with some other series $\sum_{n\in\mathbb{N}}\frac{1}{n^\lambda}$. Find this lambda.
Here is how:
Spoiler:
We need $\lim_{n\to\infty}\frac{1-n\sin\left(\tfrac{1}{n}\right)}{\frac{1}{n^{\lambd a}}}=C$. Make the substitution $\frac{1}{n}=z$ to get the limit $\lim_{z\to0}\frac{1-\frac{\sin(z)}{z}}{z^{\lambda}}$. Or, equivalently $\lim_{z\to0}\frac{z-\sin(z)}{z^{\lambda+1}}$. Applying L'hopital's (since I assume that is your most familiar means of computing limits) we see that $\frac{1-\cos(z)}{\left(\lambda+1\right)z^{\lambda}}=C$. Now, your experience with limits should bring the limit $\lim_{x\to0}\frac{1-\cos(x)}{x^2}=\frac{1}{2}$ to let you conclude that $\lambda=2$. Lo and behold $\lim_{n\to\infty}\frac{1-n\sin\left(\tfrac{1}{n}\right)}{\frac{1}{n^2}}=\fr ac{1}{6}$. Now, just apply the limit comparison test
Thanks all but all soultions are too advanced to me.
this one is good
but my experience did not help me
how did you make it $\lim_{x\to0}\frac{1-\cos(x)}{x^2}$ ??
9. Thanks.
I got it
10. Originally Posted by TWiX
Thanks.
I got it
Are you good then with this problem?
11. Originally Posted by Drexel28
Are you good then with this problem?
Yeah, but using the known tests.
not O notation and complicated solutions
I love testing series for convergence.
In meduim level.
not the series like 1/(ln n)^9 from 2 to infinity
I hate it
Can you test it?
Sorry I should open new thread, but anyway I can ask 2 question in 1 thread .
Limit comparison test with $\frac{1}{ n^{ \frac{9}{11} } }$ will show it diverges.
But I want another solution.
12. Originally Posted by TWiX
Yeah, but using the known tests.
not O notation and complicated solutions
I love testing series for convergence.
In meduim level.
not the series like 1/(ln n)^9 from 2 to infinity
I hate it
Can you test it?
Sorry I should open new thread, but anyway I can ask 2 question in 1 thread .
Limit comparison test with $\frac{1}{ n^{ \frac{9}{11} } }$ will show it diverges.
But I want another solution.
Cauchy's condensation test. Relatively easy to prove. It says that under certain conditions (that this series satisfies) we have that $\sum_{n\in\mathbb{N}}a_n\text{ converges }\Longleftrightarrow \sum_{n\in\mathbb{N}}2^n a_{2^n}\text{ converges}$
P.S. Yout might be interested in knowing that the integral brought up in the integral test post actually has a nice solution. $\int_0^{\infty}\left\{1-n\sin\left(\tfrac{1}{n}\right)\right\}\text{ }dn=\frac{\pi}{4}$. Not sure if you cared...you can prove it a couple of ways...if you're interested.
13. Also, notice alternatively that $\lim_{n\to\infty}\frac{\ln(n)}{n^{\frac{1}{9}}}=0$ so it follows that eventually $\ln(n)\leqslant n^{\frac{1}{9}}\implies \frac{1}{n^{\frac{1}{9}}}\leqslant\frac{1}{\ln(n)} \implies \frac{1}{n}\leqslant\frac{1}{\ln^9(n)}$
14. Originally Posted by Drexel28
Cauchy's condensation test. Relatively easy to prove. It says that under certain conditions (that this series satisfies) we have that $\sum_{n\in\mathbb{N}}a_n\text{ converges }\Longleftrightarrow \sum_{n\in\mathbb{N}}2^n a_{2^n}\text{ converges}$
P.S. Yout might be interested in knowing that the integral brought up in the integral test post actually has a nice solution. $\int_0^{\infty}\left\{1-n\sin\left(\tfrac{1}{n}\right)\right\}\text{ }dn=\frac{\pi}{4}$. Not sure if you cared...you can prove it a couple of ways...if you're interested.
Although I dont know this test, but it leads to (1/ln2)Sigma (2^n/n)
which diverges by Ratio,B.C.T and maybe L.C.T
Sorry but I dont care about the first one.
Although this integral seems not easy to solve.
maybe x=1/n can solve it but am not sure.
Forget it!
Do you have another solution for my second series?
15. Originally Posted by TWiX
Although I dont know this test, but it leads to (1/ln2)Sigma (2^n/n)
which diverges by Ratio,B.C.T and maybe L.C.T
Sorry but I dont care about the first one.
Although this integral seems not easy to solve.
maybe x=1/n can solve it but am not sure.
Forget it!
Do you have another solution for my second series?
I already posted a second solution, son. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 49, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9780043959617615, "perplexity": 1552.2806466783009}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435376073161.33/warc/CC-MAIN-20150627033433-00117-ip-10-179-60-89.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/573840/abelian-group-not-finitely-generated | # Abelian Group Not Finitely Generated
The Structure Theorem: Every finitely generated abelian group is isomorphic to a direct product of cyclic groups $C_{d_0}\times C_{d_1}\times \ldots \times C_{d_k} \times L$ such that $d_i | d_{i+1} \forall \space 0\le i\le {k-1}$ and $L$ is a free abelien groups (i.e $\mathbb{Z}^r$ for some $r$). One method to prove this fact is using Smith Normal Form. What is an example of a abelian groups that is NOT finitely generated? What can we say about about the isomorphism classes of infinitly generated abelien groups?
-
$\mathbb{Q}$... – Tyler Nov 19 '13 at 22:29
As Nicky Hekster's answer shows, the rational numbers $\mathbb{Q}$ under addition provide an example of a non-finitely generated abelian group. The classification of non-finitely generated abelian groups is an open problem. See Status of the classification of non-finitely generated abelian groups..
(A different way of seeing why $(\mathbb{Q}, +)$ is not finitely generated is to note that if $\{p_1/q_1, \ldots, p_n/q_n\}$ is a finite set of rational numbers then the additive subgroup it generates is contained in the subgroup generated by $1/q$ where $q$ is the least common multiple of the $q_i$ and hence cannot be the whole of $\mathbb{Q}$.)
Assume that $(\mathbb Q,+)$ is finitely generated. Then, by the structure theorem there exist $m,n\ge 0$ and $d_1\mid\cdots\mid d_m$ with $d_i>1$ such that $\mathbb Q\simeq \mathbb Z/d_1\mathbb Z\oplus\cdots\oplus\mathbb Z/d_m\mathbb Z\oplus \mathbb Z^n$. If $m\ge 1$, then there exists $x\in\mathbb Q$, $x\neq 0$, such that $d_1x=0$, a contradiction. Thus we get $m=0$. Then $\mathbb Q\simeq \mathbb Z^n$. If $n\ge 2,$ then there exist $x_1,x_2\in\mathbb Q$ which are linearly independent over $\mathbb Z$. But $x_1=a_1/b_1$ and $x_2=a_2/b_2$ give $(b_1a_2)x_1+(-b_2a_1)x_2=0$, a contradiction. So we must have $n=1$, that is, $\mathbb Q$ is cyclic. Assume that it is generated by $a/b$ with $b\ge 1$. Then $\frac{1}{b+1}$ can not be written as $\frac{ka}{b}$, and again a contradiction. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9784980416297913, "perplexity": 36.26695605912089}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860111396.55/warc/CC-MAIN-20160428161511-00205-ip-10-239-7-51.ec2.internal.warc.gz"} |
https://lammps.sandia.gov/doc/pair_spin_dmi.html | pair_style spin/dmi command
Syntax
pair_style spin/dmi cutoff
• cutoff = global cutoff pair (distance in metal units)
Examples
pair_style spin/dmi 4.0
pair_coeff * * dmi 2.6 0.001 1.0 0.0 0.0
pair_coeff 1 2 dmi 4.0 0.00109 0.0 0.0 1.0
Description
Style spin/dmi computes the Dzyaloshinskii-Moriya (DM) interaction between pairs of magnetic spins. According to the expression reported in (Rohart), one has the following DM energy:
where si and sj are two neighboring magnetic spins of two particles, eij = (ri - rj)/|ri-rj| is the unit vector between sites i and j, and D is the DM vector defining the intensity (in eV) and the direction of the interaction.
In (Rohart), D is defined as the direction normal to the film oriented from the high spin-orbit layer to the magnetic ultra-thin film.
The application of a spin-lattice Poisson bracket to this energy (as described in (Tranchida)) allows to derive a magnetic torque omega, and a mechanical force F (for spin-lattice calculations only) for each magnetic particle i:
More details about the derivation of these torques/forces are reported in (Tranchida).
For the spin/dmi pair style, the following coefficients must be defined for each pair of atoms types via the pair_coeff command as in the examples above, or in the data file or restart files read by the read_data or read_restart commands, and set in the following order:
• rc (distance units)
• |D| (energy units)
• Dx, Dy, Dz (direction of D)
Note that rc is the radius cutoff of the considered DM interaction, |D| is the norm of the DM vector (in eV), and Dx, Dy and Dz define its direction.
None of those coefficients is optional. If not specified, the spin/dmi pair style cannot be used.
Restrictions
All the pair/spin styles are part of the SPIN package. These styles are only enabled if LAMMPS was built with this package, and if the atom_style “spin” was declared. See the Build package doc page for more info. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9257950782775879, "perplexity": 3629.8471513766904}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912203093.63/warc/CC-MAIN-20190323221914-20190324003914-00197.warc.gz"} |
http://www.bowaggoner.com/blog/2016/10-02-risk-aversion-entropy/index.html | # The Tiger's Stripes
A technical blog on math, computer science, and game theory.
Author: Bo Waggoner RSS feed
# Risk Aversion and Max-Entropy
Posted: 2016-10-02.
I want to share a nice little problem that came up in discussion between Yiling Chen, Madhu Sudan, and myself. It turns out to have a cute solution that connects the geometry of proper scoring rules with a "max-entropy" rule.
A primer on proper scoring rules will be useful first!
## The Tale of Rita
Once upon a time, Risk-Averse Rita worked as a weather reporter in Incentiveland playing the following game each day:
1. Rita is given a set $Q$ of possible probability distributions over the day's weather outcome, e.g. $Q \subseteq \Delta_{\{\text{rain, cloud, sun}\}}$.
2. Rita makes a prediction $p$, a distribution over the day's weather outcome.
3. Nature picks a true distribution $q$ from $Q$, perhaps adversarially depending on Rita's prediction.
4. The true weather outcome $e$ is drawn according to $q$.
5. Rita is scored with a proper scoring rule $S(p,e)$.
In the case where $|Q| = 1$, i.e. Rita knows the true distribution $q$ of nature, then properness of the scoring rule implies that she should just report $q$. Now, if Rita is risk averse and wishes to maximizes her minimum expected score, how should she play? In math, she wants to solve
$\max_p \min_{q \in Q} \mathbb{E}_{e\sim q} S(p, e) .$ Recalling the notation $S(p;q)$ for expected score of report $p$ when $e \sim q$, this is $\max_p \min_{q \in Q} S(p;q) .$
## The Solution
Luckily, Rita found an optimal strategy. The intuitive statement of the solution is:
If $Q$ is a convex set, then the optimal strategy is to assume the adversary will pick the "max-entropy" distribution in $Q$ and best-respond accordingly.
Furthermore, in this case, the adversary cannot improve on actually picking the "max-entropy" distribution.
Why do I have quotes around "max-entropy"? Because the measure of entropy changes depending on the scoring rule.
If you read my post on generalized entropies, you know that:
1. Every proper scoring rule maps to a convex function $G$ where $G(q)$ is the expected score for optimally (truthfully) reporting given belief $q$.
2. The concave function $F = -G$ can be interpreted as a "generalized entropy" function under an axiomatic approach.
For example, the log scoring rule is $S(p,e) = \log p(e)$, so its expected truthful score function is $\sum_e p(e) \log p(e) = -H(p)$, so its corresponding generalized entropy is just Shannon entropy. For another example, the quadratic scoring rule is $S(p,e) = 2p(e) - \sum_{e'} p(e')^2$ and has corresponding $G(p) = \sum_e p(e)^2$, or generalized entropy $-\|p\|_2^2$, i.e. the collision probability.
Theorem. Let $Q$ be any convex set of distributions and suppose Rita faces scoring rule $S$ with differentiable convex expected score $G$. To solve the problem $\arg\max_p \min_{q \in Q} S(p;q) ,$ Rita's optimal strategy is to report $q^* := \arg\min_{q \in Q} G(Q) .$
Proof. It may be helpful to first see the pictures below. First, we will show that if Rita picks $p=q^*$, then her expected score is at least $G(q^*)$ for any choice of nature (achieved when nature chooses $q^*$). Then, we will show that any other report $p \neq q^*$ results in a worse score.
Claim 1. $\min_{q\in Q} S(q^*;q) = G(q^*)$.
Proof of claim: We just want to show for all $q \in Q$ \begin{align} S(q^*; q) - G(q^*) &\geq 0 \\ \iff ~ \langle dG(q^*), q-q^* \rangle &\geq 0 \end{align} using the scoring rule characterization $S(p;q) = G(p) + \langle dG(p), q-p \rangle$ where $dG(p)$ is the subgradient (i.e. gradient) of $G$ at point $p$.
Now we just want to prove $\langle dG(q^*), q-q^* \rangle \geq 0$ for all $q \in Q$. But since $G$ is differentiable, this is actually equal to the directional derivative of $G$ in the direction $q-q^*$: $\langle dG(q^*), q-q^* \rangle = \lim_{h\to 0} \frac{G(q^* + h(q-q^*)) - G(q^*)}{h} .$ Now, we assumed $Q$ is a convex set, and both $q^*,q \in Q$. So for every $0 \leq h \leq 1$, the point $q' := q^* + h(q-q^*)$ is in $Q$. Furthermore, since we assumed $q^* = \arg\min_{p \in Q} G(p)$, we have $G(q') - G(q) \geq 0$.
So the right side is a limit over nonnegative terms, so the left side cannot be negative.
Claim 2. For all $p$, $\min_{q \in Q} S(p;q) \leq G(q^*)$.
Proof of claim: In particular, $\min_{q \in Q} \leq S(p;q^*)$ because $q^* \in Q$. Now by definition of a proper scoring rule, $S(p;q^*) \leq S(q^*;q^*) = G(q^*)$.
## Questions
Here are some exercises you might enjoy followed by open questions that would be nice to resolve, at least for Rita's sake.
Exercise 1.
Rita's cousin, Risk-Neutral Rick, has a "meta-belief" in the form of a distribution $r$ over $Q$, i.e. $r$ assigns a probability to each $q \in Q$ as being the correct belief. How should Rick report to maximize expected score, i.e. what $p$ solves
$\max_p \mathbb{E}_{q \sim r} \mathbb{E}_{e \sim q} S(p, e) ~ ?$
Exercise 2.
Suppose $Q$ is not a convex set; give a counterexample to the theorem.
Open question 1.
Suppose $Q$ is not a convex set. I would guess that Rita's optimal report is the max-entropy $q$ in the convex hull of $Q$. Prove me right, or disprove it and find Rita's optimal strategy instead.
Open question 2.
What if $G$ is convex, but not differentiable?
The theorem probably still holds, but the proof needs to be much more careful. What happens when $G$ is non-differentiable is that, at some points $q$, there are multiple choices of subgradient $dG(q)$. If we use the wrong one when constructing the scoring rule $S$, then the theorem won't hold; to see this, picture $G$ with a kink at its minimum (such as $G(x) = |x|$) and convince yourself that, if $Q$ contains a ball around the global minimum, then the subgradient chosen at the minimum must be $0$ for the theorem to hold. Maybe elsewhere you don't need to be so careful, and can just use that the subgradients at $p$ form a convex set; I'm not sure.
Open question / challenge 3.
Extend this theorem, as far as possible, to the case of general decision problems rather than just proper scoring rules.
## Comments
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Please allow a few seconds after clicking "Submit" for it to go through. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9663591384887695, "perplexity": 618.4131747407079}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039744348.50/warc/CC-MAIN-20181118093845-20181118115845-00362.warc.gz"} |
http://mathhelpforum.com/calculus/80978-differentials-estimate-maximum-error.html | # Thread: Differentials to estimate the maximum error
1. ## Differentials to estimate the maximum error
hey guys, i really don't know where to start where to end. plz help me solve this problem
The dimensions of a closed rectangular box are measured as 100 centimeters, 80 centimeters, and 100 centimeters, respectively, with the error in each measurement at most .2 centimeters. Use differentials to estimate the maximum error in calculating the surface area of the box.
___________________ square centimeters
2. Originally Posted by DMDil
hey guys, i really don't know where to start where to end. plz help me solve this problem
The dimensions of a closed rectangular box are measured as 100 centimeters, 80 centimeters, and 100 centimeters, respectively, with the error in each measurement at most .2 centimeters. Use differentials to estimate the maximum error in calculating the surface area of the box.
___________________ square centimeters
the surface area of the box is $S=2xy + 2yz + 2zx.$ thus $dS=2(y+z)dx + 2(x+z)dy + 2(x+y)dz.$ so: the maximum error in calculating $S \approx 0.2 \times (360 + 400 + 360) =224 \ \text{cm}^2$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 3, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9907483458518982, "perplexity": 617.3270286050465}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-36/segments/1471982293615.23/warc/CC-MAIN-20160823195813-00207-ip-10-153-172-175.ec2.internal.warc.gz"} |
http://advancedintegrals.com/tag/zeta/ | # Tag Archives: Zeta
## Integral representation of the zeta function proof
$$\zeta(s) = \frac{1}{\Gamma(s)}\int^\infty_0\frac{t^{s-1}}{e^t-1}dt$$ $$\textit{proof}$$ Start by the integral representation $$\int^\infty_0 \frac{e^{-t}t^{s-1}}{1-e^{-t}}\,dt$$ Using the power expansion $$\frac{1}{1-e^{-t}} = \sum_{n=0}^\infty e^{-nt}$$ Hence we have $$\int^\infty_0\,e^{-t}t^{s-1}\left(\sum_{n=0}^\infty e^{-nt}\right)\,dt$$ By swapping the series and integral $$\sum_{n=0}^\infty\int^\infty_0\,t^{s-1}e^{-(n+1)t}\,dt = \Gamma(s) \sum_{n=0}^\infty \frac{1}{(n+1)^s}=\Gamma(s)\zeta(s)\,$$
$\forall \,\, n\geq 1$ $$\psi_{n}(z) \, = \, (-1)^{n+1}n!\,\zeta(n+1,z)$$ $$\textit{proof}$$ Use the series representation of the digamma $$\psi_{0}(z) = -\gamma-\frac{1}{z}+ \sum_{n=1}^\infty\frac{z}{n(n+z)}$$ This can be written as the following $$\psi_{0}(z) = -\gamma + \sum_{k=0}^\infty\frac{1}{k+1}-\frac{1}{k+z}$$ By differentiating with respect to … Continue reading
$$\zeta(2k) \, = \, (-1)^{k-1} B_{2k} \frac{2^{2k-1}}{(2k)!}{\pi}^{2k}$$ $$\textit{proof}$$ We start by the product formula of the sine function $$\frac{\sin(z)}{z} = \prod_{n=1}^\infty \left(1-\frac{z^2}{n^2 \, \pi^2} \right)$$ Take the logarithm to both sides \log(\sin(z)) – \log(z) = \sum_{n=1}^\infty \log \left(1-\frac{z^2}{n^2 \, \pi^2} … Continue reading | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9968951344490051, "perplexity": 347.15406699362563}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247512461.73/warc/CC-MAIN-20190222013546-20190222035546-00186.warc.gz"} |
https://topologicalmusings.wordpress.com/tag/definite-integrals/ | You are currently browsing the tag archive for the ‘definite integrals’ tag.
Okay, I thought earlier that part 3 of the above series of posts would be my last one. For some reason, this series has turned out to be a somewhat popular one when considering the fact that a big chunk of blog visitors visit it. Probably, this is due to the fact that the 2008 MIT Integration Bee is going to be held sometime soon – I have no idea exactly when – or perhaps, there are other Integration Bees that are going to be held in other colleges/universities sometime soon. If I get some more feedback/interest, then I will consider posting more results/identities/tricks on this same topic.
As you might have noticed from the title of the post, this identity isn’t related to definite integrals; it is related to indefinite integrals. Of course, since definite integrals form a “subset” of indefinite integrals, we can apply this identity to either one of them.
Let me begin by posing two problems, which I ask you to solve in your head. If you are able to do so, then you probably know the trick that is stated below, and hence you may stop reading this post; else, continue reading.
Problem 1: Evaluate $\displaystyle \int e^x (\frac{x-2}{x^3}) \, dx$.
Problem 2: Evaluate $\displaystyle \int e^x (\ln x + \frac1{x}) \, dx$.
And, here’s our identity.
$(4) \displaystyle \int e^x (f(x) + f'(x)) \, dx = e^x f(x) + C$,
where $C$ is the constant of integration.
Proof: We use integration by parts. Recall, $\displaystyle \int u\, dv = uv - \int v\, du$, where $u \equiv f(x)$ and $v \equiv g(x)$. So, if we let $\displaystyle v = e^x$, then we have $\displaystyle \int e^x f(x) \, dx = \int f(x) \, d(e^x) = f(x) e^x - \int e^x d(f'(x))$, which leads us to our identity.
Now, you should be able to solve the above problems in your head in just a couple of seconds if not less.
Solution 1: $\displaystyle e^x/x^2 + C$.
Solution 2: $e^x \ln x + C$.
If you have found this particular series of posts useful, drop me a comment. Doing so will provide me the motivation to post more stuff on this topic in the near future.
Okay, this is the final part in the above series of posts on some identities related to definite integrals (before I get too lazy and forget to post the same).
So, what is the magic identity? Here it is.
$\displaystyle (3) \int_{-a}^{a} f(x) \, dx = \int_0^a \left( f(x) + f(-x)\right) \, dx$
Proof: Let $t = -x$ in the second integral on the right hand side. Then, we have $\displaystyle \int_0^a f(-x) \, dx = - \int_0^{-a} f(t) \, dt = \int_{-a}^0 f(x) \, dx$, and combining this with the first integral on the right hand side yields the desired result.
Now, apply the above identity to the “difficult” integral in problem $(6)$ from the Integration Bee, Challenging Integrals post to evaluate the integral. The solution turns out to be an easy one. The answer is $\pi /4$, just in case you need to verify.
Here’s our second important identity which is a generalization of the first one.
$(2) \displaystyle \int_a^b f(x) \, dx = \int_a^b f(a+b-x) \, dx$
Proof: Let $t = a+b-x$. Then, $dt = - dx$. Therefore,
$\displaystyle \int_a^b f(a+b-x) \, dx = - \int_b^a f(t) \, dt = \int_a^b f(t) \, dt$. And, we are done.
Let us now look at an integral that is quite easy to solve, though it looks quite formidable at first glance.
Problem 1. (Putnam 1987/B1) Evaluate $\displaystyle \int_2^4 \frac{\sqrt{\ln (9-x)}}{\sqrt{\ln (9-x)} + \sqrt{\ln (3+x)}} \, dx$.
Solution. Let us denote the given integral by $I$. Then applying identity (2) to $I$, we obtain
$\displaystyle I = \int_2^4 \frac{\sqrt{\ln (9-(6-x))}}{\sqrt{\ln (9-(6-x))} + \sqrt{\ln (3+(6-x))}} \, dx$, which implies
$\displaystyle I = \int_2^4 \frac{\sqrt{\ln (3+x)}}{\sqrt{\ln (3+x)} + \sqrt{\ln (9-x)}} \, dx$, which implies
$\displaystyle I + I = \int_2^4 \frac{\sqrt{\ln (9-x)} + \sqrt{\ln (3+x)}}{\sqrt{\ln (3+x)} + \sqrt{\ln (9-x)}}\, dx = \int_2^4 \, dx = 2$.
Hence, $I = 1$. Now, that was very easy!
(I might post a few more problems later, but you get the idea now.)
I will discuss a list of some “identities” that one may employ in evaluating certain types of definite integrals. Without knowing them, it may virtually be impossible to integrate certain functions. The knowledge of such identities greatly enhances one’s ability to integrate!
(Below, $a$ and $b$ are real numbers and $f$ is some “suitable” integrable function in the Riemannian sense.)
Here’s our first one.
$(1) \displaystyle \int_0^a f(x)\, dx = \int_0^a f(a-x)\, dx$.
Proof: Let $t = a-x$. Then, $dt = - dx$. Therefore, $\displaystyle \int_0^a f(a-x)\, dx = - \int_a^0 f(t)\, dt = \int_0^a f(t)\, dt$. And, we are done.
Let us now solve the following integral.
Problem 1. Evaluate $\displaystyle \int_0^{\pi /2} \frac{\sin x}{\sin x + \cos x} \, dx$.
Solution. Let $\displaystyle I = \int_0^{\pi /2} \frac{\sin x}{\sin x + \cos x} \, dx$. Then, applying identity (1) to the above integral, we obtain
$\displaystyle I = \int_0^{\pi /2} \frac{\sin (\pi /2 - x)}{\sin (\pi /2 - x) + \cos (\pi /2 - x)} \, dx = \int_0^{\pi /2} \frac{\cos x}{\cos x + \sin x} \, dx$.
Therefore, $\displaystyle I + I = \int_0^{\pi /2} \frac{\sin x + \cos x}{\sin x + \cos x} \, dx = \int_0^{\pi /2} \, dx = \pi /2$.
Hence, $I = \pi /4$, which is our answer.
Okay, let us now evaluate a more difficult integral that appeared on the Putnam contest in 2005.
Problem 2. (Putnam 2005/A5) Evaluate $\displaystyle \int_0^1\frac {\ln(x + 1)}{x^2 + 1}\, dx.$
Solution. There are several ways of solving this problem, but the easiest way is the one that employs identity (1).
First, we use a “natural” trigonometric substitution, viz. $x = \tan t$. Then, $dx = \sec^2 t \, dt$. Denoting the given integral by $I$, we thus have
$\displaystyle I = \int_0^{\pi / 4} \ln (1 + \tan t) \, dt = \int_0^{\pi / 4} \ln (1 + \tan (\pi /4 - t)) \, dt$
$\ldots$ (applying identity (1) to $I$)
Now, using the identity $\displaystyle \tan (a-b) = \frac{\tan a - \tan b}{1 + \tan a \tan b}$, we get
$\displaystyle I = \int_0^{\pi /4} \ln (\frac{2}{1 + \tan t}) \, dx = \int_0^{\pi /4} \ln 2 \, dx- I$, which implies
$\displaystyle 2I = \ln 2 \int_0^{\pi /4}\, dx$. Hence, $\displaystyle I = \frac{\pi}{8} \ln 2$.
The second identity is a generalization of the first one, and I will discuss it (along with some sample problems) in my next post.
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http://www.maplesoft.com/support/help/MapleSim/view.aspx?path=ModelonHydraulics/Fluids/Constant/Oil | Oil $—$ Hydraulic medium with constant properties
A hydraulic medium with constant properties.
Equations
Density ${\mathrm{\rho }}_{\mathrm{mix}}=\frac{{v}_{\mathrm{gas}\left(\mathrm{lim}\right)}{\mathrm{\rho }}_{\mathrm{gas}\left(\mathrm{ref}\right)}+\left(1-{v}_{\mathrm{gas}\left(\mathrm{lim}\right)}\right){\mathrm{\rho }}_{\mathrm{hyd}\left(\mathrm{ref}\right)}}{\frac{\left(1-{v}_{\mathrm{gas}\left(\mathrm{lim}\right)}\right){\mathrm{\rho }}_{\mathrm{hyd}\left(\mathrm{ref}\right)}}{{\mathrm{\rho }}_{\mathrm{hyd}}}+\frac{{v}_{\mathrm{gas}\left(\mathrm{lim}\right)}T{p}_{\mathrm{ref}}y}{{T}_{\mathrm{ref}}{p}_{\mathrm{lim}}}}$ ${\mathrm{\rho }}_{\mathrm{hyd}}=\mathrm{\rho }$ ${\mathrm{\rho }}_{\mathrm{hyd}\left(\mathrm{ref}\right)}=\mathrm{\rho }$
Bulk Modulus ${\mathrm{\beta }}_{\mathrm{mix}}=\frac{\frac{{C}_{1}}{{\mathrm{\rho }}_{\mathrm{hyd}}}+\frac{{C}_{2}y}{{p}_{\mathrm{lim}}}}{\frac{{C}_{1}}{{\mathrm{\rho }}_{\mathrm{hyd}}{\mathrm{\beta }}_{\mathrm{hyd}}}+\frac{{C}_{2}\left(\frac{y}{{p}_{\mathrm{lim}}}-\left(\frac{{\partial }}{{\partial }p}y\right)\right)}{{p}_{\mathrm{lim}}}}$ ${\mathrm{\beta }}_{\mathrm{hyd}}={\mathrm{\beta }}_{0}$ ${C}_{1}={\mathrm{\rho }}_{\mathrm{hyd}}\left(1-{v}_{\mathrm{gas}\left(\mathrm{lim}\right)}\right)$ ${C}_{2}=\frac{{v}_{\mathrm{gas}\left(\mathrm{lim}\right)}T{p}_{\mathrm{ref}}}{{T}_{\mathrm{ref}}}$
Dynamic Viscosity The dynamic viscosity is $\mathrm{\eta }=\mathrm{\nu }\mathrm{\rho }$.
Kinematic Viscosity ${\mathrm{\nu }}_{\mathrm{mix}}=\frac{{\mathrm{\nu }}_{\mathrm{gas}}{V}_{\mathrm{gasFree}\left(\mathrm{nom}\right)}+{\mathrm{\nu }}_{\mathrm{hyd}}{V}_{\mathrm{hyd}\left(\mathrm{nom}\right)}}{{V}_{\mathrm{tot}\left(\mathrm{nom}\right)}}$ ${V}_{\mathrm{tot}\left(\mathrm{nom}\right)}={V}_{\mathrm{gasFree}\left(\mathrm{nom}\right)}+{V}_{\mathrm{hyd}\left(\mathrm{nom}\right)}$ ${V}_{\mathrm{hyd}\left(\mathrm{nom}\right)}=\frac{{\mathrm{\rho }}_{\mathrm{hyd}\left(\mathrm{ref}\right)}\left(1-{v}_{\mathrm{gas}\left(\mathrm{lim}\right)}\right)}{{\mathrm{\rho }}_{\mathrm{hyd}}}$ ${V}_{\mathrm{gasFree}\left(\mathrm{nom}\right)}=\frac{{v}_{\mathrm{gas}\left(\mathrm{lim}\right)}T{p}_{\mathrm{ref}}y}{{T}_{\mathrm{ref}}{p}_{\mathrm{lim}}}$ ${\mathrm{\nu }}_{\mathrm{hyd}}={\mathrm{\nu }}_{0}$ ${\mathrm{\nu }}_{0}=\mathrm{\nu }$
Miscellaneous ${p}_{\mathrm{lim}}=\mathrm{max}\left({p}_{\mathrm{abs}},{p}_{abs\left(\mathrm{min}\right)}\right)$ ${p}_{abs\left(\mathrm{min}\right)}=0.001$ ${p}_{\mathrm{sat}}=\frac{{v}_{\mathrm{gas}}{p}_{\mathrm{ref}}}{\mathrm{av}}$ ${v}_{\mathrm{gas}\left(\mathrm{lim}\right)}=\mathrm{min}\left(\mathrm{max}\left(0.000001,{v}_{\mathrm{gas}}\right),1\right)$ $y={\left(1-z\right)}^{5}\left(70{z}^{4}+35{z}^{3}+15{z}^{2}+5z+1\right)$ $z={\begin{array}{cc}{\begin{array}{cc}1& {p}_{\mathrm{sat}}<{p}_{\mathrm{abs}}\\ 0& {p}_{\mathrm{abs}}<{p}_{\mathrm{vap}}\\ \mathrm{max}\left(0,\frac{{p}_{\mathrm{abs}}-{p}_{\mathrm{vap}}}{{p}_{\mathrm{sat}}-{p}_{\mathrm{vap}}}\right)& \mathrm{otherwise}\end{array}& {p}_{\mathrm{vap}}<{p}_{\mathrm{sat}}\\ {\begin{array}{cc}1& {p}_{\mathrm{sat}}<{p}_{\mathrm{abs}}\\ \mathrm{max}\left(0,\frac{{p}_{\mathrm{abs}}}{\mathrm{max}\left(0.0001,{p}_{\mathrm{sat}}\right)}\right)& \mathrm{otherwise}\end{array}& \mathrm{otherwise}\end{array}$ The variables $y$ and $z$ are, respectively, the fractions of undissolved and dissolved gas in the mixture.
General Parameters
Name Default Units Description Modelica ID ${T}_{0}$ $293.15$ $K$ Working temperature T0 ${p}_{0}$ ${10}^{7}$ $\mathrm{Pa}$ Reference pressure p0 ${p}_{\mathrm{vapour}}$ $100$ $\mathrm{Pa}$ Absolute vapour pressure p_vapour ${p}_{\mathrm{atm}}$ ${10}^{5}$ $\mathrm{Pa}$ Atmospheric pressure: in case change is wanted for high altitudes p_atm display labels $\mathrm{false}$ Display labels display_labels $\mathrm{\nu }$ $4.6·{10}^{-5}$ $\frac{{m}^{2}}{s}$ Kinematic viscosity nu $\mathrm{\beta }$ $\mathrm{Pa}$ Bulk modulus beta $\mathrm{\rho }$ $\frac{\mathrm{kg}}{{m}^{3}}$ Density rho
Gas Parameters
Name Default Units Description Modelica ID ${v}_{\mathrm{gas}}$ $1·{10}^{-6}$ $1$ Gas/(hydraulic medium) volume fraction at atmospheric pressure and 0 degC v_gas $\mathrm{av}$ $6.8$ Bunsen coefficient av ${\mathrm{\rho }}_{\mathrm{gas}\left(0\right)}$ $1.28$ $\frac{\mathrm{kg}}{{m}^{3}}$ Gas density at atmospheric pressure and 0 degC rho_gas_0 ${\mathrm{\nu }}_{\mathrm{gas}}$ $1.5·{10}^{-5}$ $\frac{{m}^{2}}{s}$ Gas kinematic viscosity nu_gas
Constant Parameters
Name Default Units Description Modelica ID $A$ $0$ Coefficient for temperature dependent viscosity A $B$ $0$ Coefficient for temperature dependent viscosity B $\mathrm{\alpha }$ $0$ Coefficient for pressure dependent viscosity alpha
Constants
Name Value Units Description Modelica ID ${T}_{\mathrm{ref}}$ $273.15$ $K$ Reference temperature for v_air T_ref ${p}_{\mathrm{ref}}$ $1·{10}^{5}$ $\mathrm{Pa}$ Reference pressure for v_air p_ref | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 66, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.963082492351532, "perplexity": 4410.369043901904}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917119120.22/warc/CC-MAIN-20170423031159-00081-ip-10-145-167-34.ec2.internal.warc.gz"} |
https://infoscience.epfl.ch/record/77721 | Infoscience
Journal article
On iterative algorithms for linear least squares problems with bound constraints
Three new iterative methods for the solution of the linear least squares problem with bound constraints are presented and their performance analyzed. The first is a modification of a method proposed by Lötstedt, while the two others are characterized by a technique allowing for fast active set changes, resulting in noticeable improvements in the speed with which constraints active at the solution are identified. The numerical efficiency of those algorithms is experimentally studied, with particular emphasis on the dependence on the starting point and the use of preconditioning for ill-conditioned problems. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9168418645858765, "perplexity": 290.98912703474736}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560281332.92/warc/CC-MAIN-20170116095121-00007-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://en.wikiversity.org/wiki/Microeconomics/Consumer_Theory | # Microeconomics/Consumer Theory
## Consumer Theory
The aim of this section is to explain a fundamental problem in economics, the derivation of a consumer’s demand function, in a very simple way. The article is organized as follows:
• Conceptual review of assumptions in demand theory
• Description of the Utility Maximization Problem
• Derivation of the Expenditure Minimization Problem
• Relationship between both problems
## Assumptions
The consumer theory assumes that the consumer is rational. This implies that his preferences satisfy the following properties: 1. They are complete; that is, given any set of possible bundles of goods, the consumer is always capable of deciding which one is preferable to the others and then ranking them in terms of preference.
2. They are reflexive; it means that any bundle is at least as good as itself.
3. They are transitive; meaning that if a bundle ${\displaystyle A\,}$ is preferred to a bundle ${\displaystyle B\,}$, and this bundle ${\displaystyle B\,}$ is preferable to a third bundle ${\displaystyle C\,}$, then it is implied that the first bundle ${\displaystyle A\,}$ will be preferred to the bundle ${\displaystyle C\,}$ .
4. They are continuous; there are no big jumps in the ranking of alternatives.
The fulfillment of these properties ensures that consumer’s preferences are consistent and can be represented by an utility function, ${\displaystyle U(.)\,}$ such that if bundle ${\displaystyle A\,}$ is preferred to bundle ${\displaystyle B\,}$, then ${\displaystyle U(A)>U(B)\,}$
The locus of all bundles that give a certain level of utility to the consumer constitutes an indifference curve (or level curve), which is the usual way of representing preferences. Nevertheless, in spite of these four properties, there is still the possibility of having “special cases” such as the existence of perfect substitutes or perfect complements, among others, which lead to special shapes for the indifference curves. For avoiding these cases, two additional properties are assumed:
5. Preferences are monotonic, or “more is preferred to less”; this implies that, given any set of two bundles, if one of them contains at least as much of all goods and more of one good than the other, then the first bundle will be preferred to the second.
6. Preferences are convex; that is, any combination of two equally preferable bundles will be more desirable than these bundles.
These five properties confer a special shape to level curves: they are downward slopping and convex.
## Utility Maximization Problem
This section develops the Utility Maximization Problem (UMP) for the simplest case of only two goods. The model can easily be generalized to ${\displaystyle N\,}$ goods.
Assume that there are two goods, ${\displaystyle x_{1}\,}$ and ${\displaystyle x_{2}\,}$, whose prices are ${\displaystyle p_{1}\,}$ and ${\displaystyle p_{2}\,}$, respectively. The consumer has a fixed amount of income, ${\displaystyle m\,}$, for spending on consumption, and his preferences are represented by a generic utility function, ${\displaystyle U(x_{1},x_{2})\,}$ , with ${\displaystyle U_{1}\,>0}$, ${\displaystyle U_{2}\,>0}$. The consumer’s aim is to obtain the maximum possible utility but he is constrained by his level of income. He cannot spend more than ${\displaystyle m\,}$, thus he faces a budget constraint: ${\displaystyle p_{1}x_{1}+p_{2}x_{2}=m\,}$1
Formally, the problem can be formulated as follows:
Max ${\displaystyle U(x_{1},x_{2})\,}$ subject to ${\displaystyle p_{1}x_{1}+p_{2}x_{2}=m\,}$
And it can be solved by the Lagrange Multipliers method:
Max ${\displaystyle L=U(x_{1},x_{2})+\lambda \left({m-p_{1}x_{1}-p_{2}x_{2}}\right)\,}$
${\displaystyle \left\{{x_{1},x_{2},\lambda }\right\}}$
The first order conditions (FOC) are:
${\displaystyle (1)L_{1}=U_{1}-\lambda p_{1}=0\,}$
${\displaystyle (2)L_{2}=U_{2}-\lambda p_{2}=0\,}$
${\displaystyle (3)L_{\lambda }=m-p_{1}x_{1}-p_{2}x_{2}=0\,}$
Note that conditions ${\displaystyle (1)\,}$ and ${\displaystyle (2)\,}$ imply that ${\displaystyle {\frac {U_{1}}{U_{2}}}={\frac {p_{1}}{p_{2}}}}$. That is, the marginal rate of substitution (MRS) must be equal to the relation of prices, and it means that the indifference curve must be tangent to the budget constraint.
The second order conditions (SOC) are:
${\displaystyle (4)\left|H_{U}\right|={\begin{vmatrix}U_{11}&U_{12}&-p_{1}\\U_{21}&U_{22}&-p_{2}\\-p_{1}&-p_{2}&0\end{vmatrix}}>0}$
It can be demonstrated that the SOC imply that indifference curves are convex. The reciprocal is true only for the case of two goods.
The solutions to the FOC are ${\displaystyle x_{1}^{M}}$,${\displaystyle x_{2}^{M}}$,${\displaystyle \lambda ^{M}\,}$.They depend on prices and income, thus they can be written as ${\displaystyle x_{1}^{M}(p_{1},p_{2},m)}$,${\displaystyle x_{2}^{M}(p_{1},p_{2},m)}$,${\displaystyle \lambda ^{M}(p_{1},p_{2},m)\,}$.The functions ${\displaystyle x_{1}^{M}}$ and ${\displaystyle x_{2}^{M}}$,< are the Marshallian Demand Functions. They represent the amount of goods ${\displaystyle x_{1}\,}$ and ${\displaystyle x_{2}\,}$, that the consumer is willing to purchase given their prices, income and tastes.
Another concept that emerges from the UMP is the Indirect Utility Function, and it can be obtained by replacing the Marshallian demands into the utility function. By definition, it also is a function of prices and income, then it can be written as ${\displaystyle U^{*}(p_{1},p_{2},m)\equiv U(x_{1}^{M}(p_{1},p_{2},m),x_{2}^{M}(p_{1},p_{2},m))}$. Intuitively, it represents the maximum utility that the consumer can achieve for any given values of ${\displaystyle p_{1},p_{2},m\,}$.
Note that, because of the Envelop Theorem, it must be the case that ${\displaystyle {\frac {\partial U^{*}}{\partial m}}=\lambda ^{M}(p_{1},p_{2},m)}$. It implies that the Lagrange multiplier can be thought as the marginal utility of income. That is, it represents the rate of change of the maximum utility that is derived from an infinitesimal rise in income.
1 Striclty, the constraint is ${\displaystyle p_{1}x_{1}+p_{2}x_{2}\leq m}$, but the monotonicity assumption ensures that he will spend all his income.
## Expenditure Minimization Problem
The Expenditure Minimization Problem (EPM) is the dual problem of the UMP and it can be thought as follows. Consider a consumer who gets utility through the consumption of the two goods. In this case, there is no restriction on the income to be spent, but the consumer must be on a certain level curve, ${\displaystyle U^{0}\,}$. Given this constraint, his objective is to reach this indifference curve with the minimum possible expenditure. Therefore, the problem is:
Min ${\displaystyle G=p_{1}x_{1}+p_{2}x_{2}\,}$ subject to ${\displaystyle U(x_{1},x_{2})=U^{0}\,}$
Again, this constrained optimization can be solved by the Lagrange Multipliers method:
Min ${\displaystyle L=p_{1}x_{1}+p_{2}x_{2}+\mu \left({U^{0}-U(x_{1},x_{2})}\right)\,}$
${\displaystyle \left\{{x_{1},x_{2},\mu }\right\}}$
The FOC of this program are:
${\displaystyle (5)L_{1}=p_{1}-\mu U_{1}=0\,}$
${\displaystyle (6)L_{2}=p_{2}-\mu U_{2}=0\,}$
${\displaystyle (7)L_{\mu }=U^{0}-U(x_{1},x_{2})=0\,}$
And the SOC are:
${\displaystyle (8)\left|H_{E}\right|={\begin{vmatrix}-\mu U_{11}&-\mu U_{12}&-U_{1}\\-\mu U_{21}&-\mu U_{22}&-U_{2}\\-U_{1}&-U_{2}&0\end{vmatrix}}<0}$
Note that conditions ${\displaystyle (5)\,}$ and ${\displaystyle (6)\,}$ imply the same tangency condition than the UMP: ${\displaystyle {\frac {U_{1}}{U_{2}}}={\frac {p_{1}}{p_{2}}}}$. In this program, it means that the expenditure function must be tangent to the indifference curve ${\displaystyle U^{0}\,}$.
Solving equations ${\displaystyle (5)\,}$ to ${\displaystyle (7)\,}$ gives the optimal levels of ${\displaystyle x_{1}^{H}}$,${\displaystyle x_{2}^{H}}$,${\displaystyle \mu ^{H}\,}$.The demand functions ${\displaystyle x_{1}^{H}}$ and ${\displaystyle x_{2}^{H},}$ are the Hicksian (or Compensated) Demand Functions. Note that these demands depend on prices and the utility level, therefore they are denoted ${\displaystyle x_{1}^{H}(p_{1},p_{2},U^{0})}$,${\displaystyle x_{2}^{H}(p_{1},p_{2},U^{0})}$,${\displaystyle \mu ^{H}(p_{1},p_{2},U^{0})\,}$.
The function resulting from replacing the Hicksian demands into the expenditure function gives the minimum expenditure necessary to reach ${\displaystyle U^{0}\,}$ for any given values of ${\displaystyle p_{1},p_{2},U^{0}}$. It is called the Indirect Expenditure Function and is denoted ${\displaystyle E^{*}\equiv p_{1}x_{1}^{H}(p_{1},p_{2},U^{0})+p_{2}x_{2}^{H}(p_{1},p_{2},U^{0})}$.
Again, the Lagrange multiplier has a special interpretation. The Envelop Theorem implies that ${\displaystyle {\frac {\partial E^{*}}{\partial U^{0}}}=\mu ^{H}(p_{1},p_{2},U^{0})}$, meaning that the Lagrange multiplier represents the rate of change of the expenditure function given a change in the utility level to reach.
## References
Nicholson, W "Microeconomic Theory"
Chiang, A. Métodos fundamentales de economía matemática. Mc Graw-Hill. 2006.
Silberberg, E. The Structure of Economics. McGraw Hill, 3rd Edition. 2000.
Varian, H. Microeconomía intermedia. Antoni Bosch, 5ta edición. 2004.
Varian, H. Microeconomic Analysis. W.W. Norton, 3rd Edition. 1992. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 73, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9513870477676392, "perplexity": 579.3590750305029}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128323895.99/warc/CC-MAIN-20170629084615-20170629104615-00024.warc.gz"} |
https://mathoverflow.net/questions/344391/delooping-a-fibration-sequence-with-loopspace-fiber-and-finite-cw-complexes | # Delooping a fibration sequence with loopspace fiber and finite CW complexes
The following question is somewhat similar to a previous one on MathOverflow, except that my application does not directly involve Eilenberg-MacLane spaces $$K(G,n)$$, and so I don't see the immediate need for $$n$$-simple maps as in the theory of Postnikov $$k$$-invariants.
Suppose that $$\Omega T \longrightarrow F \longrightarrow E$$ is a homotopy fibration sequence, where $$T$$ and $$F$$ are connected finite CW complexes. Is there a quantification in some sort of obstruction theory for extending this sequence further to the right in the form $$F \longrightarrow E \longrightarrow T$$ ? This is a 1950s-era topology question.
Of course, the original sequence extends further to the right as $$F \longrightarrow E \longrightarrow B hAut(\Omega T)$$. First, this is by Stasheff's paper for finite CW complexes as the fiber and any CW base, working with products in the full topological category [Sta63]. Later, this is by May's simplicial upgrade to infinite CW complexes as the fiber, working in the compactly generated category [9.5, 9.8][May75].
In my application, $$T$$ happens to have the structure of a Lie group, but this is a happy accident of its dimension, and I'm hoping for an answer that can work independent of this fact.
[Sta63]: James D Stasheff, A classification theorem for fibre spaces, Topology 2, 1963
[May75]: J Peter May, Classifying spaces and fibrations, Memoirs AMS 1:155, 1975
EDIT: As suggested, I renamed the base space $$B$$ to $$T$$, in order to not confuse it with the classifying space functor. Also, I added two citations.
• (I rename $B$ to $T$ to avoid confusion) I think the only obstruction is factorizing action map $E \to BAut(\Omega T)$ through delooping of tautological action $T \to BAut(\Omega T)$ which is done by usual obstruction theory. – Denis T. Oct 22 '19 at 21:50
• @Denis T. : Любезно, what is the explicit formula for this tautological action? Does it it assume that $T$ is a topological group, say by pointwise-conjugating a loop by an element of $T$? If so, isn't this instead a map $T \longrightarrow Aut(\Omega T)$? – Qayum Khan Oct 22 '19 at 22:49
• @QayumKhan The tautological action is the one corresponding to the pathspace fibration $\Omega T\to P T\to T$ (morally it is the action of $\Omega T$ on itself by left(?) multiplication) – Denis Nardin Oct 23 '19 at 7:48
• @QayumKhan DenisNardin is right, I mean just an application of classifying space functor to map $\Omega T \to Aut(\Omega T)$ representing left (more precisely, the side $\pi_1$ of base acts on fiber in your preferrable conventions) multiplication. Also possibly you want to use Moore loops for that to avoid some nuances with non-strictly associative actions etc. – Denis T. Oct 23 '19 at 16:55
This question is addressed in the paper
Ganea, T., Induced fibrations and cofibrations, Trans. Am. Math. Soc. 127, 442-459 (1967). ZBL0149.40901.
A first observation is that $$\Omega T\to F\to E$$ extends to the right if and only if it is induced from the based path fibration $$\Omega T\to PT\to T$$ by a map $$p:E\to T$$. In Section 2, various sufficient conditions for such a fibration to be induced are given.
Two sample results:
Corollary 2.5: Suppose that $$\pi_q(E)\neq 0$$ only if $$m\le q\le n + m-1$$ and that $$\pi_q(\Omega T) \neq 0$$ only if $$n\le q\le n + m-1$$, where $$n\ge m\ge 2$$. If the Whitehead product pairing $$W:\pi_n(\Omega T) \otimes \pi_m(F)\to \pi_{n + m-1}(F)$$ vanishes, then $$\Omega T\to F\to E$$ is induced.
Theorem 2.10: Suppose $$\Omega T$$, $$F$$ and $$E$$ all have the homotopy type of aspherical CW complexes. Then $$\Omega T\to F\to E$$ is induced if and only if the image of the induced map $$\pi_1(\Omega T)\to \pi_1(F)$$ lies in the center.
• I would be very interested in knowing the level of generality in which these results hold. Do they work, for instance, in any $\infty$-topos? – skd Oct 28 '19 at 0:26
• @MarkGrant : I appreciate the partial answer; however, my specific application (which I didn't specify in the question) fails to satisfy the connectivity hypotheses of your answer here and also the previously linked MathOverflow question. Specifically, all of my finite CW complexes $T, F, E$ are non-simply connected and have homotopy groups in infinitely many dimensions. My opinion is that the results in Ganea's paper are more-or-less a 'thickening' of the preexisting result on Eilenberg--MacLane spaces that he advances. – Qayum Khan Oct 28 '19 at 17:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 26, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9071066379547119, "perplexity": 451.42242354792126}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655883439.15/warc/CC-MAIN-20200703215640-20200704005640-00128.warc.gz"} |
https://m-phi.blogspot.co.uk/2013/05/nominalism-vs-syntax.html | ## Friday, 24 May 2013
### Nominalism vs. Syntax
It is difficult to maintain, consistently, the following two claims:
(i) There are no abstracta.
(ii) There are syntactical entities (and they behave as our standard accounts say they do).
Consider, for example, how one defines a language $L$. Beginning with two building blocks, $A$ and $B$, we say that $\{A,B\}$ is the alphabet. It's usually implicit, but sometimes needs to be stated, that $A \neq B$. (One has to state this in a formalized theory of syntax.)
For the many of the usual purposes of syntactical theory, it does not matter what these building blocks $A$ and $B$ are. They could be two eggs. They could be the numbers 7 and $\aleph_{57}$. They could be the letter types "a" and "$\aleph$". Or they could be two of my guitars. Or they could be two tokens of the letters "a" and "$\aleph$''. It does not matter. And the fact that it doesn't matter plays an important role in Gödel's incompleteness results, where the leading ideas involve the structural interplay between the properties of numbers, sequences, syntactical entities and finitary computations (plus, times, exponentation, and so on).
Let our alphabet $\Sigma = \{A,B\}$. Next we consider the set $\Sigma^{\ast}$ of finite sequences drawn from $\Sigma$. Finite sequences drawn from an alphabet are usually called,
• strings
• words
• expressions
These are the syntactical entities that one is discussing, quantifying over, referring to, etc. The crucial point is that these are sequences from the alphabet. In particular, $\Sigma^{\ast}$ is closed under sequence concatenation. So,
if $\alpha, \beta \in \Sigma^{\ast}$, then $\alpha ^{\frown} \beta \in \Sigma^{\ast}$.
And:
$|\Sigma^{\ast}| = \aleph_0$.
This means that there are $\aleph_0$-many syntactical entities. The terms $a_0, a_2, \dots, a_n$ occurring in a sequence $\alpha = (a_0, a_1, \dots, a_n)$ may well be concreta. But the sequence $\alpha$ itself is a (possibly mixed) abstractum. More exactly, a sequence is usually understood as a function:
$\alpha : \{0,1,\dots,n\} \to \Sigma$.
This is not mandated. What is mandated is that sequences are individuated in a certain way:
$\alpha = \beta$ if and only if $\alpha$ and $\beta$ have the same terms, in the same order.
So, e.g,, if $(a_0, a_1, \dots, a_n) = (b_0, b_1, \dots, b_k)$, then $n = k$, and $a_0 = b_0$, $a_1 = b_1$, and so on.
Normally, one goes on to define certain special subsets $X, Y, \dots$ of $\Sigma^{\ast}$. Perhaps these are the formulas, or terms, and whatnot. Usually, the definitions satisfy certain computational constraints: e.g., perhaps an inductive definition. So, $X$ might be, e.g., a recursive set or a recursively enumerable set. But for this discussion here, these subsets don't matter. They're subsets, and we discussing the enclosing set, of all strings from the alphabet.
Return to (i) and (ii). Suppose (i) is true. So, there are no abstracta. Hence, there are, a fortiori, no mixed abstracta; and therefore, there are no sequences; and, therefore, there are no strings; and therefore no syntactical entities, except a very, very small number of tokens, which are not closed under concatenation. Hence, (ii) is false.
One might suggest that these claims (i) and (ii) are "really" consistent under some reinterpretation $I$. But what exactly is this $I$? How is $I$ defined? Is it a secret?
I think that the optimal nominalistic responses to the inconsistency of (i) and (ii) are:
• either to accept the inconsistency and thus simply accept that (ii) (i.e., syntax) is false (see Quine & Goodman 1947, "Steps Toward a Constructive Nominalism"),
• or to reinterpret (ii,) to make it "true under a reinterpretation", so that "syntactical entity" refers perhaps to possibilia (i.e., possible concrete tokens: see Burgess & Rosen 1997, A Subject with No Object, for some discussion of this) or perhaps to some kind of physical entity (such as perhaps spacetime regions), assuming there are sufficiently many.
I'm not optimistic about either kind of approach. The first approach is an error theory, and is too damaging to science. An error theory for morality is one thing; an error theory for science is another! The second, "hermeneutic", approach invokes possibilia and this raises similar sceptical and metaphysical worries as abstracta do. (See the final chapter of Shapiro 1997, Philosophy of Mathematics: Structure and Ontology.) It also raises the question of what grounds one might give for the reinterpretation. A classic discussion of some of these topics is Burgess 1983, "Why I am not a Nominalist".
So far as I can tell, the more recent "weaseling" approach to nominalism---which I think is extremely interesting---proposed by Melia 2000 ("Weaseling Away the Indispensability Argument", Mind) and endorsed and developed recently by Yablo 2012 ("Explanation, Extrapolation and Existence", Mind) doesn't seem to apply in the syntactic case. But I'm not sure.
#### 1 comment:
1. I'm skeptical of your reasoning.
When we study a formal language, or try to define it, then we are doing mathematics and what we are defining is a mathematical idealization of language. So, of course, the entities in this idealization are mathematical entities and count as abstracta.
The question of the existence of abstracta, including those you discuss, still seems entirely distinct from the question of the existence of syntactic entities - the actual syntactic entities that are used, rather than their mathematical idealizations. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8913487195968628, "perplexity": 1020.7976253517711}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891815934.81/warc/CC-MAIN-20180224191934-20180224211934-00541.warc.gz"} |
http://tex.stackexchange.com/questions/34942/use-tabularx-in-lyx-instead-of-default-tabular | # Use tabularx in LyX instead of default tabular
I recently got into LyX and I’m getting along quite well.
The only thing that bothers me is that LyX’s table feature produces really sloppy-looking tables by default, i.e. double borders and stuff like that.
I would love to tell LyX to use the tabularx environment by default for the tables I enter via the GUI element, so I can get tables with 100% width matching the justified paragraphs. Concerning the borders, I would like to remove all of them but a \toprule above and beneath the first row of the table and a \bottomrule beneath the last row. These features are added by the ctable package which is included in my document preamble. Is there any way I can achieve these things without performing crazy stuff like regex?
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Welcome to TeX.sx! A tip: You can use backticks to mark your inline code as I did in my edit. – Joseph Wright Nov 15 '11 at 17:15
write into the preamble of your document (document->settings->preamble)
\usepackage{array}
\def\tabular{%
\setlength\dimen@{\linewidth}%
\edef\@halignto{to\the\dimen@}\@tabular}
\newcolumntype{C}{@{\extracolsep{\fill}}c}
then all tables are by default converted into tabular*. Choose for the first column the column type C or write alternetively the definition @{\extracolsep{\fill}}c for the first column via the tabular menu
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This seems to be a good starting point, but instead of evenly spacing out the colums of my 3-column-table, this just adds a fourth, empty column covering the remaining text area. Note that the author line in the heading is awkwardly aligned as well. – Christoph Nov 15 '11 at 18:47
with tabular* you can not stretch all columns. See edited example. However, I suppose, that you want the tabularx instead of tabular* – Herbert Nov 15 '11 at 19:03
Thank you for your comment, it was really the tabularx environment I was looking for. I used \begin{tabularx}{\textwidth}{XXX} without your custom preamble to create the effect I was looking for. But nevertheless, LyX won’t format tables inserted via the GUI this way and I couldn’t wrap my head around the border issue, yet. Seems like I will stick to entering code for my tables instead of using the GUI. If only one could use CSS to manipulate the PDF output, I would know exactly what to do! But anyways, thank you very much for your help. – Christoph Nov 15 '11 at 19:44
I guess you could perform the following two global replacement operations:
• replace all "\end{tabular}" strings with "\end{tabular*}"
• replace all "\begin{tabular}{" strings with "\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}"
Regarding the replacment of the \hline commands with \toprule, \midrule, and \bottomrule: Whether it's possible to do a global search-and-replace will depend greatly on how you've entered the \hline` commands so far. Without detailed knowledge of this aspect, I can't give a suggestion for how to perform this step.
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Thank you for your answer but I think you misinterpreted / I misformulated my question. I can of course enter the right TeX code to produce the tables the way I want them to look, but my intention is to „tell“ LyX to use this code for the tables I entered via the GUI automatically, i.e. without having to enter any TeX code. I will update my question to reflect these goals. – Christoph Nov 15 '11 at 18:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8950557112693787, "perplexity": 1165.4890737795245}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701145578.23/warc/CC-MAIN-20160205193905-00247-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://optimization-online.org/author/vavasis/ | ## MGProx: A nonsmooth multigrid proximal gradient method with adaptive restriction for strongly convex optimization
We study the combination of proximal gradient descent with multigrid for solving a class of possibly nonsmooth strongly convex optimization problems. We propose a multigrid proximal gradient method called MGProx, which accelerates the proximal gradient method by multigrid, based on utilizing hierarchical information of the optimization problem. MGProx applies a newly introduced adaptive restriction operator … Read more
## Computational complexity of decomposing a symmetric matrix as a sum of positive semidefinite and diagonal matrices
We study several variants of decomposing a symmetric matrix into a sum of a low-rank positive semidefinite matrix and a diagonal matrix. Such decompositions have applications in factor analysis and they have been studied for many decades. On the one hand, we prove that when the rank of the positive semidefinite matrix in the decomposition … Read more
## Nonlinear conjugate gradient for smooth convex functions
The method of nonlinear conjugate gradients (NCG) is widely used in practice for unconstrained optimization, but it satisfies weak complexity bounds at best when applied to smooth convex functions. In contrast, Nesterov’s accelerated gradient (AG) method is optimal up to constant factors for this class. However, when specialized to quadratic function, conjugate gradient is optimal … Read more
## On identifying clusters from sum-of-norms clustering computation
Sum-of-norms clustering is a clustering formulation based on convex optimization that automatically induces hierarchy. Multiple algorithms have been proposed to solve the optimization problem: subgradient descent by Hocking et al.\ \cite{hocking}, ADMM and ADA by Chi and Lange\ \cite{Chi}, stochastic incremental algorithm by Panahi et al.\ \cite{Panahi} and semismooth Newton-CG augmented Lagrangian method by Yuan … Read more
## Provable Overlapping Community Detection in Weighted Graphs
Community detection is a widely-studied unsupervised learning problem in which the task is to group similar entities together based on observed pairwise entity interactions. This problem has applications in diverse domains such is social network analysis and computational biology. There is a significant amount of literature studying this problem under the assumption that the communities … Read more
## A termination criterion for stochastic gradient descent for binary classification
We propose a new, simple, and computationally inexpensive termination test for constant step-size stochastic gradient descent (SGD) applied to binary classification on the logistic and hinge loss with homogeneous linear predictors. Our theoretical results support the effectiveness of our stopping criterion when the data is Gaussian distributed. This presence of noise allows for the possibility … Read more
## Potential-based analyses of first-order methods for constrained and composite optimization
We propose potential-based analyses for first-order algorithms applied to constrained and composite minimization problems. We first propose “idealized” frameworks for algorithms in the strongly and non-strongly convex cases and argue based on a potential that methods following the framework achieve the best possible rate. Then we show that the geometric descent (GD) algorithm by Bubeck … Read more
## Recovery of a mixture of Gaussians by sum-of-norms clustering
Sum-of-norms clustering is a method for assigning $n$ points in $\R^d$ to $K$ clusters, $1\le K\le n$, using convex optimization. Recently, Panahi et al.\ proved that sum-of-norms clustering is guaranteed to recover a mixture of Gaussians under the restriction that the number of samples is not too large. The purpose of this note is to … Read more
## A single potential governing convergence of conjugate gradient, accelerated gradient and geometric descent
Nesterov’s accelerated gradient (AG) method for minimizing a smooth strongly convex function $f$ is known to reduce $f({\bf x}_k)-f({\bf x}^*)$ by a factor of $\epsilon\in(0,1)$ after $k=O(\sqrt{L/\ell}\log(1/\epsilon))$ iterations, where $\ell,L$ are the two parameters of smooth strong convexity. Furthermore, it is known that this is the best possible complexity in the function-gradient oracle model of … Read more
Nesterov’s accelerated gradient method for minimizing a smooth strongly convex function $f$ is known to reduce $f(\x_k)-f(\x^*)$ by a factor of $\eps\in(0,1)$ after $k\ge O(\sqrt{L/\ell}\log(1/\eps))$ iterations, where $\ell,L$ are the two parameters of smooth strong convexity. Furthermore, it is known that this is the best possible complexity in the function-gradient oracle model of computation. The … Read more | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.897094190120697, "perplexity": 541.0747754886559}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296943749.68/warc/CC-MAIN-20230322020215-20230322050215-00022.warc.gz"} |
https://goldbook.iupac.org/terms/view/H02929 | Wikipedia - Гіпсохромний зсув (uk) hypsochromic shift
https://doi.org/10.1351/goldbook.H02929
Shift of a spectral band to higher frequency or shorter @[email protected] upon substitution or change in medium. It is informally referred to as @[email protected] | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9679793119430542, "perplexity": 4165.716428125758}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710980.82/warc/CC-MAIN-20221204204504-20221204234504-00288.warc.gz"} |
http://emathematics.matematicas.us/similarity.php?tp=2 | User:
Similarity
Similar Polygons
Two polygons are similar if these two facts both must be true:
• Corresponding angles are equal.
• The ratios of pairs of corresponding sides must be equal.
(in other words, if they are proportional).
The symbol for "is similar to" is $\sim$
quadrilateral ABCD $\sim$ quadrilateral EFGH
This means:
m<A = m<E, m<B=m<F, m<C=m<G, m<D=m<H and
$\frac{AB}{EF}=\frac{BC}{GH}=\frac{CD}{GH}=\frac{AD}{EH}$
It is possible for a polygon to have one of the above facts true without having the other fact true. The following two examples show how that is possible:
Quadrilaterals that are not similar to one another.
Even though the ratios of corresponding sides are equal, corresponding angles are not equal $(90^o\;\neq\;120^o,\;90^o\;\neq\;60^o)$
Quadrilaterals that are not similar to one another.
Even though corresponding angles are equal, the ratios of each pair of corresponding sides are not equal $(\frac{3}{3}\;\neq\;\frac{5}{3})$
Typically, problems with similar polygons ask for missing sides. To solve for a missing length, find two corresponding sides whose lengths are known. After we do this, we set the ratio equal to the ratio of the missing length and its corresponding side and solve for the variable.
Given that polygon WXYZ $\sim$ polygon ABCD, find the missing measure:
The missing measure m is the lenght of $\bar{XY}$. Write a proportion.
$\frac{m}{12}=\frac{15}{10}$ XY=m, BC=12, YZ=15, and CD=10 m·10=12·15 Find the cross products. 10m=180 Multiply m=18 Divide each side by 10
Find the scale factor from polygon WXYZ to polygon ABCD
scale factor: $\frac{YZ}{CD}=\frac{15}{10}\;or\;\frac{3}{2}$ The scale factor is the constant of proportionality
A length on polygon WXYZ is $\frac{3}{2}$ times as long as a corresponding length on polygon ABCD.
Let m represent the measure of $\bar{XY}$:
$m=\frac{3}{2}\;\cdot\;12$
m=18
Congruent Versus Similar polygons
Congruent polygons have the same shape and the same size, while similar figures have the same shape buy may have different sizes. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 12, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8986557126045227, "perplexity": 1235.9864026930115}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463613135.2/warc/CC-MAIN-20170529223110-20170530003110-00261.warc.gz"} |
http://math.stackexchange.com/questions/365269/intuition-behind-the-axiom-of-choice?answertab=oldest | # Intuition behind the Axiom of Choice
Why is it different to make one choice or many choices than to make infinite choices from a theoretical point of view in which indeed you are not going to do any?
How could that be different from making infinite additions for example $\sum_{i=0}^{+\infty}A_n$.
Thanks a lot.
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But infinite sums are very different from finite sums. For any positive integer $n$ and real numbers $x_1,\dots,x_n$ the sum $x_1+\ldots+x_n$ is defined directly in terms of the binary operation of addition, and it always exists. If $\langle x_n:n\in\Bbb N\rangle$ is an infinite sequence of real numbers, however, we can’t even define $\sum_{n=0}^\infty x_n$ using just the binary operation of addition: we also need the whole machinery of limits. Moreover, $\sum_{n=0}^\infty x_n$ doesn’t always exist. – Brian M. Scott Apr 18 '13 at 7:00
@BrianM.Scott Still I don't see why the need of AC. – Ambesh Apr 18 '13 at 7:02
What do you know about transfinite constructions? Have you read any of the other dozen threads about intuition behind the axiom of choice? – Asaf Karagila Apr 18 '13 at 7:03
I wasn’t trying to answer that; at the moment I can’t think of anything to add to what I said in answering the other question and in the subsequent comments. I was pointing out that your comparison with addition doesn’t work the way you intended, because it shows another situation in which the infinite case is very different from the finite case. – Brian M. Scott Apr 18 '13 at 7:05
Making one choice is simple, if a set $A$ is not empty, then $\exists a(a\in A)$, and therefore we can pick such $a$. This is called existential instantiation. But this choice is completely arbitrary. This is important because in mathematics we are always within the context of writing a proof (even if we only play around, we essentially prepare ourselves for such proof).
However making infinitely many arbitrary choices is something we cannot prove to be possible1. The axiom of choice asserts that we can, in fact, many infinitely many choices at once - as long as we could make each one (i.e. the sets were not empty).
Remember that arbitrary sets have absolutely no structure. We only have $\in$ in our language, and we have sets and their elements. Sometimes we are lucky and the elements of the set are nice enough to allow for a definable way of choosing from them. For example if the empty set is a member or something.
But this need not be the case. The axiom of choice allows us to uniformly endow all the sets with a particular structure from which we can define a selection.
In comparison, addition of infinitely many real numbers happens within a complete ordered field, where we have some structure, and we use it to establish a criterion when the sum is finite, and if so what is its value.
Footnotes:
1. I am being deliberately imprecise here. The axiom of choice is more than a generalization of existential instantiation for the infinite case. But the intuition which should guide you, in my opinion, is that.
To give a small taste on why things may break down, if we are working within a universe which has non-standard integers then there would be a product which is finite (from the point of view of that model) and therefore is not empty, but since its index set is a non-standard integer we cannot possible write down a formula which instantiate an element from each set.
But all this require first to understand what does internal and external mean in these contexts, and to understand what are non-standard integers and non-standard models better. So it's all pretty far along the road. It is my firm belief that one should start with the idea that the axiom of choice is indeed some sort of a generalization of existential instantiation, and then learn why it is not.
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@Asaf_Karagila Forgive my ignorance. But why this "existential instantiation" does not work when we do infinite choices instead of only some? – Ambesh Apr 18 '13 at 12:20
@Myke: Existential instantiation is part of a proof. How would you write infinitely many of them within a finite proof? You can't. (As with incompleteness, there are delicate points regarding the axiom of choice as an extension of this sort of instantiations, but I deliberately chose to ignore them for this purpose.) – Asaf Karagila Apr 18 '13 at 12:25
@Asaf_Karagila I need to buy an introductory book about this topics before I become mad. – Ambesh Apr 18 '13 at 12:27
@Myke: Which topics? Logic and set theory? An introductory book would be a reasonable idea. Taking a course is an even better idea, if your university offers one (well, unless the teacher is really bad or something like that). – Asaf Karagila Apr 18 '13 at 12:28
@Asaf_Karagila I took the course, and this is my state. Everybody hates the logics teacher in my faculty. Yes a book on logics and set theory introductory level would be great. This summer I will read it. – Ambesh Apr 18 '13 at 12:32
The axiom of choice can be seen as a generalization of the principle of induction. So, since the principle of induction is technically a lot simpler, let's ask why is there a need to accept the principle of induction.
The principle of induction, for the purposes of this answer, says that for a property $P(n)$ about a natural number $n$, if $P(0)$ holds and if $P(n)$ holds, then $P(n+1)$ holds, then in fact $P(n)$ holds for all natural numbers $n$.
This principle seems obvious enough (just like the axiom of choice (in at least one of its forms) seems obvious) so why the fuss about calling it a principle? Well, let's first agree that any proof must be a finite list of characters. Now, how does one argue to convince the skeptic about the validity of the principle of induction? One way is to say, well suppose you want to prove that $P(1)$ holds. Then here is a (finite!) proof: $P(0)$ is known. It is also known that $P(0)\implies P(1)$, thus Modus Ponens tells us that $P(1)$ holds. QED.
This of course is far from proving $\forall n\in \mathbb N \quad P(n)$. So we go on. Suppose you want to establish $P(2)$. Well, here is a (finite!) proof: $P(1)$ was already established (i.e., cut and paste prvious (finite!) proof here), and it is given that $P(1)\implies P(2)$. Thus, Modus Ponens again, gives us that $P(2)$ holds. QED.
Usually one then concludes with the not so convincing argument "and so on" to then argue that we actually established $\forall n\in \mathbb N \quad P(n)$. Well, here is the problem then. We didn't actually prove that! What we did was give a hand-wavy argument that the two assertions 1) $P(0)$ holds and 2) $P(n)\implies P(n+1)$ holds, are sufficient to convince one that one has a recipe for proving $P(n)$ for all $n\in \mathbb N$. In other words, one seems to be convinced that for any given $n$, one can find a (finite!) proof that $P(n)$ holds. But, do we now have a single finite proof that $\forall n\in \mathbb N\quad P(n)$ ? Well, the answer would be yes if you accept the recipe for proofs as an actual proof. In other words, if you accept the principle of induction.
So, accepting the principle of induction can be said to be the acceptance of a finite recipe of finite proofs for $P(n)$ (where the length of the proof of $P(n)$ depends on $n$ and will typically tend to infinity with $n$) as a single finite proof of all $P(n)$ in one go. It seems very reasonable to accept such a proof recipe as a proof, which is why the principle of induction is doubted by very few.
Now, the principle of induction is equivalent to the existence of a least element in any non-finite subset of $\mathbb N$, namely to $\mathbb N$ being well-ordered. The axiom of choice, is equivalent to the existence of a well-ordering on any non-finite set. So the axiom of choice allows for more intricate recipes of proofs and is no longer so easily accepted.
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Transfinite induction requires no choice whatsoever. Many of the particular constructions we make require us to make arbitrary choices. And even then often we run into problems only at certain points in the construction (often in limit stages). – Asaf Karagila Apr 18 '13 at 7:54
Also in the second paragraph you have a lowercase $p$ where it should be an uppercase $P$; and in the fourth paragraph you wrote "finte!" instead of "finite". – Asaf Karagila Apr 18 '13 at 7:56
@AsafKaragila I do think AC is a kind of local-to-global principle of the same kind as induction, however: it says, if I can make a choice locally (i.e. for each member of a set), then I can make a choice globally (i.e. for the whole set at once). – Zhen Lin Apr 18 '13 at 8:17
@AsafKaragila thanks for the comments. I changed the last paragraph in a way that (I believe) is correct. – Ittay Weiss Apr 18 '13 at 8:21
@DanChristensen I'm afraid I find your blog entry is highly imprecise. See the comment I left there. Basically, your approach depends on a pre-existing model of PA with induction (in order to rigorously defined the chain you mention there, giving rise to the natural-number-like set). It is then a tautology that what you get is a model of PA with induction, since you build it as an isomorphic copy of a given model of PA with induction. – Ittay Weiss Apr 18 '13 at 22:46 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9032484292984009, "perplexity": 205.5703193750586}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860118790.25/warc/CC-MAIN-20160428161518-00145-ip-10-239-7-51.ec2.internal.warc.gz"} |
https://mathshistory.st-andrews.ac.uk/OfTheDay/oftheday-07-17/ | ## Mathematicians Of The Day
### 17th July
On this day in 1935, the first entry was made in The Scottish Book. See THIS LINK.
Click on for a poster.
#### Quotation of the day
##### From Henri Poincaré
What is it indeed that gives us the feeling of elegance in a solution, in a demonstration? It is the harmony of the diverse parts, their symmetry, their happy balance; in a word it is all that introduces order, all that gives unity, that permits us to see clearly and to comprehend at once both the ensemble and the details.
Quoted in N Rose Mathematical Maxims and Minims (Raleigh N C 1988). | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8091103434562683, "perplexity": 1925.5365522536463}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882572898.29/warc/CC-MAIN-20220817092402-20220817122402-00237.warc.gz"} |
http://boolesrings.org/krautzberger/tag/stone-cech-compactification/ | # Eternal preliminaries part 2, filters and ultrafilters
Last time I wrote about the basic structures, (partial) semigroups. But algebra in the Stone-Cech compactification deals, well, with the Stone-Cech compactification. I will try to ignore the general theory of compactifications because we only deal with a very simple case — discrete spaces. Suffice it to say that any elementary topology book should have a chapter on compactifications if you want to read more.
## Filters and ultrafilters
Already at the end of the first part, I needed to refer to the notions of filters. I don’t want to talk too much about filters or ultrafilters formally because a) we’re going to talk about them all the time anyway and b) wikipedia (to which I link) is much better at giving you a concise but broad overview than I am. Let me give you a cheat sheet though.
• A family of subsets of $S$ is a filter if it does not contain $\emptyset$, is closed under taking finite intersections and supersets.
• A family of subsets of $S$ has the (strong or infinite) finite intersection property or (i)FIP if the intersection of any finite subfamily is non-empty (infinite).
• A family with (i)FIP generates a (free) filter by closing it under finite intersections and supersets.
• A filter $F$ is an ultrafilter iff it is a maximal filter (with respect to inclusion) iff it is prime $(A\cup B \in F \leftrightarrow A \in F \mbox{or} B\in F)$ iff $(\forall A\subseteq S) A\in F \mbox{or} S\setminus A \in F$.
• We identify $s\in S$ with the principal ultrafilter $\{ A\subseteq S : s\in A\}$.
Filters can be considered as 0-1-valued, finitely-additive measures (or rather the measure 1 sets of such a measure) in which case ultrafilters are complete measures which is an idea I might use in “prose” every once in a while. You can also consider them as a form of universal quantifiers which gives another intuition. A useful shorthand will be “almost all (with respect to some ultrafilter $p$)” or “$p$-many” or “$p$-large” etc. instead of, say, “there exists a set in the filter such that for every element in that set…”.
We’ll get back to this later. One final note, $F,G,H$ are usually denoting filters, $p,q,u$ ultrafilters. We’ll discuss many filters explicitly later (and in part 1 we already considered the filter $\sigma(S)$) but the existence of ultrafilters is a tricky business that requires quite a bit of the axiom of choice.
The Ultrafilter Lemma Every filter can be extended to an ultrafilter.
Proof.
• The set of filters is partially ordered by inclusion.
• The union of any chain of filters is a filter.
• Apply Zorn’s Lemma to find a maximal element.
This theorem is weaker than the axiom of choice, but very strong already in itself (looking up that link I just learned that Tarski himself proved the existence of non-principal ultrafilters in 1930; wikipedia. awesome.). Of course, the real power lies in the three characterizations of ultrafilters in the cheat sheet, so let’s prove the difficult one
A filter $F$ is maximal iff $F$ is prime, i.e., $(\forall A\subseteq S) A\in F \mbox{ or } S\setminus A \in F$
Proof.
• If $F$ is maximal, $A\subseteq S$, then either $A \in F$ or $S\setminus A \in F$.
• If there exists $B\in F$ such that $A \cap B = \emptyset$, then $B \subseteq S\setminus A$, so $S\setminus A \in F$ and we’re done.
• Otherwise every $B\in F$ has $A \cap B \neq \emptyset$ in which case $F\cup \{A\}$ has the FIP, hence generates a filter.
• But $F$ was maximal, so the generated filter cannot gain new elements.
• In other words, $A\in F$.
• If $F$ is prime, then $F$ is maximal.
• Consider any family $G$ such that $F \subseteq G$, but there exists $A \in G \setminus F$.
• Since $A \notin F$, then by primeness of $F$, $S \setminus A \in F$.
• Therefore, $A, S\setminus A$ are both in $G$.
• In other words, $G$ is not a filter — in other words, $F$ is maximal.
## The Stone-Cech compactification (apologies for missing haceks)
The set of ultrafilters is often denoted by $\beta S$ and it turns out to be the Stone-Cech compactification, i.e., the maximal compactification of $S$, because $S$ is discrete. There’s a gazillion things to be said about $\beta S$. To get started, we should celebrate the most practical and in fact characterizing property of the Stone-Cech compactification.
Universal Property of $\beta S$ If $X$ is compact and Hausdorff, $f: S \rightarrow X$ continuous (in our case, any map is), then there exists a unique continuous map $\beta f: \beta S \rightarrow X$ that extends $f$. We usually identify $\beta f$ with $f$ for convenience.
The easiest way to do this in our setting, is to take the limit along ultrafilters. But for now we don’t need to.
An interlude about extensions If $f: S \rightarrow S$, then we can describe the image quite nicely, namely $f(p) = \{ B : (\exists A \in p) f[A] \subseteq B \}.$
Often this definition is given by $f(p)= \{ B: f^{-1}[B] \in p\}$ but I think this is a perfect example of the stupid tendency of mathematicians to write a definition as efficiently as possible even though the compression does more harm than good — as a student it always confused the hell out of me and I mixed it up with preimage filters (which are more difficult to define unless $f$ is surjective). To remember: $f(p)$ is the unique ultrafilter generated by the family $(f[A])_{A\in p}$, the filter generated by the images. Yes, it’s longer to write down, it’s not as self-contained a definition, but really: it does make more sense that way, no? And who’d think the self-contained definition in itself helps anyone understand anything anyway…
## Extending the semigroup operation
We want to extend our (partial) semigroup operation to $\beta S$. The trouble is that the extension won’t be unique and from a theoretical point of view each of those non-unique extensions can be defined using different techniques (resulting in the same kind of extension). The problem of uniqueness also leads to four different descriptions when it comes to the continuity of the operation, but let’s first get started.
I “grew up” with the book by Neil Hindman and Dona Strauss, so I tend to follow their set up (regarding which kind of extension we want).
### Using the universal property of $\beta S$
• For each $s\in S$, we can consider $\lambda_s: S \rightarrow S\subseteq \beta S, t\mapsto s \cdot t$, i.e., multiplication with a fixed left-hand side.
• This is a continuous map (since any map on $S$ is), so we can extend it to $\beta \lambda_s : \beta S \rightarrow \beta S$; we simply write $s \cdot q$ for this.
• Now switch it around and for $q\in \beta S$ consider this extended multiplication with $q$ fixed on the right hand side, i.e., the map $\rho_q: S \rightarrow \beta S, q \mapsto s \cdot q$.
• Again this is a continuous map (since any map on $S$ is), so we can extend it to $\beta \rho_q : \beta S \rightarrow \beta S$; and for this we write $\rho_q(p) = p \cdot q$.
• Tada, we have our operation.
Of course, this gives us no tangible clue as to what such a product of ultrafilters actually looks like. But at least one thing is easy — multiplication with a fixed right hand side is continuous! I call this right-topological. You can see that we might start symmetrically and then we end up with a different operation (though very similar to our own). Also, some people like to call the above continuity left-topological (because its continuous in the left hand argument). So, lots of confusion… we’ll stick to this one.
### The brute force definition
There’s thankfully a way to give the same definition by brute force (which is my favourite way to write it down), but let’s think about it naively. We have two ultrafilters $p,q$ and we have our operation $\cdot$. So why not just take $A \in p$ and $B\in q$ and look at all possible products $A \cdot B$? Collect all these $\{ A \cdot B: A \in p, B \in q\}$ and we get a nice little filter. Are we done? Well, the problem is that this will pretty much never give you an ultrafilter (if it does you either have a very simple operation or (say in $\mathbb{N}$) very, very special ultrafilters).
So what do we need to do? We need to complicate things (and if you try to write down to check where the above attempt of a definition fails, this complication comes naturally). Later I’ll introduce some notation to make nicer general nonsense, but let’s take a look first.
Extending multiplication to $\beta S$ For a semigroup $(S, \cdot)$ and $p,q \in \beta S$ we define the product $p \cdot q$ by $A \in p\cdot q \Leftrightarrow (\exists V\in p)(\exists {(W_v)}_{v\in V} \mbox{ in } q) \bigcup_{v\in V} v \cdot W_v \subseteq A.$
Ok, quite a beast. Don’t despair! Remember what we tried first: sets of the form $V \cdot W = \bigcup_{v\in V} v \cdot W$. What the above definition tells us is that we need to allow the $W$ to be more flexible — possibly different for each $v$!
There is a different angle to look at this: the tensor (or Fubini) product of ultrafilters.
Tensor product of ultrafilters For $p,q \in \beta S$ define $p \otimes q \in \beta (S \times S)$ by $A\in p\otimes q \Leftrightarrow \{ s: \{ t : (s,t) \in A \} \in p \} \in q$
Not much better, eh? Let’s take a look though: the tensor product is contains sets $A$ such that the first projection of $A$ lies in $p$ and additionally almost all fibers (in the sense of $p$) of the first projection lie in $q$. So you might say that the sets are $p$-large horizontally and $q$-large vertically.
What has this to do with the product we defined before? Well, the tensor product live on $S \times S$ and the multiplication is a map from $S \times S$ to $S$. So looking at the continuous extension $\beta \cdot$, i.e., the extension to $\beta (S\times S)$ (which is different from $\beta S \times \beta S$ btw) we can simply take the image, $\beta \cdot(p \otimes q)$. If you look at the “interlude” earlier regarding such images, you’ll notice that we get exactly the ultrafilter described in the brute force definition.
Still not happy? Yeah, I know that feeling… Ok, let me offer my favourite general nonsense notation.
• For $s \in S, A\subseteq S$ define $s^{-1}A := \{ t \in S: st \in A\}$ (note: don’t have to be able to invert to define this…)
• For $A\subseteq S, q \in \beta S$ define $A^{-q} := \{ s \in S: s^{-1}A \in q \}$
• Then $A \in p \cdot q$ if and only if $A^{-q} \in p$.
Alright, much shorter now. But does it help? I don’t know. I certainly don’t claim to “really” understand this operation (but there’s a certain limit since, well, it’s on ultrafilters after all…). My notation for the set $A^{-q}$ is not standard (but there’s no notation, so I made it up for my thesis). This set consists of those elements that (inverse-)shift $A$ to make it an element of $q$. If $p$ contains it, we can expect elements in $p$ to contain elements that shift elements of $q$ into $A$ — which is maybe an idea.
One advantage is that you can check a few things more easily with the brute force definition.
• The operation is right-continuous — $A^{-q}$ is exactly the neighbourhood that shows the continuity of $\rho_q$ with respect to $A$.
• The operation is associative — just write it out.
Phew, that was a lot. But we’re finally ready to get to some real theorems!
# Van Douwen spaces
At the winterschool Alan Dow gave quite challenging tutorials. He also mentioned something about van Douwen spaces.
### Van Douwen Spaces
As formulated here
van Douwen space A countable $S$ is a van Douwen space if it is crowded (i.e. has no isolated points) and there is a 1-to-1 function from $\mathbf{N}$ to $S$ that extends to a $\leq$2-to-1 function from $\beta \mathbf{N}$ to $\beta S$.
What caught my interest was that there is an example that has something to do with idempotent ultrafilters. Let me introduce something first.
A partial order On the idempotent ultrafilters (on $\mathbf{N}$) define a partial ordering by
$p \leq_r q \Leftrightarrow q + p =p.$
### Digressing
This partial order (as well as its left counterpart and their intersection) is quite important in the algebra in the Stone-Cech-compactification. Mostly because this order has minimal idempotents which are central to the field. (pardon the pun) Oops, after ignoring its definition in my last post this is not a pun. So let me add: a set is in fact central if it is an element of a minimal idempotent. Central, get it? Ah, well…
### Strongly right maximal
For van Douwen spaces it is useful to go in the other direction. There exist many right-maximal elements in this order, but even more can be said.
Strongly right maximal idempotents An idempotent ultrafilters $p \in \beta \mathbf{N}$ is strongly right maximal if
$q+ p =p \Rightarrow q= p.$
Yevhen Zelenyuk once gave an example of a right-maximal that is not strongly right maximal assuming CH or MA (and even less). In any case these idempotents are very nice and thanks to Igor Protasov exist under ZFC alone. Nevertheless it is an open question whether consistenly all right-maximal idempotents are strongly right-maximal, i.e., if non-strongly but right-maximal idempotents exist under ZFC alone.
### Back to van Douwen spaces
Anyhow, the main point is that strongly right maximal idempotents have an orbit that is a van Douwen space!
Let $p$ be strongly right maximal. Then $\mathbf{N} + p$ is a van Douwen space.
And this is what Alan Dow mentioned. Ignoring the crowdedness, this is really easy for in fact more holds in this case.
If $p$ is strongly right maximal, then
$\rho_p: \mathbf{N} \rightarrow \beta \mathbf{N}, n \mapsto n+ p$
is injective, hence also its continuous extension to $\beta \mathbf{N}$ (which is naturally onto the orbit $\mathbf{N} +p$).
So in fact, it is not just a $\leq$2-to-one function, but an injective function. Strange, isn’t it? Strongly right maximality really only speaks of injectivity at $p$, but this is already enough.
#### Proof
The proof needs some basic stuff such as ‘multiplication with fixed right hand side is continuous’. Oh, and you need to know that natural numbers are cancelative…
• Since $( \mathbf{N} , + )$ is cancelative, the maps $\lambda_{n} = n + \cdot$ are injective for all $n$.
• Since $\lambda_n$ is continuous (on a discrete space), its extension to $\beta \mathbf{N}$ is injective as well.
• Then $\rho_p$ is injective on $\mathbf{N}$.
• If $n < k \in \mathbf{N}$ had $n+ p = k + p$, then by the above steps $k-n + p = p$.
• Since $p$ is strongly right maximal, this would imply $n-k = p$ — which is absurd since $p$ is idempotent, hence free.
• But then by continuity the whole of $\rho_p$ is injective.
I like that. Now, my favourite kind of idempotent ultrafilters are strongly summable ultrafilters. Those were the first examples of strongly right maximal idempotents, however their existence is independent of ZFC. On the other hand, they have much stronger properties and I would not be surprised if this affected their orbit, i.e., if that van Douwen space is not special somehow.
# Understanding the Central Sets Theorem
To write the first post on the new domain I thought I might just write a little about what I’ve been studying recently — the Central Sets Theorem.
This theorem dates back to the 70s and the original formulation and proof are due to Hillel Furstenberg. In its current form as found say in De, Hindman, Strauss it is probably the strongest algebraic partition theorem around. I had encountered the theorem many times before, in books, lectures, papers and talks but I never truly developed an understanding for it. Since I recently felt it might give me an edge in a problem I’m working on I decided to take a better look.
### Detour 1 — metamathematics
How do you achieve an understanding of a theorem? In an incomplete list I would include the following
• Understand its most important application or corollary
• Understand its statement
• Understand its proof
• Improve its proof
• Understand how to come up with the proof
• Give a different proof
• Improve the theorem
I would say this list is in increasing order of understanding but that’s open for discussion.
I might write about the history (and applications) of the Central Sets Theorem some other time, but here I want to focus on its formulation; in fact, I don’t even want to write about what it means to be central (sorry) except that it is a partition regular notion.
### Formulation
So, what does the usual formulation look like?
Central Sets Theorem
Imagine you are given finitely many sequences in a commutative semigroup $(S,+)$, say $\mathbf{y^0}, \ldots, \mathbf{y^\alpha}$ as well as a central set $C \subseteq S$.
Then you can find a sequence $\mathbf{a}$ in $S$ as well as a sequence $\mathbf{h}$ of non-empty, disjoint and finite subsets of $\mathbb{N}$ such that for $\beta \leq \alpha$ $FS ( {a_n} + {\sum_{i \in h_n} y_i^\beta} ) \subseteq C.$
Complicated, no? I mean, a random bunch of sequences, some strange set and you find some other sequence and some weird subsets of of the natural numbers and then the IP-set of some strange sums are in that strange set — ye what?
Let’s cut it down a little and just consider the case $\alpha = 0$.
simple Central Sets Theorem
Imagine you are given a sequence $\mathbf{y}$ in a commutative semigroup $(S,+)$ as well as a central set $C \subseteq S$.
Then you can find a sequence $\mathbf{a}$ in $S$ as well as a sequence $\mathbf{h}$ of non-empty, disjoint and finite subsets of $\mathbb{N}$ such that $FS ( {a_n} + {\sum_{i \in h_n} y_i} ) \subseteq C.$
### Detour 2 — oversimplification
Even this special case of the standard formulation somehow focuses on aspects that get me sidetracked. So I attempted to formulate it in a way that gives (me) better focus.
Now, the theorem says all kinds of complicated things about the existence of a sequence of disjoint finite subsets of $\mathbb{N}$. Can I get around this? I thought I should be able to. Let’s start with a much weaker version of the theorem.
A weak simple Central Sets Theorem
Imagine you are given a subsemigroup $T \subseteq \mathbb{N}$ as well as a central set $C \subseteq \mathbb{N}$.
Then you can find a sequence $\mathbf{a}$ in $\mathbb{N}$ as well as a sequence $\mathbf{b}$ in $T$ so that $FS ( {a_n} + {b_n} ) \subseteq C.$
I find this weaker version much easier to understand. It just says that I can always translate infinitely many elements from a given subsemigroup into the central set; additionally the finite sums stay within the set.
This is much weaker than the statement before. Of course, given a sequence $\mathbf{y}$ we could consider the generated subsemigroup and use the weaker version. But this would not guarantee the result of applying the Central Sets Theorem — Furstenberg’s theorem gives much more control over which elements are picked since there are no repititions in the sums of the generators.
### Partial Semigroups
So where does this leave us? Well, when I hear finite subsets of $\mathbb{N}$ I think of my favourite structure — in fact the favourite structure for a lot of algebra in the Stone-Cech compactification on $\mathbb{N}$, the semigroup $\delta \mathbb{F}$. But let’s step back a little. The best way to think about $\delta \mathbb{F}$ is in terms of partial semigroups.
A partial semigroup operation on a set $S$ is a map $\cdot: S \times S \rightarrow S$ such that associativity $s \cdot (t \cdot u) = (s \cdot t) \cdot u$ holds in the sense that if one side is defined so is the other and they are equal. A partial semigroup is adequate if the sets
$\sigma(s) := \{ t\in S : {s \cdot t} \mbox{ is defined} \}$
generate a filter, i.e., finitely many elements have a common compatible element.
This notion was introduced by Bergelson, Blass and Hindman in the 90s. It tells us that the operation, although partial, is associative in a strong way. Additionally, it makes sure the operation is not just empty but defined for many elements (well, ok it could be just one for all, but that’s not the point).
For ultrafilters the critical point is the following.
The semigroup $\delta S$
Given an adequate partial semigroup and $p,q$ ultrafilters containing all $\sigma(s)$. Then the operation
$p \cdot q = \{ A \subseteq S : \{ s : \{ t : s \cdot t \in A \} \in q \} \in p \}$
is well-defined and associative and semi-continuous. In other words, $\delta S$ is a closed semi-continuous semigroup.
Now this is somewhat surprising. Even though our operation is partial, these ultrafilters are a full semigroup! With all the bells and whistles it takes to do algebra in the Stone-Cech compactification.
What does this have to do with the Central Sets Theorem?
Denote the non-empty, finite subsets of $\mathbb{N}$ by $\mathbb{F}$. Consider the restriction of $\cup$ on $\mathbb{F}$ defined by
$s + t \mbox{ defined } \Longleftrightarrow \max(s) \cap \min(t) = \emptyset.$
Then in fact this constitutes a partial semigroup, adequate at that.
This partial semigroup structure could be called the free partial semigroup in the following sense: given any sequence $\mathbf{s}$ in any semigroup $S$ we can consider the induced partial semigroup on the set of finite sums ${FS( \mathbf{s} ) }$: we only allow sums where the index sets are disjoint (so that we are closed under our partial operation). Then all $FS$-sets are naturally isomorphic (in the appropriate sense of partial semigroups).
### The weak version revisited
To come back to the weak version of the Central sets theorem — partial semigroups are exactly what it talks about. So let us reformulate,
simple Central Sets Theorem
Imagine we are given a partial subsemigroup $T$ of $(S,+)$ as well as a central set $C \subseteq \mathbb{N}$. Then we find sequences $\mathbf{a}$ in $\mathbb{N}$ and $\mathbf{t} \in T$ such that $FS ( {t_n} ) \subseteq T$ and
${FS( a_{n} + t_{n}) \in C.}$
Now this sounds much closer to the original theorem. Since any sequence generates a partial semigroup on its $FS$-set (isomorphic to $\mathbb{F}$), this is in fact the Central Sets Theorem for just one sequence.
### Leaving the simplification
However, the actual theorem is more than just some kind of induction on the above version. It is considerably stronger and here it is time to let go of the simplifications of partial semigroups again. For the theorem really does talk about $FS$-sets, i.e., partial semigroups isomorphic to $\mathbb{F}$. The strength lies in the fact that the infinite sequences can be chosen uniformly in the sense that we pick from the different partial semigroups in the same prescribed way.
Central Sets Theorem
Imagine you are given finitely many $FS$-sets in a commutative semigroup $(S,+)$, say ${FS( {\mathbf{y^0}} )}, {\ldots}, {FS( {\mathbf{y^\alpha}} )}$ as well as a central set $C \subseteq S$.
Then you can find a sequence $\mathbf{a}$ in $S$ as well as one disjoint sequence $\mathbf{h}$ in $\mathbb{F}$ such that for all $\beta \leq \alpha$ $FS ( {a_n} + {\sum_{i \in h_n} y_i^\beta} ) \subseteq C.$
To see this strength at work it is time to look at the classical application.
Central sets in $( \mathbb{N},+)$ contain arbitrarily long arithmetic progressions
Take $\mathbf{y^\beta}$ to be the multiples of $\beta$ (for $\beta \leq \alpha$). Then the central set theorem guarantees we find $a_1, h_1$ such that for all $\beta \leq \alpha$ $(a_1 + \beta \cdot \sum_{i\in h_1} i) \in C.$
For this application is obviously critical that the to-be-translated elements can be chosen uniformly. That’s all for now but I hope I can write a follow up some other time.
# Matrices vs. idempotent ultrafilters part 2.5
Note: there seems to be some problematic interaction between the javascripts I use and blogspot’s javascripts which prevents longer posts from being displayed correctly. As long as I don’t understand how to fix this, I will simply split the posts.
We can also describe size and the algebraic structure.
1. $A$ with $F_1$ ($F_2$) generates a right (left) zero semigroup (hence of size $2$, except for $x=0$).
2. $A$ with $F_3$ or $F_4$ generates a semigroup with $AB$ nilpotent (of size $4$, except for $x=0$, where we have the null semigroup of size $3$).
3. $A$ with $G_i$ generate (isomorphic) semigroups of size $8$. These contain two disjoint right ideals, two disjoint left ideals generated by $A$ and $B$ respectively.
Luckily enough, we get something very similar from our alternative for $A$.
Proposition In case $A = \begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix}$ the solutions for $B$ being of rank one consist of five one – dimensional families namely (for $x\in \mathbf{Q}$)
$H_1(x) = \begin{pmatrix} 1 & x \\ 0 & 0 \end{pmatrix}, H_2(x) = \begin{pmatrix} x+1 & x \\ ( – x – 1) & – x \end{pmatrix}, H_3(x) = \begin{pmatrix} 0 & x \\ 0 & 1 \end{pmatrix}, H_4(x) = \begin{pmatrix} ( – x+1) & ( – x+1) \\ x & x \end{pmatrix},$
$H_5(x) = \begin{pmatrix} ( – x+1) & ( – x – 1 – \frac{2}{x – 2}) \\ x – 2 & x \end{pmatrix} , x \neq 2.$
As before we can describe size and structure.
1. $A$ with $H_1$ ($H_2$) generates a right (left) zero semigroup (as before).
2. $A$ with $H_3$ or $H_4$ generates a semigroup with $AB$ nilpotent (as before).
3. $A$ with $H_5$ generates the same $8$ element semigroup (as before).
Finally, it might be worthwhile to mention that the seemingly missing copies of the $8$ element semigroup are also dealt with; e.g. $– G_i$ generates the same semigroup as $G_i$ etc.
It is striking to see that the orders of all finite semigroups generated by rational idempotent two by two matrices are either $2^k,2^k + 1$ or $2^k + 2$.
At first sight it seems strange that we cannot find other semigroups with two generators like this. As another friend commented, there’s just not enough space in the plane. I would love to get some geometric idea of what’s happening since my intuition is very poor. But that’s all for today. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9569845199584961, "perplexity": 345.74673674342637}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368711406217/warc/CC-MAIN-20130516133646-00001-ip-10-60-113-184.ec2.internal.warc.gz"} |
http://experiment-ufa.ru/64-is-what-percent-of-278 | # 64 is what percent of 278 - step by step solution
## Simple and best practice solution for 64 is what percent of 278. Check how easy it is, and learn it for the future. Our solution is simple, and easy to understand, so don`t hesitate to use it as a solution of your homework.
If it's not what You are looking for type in the calculator fields your own values, and You will get the solution.
To get the solution, we are looking for, we need to point out what we know.
1. We assume, that the number 278 is 100% - because it's the output value of the task.
2. We assume, that x is the value we are looking for.
3. If 100% equals 278, so we can write it down as 100%=278.
4. We know, that x% equals 64 of the output value, so we can write it down as x%=64.
5. Now we have two simple equations:
1) 100%=278
2) x%=64
where left sides of both of them have the same units, and both right sides have the same units, so we can do something like that:
100%/x%=278/64
6. Now we just have to solve the simple equation, and we will get the solution we are looking for.
7. Solution for 64 is what percent of 278
100%/x%=278/64
(100/x)*x=(278/64)*x - we multiply both sides of the equation by x
100=4.34375*x - we divide both sides of the equation by (4.34375) to get x
100/4.34375=x
23.=x
x=23.
now we have:
64 is 23.% of 278
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http://www.math.wpi.edu/Course_Materials/MA1022C07/volrev/node1.html | Subsections
# Solids of Revolution
## Introduction
The purpose of this lab is to use Maple to study solids of revolution. Solids of revolution are created by rotating curves in the x-y plane about an axis, generating a three dimensional object.
## Background
So far we have used the integral mainly to to compute areas of plane regions. It turns out that the definite integral can also be used to calculate the volumes of certain types of three-dimensional solids. The class of solids we will consider in this lab are called Solids of Revolution because they can be obtained by revolving a plane region about an axis.
As a simple example, consider the graph of the function for , which appears below.
If we take the region between the graph and the x-axis and revolve it about the x-axis, we obtain the solid pictured in the next graph.
To help you in plotting surfaces of revolution, A Maple procedure called revolve has been written. The commands used to produce the graphs are shown below. The revolve procedure, as well as the RevInt, LeftInt, and LeftDisk procedures described below are all part of the CalcP7 package, which must be loaded first. The last line in the example below shows the optional argument for revolving the graph of about the line instead of the default .
> with(CalcP7):
> f := x -> x^2+1;
> plot(f(x),x=-2..2);
> revolve(f(x),x=-2..2);
> revolve(f(x),x=-2..2,y=-2);
The revolve command has other options that you should read about in the help screen. For example, you can speed the command up by only plotting the surface generated by revolving the curve with the nocap argument, and you can also plot a solid of revolution formed by revolving the area between two functions. Try the following examples. (Note: The last example shows how to use revolve with a piecewise defined function using the piecewise command.)
> revolve({f(x),0.5},,x=-2..2,y=-1);
> revolve(cos(x),x=0..4*Pi,y=-2,nocap);
> revolve({5,x^2+1},,x=-2..2);
> g := x-> piecewise(x<0,-x+1/2,x^2-x+1/2);
> revolve(g(x),x=-1..2);
It turns out that the volume of the solid obtained by revolving the region between the graph and the -axis about the -axis can be determined from the integral
to have the value . More generally, if you revolve the area under the graph of for about the -axis, the volume is given by
Where does this formula come from? To help you understand it, two more Maple procedures, RevInt and LeftDisk, have been written. The procedure RevInt sets up the integral for the volume of a solid of revolution as shown below. The Maple commands evalf and value can be used to obtain a numerical or analytical value.
The integral formula given above for the volume of a solid of revolution comes, as usual, from a limit process. Recall the rectangular approximations we used for plane regions. If you think of taking one of the rectangles and revolving it about the x-axis, you get a disk whose radius is the height of the rectangle and thickness is , the width of the rectangle. The volume of this disk is . If you revolve all of the rectangles in the rectangular approximation about the x-axis, you get a solid made up of disks that approximates the volume of the solid of revolution obtained by revolving the plane region about the x-axis.
To help you visualize this approximation of the volume by disks, the LeftDisk procedure has been written. The syntax for this command is similar to that for revolve, except that the number of subintervals must be specified. The examples below produce approximations with five and ten disks. The latter approximation is shown in the graph below.
> LeftDisk(f(x),x=-2..2,5);
> LeftDisk(f(x),x=-2..2,10);
## Finding Volumes of Revolution
In order to calculate the volume of a solid of revolution, you can either use the int command implementing the formula above or use the Maple procedure RevInt which sets up the integral for you. Try the examples below to see the different types of output.
> Pi*int(f(x)^2,x=-2..2);
> evalf(Pi*int(f(x)^2,x=-2..2));
> RevInt(f(x),x=-2..2);
> value(RevInt(f(x),x=-2..2));
> evalf(RevInt(f(x),x=-2..2));
## Exercises
1. For the function over the interval ,
A)
Plot over the given interval.
B)
Plot the approximation of the solid of revolution using LeftDisk with 9 disks.
C)
Plot the solid formed by revolving about the -axis.
D)
Plot the solid formed by revolving about the line .
E)
Find the exact volume of the solid of revolution about the -axis using the RevInt command and label your output exact.
F)
Find the number of subintervals needed to approximate the volume of the solid of revolution about the -axis using LeftInt with error no greater than 0.1 and then again with error no greater than 0.01.
2. Last week, you used a definite integral to prove that the area of a circle of radius is . This week, use the RevInt command and the same function from last week to prove that the volume of a sphere is .
3. A brass finial is to be made in the shape of the solid obtained by revolving the function about the axis over the interval . (A finial is a decorative cap or projection often seen on top of fence posts or staircase posts.) Plot the function revolved about the -axis over the given interval. If the dimensions of the solid are all in inches, determine how many cubic inches of brass will be needed to make of these finials. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9328546524047852, "perplexity": 387.714085386011}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084891196.79/warc/CC-MAIN-20180122073932-20180122093932-00767.warc.gz"} |
https://www.chemeurope.com/en/encyclopedia/H-theorem.html | My watch list
my.chemeurope.com
# H-theorem
In thermodynamics, the H-theorem, introduced by Boltzmann in 1872, describes the increase in the entropy of an ideal gas in an irreversible process, by considering the Boltzmann equation.
It appears to predict an irreversible increase in entropy, despite microscopically reversible dynamics. This has led to much discussion.
## Boltzmann's H-theorem
The quantity H is defined as the integral over velocity space :
$\displaystyle H \ \stackrel{\mathrm{def}}{=}\ \int { P ({\ln P}) d^3 v} = \left\langle { \ln P } \right\rangle$ (1)
where P(v) is the probability. H is a forerunner of Shannon's information entropy.
The article on Shannon's information entropy contains a good explanation of the discrete counterpart of the quantity $\displaystyle H$, known as the information entropy or information uncertainty (with a minus sign). By extending the discrete information entropy to the continuous information entropy, also called differential entropy, one obtains the expression in Eq.(1), and thus a better feel for the meaning of $\displaystyle H$.
Using the Boltzmann equation one can prove that H can only decrease.
For a system of N statistically independent particles, H is related to the thermodynamic entropy S through:
$S \ \stackrel{\mathrm{def}}{=}\ - N k H$
so, according to the H-theorem, S can only increase.
However, Loschmidt objected that it should not be possible to deduce an irreversible process from time-symmetric dynamics and a time-symmetric formalism: something must be wrong (Loschmidt's paradox). The answer is that the theorem is based on Boltzmann's assumption of "molecular chaos", i.e., that it is acceptable for all the particles to be considered independent and uncorrelated. This in fact breaks time reversal symmetry and therefore begs the question.
## Quantum mechanical H-theorem
The following quantum-mechanical analogue of Boltzmann's H-theorem is sometimes given (e.g., Waldram (1985), p.39).
Starting from the Gibbs definition of thermodynamic entropy,
$S = - k \sum_i p_i \ln p_i \,$
differentiating gives
$\frac{dS}{dt} = - k \sum_i \ln p_i \frac{dp_i}{dt}$
(using the fact that ∑ dpi/dt = 0, since ∑ pi = 1).
Now Fermi's golden rule gives a master equation for the probability of quantum jumps from state α to β; and from state β to α. For an isolated system the jumps will make a contribution ναβ(pβ-pα) to dpα/dt, and a contribution ναβ(pα-pβ) to dpβ/dt; the micro-reversibility of the dynamics ensuring that the same transition constant ναβ appears in both expressions.
Thus
$\frac{dS}{dt} = \frac{1}{2} k \sum_{\alpha\beta} \nu_{\alpha\beta}(\ln p_{\beta}-\ln p_{\alpha})(p_{\beta}- p_{\alpha}).$
But the two brackets will have the same sign, so each contribution to dS/dt cannot be negative.
Therefore
$\Delta S \geq 0$
for an isolated system.
The same mathematics is sometimes also presented for classical systems, considering probability flows between coarse-grained cells in the phase space (e.g., Tolman (1938)).
### Critique
Several criticisms can be made of the above "proof", for example by Gull (1989):
1. It relies on the use of approximate quantum mechanics (Fermi's golden rule), not necessarily valid for large perturbations.
2. Are the probabilities to be considered as representing N independent systems of 1 particle, or as applying to 1 system of N particles? If it is the former, then it is ignoring the inter-particle correlations between the systems after collisions, explaining the information loss. The 1-particle entropy also ignores many-body effects in the potential energy, so bears little relation to the entropy of any real gas.
3. On the other hand, treated properly, an N-particle system has N-particle states. An isolated system will presumably sit in one of its N-particle microstates and make no transitions at all.
## Analysis
At the heart of the H-theorem is the replacement of 1-state to 1-state deterministic dynamics by many-state to many-state Markovian mixing, with information lost at each Markovian transition.
Gull is correct that, with the powers of Laplace's demon, one could in principle map forward exactly the ensemble of the original possible states of the N-particle system exactly, and lose no information. But this would not be very interesting. Part of the program of statistical mechanics, not least the MaxEnt school of which Gull is an enthusiastic proponent, is to see just how much of the detail information in the system one can ignore, and yet still correctly predict experimentally reproducible results.
The H-theorem's program of regularly throwing information away, either by systematically ignoring detailed correlations between particles, or between particular sub-systems, or through systematic regular coarse-graining, leads to predictions such as those from the Boltzmann equation for dilute ideal gases or from the recent entropy-production fluctuation theorem, which are useful and reproducibly observable. They also mean that we have learnt something qualitative about the system, and which parts of its information are useful for which purposes, which is additional beyond even the full specification of the microscopic dynamical particle trajectories.
(It may be interesting that having rounded on the H-theorem for not considering the microscopic detail of the microscopic dynamics, Gull then chooses to demonstrate the power of the extended-time MaxEnt/Gibbsian method by applying it to a Brownian motion example - a not so dissimilar replacement of detailed deterministic dynamical information by a simplified stochastic/probabilistic summary!)
However, it is an assumption that the H-theorem's coarse-graining is not getting rid of any 'interesting' information. With such an assumption, one moves firmly into the domain of predictive physics: if the assumption goes wrong, it may produce predictions which are systematically and reproducibly wrong. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 8, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.890425443649292, "perplexity": 1054.501745072555}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655896905.46/warc/CC-MAIN-20200708062424-20200708092424-00129.warc.gz"} |
http://mathhelpforum.com/geometry/180076-vectors-3d-geometry.html | # Math Help - Vectors and 3D Geometry
1. ## Vectors and 3D Geometry
Hi guys, I am new to this forum, hope I posted this thread in the correct place...
Anyway,I got a question on vectors which I need help in:
The vector q is inclined at angles 135, 60 and \alpha where \alpha is an obtuse angle, to the x-,y- and z-axes respectively. The line L passes through the point A(2,-2,3) and is parallel to q. The plane 'pie' passes through B(5,-3,2) and contains the y-axis.
(i) Find a vector EQN of line L.
(ii) Find the cartesian EQN of plane 'pie'
(iii) Find the acute angle b/w L and 'pie'
(iv) Find the length of projection of AB onto 'pie'
I cant solve (ii) and (iv). By the way, the ans to (i) is r = (2,-2,3) + \lambda (-2^1/2,1,-1) if it is of any help.
From what i understand from 'contains y-axis', I think it means the plane is parallel to y-axis and this would mean the direction vectors of x- and z-axes is 0 right? But what does it mean? I am bad with planes + axes... I think (ii) and (iv) are related thats why I can't do (iv) as well but I do know about length of projection.
Sorry if my post is messy. Thank you!
2. Originally Posted by Blizzardy
Hi guys, I am new to this forum, hope I posted this thread in the correct place...
Anyway,I got a question on vectors which I need help in:
The vector q is inclined at angles 135, 60 and \alpha where \alpha is an obtuse angle, to the x-,y- and z-axes respectively. The line L passes through the point A(2,-2,3) and is parallel to q. The plane 'pie' passes through B(5,-3,2) and contains the y-axis.
...
(ii) Find the cartesian EQN of plane 'pie'
...
From what i understand from 'contains y-axis', I think it means the plane is parallel to y-axis and this would mean the direction vectors of x- and z-axes is 0 right? But what does it mean? I am bad with planes + axes... I think (ii) and (iv) are related thats why I can't do (iv) as well but I do know about length of projection.
Sorry if my post is messy. Thank you!
1. You know about the plane pi (I assume that you don't refer to a kind of pastry(?)) that
• the point B(5, -3, 2) belongs to pi
• the y-axis belong to pi, so the origin belongs to pi too
2. Use the origin as a fixed point of pi, then the vectors
$\vec u = \langle 0,1,0 \rangle , \vec v = \langle 5,-3,2 \rangle$
span the plane pi.
3. A parametric equation of pi could be:
$\pi:\langle x,y,z \rangle = s \cdot \vec u + t \cdot \vec v$
Plug in the appropriate value. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8450931310653687, "perplexity": 933.0913127270906}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440645330174.85/warc/CC-MAIN-20150827031530-00151-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/lebesgue-integration.425046/ | # Lebesgue Integration
1. Aug 29, 2010
### wayneckm
Hello all,
I am wondering the implication between almost everywhere bounded function and Lebesgue integrable.
In the theory of Lebesgue integration, if a non-negative function $$f$$ is bounded a.e., then it should be Lebesgue integrable, i.e. $$\int f d\mu < \infty$$ because we do not take into account the unboundedness of $$f$$ in a null set when approximate by sequence of simple function, am I correct? So this means a.e. boundedness implies Lebesgue integrable?
And seems there is a counterexample on the reverse implication, http://planetmath.org/encyclopedia/AnIntegrableFunctionWhichDoesNotTendToZero.html [Broken], so that means Lebesgue integrable does not imply bounded a.e.
So is this because in finding the Lebesgue integral, it is indeed an infinite series of products, which is $$\sum s_{n} \cdot \mu(A_{n})$$, so as long as the increase in $$s_{n}$$ is not faster than the decrease in $$\mu(A_{n})$$, it is possible to have a finite value of this infinite sum? So in this way we may end up with a non-bounded a.e. function but Lebesgue integrable?
Wayne
Last edited by a moderator: May 4, 2017
2. Aug 29, 2010
### Hurkyl
Staff Emeritus
One standard example of a non-Lebesgue integrable function is the characteristic function of a non-measurable set.
While the Lebesgue integral (and the Riemann integral!) are limits of finite sums, neither is an infinite sum. (In the usual formulations, anyways)
3. Aug 29, 2010
### mathman
This assumes that the total measure of the space is finite.
4. Aug 29, 2010
### wayneckm
Thanks so much.
So, in conclusion, relationship between a.e. boundedness and Lebesgue integrable is not definitive in general, right?
Regarding the characteristic function of a non-measurable set, it is then a non-measurable function, hence, its Lebesgue integral is not well-defined? Or is there any reference about this?
Thanks. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9886641502380371, "perplexity": 568.7172828304105}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267863411.67/warc/CC-MAIN-20180620031000-20180620051000-00637.warc.gz"} |
http://math.stackexchange.com/questions/316825/discontinuities-of-a-function-whose-graph-is-invariant-under-rotation-by-90-degr | # Discontinuities of a function whose graph is invariant under rotation by 90 degrees
Prove that there is no function on open interval $(-1,1)$, which has only finite number of discontinuity point, such that its graph is invariant under rotation by the right angle around the origin.
-
nothing good really. – user64370 Feb 28 '13 at 13:13
What are some functions that are invariant under rotation by right angle about origin? except disc? – user45099 Feb 28 '13 at 13:47
Maybe it helps that such a function will satisfy $f(f(t))=-t$. – Berci Feb 28 '13 at 15:20
Any $f: \mathbb R \to \mathbb R$ that satisfies $f(f(t))=-t$ for all $t \in \mathbb R$ has infinitely many points of discontinuity. This is a problem from the 1985 Vietnam Team Selection Tests for the IMO (source: artofproblemsolving.com/Forum/…). – marlu Feb 28 '13 at 20:55
can't figure out the proof, bu thanks – user64370 Feb 28 '13 at 20:55
Let $f:(-1,1)\to\mathbb{R}$ be a function such that its graph is invariant under rotation by the right angle around the origin. It implies that if $(x,y)$ is the graph, then $(y,-x)$ is also in the graph, i.e. if $x\in(-1,1)$ and $y=f(x)$, then $y\in(-1,1)$ and $-x=f(y)$. It follows that $f$ maps $(-1,1)$ to itself, and $$f^{\circ 2}(x)=-x,\quad\forall x\in(-1,1).\tag{1}$$ From $(1)$ we know that $f$ must be bijective on $(-1,1)$, i.e. $f$ is $1$ to $1$ and onto. Therefore, $f$ cannot be continuous on $(-1,1)$, because if $f$ is continuous and injective on $(-1,1)$, $f$ must be monotone on $(-1,1)$, and hence $f^{\circ 2}$ must be increasing, contradicting to $(1)$.
Now suppose that $f$ has finitely many discontinuity points, which are $a_1<a_2<\dots<a_n$. Denote $I_i=(a_i,a_{i+1})$, $0\le i\le n$, where $a_0=-1$ and $a_{n+1}=1$. Moreover, denote $C=\cup_{i=0}^n I_i$, the collection of continuity points of $f$, and $D=\{a_i:1\le i\le n\}$, the collection of discontinuity points of $f$.
Since for each open interval $I_i$, $f$ is continuous and injective on $I_i$, $f(I_i)$ is also an open interval, and $f$ has a continuous inverse $g_i:f(I_i)\to I_i$. Since on $f(I_i)$, $f= f\circ f\circ g_i=-g_i$, $f$ is continuous on $f(I_i)$. That is to say, $f$ maps continuity points to continuity points. Combining this fact with $f$ being bijective, we have $$f(C)=C \quad\text{and}\quad f(D)=D.\tag{2}$$
In particular, for every $0\le i\le n$, there exists $0\le j\le n$, such that $f(I_i)\subset I_j$. Since $I_j\subset C$ and $C=\cup_kf(I_k)$, we know that $I_j=\cup_k(f(I_k)\cap I_j)$. Then by the connectedness of $I_j$, in fact $f(I_i)=I_j$. As a result, $f$ defines a permutation on $\mathcal{I}:=\{I_i:0\le i\le n\}$. Note that $f^{\circ 2 }(I_i)=-I_i$, so there are two cases. First, if $I_i\ne -I_i$, i.e. $0\notin I_i$, then $f^{\circ k}(I_i)$ are pairwise different, $k=0,1,2,3$. Second, if $I_i=-I_i$, i.e. $0\in I_i$, then $f(I_i)=I_i$, because $$f(I_i)=f(-I_i)=f^{\circ 3}(I_i)=f^{\circ 2}(f(I_i))=-f(I_i)\Rightarrow 0\in f(I_i)\Rightarrow I_i=f(I_i).$$ However, the second case cannot happen, and the reason is the same as in the first paragragh: if $f:I_i\to I_i$ is injective and continuous, then $f^{\circ 2}$ must be increasing. Finally, we can conclude that: (i) $0$ is a discontinuity point, and (ii), $\mathcal{I}$ is a disjoint union of the $f$-orbits, and each orbit is of length $4$, i.e.
$$n+1=\#\mathcal{I}\equiv 0 \mod 4.\tag{3}$$
A similar argument can be applied to $f:D\to D$. For each $a_i\in D$, if $a_i\ne 0$, then $f^{\circ k}(a_i)$ are pairwise different, $k=0,1,2,3$; if $a_i=0$, then $f(a_i)=a_i$. Since $0\in D$, we can also conclude that $$n=\# D\equiv 1 \mod 4.\tag{4}$$
The contradiction between $(3)$ and $(4)$ completes the proof.
-
Is it possible to explain some of the terms such as bijection, and to phrase this in a simpler way, for those who do not have much calculus experience? This is a good answer, but read the bounty conditions carefully. – cuabanana Apr 21 '13 at 1:46
@cuabanana: I edited my answer a little. Hope it looks clearer now. Since the answer is already very long in words, I don't want to expand it too much. Due to my poor English, this is the best I can do. – 23rd Apr 21 '13 at 8:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9896741509437561, "perplexity": 93.88991480427192}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049274059.91/warc/CC-MAIN-20160524002114-00068-ip-10-185-217-139.ec2.internal.warc.gz"} |
https://www.science.gov/topicpages/a/accretion+disk+instabilities.html | #### Sample records for accretion disk instabilities
1. Accretion disk thermal instability in galactic nuclei
Mineshige, S.; Shields, G. A.
1990-03-01
The nonlinear evolution and spatial propagation of the thermal instability in accretion disks in galactic nuclei are investigated. Integrations of the vertical structure of the disks are described for different alpha prescriptions, and the thermal stability is examined. Global time-dependent calculations of the unstable disks are performed which show that there are two distinct types of behavior according to the assumed prescription for the viscosity parameter: the 'purr' type and the 'roar' type. The roar type is analyzed in some detail.
2. Magnetic interchange instability in accretion disks
Lubow, Stephen H.; Spruit, Hendrik C.
1995-05-01
We investigate the stability of a disk to magnetic interchange in the disk plane, when a poloidal magentic field provides some radial support of the disk. The disk is assumed to be geometrically thin and may possess rotation and shear. We assume the unperturbed magnetic field vertically threads the disk and has a comparable radial component at the disk surface. We formulate the linear stability problem as an initial value problem in shearing coordinates and ignore any effects of winds. Shear stabilizes the interchange instability strongly compared to the uniformly rotating case studied previously and makes the growth algebraic rather than exponential. A second form of instability with long wavelengths is identified, whose growth appears to be transient. If the field strength is measured by the travel time tauA of an Alfven wave across the disk thickness, significant amplification for both forms of instability requires (tauA Omega)-2 greater than or approximately equal to L/H, where L is the radial length scale of the field gradient and H is the disk thickness. Field strengths such that 1 less than or approximately equal (tauA Omega)-2 less than or approximately equal L/H are stable to these instabilities as well as the instability recently investigated by Balbus & Hawley (1991). The results suggest that in dark environments in which the magnetic energy density is greater than the thermal energy density, disks are stable over a substantial range of parameter space, with radial advection of magnetic flux limited by the interchange instability possibly near the disk center. Such environments may be relevant for the production of magnetic winds or jets in young stars or active galactic nuclei.
3. PARTICLE ACCELERATION DURING MAGNETOROTATIONAL INSTABILITY IN A COLLISIONLESS ACCRETION DISK
SciTech Connect
Hoshino, Masahiro
2013-08-20
Particle acceleration during the magnetorotational instability (MRI) in a collisionless accretion disk was investigated by using a particle-in-cell simulation. We discuss the important role that magnetic reconnection plays not only on the saturation of MRI but also on the relativistic particle generation. The plasma pressure anisotropy of p > p{sub ||} induced by the action of MRI dynamo leads to rapid growth in magnetic reconnection, resulting in the fast generation of nonthermal particles with a hard power-law spectrum. This efficient particle acceleration mechanism involved in a collisionless accretion disk may be a possible model to explain the origin of high-energy particles observed around massive black holes.
4. Irradiation instability at the inner edges of accretion disks
SciTech Connect
Fung, Jeffrey; Artymowicz, Pawel
2014-07-20
An instability can potentially operate in highly irradiated disks where the disk sharply transitions from being radially transparent to opaque (the 'transition region'). Such conditions may exist at the inner edges of transitional disks around T Tauri stars and accretion disks around active galactic nuclei. We derive the criterion for this instability, which we term the 'irradiation instability', or IRI. We also present the linear growth rate as a function of β, the ratio between radiation force and gravity, and c{sub s}, the sound speed of the disk, obtained using two methods: a semi-analytic analysis of the linearized equations and a numerical simulation using the GPU-accelerated hydrodynamical code PEnGUIn. In particular, we find that IRI occurs at β ∼ 0.1 if the transition region extends as wide as ∼0.05r, and at higher β values if it is wider. This threshold value applies to c{sub s} ranging from 3% of the Keplerian orbital speed to 5%, and becomes higher if c{sub s} is lower. Furthermore, in the nonlinear evolution of the instability, disks with a large β and small c{sub s} exhibit 'clumping', extreme local surface density enhancements that can reach over 10 times the initial disk surface density.
5. Lightman-Eardley instabilities and accretion disk thickening. [for compact astronomical objects
NASA Technical Reports Server (NTRS)
Stoeger, W. R.
1979-01-01
After reviewing the role of Compton scattering in accretion disks around black holes, it is discussed whether Lightman-Eardley (LE) secular instabilities can trigger and maintain Pringle-Rees (PR) thermal instabilities. The radiative-transfer-equation and equation-of-state criteria for LE stability in alpha-viscosity-law disk models and dynamic viscosity criteria for more general situations is derived. On the basis of these considerations the LE instability is insufficient for inducing PR instabilities and hot thick inner regions important in accretion-disk models of compact hard X-ray sources. The density thinning due to radial velocity gradients in the accretion flow is suggested as a more likely and satisfactory mechanism.
6. Black hole accretion disks - Coronal stabilization of the Lightman-Eardley instability
NASA Technical Reports Server (NTRS)
Ionson, J. A.; Kuperus, M.
1984-01-01
Physical processes by which the presence of a corona around a black hole can raise the threshold of onset of the Lightman-Eardley (L-E, 1976) instability are explored analytically. The L-E model predicts that an optically thick disk becomes unstable when the disk radiation pressure exceeds the disk gas pressure. The model has important implications for the validity of either the coronal disk or two-temperature disk models for accretion zones around black holes. It is shown that a corona can dissipate accreting gravitational energy through radiative cooling. Specific ratios of hard/soft X-rays are quantified for stable and unstable conditions. X-ray spectra from Cyg X-1 are cited as residing below the instability threshold value and thus are supportive of the coronal disk model.
7. Accretion disk evolution in dwarf novae through outbursts: disk instability and mass-transfer instability cases
Baptista, R.
I discuss a set of observations of eclipsing dwarf novae through outbursts which allow fundamental tests of the predictions of the two models proposed to explain their outbursts. The observational picture which emerges from these tests indicate that there are two distinct groups of dwarf novae. While the outbursts of one group can be understood in the framework of the thermal-viscous disc instability model, those of the other group can only be explained in terms of the mass-transfer instability model. I also show that morphological differences in the orbital light curves of eclipsing dwarf novae can be useful to distinguish members of each group.
8. REVIEWS OF TOPICAL PROBLEMS: The nature of accretion disks of close binary stars: overreflection instability and developed turbulence
Fridman, A. M.; Bisikalo, D. V.
2008-06-01
The current status of the physics of accretion disks in close binary stars is reviewed, with an emphasis on the hydrodynamic overreflection instability, which is a factor leading to the accretion disk turbulence. The estimated turbulent viscosity coefficients are in good agreement with observations and explain the high angular momentum transfer rate and the measured accretion rate. Based on the observations, a power-law spectrum for the developed turbulence is obtained.
9. Instability of high-frequency acoustic waves in accretion disks with turbulent viscosity
Khoperskov, A. V.; Khrapov, S. S.
1999-05-01
The dynamics of linear perturbations in a differentially rotating accretion disk with a non-homogeneous vertical structure is investigated. We find that turbulent viscosity results in instability of both pinching oscillations, and bending modes. Not only the low-frequency fundamental modes, but also the high-frequency reflective harmonics appear to be unstable. The question of the limits of applicability of the thin disk model (MTD) is also investigated. Some differences in the dispersion properties of the MTD and of the three-dimensional model appear for wave numbers k <~ (1-3)/h (h is the half-thickness of a disk). In the long-wavelength limit, the relative difference between the eigenfrequencies of the unstable acoustic mode in the 3D-model and the MTD is smaller than 5%. In the short wavelength case (kh > 1) these differences are increased.
10. Variability of accretion disks surrounding black holes: The role of inertial-acoustic mode instabilities
NASA Technical Reports Server (NTRS)
Chen, Xingming; Taam, Ronald E.
1995-01-01
The global nonlinear time-dependent evolution of the inertial-acoustic mode instability in accretion disks surrounding black holes has been investigated. The viscous stress is assumed to be proportional to the gas pressure only, i.e., tau = alphap(sub g). It is found that an oscillatory nonsteady behavior exists in the inner regions of disks (r is less than 10r(sub g) where r(sub g) is the Schwarzschild radius) for sufficiently large alpha(greater than or approximately equal to 0.2) and for mass accretion rates less than about 0.3 times the Eddington value. The variations of the integrated bolometric luminosity from the disk, Delta L/L, are less than 3%. A power spectrum analysis of these variations reveals a power spectrum which can be fitted to a power-law function of the frequency Pis proportional to f(exp -gamma), with index gamma = 1.4-2.3 and a low-frequency feature at about 4 Hz in one case. In addition, a narrow peak centered at a frequency corresponding to the maximum epicyclic frequency of the disk at approximately 100-130 Hz and its first harmonic is also seen. The low-frequency modulations are remarkably similar to those observed in black hole candidate systems. The possible existence of a scattering corona in the inner region of the disk and/or other processes contributing to the power at high frequencies in the inner region of the accretion disk may make the detection of the high-frequency component difficult.
11. Local Axisymmetric Simulations of Magnetorotational Instability in Radiation-dominated Accretion Disks
Turner, N. J.; Stone, J. M.; Sano, T.
2002-02-01
We perform numerical simulations of magnetorotational instability in a local patch of accretion disk in which radiation pressure exceeds gas pressure. Such conditions may occur in the central regions of disks surrounding compact objects in active galactic nuclei and Galactic X-ray sources. We assume axisymmetry and neglect vertical stratification. The growth rates of the instability on initially uniform magnetic fields are consistent with the linear analysis of Blaes & Socrates (2001). As is the case when radiation effects are neglected, the nonlinear development of the instability leads to transitory turbulence when the initial magnetic field has no net vertical flux. During the turbulent phase, angular momentum is transported outward. The Maxwell stress is a few times the Reynolds stress, and their sum is about 4 times the mean pressure in the vertical component of the magnetic field. For magnetic pressure exceeding gas pressure, turbulent fluctuations in the field produce density contrasts about equal to the ratio of magnetic to gas pressure. These are many times larger than in the corresponding gas pressure-dominated situation and may have profound implications for the steady state vertical structure of radiation-dominated disks. Diffusion of radiation from compressed regions damps turbulent motions, converting kinetic energy into photon energy.
12. A Pure Hydrodynamic Instability in Shear Flows and Its Application to Astrophysical Accretion Disks
2016-10-01
We provide a possible resolution for the century-old problem of hydrodynamic shear flows, which are apparently stable in linear analysis but shown to be turbulent in astrophysically observed data and experiments. This mismatch is noticed in a variety of systems, from laboratory to astrophysical flows. There are so many uncountable attempts made so far to resolve this mismatch, beginning with the early work of Kelvin, Rayleigh, and Reynolds toward the end of the nineteenth century. Here we show that the presence of stochastic noise, whose inevitable presence should not be neglected in the stability analysis of shear flows, leads to pure hydrodynamic linear instability therein. This explains the origin of turbulence, which has been observed/interpreted in astrophysical accretion disks, laboratory experiments, and direct numerical simulations. This is, to the best of our knowledge, the first solution to the long-standing problem of hydrodynamic instability of Rayleigh-stable flows.
13. Angular momentum transport and particle acceleration during magnetorotational instability in a kinetic accretion disk.
PubMed
Hoshino, Masahiro
2015-02-13
Angular momentum transport and particle acceleration during the magnetorotational instability (MRI) in a collisionless accretion disk are investigated using three-dimensional particle-in-cell simulation. We show that the kinetic MRI can provide not only high-energy particle acceleration but also enhancement of angular momentum transport. We find that the plasma pressure anisotropy inside the channel flow with p(∥)>p(⊥) induced by active magnetic reconnection suppresses the onset of subsequent reconnection, which, in turn, leads to high-magnetic-field saturation and enhancement of the Maxwell stress tensor of angular momentum transport. Meanwhile, during the quiescent stage of reconnection, the plasma isotropization progresses in the channel flow and the anisotropic plasma with p(⊥)>p(∥) due to the dynamo action of MRI outside the channel flow contribute to rapid reconnection and strong particle acceleration. This efficient particle acceleration and enhanced angular momentum transport in a collisionless accretion disk may explain the origin of high-energy particles observed around massive black holes.
14. Magnetic viscosity by localized shear flow instability in magnetized accretion disks
SciTech Connect
Matsumoto, R.; Tajima, T.
1995-01-01
Differentially rotating disks are subject to the axisymmetric instability for perfectly conducting plasma in the presence of poloidal magnetic fields. For nonaxisymmetric perturbations, the authors find localized unstable eigenmodes whose eigenfunction is confined between two Alfven singularities at {omega}{sub d} = {+-} {omega}{sub A}, where {omega}{sub d} is the Doppler-shifted wave frequency, and {omega}{sub A} = k{parallel}v{sub A} is the Alfven frequency. The radial width of the unstable eigenfunction is {Delta}x {approximately} {omega}{sub A}/(Ak{sub y}), where A is the Oorts constant, and k{sub y} is the azimuthal wave number. The growth rate of the fundamental mode is larger for smaller value of k{sub y}/k{sub z}. The maximum growth rate when k{sub y}/k{sub z} {approximately} 0.1 is {approximately} 0.2{Omega} for the Keplerian disk with local angular velocity {Omega}. It is found that the purely growing mode disappears when k{sub y}/k{sub z} > 0.12. In a perfectly conducting disk, the instability grows even when the seed magnetic field is infinitesimal. Inclusion of the resistivity, however, leads to the appearance of an instability threshold. When the resistivity {eta} depends on the instability-induced turbulent magnetic fields {delta}B as {eta}([{delta}B{sup 2}]), the marginal stability condition self-consistently determines the {alpha} parameter of the angular momentum transport due to the magnetic stress. For fully ionized disks, the magnetic viscosity parameter {alpha}{sub B} is between 0.001 and 1. The authors three-dimensional MHD simulation confirms these unstable eigenmodes. It also shows that the {alpha} parameter observed in simulation is between 0.01 and 1, in agreement with theory. The observationally required smaller {alpha} in the quiescent phase of accretion disks in dwarf novae may be explained by the decreased ionization due to the temperature drop.
15. Numerical experiments in galactic disks: Gravitational instability, stochastic accretion, and galactic winds
Forbes, John C.
Using 0D, 1D, and 3D models of galaxies, I explore different problems in galaxy evolution most suited to each technique. In the simplest case, a galaxy is described by a few numbers integrated via coupled ordinary differential equations. By allowing the galaxies to respond to a stochastic accretion rate, I show a natural way of generating the finite scatter observed in several galaxy scaling relations: the correlation between a galaxy's stellar mass and its star formation rate or metallicity. By comparing this simple model to observations, we constrain the process by which gas accretes onto galaxies, which must occur, but is essentially impossible to observe directly. Adding an additional dimension to the models, we explore the structure of galactic disks as a function of radius. We find that turbulence driven by gravitational instability in the disks and the resulting migration of gas can explain a wide variety of phenomena, including the age-velocity dispersion correlation of stars in the solar neighborhood, the central quenching star formation in disk galaxies, rings of star formation, and the observed radial profile of gas column densities. Finally, we run a set of fully three-dimensional galaxy simulations to try to understand what physics is responsible for basic properties of galaxies, including the rate at which they form stars, and the rate at which they eject mass in large-scale winds. We find that supernovae are capable of driving moderate metal-enhanced winds, but surprisingly they have very little effect on the star formation rates of dwarf galaxies. Instead, ordinary photoelectric heating dominates the star formation law in low-mass galaxies.
16. Theory of protostellar accretion disks
NASA Technical Reports Server (NTRS)
Ruden, S.
1994-01-01
I will present an overview of the current paradigm for the theory of gaseous accretion disks around young stars. Protostellar disks form from the collapse of rotating molecular cloud cores. The disks evolve via outward angular momentum transport provided by several mechanisms: gravitational instabilities, thermal convective turbulence, and magnetic stresses. I will review the conditions under which these mechanisms are efficient and consistent with the observed disk evolutionary timescales of several million years. Time permitting, I will discuss outbursts in protostellar disks (FU Orionis variables), the effect of planet formation on disk structure, and the dispersal of remnant gas.
17. Self-destructing Spiral Waves: Global Simulations of a Spiral-wave Instability in Accretion Disks
Bae, Jaehan; Nelson, Richard P.; Hartmann, Lee; Richard, Samuel
2016-09-01
We present results from a suite of three-dimensional global hydrodynamic simulations that shows that spiral density waves propagating in circumstellar disks are unstable to the growth of a parametric instability that leads to break down of the flow into turbulence. This spiral wave instability (SWI) arises from a resonant interaction between pairs of inertial waves, or inertial-gravity waves, and the background spiral wave. The development of the instability in the linear regime involves the growth of a broad spectrum of inertial modes, with growth rates on the order of the orbital time, and results in a nonlinear saturated state in which turbulent velocity perturbations are of a similar magnitude to those induced by the spiral wave. The turbulence induces angular momentum transport and vertical mixing at a rate that depends locally on the amplitude of the spiral wave (we obtain a stress parameter α ˜ 5 × 10-4 in our reference model). The instability is found to operate in a wide range of disk models, including those with isothermal or adiabatic equations of state, and in viscous disks where the dimensionless kinematic viscosity ν ≤ 10-5. This robustness suggests that the instability will have applications to a broad range of astrophysical disk-related phenomena, including those in close binary systems, planets embedded in protoplanetary disks (including Jupiter in our own solar system) and FU Orionis outburst models. Further work is required to determine the nature of the instability and to evaluate its observational consequences in physically more complete disk models than we have considered in this paper.
18. PROPERTIES OF GRAVITOTURBULENT ACCRETION DISKS
SciTech Connect
Rafikov, Roman R.
2009-10-10
We explore the properties of cold gravitoturbulent accretion disks-non-fragmenting disks hovering on the verge of gravitational instability (GI)-using a realistic prescription for the effective viscosity caused by gravitational torques. This prescription is based on a direct relationship between the angular momentum transport in a thin accretion disk and the disk cooling in a steady state. Assuming that opacity is dominated by dust we are able to self-consistently derive disk properties for a given M-dot assuming marginal gravitational stability. We also allow external irradiation of the disk and account for a non-zero background viscosity, which can be due to the magneto-rotational instability. Spatial transitions between different co-existing disk states (e.g., between irradiated and self-luminous or between gravitoturbulent and viscous) are described and the location of the boundary at which the disk must fragment is determined in a variety of situations. We demonstrate in particular that at low enough M-dot external irradiation stabilizes the gravitoturbulent disk against fragmentation to very large distances thus providing means of steady mass transport to the central object. Implications of our results for the possibility of planet formation by GI in protoplanetary disks and star formation in the Galactic center and for the problem of feeding supermassive black holes in galactic nuclei are discussed.
19. WIND-DRIVEN ACCRETION IN PROTOPLANETARY DISKS. I. SUPPRESSION OF THE MAGNETOROTATIONAL INSTABILITY AND LAUNCHING OF THE MAGNETOCENTRIFUGAL WIND
SciTech Connect
Bai Xuening; Stone, James M.
2013-05-20
We perform local, vertically stratified shearing-box MHD simulations of protoplanetary disks (PPDs) at a fiducial radius of 1 AU that take into account the effects of both Ohmic resistivity and ambipolar diffusion (AD). The magnetic diffusion coefficients are evaluated self-consistently from a look-up table based on equilibrium chemistry. We first show that the inclusion of AD dramatically changes the conventional picture of layered accretion. Without net vertical magnetic field, the system evolves into a toroidal field dominated configuration with extremely weak turbulence in the far-UV ionization layer that is far too inefficient to drive rapid accretion. In the presence of a weak net vertical field (plasma {beta} {approx} 10{sup 5} at midplane), we find that the magnetorotational instability (MRI) is completely suppressed, resulting in a fully laminar flow throughout the vertical extent of the disk. A strong magnetocentrifugal wind is launched that efficiently carries away disk angular momentum and easily accounts for the observed accretion rate in PPDs. Moreover, under a physical disk wind geometry, all the accretion flow proceeds through a strong current layer with a thickness of {approx}0.3H that is offset from disk midplane with radial velocity of up to 0.4 times the sound speed. Both Ohmic resistivity and AD are essential for the suppression of the MRI and wind launching. The efficiency of wind transport increases with increasing net vertical magnetic flux and the penetration depth of the FUV ionization. Our laminar wind solution has important implications on planet formation and global evolution of PPDs.
20. Accretion disk electrodynamics
NASA Technical Reports Server (NTRS)
Coroniti, F. V.
1985-01-01
Accretion disk electrodynamic phenomena are separable into two classes: (1) disks and coronas with turbulent magnetic fields; (2) disks and black holes which are connected to a large-scale external magnetic field. Turbulent fields may originate in an alpha-omega dynamo, provide anomalous viscous transport, and sustain an active corona by magnetic buoyancy. The large-scale field can extract energy and angular momentum from the disk and black hole, and be dynamically configured into a collimated relativistic jet.
1. Constraining the Physics of AM Canum Venaticorum Systems with the Accretion Disk Instability Model
NASA Technical Reports Server (NTRS)
Cannizzo, John K.; Nelemans, Gijs
2015-01-01
Recent work by Levitan et al. has expanded the long-term photometric database for AM CVn stars. In particular, their outburst properties are well correlated with orbital period and allow constraints to be placed on the secular mass transfer rate between secondary and primary if one adopts the disk instability model for the outbursts. We use the observed range of outbursting behavior for AM CVn systems as a function of orbital period to place a constraint on mass transfer rate versus orbital period. We infer a rate approximately 5 x 10(exp -9) solar mass yr(exp -1) ((P(sub orb)/1000 s)(exp -5.2)). We show that the functional form so obtained is consistent with the recurrence time-orbital period relation found by Levitan et al. using a simple theory for the recurrence time. Also, we predict that their steep dependence of outburst duration on orbital period will flatten considerably once the longer orbital period systems have more complete observations.
2. Gravitational instabilities in protostellar disks
NASA Technical Reports Server (NTRS)
Tohline, J. E.
1994-01-01
The nonaxisymmetric stability of self-gravitating, geometrically thick accretion disks has been studied for protostellar systems having a wide range of disk-to-central object mass ratios. Global eigenmodes with four distinctly different characters were identified using numerical, nonlinear hydrodynamic techniques. The mode that appears most likely to arise in normal star formation settings, however, resembles the 'eccentric instability' that was identified earlier in thin, nearly Keplerian disks: It presents an open, one-armed spiral pattern that sweeps continuously in a trailing direction through more than 2-pi radians, smoothly connecting the inner and outer edges of the disk, and requires cooperative motion of the point mass for effective amplification. This particular instability promotes the development of a single, self-gravitating clump of material in orbit about the point mass, so its routine appearance in our simulations supports the conjecture that the eccentric instability provides a primary route to the formation of short-period binaries in protostellar systems.
3. ACCRETION OUTBURSTS IN CIRCUMPLANETARY DISKS
SciTech Connect
Lubow, S. H.; Martin, R. G.
2012-04-20
We describe a model for the long-term evolution of a circumplanetary disk that is fed mass from a circumstellar disk and contains regions of low turbulence (dead zones). We show that such disks can be subject to accretion-driven outbursts, analogous to outbursts previously modeled in the context of circumstellar disks to explain FU Ori phenomena. Circumplanetary disks around a proto-Jupiter can undergo outbursts for infall accretion rates onto the disks in the range M-dot{sub infall} approx. 10{sup -9} to 10{sup -7} M{sub Sun} yr{sup -1}, typical of accretion rates in the T Tauri phase. During outbursts, the accretion rate and disk luminosity increases by several orders of magnitude. Most of the planet mass growth during planetary gas accretion may occur via disk outbursts involving gas that is considerably hotter than predicted by steady state models. For low infall accretion rates M-dot{sub infall} {approx}< 10{sup -10} M{sub sun} yr{sup -1} that occur in late stages of disk accretion, disk outbursts are unlikely to occur, even if dead zones are present. Such conditions are favorable for the formation of icy satellites.
4. Local model for angular-momentum transport in accretion disks driven by the magnetorotational instability.
PubMed
Pessah, Martin E; Chan, Chi-Kwan; Psaltis, Dimitrios
2006-12-01
We develop a local model for the exponential growth and saturation of the Reynolds and Maxwell stresses in turbulent flows driven by the magnetorotational instability. We first derive equations that describe the effects of the instability on the growth and pumping of the stresses. We highlight the relevance of a new type of correlations that couples the dynamical evolution of the Reynolds and Maxwell stresses and plays a key role in developing and sustaining the magnetorotational turbulence. We then supplement these equations with a phenomenological description of the triple correlations that lead to a saturated turbulent state. We show that the steady-state limit of the model describes successfully the correlations among stresses found in numerical simulations of shearing boxes.
5. Hydrodynamic Viscosity in Accretion Disks
Duschl, Wolfgang J.; Strittmatter, Peter A.; Biermann, Peter L.
We propose a generalized accretion disk viscosity prescription based on hydrodynamically driven turbulence at the critical effective Reynolds number. This approach is consistent with recent re-analysis by Richard & Zahn (1999) of experimental results on turbulent Couette-Taylor flows. This new β-viscosity formulation applies to both selfgravitating and non-selfgravitating disks and is shown to yield the standard α-disk prescription in the case of shock dissipation limited, non-selfgravitating disks.
6. ACCRETING CIRCUMPLANETARY DISKS: OBSERVATIONAL SIGNATURES
SciTech Connect
Zhu, Zhaohuan
2015-01-20
I calculate the spectral energy distributions of accreting circumplanetary disks using atmospheric radiative transfer models. Circumplanetary disks only accreting at 10{sup –10} M {sub ☉} yr{sup –1} around a 1 M{sub J} planet can be brighter than the planet itself. A moderately accreting circumplanetary disk ( M-dot ∼10{sup −8} M{sub ⊙} yr{sup −1}; enough to form a 10 M{sub J} planet within 1 Myr) around a 1 M{sub J} planet has a maximum temperature of ∼2000 K, and at near-infrared wavelengths (J, H, K bands), this disk is as bright as a late-M-type brown dwarf or a 10 M{sub J} planet with a ''hot start''. To use direct imaging to find the accretion disks around low-mass planets (e.g., 1 M{sub J} ) and distinguish them from brown dwarfs or hot high-mass planets, it is crucial to obtain photometry at mid-infrared bands (L', M, N bands) because the emission from circumplanetary disks falls off more slowly toward longer wavelengths than those of brown dwarfs or planets. If young planets have strong magnetic fields (≳100 G), fields may truncate slowly accreting circumplanetary disks ( M-dot ≲10{sup −9} M{sub ⊙} yr{sup −1}) and lead to magnetospheric accretion, which can provide additional accretion signatures, such as UV/optical excess from the accretion shock and line emission.
7. Anisotropic hydrodynamic turbulence in accretion disks
Stoll, Moritz H. R.; Kley, Wilhelm; Picogna, Giovanni
2017-03-01
Recently, the vertical shear instability (VSI) has become an attractive purely hydrodynamic candidate for the anomalous angular momentum transport required for weakly ionized accretion disks. In direct three-dimensional numerical simulations of VSI turbulence in disks, a meridional circulation pattern was observed that is opposite to the usual viscous flow behavior. Here, we investigate whether this feature can possibly be explained by an anisotropy of the VSI turbulence. Using three-dimensional hydrodynamical simulations, we calculate the turbulent Reynolds stresses relevant for angular momentum transport for a representative section of a disk. We find that the vertical stress is significantly stronger than the radial stress. Using our results in viscous disk simulations with different viscosity coefficients for the radial and vertical direction, we find good agreement with the VSI turbulence for the stresses and meridional flow; this provides additional evidence for the anisotropy. The results are important with respect to the transport of small embedded particles in disks.
8. Disk Accretion of Tidally Disrupted Rocky Bodies onto White Dwarfs
Feng, W.; Desch, S.
2017-03-01
The prevailing model for the pollution of white dwarf photospheres invokes accretion from a disk of gas and solid particles, fed by tidal disruption of rocky bodies inside the Roche radius. Current models can successfully explain the accretion rates of metals onto white dwarfs, provided the gaseous disks viscously spread at rates consistent with a partially suppressed magnetorotational instability (Metzger et al. 2012); however, these models do not explore the extent of the magnetorotational instability in disks by calculating the degree of ionization. We present ionization fractions for thermal and non-thermal processes to assess the extent of the magnetorotational instability in white dwarf disks. We determine that the disk viscosity parameter α can be as high as 0.1 in white disks, implying that the magnetorotational instability must be carefully modeled.
9. RINGED ACCRETION DISKS: EQUILIBRIUM CONFIGURATIONS
SciTech Connect
Pugliese, D.; Stuchlík, Z. E-mail: [email protected]
2015-12-15
We investigate a model of a ringed accretion disk, made up by several rings rotating around a supermassive Kerr black hole attractor. Each toroid of the ringed disk is governed by the general relativity hydrodynamic Boyer condition of equilibrium configurations of rotating perfect fluids. Properties of the tori can then be determined by an appropriately defined effective potential reflecting the background Kerr geometry and the centrifugal effects. The ringed disks could be created in various regimes during the evolution of matter configurations around supermassive black holes. Therefore, both corotating and counterrotating rings have to be considered as being a constituent of the ringed disk. We provide constraints on the model parameters for the existence and stability of various ringed configurations and discuss occurrence of accretion onto the Kerr black hole and possible launching of jets from the ringed disk. We demonstrate that various ringed disks can be characterized by a maximum number of rings. We present also a perturbation analysis based on evolution of the oscillating components of the ringed disk. The dynamics of the unstable phases of the ringed disk evolution seems to be promising in relation to high-energy phenomena demonstrated in active galactic nuclei.
10. Disk tides and accretion runaway
NASA Technical Reports Server (NTRS)
Ward, William R.; Hahn, Joseph M.
1995-01-01
It is suggested that tidal interaction of an accreting planetary embryo with the gaseous preplanetary disk may provide a mechanism to breach the so-called runaway limit during the formation of the giant planet cores. The disk tidal torque converts a would-be shepherding object into a 'predator,' which can continue to cannibalize the planetesimal disk. This is more likely to occur in the giant planet region than in the terrestrial zone, providing a natural cause for Jupiter to predate the inner planets and form within the O(10(exp 7) yr) lifetime of the nebula.
11. Strongly Magnetized Accretion Disks Around Black Holes
Salvesen, Greg; Armitage, Philip J.; Simon, Jacob B.; Begelman, Mitchell C.
2017-01-01
Recent observations are suggestive of strongly magnetized accretion disks around black holes. Performing local (shearing box) simulations of accretion disks, we investigate how a strong magnetization state can develop and persist. We demonstrate that poloidal flux is a necessary prerequisite for the sustainability of strongly magnetized accretion disks. We also show that black hole spin measurements can become unconstrained if magnetic fields provide a significant contribution to the vertical pressure support of the accretion disk atmosphere.
12. Accretion disks around black holes
NASA Technical Reports Server (NTRS)
Abramowicz, M. A.
1994-01-01
The physics of accretion flow very close to a black hole is dominated by several general relativistic effects. It cannot be described by the standard Shakura Sunyaev model or by its relativistic version developed by Novikov and Thome. The most important of these effects is a dynamical mass loss from the inner edge of the disk (Roche lobe overflow). The relativistic Roche lobe overflow induces a strong advective cooling, which is sufficient to stabilize local, axially symmetric thermal and viscous modes. It also stabilizes the non-axially-symmetric global modes discovered by Papaloizou and Pringle. The Roche lobe overflow, however, destabilizes sufficiently self-gravitating accretion disks with respect to a catastrophic runaway of mass due to minute changes of the gravitational field induced by the changes in the mass and angular momentum of the central black hole. One of the two acoustic modes may become trapped near the inner edge of the disk. All these effects, absent in the standard model, have dramatic implications for time-dependent behavior of the accretion disks around black holes.
13. Black hole accretion disks with coronae
NASA Technical Reports Server (NTRS)
Svensson, Roland; Zdziarski, Andrzej A.
1994-01-01
Observations suggest the existence of both hot and cold dark matter in the centers of active galactic nuclei. Recent spectral models require a major fraction of power to be dissipated in the hot matter. We study the case when the hot matter forms a corona around a standard cold alpha-disk. In particular, we investigate the case when a major fraction, f, of the power released when the cold matter accretes is transported to and dissipated in the corona. This has major effects on the cold disk, making it colder, more geometrically thin, denser, and having larger optical depths. One important consequence is the disappearance of the effectively optically thin zone as well as of the radiation pressure dominated zone for values of f sufficiently closed to unity. The disappearance of the radiation pressure dominated zone will result in a cold disk with only a gas pressure dominated zone that is stable against thermal and viscous instabilities. We also show that the pressure ( and the radiation) from the corona will only affect the surface layers of the cold disk. Our results disagree with those of other recent work on accretion disks with coronae. We find those works to be based on unphysical assumptions.
14. Accretion Disks around Young Stars
D'Alessio, Paola
1996-04-01
A method to calculate the structure and brightness distribution of accretion disks surrounding low and intermediate mass young stars is introduced and discussed. The method includes a realistic treatment of the energy transport mechanisms and disk heating by radiation from external sources. The disk is assumed steady, geometrically thin and in vertical hydrostatic equilibrium. The turbulent viscosity coefficient is expressed using the α prescription and the α parameter and the mass accretion rate are assumed to be constant through the disk. Energy is transported in the vertical direction by: (a) a turbulent flux, computed self-consistently with the viscosity coefficient used to describe the viscous energy dissipation, (b) radiation, using the first moments of the transfer equation, the Eddington approximation, and the Rosseland and Planck Mean Opacities, and (c) convection, taking into account that the convective elements, not necessarily optically thick, lose energy by radiation and turbulent flux. This treatment of the energy transport mechanisms differs from previous work in this field, allowing one to extend, with confidence, the calculation of the disk structure to optically thin regimes. The heating mechanisms considered, which affect the disk's structure and emission, are stellar radiation and a circumstellar envelope which reprocesses and scatters radiation from the star and from the disk itself. In addition to a detailed numerical calculation, an analytical self-consistent formulation of the irradiation of the disk is given. This analytical formulation allows one to understand and extend the numerical results. To evaluate the potential of the method presented in this thesis, a set of models of viscous non-irradiated and irradiated disks are computed. Their predictions are compared with observations of young stellar sources likely to have disks. Given the disk structure and specifying its orientation with respect to the line of sight, the specific
15. Accretion Disks in Algols: Progenitors and Evolution
van Rensbergen, W.; de Greve, J. P.
2017-02-01
There are only a few Algols with derived accretion disk parameters. These measurements provide additional constraints for tracing the origin of individual systems. With a modified binary evolution code, series of close binary evolution were calculated. For six Algols with accretion disks we found initial systems that evolve closely into the presently observed system parameters and disk characteristics.
16. Photon Bubbles and the Vertical Structure of Accretion Disks
Begelman, Mitchell C.
2006-06-01
We consider the effects of photon bubble'' shock trains on the vertical structure of radiation pressure-dominated accretion disks. These density inhomogeneities are expected to develop spontaneously in radiation-dominated accretion disks where magnetic pressure exceeds gas pressure, even in the presence of magnetorotational instability (MRI). They increase the rate at which radiation escapes from the disk and may allow disks to exceed the Eddington limit by a substantial factor without blowing themselves apart. To refine our earlier analysis of photon bubble transport in accretion disks, we generalize the theory of photon bubbles to include the effects of finite optical depths and radiation damping. Modifications to the diffusion law at low τ tend to fill in'' the low-density regions of photon bubbles, while radiation damping inhibits the formation of photon bubbles at large radii, small accretion rates, and small heights above the equatorial plane. Accretion disks dominated by photon bubble transport may reach luminosities from 10 to >100 times the Eddington limit (LEdd), depending on the mass of the central object, while remaining geometrically thin. However, photon bubble-dominated disks with α-viscosity are subject to the same thermal and viscous instabilities that plague standard radiation pressure-dominated disks, suggesting that they may be intrinsically unsteady. Photon bubbles can lead to a core-halo'' vertical disk structure. In super-Eddington disks the halo forms the base of a wind, which carries away substantial energy and mass, but not enough to prevent the luminosity from exceeding LEdd. Photon bubble-dominated disks may have smaller color corrections than standard accretion disks of the same luminosity. They remain viable contenders for some ultraluminous X-ray sources and may play a role in the rapid growth of supermassive black holes at high redshift.
17. Diskoseismology: Probing relativistic accretion disks
Nowak, Michael Allen
1992-08-01
Helioseismology has provided a wealth of information about the structure of the solar atmosphere. Little is known, however, about the structure of accretion disks that are thought to exist around black holes and neutron stars. In this thesis we present calculations of modes that are trapped in thin Keplerian accretion disks. We hope to use observations of thes modes to elucidate the structure of the inner relativistic regions of accretion disks. Our calculations assume that the thin disk is terminated by an innermost stable orbit, as would occur around a slowly rotating black hole or weakly magnetized compact neutron star. The dominant relativistic effects, which allow modes to be trapped within the inner region of the disk, are approximated via a modified Newtonian potential. Using the Lagrangian formulation of Friedman and Schutz, we develop a general formalism for investigating the adiabatic oscillations of arbitrary unperturbed disk models. First we consider the special case of acoustic waves in disks with isothermal atmospheres. Next we describe the Lagrangian perturbation vectors in terms of the derivatives of a scalar potential, as has been done by Ipser and Lindblom. Using this potential, we derive a single partial differential equation governing the oscillations of a disk. The eigenfunctions and eigenfrequencies of a variety of disk models are found to fall into two main classes which are analogous to the p-modes and g-modes in the sun. Specifically we use the potential formalism to compute the g-modes for disks with isothermal atmospheres. Physical arguments show that both the p-modes and g-modes belong to the same family of modes as the p-modes and g-modes in the sun, just viewed in a different parameter regime. With the aid of the Lagrangian formalism we consider possible growth or damping mechanisms and compute the (assumed) relatively small rates of growth or damping of the modes. Specifically, we consider gravitational radiation reaction and
18. Conservative GRMHD simulations of moderately thin, tilted accretion disks
SciTech Connect
Teixeira, Danilo Morales; Fragile, P. Chris; Zhuravlev, Viacheslav V.; Ivanov, Pavel B.
2014-12-01
This paper presents our latest numerical simulations of accretion disks that are misaligned with respect to the rotation axis of a Kerr black hole. In this work, we use a new, fully conservative version of the Cosmos++ general relativistic magnetohydrodynamics (GRMHD) code, coupled with an ad hoc cooling function designed to control the thickness of the disk. Together these allow us to simulate the thinnest tilted accretion disks ever using a GRMHD code. In this way, we are able to probe the regime where the dimensionless stress and scale height of the disk become comparable. We present results for both prograde and retrograde cases. The simulated prograde tilted disk shows no sign of Bardeen-Petterson alignment even in the innermost parts of the disk. The simulated retrograde tilted disk, however, does show modest alignment. The implication of these results is that the parameter space associated with Bardeen-Petterson alignment for prograde disks may be rather small, only including very thin disks. Unlike our previous work, we find no evidence for standing shocks in our simulated tilted disks. We ascribe this to the black hole spin, tilt angle, and disk scale height all being small in these simulations. We also add to the growing body of literature pointing out that the turbulence driven by the magnetorotational instability in global simulations of accretion disks is not isotropic. Finally, we provide a comparison between our moderately thin, untilted reference simulation and other numerical simulations of thin disks in the literature.
19. Foundations of Black Hole Accretion Disk Theory.
PubMed
Abramowicz, Marek A; Fragile, P Chris
2013-01-01
This review covers the main aspects of black hole accretion disk theory. We begin with the view that one of the main goals of the theory is to better understand the nature of black holes themselves. In this light we discuss how accretion disks might reveal some of the unique signatures of strong gravity: the event horizon, the innermost stable circular orbit, and the ergosphere. We then review, from a first-principles perspective, the physical processes at play in accretion disks. This leads us to the four primary accretion disk models that we review: Polish doughnuts (thick disks), Shakura-Sunyaev (thin) disks, slim disks, and advection-dominated accretion flows (ADAFs). After presenting the models we discuss issues of stability, oscillations, and jets. Following our review of the analytic work, we take a parallel approach in reviewing numerical studies of black hole accretion disks. We finish with a few select applications that highlight particular astrophysical applications: measurements of black hole mass and spin, black hole vs. neutron star accretion disks, black hole accretion disk spectral states, and quasi-periodic oscillations (QPOs).
20. Transonic disk accretion onto black holes
NASA Technical Reports Server (NTRS)
Liang, E. P. T.; Thompson, K. A.
1980-01-01
The solution for the radial drift velocity of thin disk accretion onto black holes must be transonic, and is analogous to the critical solution in spherical Bondi accretion, except for the presence of angular momentum. The transonic requirement yields a correct treatment of the inner region of the disk not found in the conventional Keplerian models and may lead to significantly different overall disk structures. Possible observational consequences, relevant to the black hole hypothesis for Cyg X-1 and other candidates, are discussed.
1. Accretion disks in Algols: Progenitors and evolution
Van Rensbergen, W.; De Greve, J. P.
2016-08-01
Context. There are only a few Algols with measured accretion disk parameters. These measurements provide additional constraints for tracing the origin of individual systems, narrowing down the initial parameter space. Aims: We investigate the origin and evolution of six Algol systems with accretion disks to find the initial parameters and evolutionary constraints for them. Methods: With a modified binary evolution code, series of close binary evolution are calculated to obtain the best match for observed individual systems. Results: Initial parameters for six Algol systems with accretion disks were determined matching both the present system parameters and the observed disk characteristics. Conclusions: When Roche lobe overflow (RLOF) starts during core hydrogen burning of the donor, the disk lifetime was found to be short. The disk luminosity is comparable to the luminosity of the gainer during a large fraction of the disk lifetime.
2. ON HYDROMAGNETIC STRESSES IN ACCRETION DISK BOUNDARY LAYERS
SciTech Connect
Pessah, Martin E.; Chan, Chi-kwan E-mail: [email protected]
2012-05-20
Detailed calculations of the physical structure of accretion disk boundary layers, and thus their inferred observational properties, rely on the assumption that angular momentum transport is opposite to the radial angular frequency gradient of the disk. The standard model for turbulent shear viscosity satisfies this assumption by construction. However, this behavior is not supported by numerical simulations of turbulent magnetohydrodynamic (MHD) accretion disks, which show that angular momentum transport driven by the magnetorotational instability (MRI) is inefficient in disk regions where, as expected in boundary layers, the angular frequency increases with radius. In order to shed light on physically viable mechanisms for angular momentum transport in this inner disk region, we examine the generation of hydromagnetic stresses and energy density in differentially rotating backgrounds with angular frequencies that increase outward in the shearing-sheet framework. We isolate the modes that are unrelated to the standard MRI and provide analytic solutions for the long-term evolution of the resulting shearing MHD waves. We show that, although the energy density of these waves can be amplified significantly, their associated stresses oscillate around zero, rendering them an inefficient mechanism to transport significant angular momentum (inward). These findings are consistent with the results obtained in numerical simulations of MHD accretion disk boundary layers and challenge the standard assumption of efficient angular momentum transport in the inner disk regions. This suggests that the detailed structure of turbulent MHD accretion disk boundary layers could differ appreciably from those derived within the standard framework of turbulent shear viscosity.
3. Gravitational Instabilities in Circumstellar Disks
Kratter, Kaitlin; Lodato, Giuseppe
2016-09-01
Star and planet formation are the complex outcomes of gravitational collapse and angular momentum transport mediated by protostellar and protoplanetary disks. In this review, we focus on the role of gravitational instability in this process. We begin with a brief overview of the observational evidence for massive disks that might be subject to gravitational instability and then highlight the diverse ways in which the instability manifests itself in protostellar and protoplanetary disks: the generation of spiral arms, small-scale turbulence-like density fluctuations, and fragmentation of the disk itself. We present the analytic theory that describes the linear growth phase of the instability supplemented with a survey of numerical simulations that aim to capture the nonlinear evolution. We emphasize the role of thermodynamics and large-scale infall in controlling the outcome of the instability. Despite apparent controversies in the literature, we show a remarkable level of agreement between analytic predictions and numerical results. In the next part of our review, we focus on the astrophysical consequences of the instability. We show that the disks most likely to be gravitationally unstable are young and relatively massive compared with their host star, Md/M*≥0.1. They will develop quasi-stable spiral arms that process infall from the background cloud. Although instability is less likely at later times, once infall becomes less important, the manifestations of the instability are more varied. In this regime, the disk thermodynamics, often regulated by stellar irradiation, dictates the development and evolution of the instability. In some cases the instability may lead to fragmentation into bound companions. These companions are more likely to be brown dwarfs or stars than planetary mass objects. Finally, we highlight open questions related to the development of a turbulent cascade in thin disks and the role of mode-mode coupling in setting the maximum angular
4. Observational Tests of the Picture of Disk Accretion
Maccarone, Thomas J.
2014-09-01
In this chapter, I present a summary of observational tests of the basic picture of disk accretion. An emphasis is placed on tests relevant to black holes, but many of the fundamental results are drawn from studies of other classes of systems. Evidence is discussed for the basic structures of accretion flows. The cases of systems with and without accretion disks are discussed, as is the evidence that disks actually form. Also discussed are the hot spots where accretion streams impact the disks, and the boundary layers in the inner parts of systems where the accretors are not black holes. The nature of slow, large amplitude variability is discussed. It is shown that some of the key predictions of the classical thermal-viscous ionization instability model for producing outbursts are in excellent agreement with observational results. It is also show that there are systems whose outbursts are extremely difficult to explain without invoking variations in the rate of mass transfer from the donor star into the outer accretion disk, or tidally induced variations in the mass transfer rates. Finally, I briefly discuss recent quasar microlensing measurements which give truly independent constraints on the inner accretion geometry around black holes.
5. Generalized Similarity for Accretion/Decretion Disks
Rafikov, Roman R.
2016-10-01
Decretion (or external) disks are gas disks freely expanding to large radii due to their internal stresses. They are expected to naturally arise in tidal disruption events, around Be stars, in mass-losing post-main-sequence binaries, as a result of supernova fallback, etc. Their evolution is theoretically understood in two regimes: when the central object does not exert torque on the disk (a standard assumption for conventional accretion disks) or when no mass inflow (or outflow) occurs at the disk center. However, many astrophysical objects—circumbinary disks, Be stars, neutron stars accreting in a propeller regime, etc.—feature non-zero torque simultaneously with the non-zero accretion (or ejection of mass) at the disk center. We provide a general description for the evolution of such disks (both linear and nonlinear) in the self-similar regime, to which the disk should asymptotically converge with time. We identify a similarity parameter λ, which is uniquely related to the degree, to which the central mass accretion is suppressed by the non-zero central torque. The known decretion disk solutions correspond to the two discrete values of λ, while our new solutions cover a continuum of its physically allowed values, corresponding to either accretion or mass ejection by the central object. A direct relationship between λ and central \\dot{M} and torque is also established. We describe the time evolution of the various disk characteristics for different λ, and show that the observable properties (spectrum and luminosity evolution) of the decretion disks, in general, are different from the standard accretion disks with no central torque.
6. Roche Potential with Luminous Accretion Disks
Fukue, Jun; Hanamoto, Keishi
2002-12-01
The radiative environments of an interacting binary, which contains a luminous accretion disk, are investigated. The disk radiation field has no effect in the direction of the orbital plane, while it significantly affects in the polar direction. As the disk luminosity increases, the Roche potential around the compact star becomes hollow in the polar direction. It further crashes toward the pole, and a cone of avoidance appears at the normalized luminosity Γd ≡ Ld/LE = 0.5, where Ld is the disk luminosity and LE the Eddington luminosity of the compact star. The disk corona, the wind-fed accretion flow, and the common envelope suffer a remarkable influence by the luminous disk in active binaries. Of these, the wind-fed accretion is briefly discussed.
7. Black-Hole Accretion Disks --- Towards a New Paradigm ---
Kato, S.; Fukue, J.; Mineshige, S.
2008-03-01
8. Gas accretion from the cosmic web feeding disk galaxies
Sánchez Almeida, J.; Olmo-García, A.; Elmegreen, B. G.; Muñoz-Tuñón, C.; Elmegreen, D. M.; Filho, M. E.; Pérez-Montero, E.; Amorín, R.
2017-03-01
Disk galaxies in cosmological numerical simulations grow by accreting gas from the cosmic web. This gas reaches the external disk, and then spirals in dragged along by tidal forces and/or disk instabilities. The importance of gas infall is as clear from numerical simulations as it is obscure to observations. Extremely metal poor (XMP) galaxies seem to be the best example we have of the gas accretion process at work. They have large off-center starbursts which show significant metallicity drop compared with the host galaxy. This observation is naturally explained as a gas accretion event caught in the act. We present preliminary results of the kinematical properties of the metal poor starbursts in XMPs, which suggest that the starbursts are kinematically decoupled entities within the host galaxy.
9. Nonthermal accretion disk models around neutron stars
NASA Technical Reports Server (NTRS)
Tavani, M.; Liang, Edison P.
1994-01-01
We consider the structure and emission spectra of nonthermal accretion disks around both strongly and weakly magnetized neutron stars. Such disks may be dissipating their gravitational binding energy and transferring their angular momentum via semicontinuous magnetic reconnections. We consider specifically the structure of the disk-stellar magnetospheric boundary where magnetic pressure balances the disk pressure. We consider energy dissipation via reconnection of the stellar field and small-scale disk turbulent fields of opposite polarity. Constraints on the disk emission spectrum are discussed.
10. Gas dynamics for accretion disk simulations
NASA Technical Reports Server (NTRS)
Whitehurst, R.
1994-01-01
The behavior of accretion disks can largely be understood in terms of the basic physical processes of mass, energy, and momentum conservation. Despite this, detailed modeling of these systems using modern computational techniques is challenging and controversial. Disturbing differences exist between methods used widely in astrophysics, namely Eulerian finite-difference techniques and particle codes such as SPH. Therefore neither technique is fully satisfactory for accretion disk simulations. This paper describes a new fully Lagrangian method designed to resolve these difficulties.
11. Evolution of accretion disks in tidal disruption events
SciTech Connect
Shen, Rong-Feng; Matzner, Christopher D. E-mail: [email protected]
2014-04-01
During a stellar tidal disruption event (TDE), an accretion disk forms as stellar debris returns to the disruption site and circularizes. Rather than being confined within the circularizing radius, the disk can spread to larger radii to conserve angular momentum. A spreading disk is a source of matter for re-accretion at rates that may exceed the later stellar fallback rate, although a disk wind can suppress its contribution to the central black hole accretion rate. A spreading disk is detectible through a break in the central accretion rate history or, at longer wavelengths, by its own emission. We model the evolution of TDE disk size and accretion rate by accounting for the time-dependent fallback rate, for the influence of wind losses in the early advective stage, and for the possibility of thermal instability for accretion rates intermediate between the advection-dominated and gas-pressure-dominated states. The model provides a dynamic basis for modeling TDE light curves. All or part of a young TDE disk will precess as a solid body because of the Lense-Thirring effect, and precession may manifest itself as a quasi-periodic modulation of the light curve. The precession period increases with time. Applying our results to the jetted TDE candidate Swift J1644+57, whose X-ray light curve shows numerous quasi-periodic dips, we argue that the data best fit a scenario in which a main-sequence star was fully disrupted by an intermediate mass black hole on an orbit significantly inclined from the black hole equator, with the apparent jet shutoff at t = 500 days corresponding to a disk transition from the advective state to the gas-pressure-dominated state.
12. Evolution of Accretion Disks in Tidal Disruption Events
Shen, Rong-Feng; Matzner, Christopher D.
2014-04-01
During a stellar tidal disruption event (TDE), an accretion disk forms as stellar debris returns to the disruption site and circularizes. Rather than being confined within the circularizing radius, the disk can spread to larger radii to conserve angular momentum. A spreading disk is a source of matter for re-accretion at rates that may exceed the later stellar fallback rate, although a disk wind can suppress its contribution to the central black hole accretion rate. A spreading disk is detectible through a break in the central accretion rate history or, at longer wavelengths, by its own emission. We model the evolution of TDE disk size and accretion rate by accounting for the time-dependent fallback rate, for the influence of wind losses in the early advective stage, and for the possibility of thermal instability for accretion rates intermediate between the advection-dominated and gas-pressure-dominated states. The model provides a dynamic basis for modeling TDE light curves. All or part of a young TDE disk will precess as a solid body because of the Lense-Thirring effect, and precession may manifest itself as a quasi-periodic modulation of the light curve. The precession period increases with time. Applying our results to the jetted TDE candidate Swift J1644+57, whose X-ray light curve shows numerous quasi-periodic dips, we argue that the data best fit a scenario in which a main-sequence star was fully disrupted by an intermediate mass black hole on an orbit significantly inclined from the black hole equator, with the apparent jet shutoff at t = 500 days corresponding to a disk transition from the advective state to the gas-pressure-dominated state.
13. Instability of counter-rotating stellar disks
Hohlfeld, R. G.; Lovelace, R. V. E.
2015-09-01
We use an N-body simulation, constructed using GADGET-2, to investigate an accretion flow onto an astrophysical disk that is in the opposite sense to the disk's rotation. In order to separate dynamics intrinsic to the counter-rotating flow from the impact of the flow onto the disk, we consider an initial condition in which the counter-rotating flow is in an annular region immediately exterior the main portion of the astrophysical disk. Such counter-rotating flows are seen in systems such as NGC 4826 (known as the "Evil Eye Galaxy"). Interaction between the rotating and counter-rotating components is due to two-stream instability in the boundary region. A multi-armed spiral density wave is excited in the astrophysical disk and a density distribution with high azimuthal mode number is excited in the counter-rotating flow. Density fluctuations in the counter-rotating flow aggregate into larger clumps and some of the material in the counter-rotating flow is scattered to large radii. Accretion flow processes such as this are increasingly seen to be of importance in the evolution of multi-component galactic disks.
14. Reverberation Mapping of AGN Accretion Disks
Fausnaugh, Michael; AGN STORM Collaboration
2017-01-01
I will discuss new reverberation mapping results that allow us to investigate the temperature structure of AGN accretion disks. By measuring time-delays between broad-band continuum light curves, we can determine the size of the disk as a function of wavelength. I will discuss the detection of continuum lags in NGC 5548 reported by the AGN STORM project and implications for the accretion disk. I will also present evidence for continuum lags in two other AGN for which we recently measured black hole masses from continuum-Hbeta reverberations. The mass measurements allow us to compare the continuum lags to predictions from standard thin disk theory, and our results indicate that the accretion disks are larger than the simplest expectations.
15. Testing Convergence for Global Accretion Disks
Hawley, John F.; Richers, Sherwood A.; Guan, Xiaoyue; Krolik, Julian H.
2013-08-01
Global disk simulations provide a powerful tool for investigating accretion and the underlying magnetohydrodynamic turbulence driven by magneto-rotational instability (MRI). Using them to accurately predict quantities such as stress, accretion rate, and surface brightness profile requires that purely numerical effects, arising from both resolution and algorithm, be understood and controlled. We use the flux-conservative Athena code to conduct a series of experiments on disks having a variety of magnetic topologies to determine what constitutes adequate resolution. We develop and apply several resolution metrics: langQz rang and langQ phirang, the ratio of the grid zone size to the characteristic MRI wavelength, αmag, the ratio of the Maxwell stress to the magnetic pressure, and \\langle B_R^2\\rangle /\\langle B_\\phi ^2\\rangle, the ratio of radial to toroidal magnetic field energy. For the initial conditions considered here, adequate resolution is characterized by langQz rang >= 15, langQ phirang >= 20, αmag ≈ 0.45, and \\langle B_R^2\\rangle /\\langle B_\\phi ^2\\rangle \\approx 0.2. These values are associated with >=35 zones per scaleheight H, a result consistent with shearing box simulations. Numerical algorithm is also important. Use of the Harten-Lax-van Leer-Einfeldt flux solver or second-order interpolation can significantly degrade the effective resolution compared to the Harten-Lax-van Leer discontinuities flux solver and third-order interpolation. Resolution at this standard can be achieved only with large numbers of grid zones, arranged in a fashion that matches the symmetries of the problem and the scientific goals of the simulation. Without it, however, quantitative measures important to predictions of observables are subject to large systematic errors.
16. On the Gravitational Stability of Gravito-turbulent Accretion Disks
Lin, Min-Kai; Kratter, Kaitlin M.
2016-06-01
Low mass, self-gravitating accretion disks admit quasi-steady, “gravito-turbulent” states in which cooling balances turbulent viscous heating. However, numerical simulations show that gravito-turbulence cannot be sustained beyond dynamical timescales when the cooling rate or corresponding turbulent viscosity is too large. The result is disk fragmentation. We motivate and quantify an interpretation of disk fragmentation as the inability to maintain gravito-turbulence due to formal secondary instabilities driven by: (1) cooling, which reduces pressure support; and/or (2) viscosity, which reduces rotational support. We analyze the axisymmetric gravitational stability of viscous, non-adiabatic accretion disks with internal heating, external irradiation, and cooling in the shearing box approximation. We consider parameterized cooling functions in 2D and 3D disks, as well as radiative diffusion in 3D. We show that generally there is no critical cooling rate/viscosity below which the disk is formally stable, although interesting limits appear for unstable modes with lengthscales on the order of the disk thickness. We apply this new linear theory to protoplanetary disks subject to gravito-turbulence modeled as an effective viscosity, and cooling regulated by dust opacity. We find that viscosity renders the disk beyond ˜60 au dynamically unstable on radial lengthscales a few times the local disk thickness. This is coincident with the empirical condition for disk fragmentation based on a maximum sustainable stress. We suggest turbulent stresses can play an active role in realistic disk fragmentation by removing rotational stabilization against self-gravity, and that the observed transition in behavior from gravito-turbulent to fragmenting may reflect instability of the gravito-turbulent state itself.
17. Where a Neutron Star's Accretion Disk Ends
Kohler, Susanna
2016-03-01
In X-ray binaries that consist of a neutron star and a companion star, gas funnels from the companion into an accretion disk surrounding the neutron star, spiraling around until it is eventually accreted. How do the powerful magnetic fields threading through the neutron star affect this accretion disk? Recent observations provide evidence that they may push the accretion disk away from the neutron stars surface.Truncated DisksTheoretical models have indicated that neutron star accretion disks may not extend all the way in to the surface of a neutron star, but may instead be truncated at a distance. This prediction has been difficult to test observationally, however, due to the challenge of measuring the location of the inner disk edge in neutron-star X-ray binaries.In a new study, however, a team of scientists led by Ashley King (Einstein Fellow at Stanford University) has managed to measure the location of the inner edge of the disk in Aquila X-1, a neutron-star X-ray binary located 17,000 light-years away.Iron line feature detected by Swift (red) and NuSTAR (black). The symmetry of the line is one of the indicators that the disk is located far from the neutron star; if the inner regions of the disk were close to the neutron star, severe relativistic effects would skew the line to be asymmetric. [King et al. 2016]Measurements from ReflectionsKing and collaborators used observations made by NuSTAR and Swift/XRT both X-ray space observatories of Aquila X-1 during the peak of an X-ray outburst. By observing the reflection of Aquila X-1s emission off of the inner regions of the accretion disk, the authors were able to estimate the location of the inner edge of the disk.The authors find that this inner edge sits at ~15 gravitational radii. Since the neutron stars surface is at ~5 gravitational radii, this means that the accretion disk is truncated far from the stars surface. In spite of this truncation, material still manages to cross the gap and accrete onto the
18. Meridional circulation in optically thick accretion disks
NASA Technical Reports Server (NTRS)
Cabot, W.; Savedoff, M. P.
1982-01-01
Thermal imbalances in stars due to rotation are known to drive mass motions in the meridional plane. A preliminary analytic investigation has been made of a similar effect in optically thick accretion disks using conventional thin-disk approximations. It is found that estimated circulation times can be as short as thermal timescales, resulting in rapid transport of heat and angular momentum. This indicates that the simple approximations commonly used are incomplete with regard to detailed, two-dimensional disk structure.
19. EARTH, MOON, SUN, AND CV ACCRETION DISKS
SciTech Connect
Montgomery, M. M.
2009-11-01
Net tidal torque by the secondary on a misaligned accretion disk, like the net tidal torque by the Moon and the Sun on the equatorial bulge of the spinning and tilted Earth, is suggested by others to be a source to retrograde precession in non-magnetic, accreting cataclysmic variable (CV) dwarf novae (DN) systems that show negative superhumps in their light curves. We investigate this idea in this work. We generate a generic theoretical expression for retrograde precession in spinning disks that are misaligned with the orbital plane. Our generic theoretical expression matches that which describes the retrograde precession of Earths' equinoxes. By making appropriate assumptions, we reduce our generic theoretical expression to those generated by others, or to those used by others, to describe retrograde precession in protostellar, protoplanetary, X-ray binary, non-magnetic CV DN, quasar, and black hole systems. We find that spinning, tilted CV DN systems cannot be described by a precessing ring or by a precessing rigid disk. We find that differential rotation and effects on the disk by the accretion stream must be addressed. Our analysis indicates that the best description of a retrogradely precessing spinning, tilted, CV DN accretion disk is a differentially rotating, tilted disk with an attached rotating, tilted ring located near the innermost disk annuli. In agreement with the observations and numerical simulations by others, we find that our numerically simulated CV DN accretion disks retrogradely precess as a unit. Our final, reduced expression for retrograde precession agrees well with our numerical simulation results and with selective observational systems that seem to have main-sequence secondaries. Our results suggest that a major source to retrograde precession is tidal torques like that by the Moon and the Sun on the Earth. In addition, these tidal torques should be common to a variety of systems where one member is spinning and tilted, regardless if
20. Rossby Wave Instability in the Accretion Flows around Black Holes
Gholipour, Mahmoud
2017-01-01
The roles of the Rossby wave instability (RWI) have been significantly developed in some important processes, such as planet formation and angular momentum transport through thin accretion disks. However, their development on accretion flows with advection is insignificant. In this paper, we investigate the effect of advection in the occurrence of RWI through accretion flows around black holes (BHs). In the absence of advection, the occurrence of RWI is extremely low because of high viscosity in the accretion flows around BHs. The results of this paper show that there is a significant chance for the occurrence of RWI in some wavelengths if we consider advection even in low amounts. Therefore, the RWI can be a suitable candidate for angular momentum transport in the accretion flows around BHs. Also, the results show that the advection parameter and the ratio of heat capacity, which are special characters of advection flows, play important roles in the occurrence of RWI.
1. Electrodynamics of disk-accreting magnetic neutron stars
NASA Technical Reports Server (NTRS)
Miller, M. Coleman; Lamb, Frederick K.; Hamilton, Russell J.
1994-01-01
We have investigated the electrodynamics of magnetic neutron stars accreting from Keplerian disks and the implications for particle acceleration and gamma-ray emission by such systems. We argue that the particle density in the magnetospheres of such stars is larger by orders of magnitude than the Goldreich-Julian density, so that the formation of vacuum gaps is unlikely. We show that even if the star rotates slowly, electromotive forces (EMFs) of order 10(exp 15) V are produced by the interaction of plasma in the accretion disk with the magnetic field of the neutron star. The resistance of the disk-magnetosphere-star circuit is small, and hence these EMFs drive very large conduction currents. Such large currents are likely to produce magnetospheric instabilities, such as relativistic double layers and reconnection events, that can accelerate electrons or ions to very high energies.
2. Gravitational Instability in Planetesimal Disks
Bolin, Bryce T.; Lithwick, Yoram; Pan, Margaret; Rein, Hanno; Wu, Yanqin
2014-11-01
Gravitational instability (GI) has been proposed as a method of forming giant gas planets enhanced by disk thermodynamics in a protoplanetary disk (Boss, 1997, Science 276; Durisen et al., 2007, Protostars and Planets V) and as a method of forming planetesimals through the focusing of boulders by the interaction between solids and gases in a turbulent circumstellar disk (Johansen et al., 2007, Nature 448; Youdin & Goodman, 2005, Astrophys. J. 620). GI is mediated through a gaseous circumstellar disk in each each of these scenarios. We explore the possibility of GI occurring in a planetesimal disk devoid of gas. In this regime, mutual collisions between planetesimals are required to dissipate their orbital shear and velocity dispersion enough for collapse to occur as described by the Toomre stability criterion (Toomre, 1964, Astrophys. J. 139; Toomre, 1981, Structure and Evolution of Normal Galaxies). How frequent must collisions be between planetesimals in a gravitationally stable planetesimal disk for GI to occur? Are there collisional rates where GI is postponed indefinitely in an equilibrium state between gravitational stirring and collisional cooling? We present 3D shearing sheet simulations using the REBOUND N-body code with the symplectic epicyclic integrator (Rein & Liu, 2011, A&A 537; Rein & Tremaine, 2011, MNRAS 415) in which the candidate collision rates are within a few orders of magnitude of the disk dynamical lifetime. Our simulations suggest that collisions rate directly controls disk cooling. The shape of the disk cooling curve is independent of the collision rate when scaled to the collision time.
3. Accretion outbursts in self-gravitating protoplanetary disks
SciTech Connect
Bae, Jaehan; Hartmann, Lee; Zhu, Zhaohuan; Nelson, Richard P. E-mail: [email protected] E-mail: [email protected]
2014-11-01
We improve on our previous treatments of the long-term evolution of protostellar disks by explicitly solving disk self-gravity in two dimensions. The current model is an extension of the one-dimensional layered accretion disk model of Bae et al. We find that gravitational instability (GI)-induced spiral density waves heat disks via compressional heating (i.e., PdV work), and can trigger accretion outbursts by activating the magnetorotational instability (MRI) in the magnetically inert disk dead zone. The GI-induced spiral waves propagate well inside of the gravitationally unstable region before they trigger outbursts at R ≲ 1 AU where GI cannot be sustained. This long-range propagation of waves cannot be reproduced with the previously used local α treatments for GI. In our standard model where zero dead-zone residual viscosity (α{sub rd}) is assumed, the GI-induced stress measured at the onset of outbursts is locally as large as 0.01 in terms of the generic α parameter. However, as suggested in our previous one-dimensional calculations, we confirm that the presence of a small but finite α{sub rd} triggers thermally driven bursts of accretion instead of the GI + MRI-driven outbursts that are observed when α{sub rd} = 0. The inclusion of non-zero residual viscosity in the dead zone decreases the importance of GI soon after mass feeding from the envelope cloud ceases. During the infall phase while the central protostar is still embedded, our models stay in a 'quiescent' accretion phase with M-dot {sub acc}∼10{sup −8}--10{sup −7} M{sub ⊙} yr{sup −1} over 60% of the time and spend less than 15% of the infall phase in accretion outbursts. While our models indicate that episodic mass accretion during protostellar evolution can qualitatively help explain the low accretion luminosities seen in most low-mass protostars, detailed tests of the mechanism will require model calculations for a range of protostellar masses with some constraint on the initial core
4. Lessons from accretion disks in cataclysmic variables
Horne, Keith
1998-04-01
We survey recent progress in the interpretation of observations of cataclysmic variables, whose accretion disks are heated by viscous dissipation rather than irradiation. Many features of standard viscous accretion disk models are confirmed by tomographic imaging studies of dwarf novae. Eclipse maps indicate that steady disk temperature structures are established during outbursts. Doppler maps of double-peaked emission lines suggest disk chromospheres heated by magnetic activity. Gas streams impacting on the disk rim leave expected signatures both in the eclipses and emission lines. Doppler maps of dwarf nova IP Peg at the beginning of an outburst show evidence for tidally-induced spiral shocks. While enjoying these successes, we must still face up to the dreaded SW Sex syndrome'' which afflicts most if not all cataclysmic variables in high accretion states. The anomalies include single-peaked emission lines with skewed kinematics, flat temperature-radius profiles, shallow offset line eclipses, and narrow low-ionization absorption lines at phase 0.5. The enigmatic behavior of AE Aqr is now largely understood in terms of a magnetic propeller model in which the rapidly spinning white dwarf magnetosphere expels the gas stream out of the system before an accretion disk can form. A final piece in this puzzle is the realization that an internal shock zone occurs in the exit stream at just the right place to explain the anomalous kinematics and violent flaring of the single-peaked emission lines. Encouraged by this success, we propose that disk-anchored magnetic propellers operate in the high accretion rate systems afflicted by the SW Sex syndrome. Magnetic fields anchored in the Keplerian disk sweep forward and apply a boost that expels gas stream material flowing above the disk plane. This working hypothesis offers a framework on which we can hang all the SW Sex anomalies. The lesson for theorists is that magnetic links appear to be transporting energy and angular
5. Stability of general-relativistic accretion disks
Korobkin, Oleg; Abdikamalov, Ernazar B.; Schnetter, Erik; Stergioulas, Nikolaos; Zink, Burkhard
2011-02-01
Self-gravitating relativistic disks around black holes can form as transient structures in a number of astrophysical scenarios such as binary neutron star and black hole-neutron star coalescences, as well as the core collapse of massive stars. We explore the stability of such disks against runaway and nonaxisymmetric instabilities using three-dimensional hydrodynamics simulations in full general relativity using the Thor code. We model the disk matter using the ideal fluid approximation with a Γ-law equation of state with Γ=4/3. We explore three disk models around nonrotating black holes with disk-to-black hole mass ratios of 0.24, 0.17, and 0.11. Because of metric blending in our initial data, all of our initial models contain an initial axisymmetric perturbation which induces radial disk oscillations. Despite these oscillations, our models do not develop the runaway instability during the first several orbital periods. Instead, all of the models develop unstable nonaxisymmetric modes on a dynamical time scale. We observe two distinct types of instabilities: the Papaloizou-Pringle and the so-called intermediate type instabilities. The development of the nonaxisymmetric mode with azimuthal number m=1 is accompanied by an outspiraling motion of the black hole, which significantly amplifies the growth rate of the m=1 mode in some cases. Overall, our simulations show that the properties of the unstable nonaxisymmetric modes in our disk models are qualitatively similar to those in the Newtonian theory.
6. Stability of general-relativistic accretion disks
SciTech Connect
Korobkin, Oleg; Abdikamalov, Ernazar B.; Schnetter, Erik; Stergioulas, Nikolaos; Zink, Burkhard
2011-02-15
Self-gravitating relativistic disks around black holes can form as transient structures in a number of astrophysical scenarios such as binary neutron star and black hole-neutron star coalescences, as well as the core collapse of massive stars. We explore the stability of such disks against runaway and nonaxisymmetric instabilities using three-dimensional hydrodynamics simulations in full general relativity using the Thor code. We model the disk matter using the ideal fluid approximation with a {Gamma}-law equation of state with {Gamma}=4/3. We explore three disk models around nonrotating black holes with disk-to-black hole mass ratios of 0.24, 0.17, and 0.11. Because of metric blending in our initial data, all of our initial models contain an initial axisymmetric perturbation which induces radial disk oscillations. Despite these oscillations, our models do not develop the runaway instability during the first several orbital periods. Instead, all of the models develop unstable nonaxisymmetric modes on a dynamical time scale. We observe two distinct types of instabilities: the Papaloizou-Pringle and the so-called intermediate type instabilities. The development of the nonaxisymmetric mode with azimuthal number m=1 is accompanied by an outspiraling motion of the black hole, which significantly amplifies the growth rate of the m=1 mode in some cases. Overall, our simulations show that the properties of the unstable nonaxisymmetric modes in our disk models are qualitatively similar to those in the Newtonian theory.
7. Analytic models of relativistic accretion disks
Zhuravlev, V. V.
2015-06-01
We present not a literature review but a description, as detailed and consistent as possible, of two analytic models of disk accretion onto a rotating black hole: a standard relativistic disk and a twisted relativistic disk. Although one of these models is older than the other, both are of topical interest for black hole studies. The treatment is such that the reader with only a limited knowledge of general relativity and relativistic hydrodynamics, with little or no use of additional sources, can gain insight into many technical details lacking in the original papers.
8. Planetary accretion in circumstellar disks
NASA Technical Reports Server (NTRS)
Lissauer, Jack J.; Stewart, Glen R.
1993-01-01
The formation of terrestrial planets and the cores of Jovian planets is reviewed in the framework of the planetesimal hypothesis, wherein planets are assumed to grow via the pairwise accumulation of small solid bodies. Emphasis is placed on the dynamics of solid body accretion from kilometer size planetesimals to terrestrial type planets. This stage of planetary growth is least dependent on the characteristics of the evolutionary state of the central star. It is concluded that the evolution of the planetesimal size distribution is determined by the gravitationally enhanced collision cross-section, which favors collisions between planetesimals with smaller velocities. Runaway growth of the largest planetesimal in each accretion zone appears to be a likely outcome. The subsequent accumulation of the resulting protoplanets leads to a large degree of radial mixing in the terrestrial planet region, and giant impacts are probable.
9. Modeling Gas Distribution in Protoplanetary Accretion Disks
Kronberg, Martin; Lewis, Josiah; Brittain, Sean
2010-07-01
Protoplanetary accretion disks are disks of dust and gas which surround and feed material onto a forming star in the earliest stages of its evolution. One of the most useful methods for studying these disks is near infrared spectroscopy of rovibrational CO emission. This paper presents the methods in which synthetically generated spectra are modeled and fit to spectral data gathered from protoplanetary disks. This paper also discussed the methods in which this code can be improved by modifying the code to run a Monte Carlo analysis of best fit across the CONDOR cluster at Clemson University, thereby allowing for the creation of a catalog of protoplanetary disks with detailed information about them as gathered from the model.
10. Observational constraints on black hole accretion disks
NASA Technical Reports Server (NTRS)
Liang, Edison P.
1994-01-01
We review the empirical constraints on accretion disk models of stellar-mass black holes based on recent multiwavelength observational results. In addition to time-averaged emission spectra, the time evolutions of the intensity and spectrum provide critical information about the structure, stability, and dynamics of the disk. Using the basic thermal Keplerian disk paradigm, we consider in particular generalizations of the standard optically thin disk models needed to accommodate the extremely rich variety of dynamical phenomena exhibited by black hole candidates ranging from flares of electron-positron annihilations and quasiperiodic oscillations in the X-ray intensity to X-ray novae activity. These in turn provide probes of the disk structure and global geometry. The goal is to construct a single unified framework to interpret a large variety of black hole phenomena. This paper will concentrate on the interface between basic theory and observational data modeling.
11. LARGE-SCALE AZIMUTHAL STRUCTURES OF TURBULENCE IN ACCRETION DISKS: DYNAMO TRIGGERED VARIABILITY OF ACCRETION
SciTech Connect
Flock, M.; Dzyurkevich, N.; Klahr, H.; Turner, N.; Henning, Th.
2012-01-10
We investigate the significance of large-scale azimuthal, magnetic, and velocity modes for the magnetorotational instability (MRI) turbulence in accretion disks. We perform three-dimensional global ideal MHD simulations of global stratified protoplanetary disk models. Our domains span azimuthal angles of {pi}/4, {pi}/2, {pi}, and 2{pi}. We observe up to 100% stronger magnetic fields and stronger turbulence for the restricted azimuthal domain models {pi}/2 and {pi}/4 compared to the full 2{pi} model. We show that for those models the Maxwell stress is larger due to strong axisymmetric magnetic fields generated by the {alpha}{Omega} dynamo. Large radial extended axisymmetric toroidal fields trigger temporal magnification of accretion stress. All models display a positive dynamo-{alpha} in the northern hemisphere (upper disk). The parity is distinct in each model and changes on timescales of 40 local orbits. In model 2{pi}, the toroidal field is mostly antisymmetric with respect to the midplane. The eddies of the MRI turbulence are highly anisotropic. The major wavelengths of the turbulent velocity and magnetic fields are between one and two disk scale heights. At the midplane, we find magnetic tilt angles around 8 Degree-Sign -9 Degree-Sign increasing up to 12 Degree-Sign -13 Degree-Sign in the corona. We conclude that an azimuthal extent of {pi} is sufficient to reproduce most turbulent properties in three-dimensional global stratified simulations of magnetized accretion disks.
12. Dust Coagulation in Protoplanetary Accretion Disks
NASA Technical Reports Server (NTRS)
Schmitt, W.; Henning, Th.; Mucha, R.
1996-01-01
The time evolution of dust particles in circumstellar disk-like structures around protostars and young stellar objects is discussed. In particular, we consider the coagulation of grains due to collisional aggregation. The coagulation of the particles is calculated by solving numerically the non-linear Smoluchowski equation. The different physical processes leading to relative velocities between the grains are investigated. The relative velocities may be induced by Brownian motion, turbulence and drift motion. Starting from different regimes which can be identified during the grain growth we also discuss the evolution of dust opacities. These opacities are important for both the derivation of the circumstellar dust mass from submillimeter/millimeter continuum observations and the dynamical behavior of the disks. We present results of our numerical studies of the coagulation of dust grains in a turbulent protoplanetary accretion disk described by a time-dependent one-dimensional (radial) alpha-model. For several periods and disk radii, mass distributions of coagulated grains have been calculated. From these mass spectra, we determined the corresponding Rosseland mean dust opacities. The influence of grain opacity changes due to dust coagulation on the dynamical evolution of a protostellar disk is considered. Significant changes in the thermal structure of the protoplanetary nebula are observed. A 'gap' in the accretion disk forms at the very frontier of the coagulation, i.e., behind the sublimation boundary in the region between 1 and 5 AU.
13. Numerical Modeling of Tidal Effects in Polytropic Accretion Disks
NASA Technical Reports Server (NTRS)
Godon, Patrick
1997-01-01
A two-dimensional time-dependent hybrid Fourier-Chebyshev method of collocation is developed and used for the study of tidal effects in accretion disks, under the assumptions of a polytropic equation of state and a standard alpha viscosity prescription. Under the influence of the m = 1 azimuthal component of the tidal potential, viscous oscillations in the outer disk excite an m = 1 eccentric instability in the disk. While the m = 2 azimuthal component of the tidal potential excites a Papaloizou-Pringle instability in the inner disk (a saturated m = 2 azimuthal mode), with an elliptic pattern rotating at about a fraction (approx. = 1/3) of the local Keplerian velocity in the inner disk. The period of the elliptic mode corresponds well to the periods of the short-period oscillations observed in cataclysmic variables. In cold disks (r(Omega)/c(sub s) = M approx. = 40) we also find a critical value of the viscosity parameter (alpha approx. = 0.01), below which shock dissipation dominates and is balanced by the wave amplification due to the wave action conservation. In this case the double spiral shock propagates all the way to the inner boundary with a Mach number M(sub s) approx. = 1.3.
14. Massive accretion disks in galactic nuclei
Scoville, N. Z.
In the luminous infrared galaxies, very large masses of interstellar matter have been concentrated in the galactic nuclei at radii less than 300 pc as a result of galactic merging, while in lower luminosity systems, this material is probably concentrated by stellar bars and viscous accretion. In both cases, the nuclear region will be highly obscured by dust at visible wavelengths, forcing studies to longer wavelengths where the extinction is reduced. We review recent high resolution near infrared (HST-NICMOS) and mm-interferometric imaging of the dense gas and dust accretion disks in nearby luminous galactic nuclei. Since this nuclear ISM is the active ingredient for both starburst activity and a likely fuel for central AGNs, the nuclear accretion disks are critical to both the activity and the optical appearance of the nucleus. For a sample of 24 luminous galaxies imaged with NICMOS at 1-2μm, approximately 13 show nuclear point sources, indicating the existence of a central AGN or an intense starburst at <= 50 pc radius. Approximately 14 of the sample galaxies have apparent central dust disks. In the best studied ultraluminous IR galaxy, Arp 220, the 2μm imaging shows dust disks in both of the merging galactic nuclei and mm-CO line imaging indicates molecular gas masses ~ 109Msolar for each disk. The two gas disks in Arp 220 are counterrotating and their dynamical masses are ~ 2×109Msolar, that is, only slightly larger than the gas masses. These disks have radii ~ 100 pc and thickness 10-50 pc. The high brightness temperatures of the CO lines indicate that the gas in the disks has area filling factors ~25-50% and mean densities of >= 104 cm-3. Within these nuclear disks, the rate of massive star formation is undoubtedly prodigious and, given the high viscosity of the gas, there will also be high radial accretion rates, perhaps >= 10 Msolar yr-1. If this inflow persists to very small radii, it is enough to feed even the highest
15. Diskoseismology - Signatures of black hole accretion disks
NASA Technical Reports Server (NTRS)
Nowak, Michael; Wagoner, Robert V.
1992-01-01
General relativity requires the existence of a spectrum of oscillations which are trapped near the inner edge of accretion disks around black holes. We have developed a general formalism for analyzing the normal modes of such acoustic perturbations of arbitrary thin disk models, approximating the dominant relativistic effects via a modified Newtonian potential (these modes do not exist in Newtonian gravity). The eigenfunctions and eigenfrequencies of a variety of disk models are found to fall in to two main classes, which are analogous to the p-modes and g-modes in the sun. In this work, we compute the eigenfunctions and eigenfrequencies of isothermal disks. The (relatively small) rates of growth or damping of these oscillations due to gravitational radiation and parameterized models of viscosity are also computed.
16. FITDisk: Cataclysmic Variable Accretion Disk Demonstration Tool
Wood, Matthew A.; Dolence, J.
2013-05-01
FITDisk models accretion disk phenomena using a fully three-dimensional hydrodynamics calculation, and data can either be visualized as they are computed or stored to hard drive for later playback at a fast frame rate. Simulations are visualized using OpenGL graphics and the viewing angle can be changed interactively. Pseudo light curves of simulated systems can be plotted along with the associated Fourier amplitude spectrum. It provides an easy to use graphical user interface as well as 3-D interactive graphics. The code computes the evolution of a CV accretion disk, visualizes results in real time, records and plays back simulations, and generates and plots pseudo light curves and associated power spectra.
17. Anomalous magnetic viscosity in relativistic accretion disks
Lin, Fujun; Liu, Sanqiu; Li, Xiaoqing
2013-07-01
It has been proved that the self-generated magnetic fields by transverse plasmons in the relativistic regime are modulationally unstable, leading to a self-similar collapse of the magnetic flux tubes and resulting in local magnetic structures; highly spatially intermittent flux is responsible for generating the anomalous viscosity. We derive the anomalous magnetic viscosity coefficient, in accretion disks around compact objects, such as black holes, pulsars and quasars, where the plasmas are relativistic, in order to help clarify the nature of viscosity in the theory of accretion disks. The results indicate that, the magnetic viscosity is modified by the relativistic effects of plasmas, and its' strength would be 1015 stronger than the molecular viscosity, which may be helpful in explaining the observations.
18. Dynamics of flux tubes in accretion disks
NASA Technical Reports Server (NTRS)
Vishniac, E. T.; Duncan, R. C.
1994-01-01
The study of magnetized plasmas in astrophysics is complicated by a number of factors, not the least of which is that in considering magnetic fields in stars or accretion disks, we are considering plasmas with densities well above those we can study in the laboratory. In particular, whereas laboratory plasmas are dominated by the confining magnetic field pressure, stars, and probably accretion disks, have magnetic fields whose beta (ratio of gas pressure to magnetic field pressure) is much greater than 1. Observations of the Sun suggest that under such circumstances the magnetic field breaks apart into discrete flux tubes with a small filling factor. On the other hand, theoretical treatments of MHD turbulence in high-beta plasmas tend to assume that the field is more or less homogeneously distributed throughout the plasma. Here we consider a simple model for the distribution of magnetic flux tubes in a turbulent medium. We discuss the mechanism by which small inhomogeneities evolve into discrete flux tubes and the size and distribution of such flux tubes. We then apply the model to accretion disks. We find that the fibrilation of the magnetic field does not enhance magnetic buoyancy. We also note that the evolution of an initially diffuse field in a turbulent medium, e.g., any uniform field in a shearing flow, will initially show exponential growth as the flux tubes form. This growth saturates when the flux tube formation is complete and cannot be used as the basis for a self-sustaining dynamo effect. Since the typical state of the magnetic field is a collection of intense flux tubes, this effect is of limited interest. However, it may be important early in the evolution of the galactic magnetic field, and it will play a large role in numerical simulations. Finally, we note that the formation of flux tubes is an essential ingredient in any successful dynamo model for stars or accretion disks.
19. Exploring Stability of General Relativistic Accretion Disks
Korobkin, Oleg; Abdikamalov, Ernazar; Schnetter, Erik; Stergioulas, Nikolaos; Zink, Burkhard
2011-04-01
Self-gravitating relativistic disks around black holes can form as transient structures in a number of astrophysical scenarios, involving core collapse of massive stars and mergers of compact ob jects. I will present results on our recent study of the stability of such disks against runaway and non-axisymmetric instabilities, which we explore using three-dimensional hydrodynamics simulations in full general relativity. All of our models develop unstable non-axisymmetric modes on a dynamical timescale. We observe two distinct types of instabilities: the Papaloizou-Pringle and the so-called intermediate type instabilities. The development of the non-axisymmetric mode with azimuthal number m=1 is accompanied by an outspiraling motion of the black hole, which significantly amplifies the growth rate of the m=1 mode in some cases. We will discuss the types, growth rates and pattern speeds of the unstable modes, as well as the detectability of the gravitational waves from such objects.
20. Recent Observational Progress on Accretion Disks Around Compact Objects
Miller, Jon M.
2016-04-01
Studies of accretion disks around black holes and neutron stars over the last ten years have made remarkable progress. Our understanding of disk evolution as a function of mass accretion rate is pushing toward a consensus on thin/thick disk transitions; an apparent switching between disk-driven outflow modes has emerged; and monitoring observations have revealed complex spectral energy distributions wherein disk reprocessing must be important. Detailed studies of disk winds, in particular, have the potential to reveal the basic physical processes that mediate disk accretion, and to connect with numerical simulations. This talk will review these developments and look ahead to the potential of Astro-H.
1. Angular momentum transport in thin accretion disks and intermittent accretion.
PubMed
Coppi, B; Coppi, P S
2001-07-30
The plasma modes, transporting angular momentum in accretion disks, under minimally restrictive conditions when the magnetic energy density is significant relative to the thermal energy density, are shown to be singular if the ideal MHD approximation is adopted. A similarity with the modes producing magnetic reconnection in current carrying plasmas is established. The combined effects of finite plasma temperature, of plasma compressibility, of the gradient of the rotation frequency, and of appropriate transport processes (outside ideal MHD) are involved in the onset of these nonaxisymmetric and locally corotating modes.
2. Black Hole Advective Accretion Disks with Optical Depth Transition
SciTech Connect
Artemove, Y.V.; Bisnovatyi-Kogan, G.S.; Igumenshchev, I.V.; Novikov, I.D.
2006-02-01
We have constructed numerically global solutions of advective accretion disks around black holes that describe a continuous transition between the effectively optically thick outer and optically thin inner disk regions. We have concentrated on models of accretion flows with large mass accretion rates, and we have employed a bridging formula for radiative losses at high and low effective optical depths.
3. Nonlinear calculations of the time evolution of black hole accretion disks
NASA Technical Reports Server (NTRS)
Luo, C.
1994-01-01
Based on previous works on black hole accretion disks, I continue to explore the disk dynamics using the finite difference method to solve the highly nonlinear problem of time-dependent alpha disk equations. Here a radially zoned model is used to develop a computational scheme in order to accommodate functional dependence of the viscosity parameter alpha on the disk scale height and/or surface density. This work is based on the author's previous work on the steady disk structure and the linear analysis of disk dynamics to try to apply to x-ray emissions from black candidates (i.e., multiple-state spectra, instabilities, QPO's, etc.).
4. THE COSMIC BATTERY IN ASTROPHYSICAL ACCRETION DISKS
SciTech Connect
Contopoulos, Ioannis; Nathanail, Antonios; Katsanikas, Matthaios
2015-06-01
The aberrated radiation pressure at the inner edge of the accretion disk around an astrophysical black hole imparts a relative azimuthal velocity on the electrons with respect to the ions which gives rise to a ring electric current that generates large-scale poloidal magnetic field loops. This is the Cosmic Battery established by Contopoulos and Kazanas in 1998. In the present work we perform realistic numerical simulations of this important astrophysical mechanism in advection-dominated accretion flows, ADAFs. We confirm the original prediction that the inner parts of the loops are continuously advected toward the central black hole and contribute to the growth of the large-scale magnetic field, whereas the outer parts of the loops are continuously diffusing outward through the turbulent accretion flow. This process of inward advection of the axial field and outward diffusion of the return field proceeds all the way to equipartition, thus generating astrophysically significant magnetic fields on astrophysically relevant timescales. We confirm that there exists a critical value of the magnetic Prandtl number between unity and 10 in the outer disk above which the Cosmic Battery mechanism is suppressed.
5. ACCRETION DISK TEMPERATURES OF QSOs: CONSTRAINTS FROM THE EMISSION LINES
SciTech Connect
Bonning, E. W.; Shields, G. A.; Stevens, A. C.; Salviander, S. E-mail: [email protected] E-mail: [email protected]
2013-06-10
We compare QSO emission-line spectra to predictions based on theoretical ionizing continua of accretion disks. The observed line intensities do not show the expected trend of higher ionization with theoretical accretion disk temperature as predicted from the black hole mass and accretion rate. Consistent with earlier studies, this suggests that the inner disk does not reach temperatures as high as expected from standard disk theory. Modified radial temperature profiles, taking account of winds or advection in the inner disk, achieve better agreement with observation. The emission lines of radio-detected and radio-undetected sources show different trends as a function of the theoretically predicted disk temperature.
6. Neutrino oscillation above a black hole accretion disk
SciTech Connect
Malkus, A.; Kneller, J. P.; McLaughlin, G. C.; Surman, R.
2015-05-15
We examine neutrino oscillations in the context of an accretion disk surrounding a black hole. Because accretion disks produce large quantities of neutrinos, they may be home to interesting neutrino oscillation as well. We model accretion disks associated with stellar collapse for the sake of understanding neutrino oscillations. We find that the neutrino oscillations include phenomena seen in the protoneutron star setting as well as phenomena not seen elsewhere.
7. Asymmetric evolution of magnetic reconnection in collisionless accretion disk
SciTech Connect
Shirakawa, Keisuke Hoshino, Masahiro
2014-05-15
An evolution of a magnetic reconnection in a collisionless accretion disk is investigated using a 2.5 dimensional hybrid code simulation. In astrophysical disks, magnetorotational instability (MRI) is considered to play an important role by generating turbulence in the disk and contributes to an effective angular momentum transport through a turbulent viscosity. Magnetic reconnection, on the other hand, also plays an important role on the evolution of the disk through a dissipation of a magnetic field enhanced by a dynamo effect of MRI. In this study, we developed a hybrid code to calculate an evolution of a differentially rotating system. With this code, we first confirmed a linear growth of MRI. We also investigated a behavior of a particular structure of a current sheet, which would exist in the turbulence in the disk. From the calculation of the magnetic reconnection, we found an asymmetric structure in the out-of-plane magnetic field during the evolution of reconnection, which can be understood by a coupling of the Hall effect and the differential rotation. We also found a migration of X-point whose direction is determined only by an initial sign of J{sub 0}×Ω{sub 0}, where J{sub 0} is the initial current density in the neutral sheet and Ω{sub 0} is the rotational vector of the background Keplerian rotation. Associated with the migration of X-point, we also found a significant enhancement of the perpendicular magnetic field compared to an ordinary MRI. MRI-Magnetic reconnection coupling and the resulting magnetic field enhancement can be an effective process to sustain a strong turbulence in the accretion disk and to a transport of angular momentum.
8. Asymmetric evolution of magnetic reconnection in collisionless accretion disk
Shirakawa, Keisuke; Hoshino, Masahiro
2014-05-01
An evolution of a magnetic reconnection in a collisionless accretion disk is investigated using a 2.5 dimensional hybrid code simulation. In astrophysical disks, magnetorotational instability (MRI) is considered to play an important role by generating turbulence in the disk and contributes to an effective angular momentum transport through a turbulent viscosity. Magnetic reconnection, on the other hand, also plays an important role on the evolution of the disk through a dissipation of a magnetic field enhanced by a dynamo effect of MRI. In this study, we developed a hybrid code to calculate an evolution of a differentially rotating system. With this code, we first confirmed a linear growth of MRI. We also investigated a behavior of a particular structure of a current sheet, which would exist in the turbulence in the disk. From the calculation of the magnetic reconnection, we found an asymmetric structure in the out-of-plane magnetic field during the evolution of reconnection, which can be understood by a coupling of the Hall effect and the differential rotation. We also found a migration of X-point whose direction is determined only by an initial sign of J0×Ω0, where J0 is the initial current density in the neutral sheet and Ω0 is the rotational vector of the background Keplerian rotation. Associated with the migration of X-point, we also found a significant enhancement of the perpendicular magnetic field compared to an ordinary MRI. MRI-Magnetic reconnection coupling and the resulting magnetic field enhancement can be an effective process to sustain a strong turbulence in the accretion disk and to a transport of angular momentum.
9. Normal Modes of Black Hole Accretion Disks
SciTech Connect
Ortega-Rodriguez, Manuel; Silbergleit, Alexander S.; Wagoner, Robert V.; /Stanford U., Phys. Dept. /KIPAC, Menlo Park
2006-11-07
This paper studies the hydrodynamical problem of normal modes of small adiabatic oscillations of relativistic barotropic thin accretion disks around black holes (and compact weakly magnetic neutron stars). Employing WKB techniques, we obtain the eigen frequencies and eigenfunctions of the modes for different values of the mass and angular momentum of the central black hole. We discuss the properties of the various types of modes and examine the role of viscosity, as it appears to render some of the modes unstable to rapid growth.
10. Dead Zone Accretion Flows in Protostellar Disks
NASA Technical Reports Server (NTRS)
Turner, Neal; Sano, T.
2008-01-01
Planets form inside protostellar disks in a dead zone where the electrical resistivity of the gas is too high for magnetic forces to drive turbulence. We show that much of the dead zone nevertheless is active and flows toward the star while smooth, large-scale magnetic fields transfer the orbital angular momentum radially outward. Stellar X-ray and radionuclide ionization sustain a weak coupling of the dead zone gas to the magnetic fields, despite the rapid recombination of free charges on dust grains. Net radial magnetic fields are generated in the magnetorotational turbulence in the electrically conducting top and bottom surface layers of the disk, and reach the midplane by ohmic diffusion. A toroidal component to the fields is produced near the midplane by the orbital shear. The process is similar to the magnetization of the solar tachocline. The result is a laminar, magnetically driven accretion flow in the region where the planets form.
11. Physics-Based Spectra of Accretion Disks around Black Holes
NASA Technical Reports Server (NTRS)
Krolik, Julian H.
2005-01-01
continuum opacity sources, including Compton scattering and bound-free opacity from abundant metal species. The principal new result is that bound-free opacity is very significant in altering the continuum spectral shape, resulting for example in quite different "color correction factors" compared to those predicted previously. In addition, the models predict a relationship between luminosity and inner disk temperature that is, for the first time, in accord with that observed. The primary purpose of the grant was to incorporate more realistic accretion disk physics, learned largely from simulations, into such spectral models. The Davis et al. paper includes consideration of a vertical dissipation profile computed from radiation magneto-hydrodynamic simulations of MRI turbulence by N. J. Turner (2004). So long as the disk is effectively thick, such dissipation profiles do not affect the predicted spectrum significantly. (More work needs to be done on these simulations, however.) A potentially more serious issue is that MRI turbulence produces substantial inhomogeneities, as do photon bubble instabilities. These inhomogeneities can affect the spectra by enhancing the effects of absorption opacity over scattering opacity. We have done some preliminary Monte Carlo calculations to explore these effects.
12. Accretion Disk Outflows from Compact Object Mergers
Metzger, Brian
Nuclear reactions play a key role in the accretion disks and outflows associated with the merger of binary compact objects and the central engines of gamma-ray bursts and supernovae. The proposed research program will investigate the impact of nucleosynthesis on these events and their observable signatures by means of analytic calculations and numerical simulations. One focus of this research is rapid accretion following the tidal disruption of a white dwarf (WD) by a neutron star (NS) or black hole (BH) binary companion. Tidal disruption shreds the WD into a massive torus composed of C, O, and/or He, which undergoes nuclear reactions and burns to increasingly heavier elements as it flows to smaller radii towards the central compact object. The nuclear energy so released is comparable to that released gravitationally, suggesting that burning could drastically alter the structure and stability of the accretion flow. Axisymmetric hydrodynamic simulations of the evolution of the torus including nuclear burning will be performed to explore issues such as the mass budget of the flow (accretion vs. outflows) and its thermal stability (steady burning and accretion vs. runaway explosion). The mass, velocity, and composition of outflows from the disk will be used in separate radiative transfer calculations to predict the lightcurves and spectra of the 56Ni-decay powered optical transients from WD-NS/WD-BH mergers. The possible connection of such events to recently discovered classes of sub-luminous Type I supernovae will be assessed. The coalescence of NS-NS/NS-BH binaries also results in the formation of a massive torus surrounding a central compact object. Three-dimensional magnetohydrodynamic simulations of the long-term evolution of such accretion disks will be performed, which for the first time follow the effects of weak interactions and the nuclear energy released by Helium recombination. The nucleosynthetic yield of disk outflows will be calculated using a detailed
13. Magnetohydrodynamic Origin of Jets from Accretion Disks
NASA Technical Reports Server (NTRS)
Lovelace, R. V. E.; Romanova, M. M.
1998-01-01
A review is made of magnetohydrodynamic (MHD) theory and simulation of outflows from disks for different distributions of magnetic field threading the disk. In one limit of a relatively weak, initially diverging magnetic field, both thermal and magnetic pressure gradients act to drive matter to an outflow, while a toroidal magnetic field develops which strongly collimates the outflow. The collimation greatly reduces the field divergence and the mass outflow rate decreases after an initial peak. In a second limit of a strong magnetic field, the initial field configuration was taken with the field strength on the disk decreasing outwards to small values so that collimation was reduced. As a result, a family of stationary solutions was discovered where matter is driven mainly by the strong magnetic pressure gradient force. The collimation in this case depends on the pressure of an external medium. These flows are qualitatively similar to the analytic solutions for magnetically driven outflows. The problem of the opening of a closed field line configuration linking a magnetized star and an accretion disk is also discussed.
14. Understanding Accretion Disks through Three Dimensional Radiation MHD Simulations
Jiang, Yan-Fei
15. RESISTIVITY-DRIVEN STATE CHANGES IN VERTICALLY STRATIFIED ACCRETION DISKS
SciTech Connect
Simon, Jacob B.; Hawley, John F.; Beckwith, Kris
2011-04-01
We investigate the effect of shear viscosity, {nu}, and Ohmic resistivity, {eta}, on the magnetorotational instability (MRI) in vertically stratified accretion disks through a series of local simulations with the Athena code. First, we use a series of unstratified simulations to calibrate physical dissipation as a function of resolution and background field strength; the effect of the magnetic Prandtl number, P{sub m} = {nu}/{eta}, on the turbulence is captured by {approx}32 grid zones per disk scale height, H. In agreement with previous results, our stratified disk calculations are characterized by a subthermal, predominately toroidal magnetic field that produces MRI-driven turbulence for |z| {approx}< 2H. Above |z| {approx} 2H, the magnetic pressure dominates and the field is buoyantly unstable. Large-scale radial and toroidal fields are also generated near the mid-plane and subsequently rise through the disk. The polarity of this mean field switches on a roughly 10 orbit period in a process that is well modeled by an {alpha}-{Omega} dynamo. Turbulent stress increases with P{sub m} but with a shallower dependence compared to unstratified simulations. For sufficiently large resistivity, {eta} {approx} c{sub s} H/1000, where c{sub s} is the sound speed, MRI turbulence within 2H of the mid-plane undergoes periods of resistive decay followed by regrowth. This regrowth is caused by amplification of the toroidal field via the dynamo. This process results in large amplitude variability in the stress on 10-100 orbital timescales, which may have relevance for partially ionized disks that are observed to have high- and low-accretion states.
16. The Growth of Central Black Hole and the Ionization Instability of Quasar Disk
NASA Technical Reports Server (NTRS)
Lu, Ye; Cheng, K. S.; Zhang, S. N.
2003-01-01
A possible accretion model associated with the ionization instability of quasar disks is proposed to address the growth of the central black hole harbored in the host galaxy. The evolution of quasars in cosmic time is assumed to change from a highly active state to a quiescent state triggered by the S-shaped ionization instability of the quasar accretion disk. For a given external mass transfer rate supplied by the quasar host galaxy, ionization instability can modify accretion rate in the disk and separates the accretion flows of the disk into three different phases, like a S-shape. We suggest that the bright quasars observed today are those quasars with disks in the upper branch of S-shaped instability, and the faint or 'dormant' quasars are simply the system in the lower branch. The middle branch is the transition state which is unstable. We assume the quasar disk evolves according to the advection-dominated inflow-outflow solutions (ADIOS) configuration in the stable lower branch of S-shaped instability, and Eddington accretion rate is used to constrain the accretion rate in each phase. The mass ratio between black hole and its host galactic bulge is a nature consequence of ADIOS. Our model also demonstrates that a seed black hole (BH) similar to those found in spiral galaxies today is needed to produce a BH with a final mass 2 x 10(exp 8) solar mases.
17. Turbulent Transport In Global Models of Magnetized Accretion Disks
Sorathia, Kareem
The modern theory of accretion disks is dominated by the discovery of the magnetorotational instability (MRI). While hydrodynamic disks satisfy Rayleigh's criterion and there exists no known unambiguous route to turbulence in such disks, a weakly magnetized disk of plasma is subject to the MRI and will become turbulent. This MRI-driven magnetohydrodnamic turbulence generates a strong anisotropic correlation between the radial and azimuthal magnetic fields which drives angular momentum outwards. Accretion disks perform two vital functions in various astrophysical systems: an intermediate step in the gravitational collapse of a rotating gas, where the disk transfers angular momentum outwards and allows material to fall inwards; and as a power source, where the gravitational potential energy of infalling matter can be converted to luminosity. Accretion disks are important in astrophysical processes at all scales in the universe. Studying accretion from first principles is difficult, as analytic treatments of turbulent systems have proven quite limited. As such, computer simulations are at the forefront of studying systems this far into the non-linear regime. While computational work is necessary to study accretion disks, it is no panacea. Fully three-dimensional simulations of turbulent astrophysical systems require an enormous amount of computational power that is inaccessible even to sophisticated modern supercomputers. These limitations have necessitated the use of local models, in which a small spatial region of the full disk is simulated, and constrain numerical resolution to what is feasible. These compromises, while necessary, have the potential to introduce numerical artifacts in the resulting simulations. Understanding how to disentangle these artifacts from genuine physical phenomena and to minimize their effect is vital to constructing simulations that can make reliable astrophysical predictions and is the primary concern of the work presented here. The use
18. Wind-accretion Disks in Wide Binaries, Second-generation Protoplanetary Disks, and Accretion onto White Dwarfs
Perets, Hagai B.; Kenyon, Scott J.
2013-02-01
Mass transfer from an evolved donor star to its binary companion is a standard feature of stellar evolution in binaries. In wide binaries, the companion star captures some of the mass ejected in a wind by the primary star. The captured material forms an accretion disk. Here, we study the evolution of wind-accretion disks, using a numerical approach which allows us to follow the long-term evolution. For a broad range of initial conditions, we derive the radial density and temperature profiles of the disk. In most cases, wind accretion leads to long-lived stable disks over the lifetime of the asymptotic giant branch donor star. The disks have masses of a few times 10-5-10-3 M ⊙, with surface density and temperature profiles that follow broken power laws. The total mass in the disk scales approximately linearly with the viscosity parameter used. Roughly, 50%-80% of the mass falling into the disk accretes onto the central star; the rest flows out through the outer edge of the disk into the stellar wind of the primary. For systems with large accretion rates, the secondary accretes as much as 0.1 M ⊙. When the secondary is a white dwarf, accretion naturally leads to nova and supernova eruptions. For all types of secondary star, the surface density and temperature profiles of massive disks resemble structures observed in protoplanetary disks, suggesting that coordinated observational programs might improve our understanding of uncertain disk physics.
19. Convective overstability in radially stratified accretion disks under thermal relaxation
SciTech Connect
Klahr, Hubert; Hubbard, Alexander
2014-06-10
This paper expands the stability criterion for radially stratified, vertically unstratified accretion disks incorporating thermal relaxation. We find a linear amplification of epicyclic oscillations in these disks that depends on the effective cooling time, i.e., an overstability. The growth rates of the overstability vanish for both extreme cases, e.g., infinite cooling time and instantaneous cooling, i.e., the adiabatic and fully isothermal cases. However, for thermal relaxation times τ on the order of the orbital frequency, τΩ ∼ 1, modes grow at a rate proportional to the square of the Brunt-Väisälä frequency. The overstability is based on epicyclic motions, with the thermal relaxation causing gas to heat while radially displaced inward and cool while radially displaced outward. This causes the gas to have a lower density when moving outward compared to when it moves inward, so it feels the outward-directed pressure force more strongly on that leg of the journey. We suggest the term 'convective overstability' for the phenomenon which has already been studied numerically in the nonlinear regime in the context of amplifying vortices in disks under the name 'subcritical baroclinic instability'. The aim of the present paper is to make clear that vortex formation in three-dimensional disks is not necessarily subcritical, i.e., does not need a finite perturbation, nor is it baroclinic in the sense of geophysical fluid dynamics, which requires on vertical shear. We find that convective overstability is a linear instability that will operate under a wide range of physical conditions for circumstellar disks.
20. Compact stars and accretion disks: Workshop summary
Li, J.
1998-07-01
A workshop on Compact Stars and Accretion Disks' was held on 11-12 August 1997 at the Australian National University. The workshop was opened by Professor Jeremy Mould, the Director of Mount Stromlo Observatory. The workshop was organised to coincide with visits to the ANU Astrophysical Theory Centre by Professor Ron Webbink from the University of Illinois, Professor Rainer Wehrse from the University of Heidelberg and Dr Chris Tout from the University of Cambridge. The workshop attracted over 25 participants nationwide. Participants included members of the Special Research Centre for Theoretical Astrophysics, University of Sydney, led by Professor Don Melrose, Professor Dick Manchester from the ATNF, Professor Ravi Sood from ADFA, Dr John Greenhill from the University of Tasmania and Dr Rosemary Mardling from Monash University. Dr Helen Johnston from AAO and Dr Kurt Liffman from AFDL also attended the workshop. The abstracts of twelve of the workshop papers are presented in this summary.
1. The magnetic nature of disk accretion onto black holes.
PubMed
Miller, Jon M; Raymond, John; Fabian, Andy; Steeghs, Danny; Homan, Jeroen; Reynolds, Chris; van der Klis, Michiel; Wijnands, Rudy
2006-06-22
Although disk accretion onto compact objects-white dwarfs, neutron stars and black holes-is central to much of high-energy astrophysics, the mechanisms that enable this process have remained observationally difficult to determine. Accretion disks must transfer angular momentum in order for matter to travel radially inward onto the compact object. Internal viscosity from magnetic processes and disk winds can both in principle transfer angular momentum, but hitherto we lacked evidence that either occurs. Here we report that an X-ray-absorbing wind discovered in an observation of the stellar-mass black hole binary GRO J1655 - 40 (ref. 6) must be powered by a magnetic process that can also drive accretion through the disk. Detailed spectral analysis and modelling of the wind shows that it can only be powered by pressure generated by magnetic viscosity internal to the disk or magnetocentrifugal forces. This result demonstrates that disk accretion onto black holes is a fundamentally magnetic process.
2. Magnetic fields in primordial accretion disks
Latif, M. A.; Schleicher, D. R. G.
2016-01-01
Magnetic fields are considered a vital ingredient of contemporary star formation and may have been important during the formation of the first stars in the presence of an efficient amplification mechanism. Initial seed fields are provided via plasma fluctuations and are subsequently amplified by the small-scale dynamo, leading to a strong, tangled magnetic field. We explore how the magnetic field provided by the small-scale dynamo is further amplified via the α-Ω dynamo in a protostellar disk and assess its implications. For this purpose, we consider two characteristic cases, a typical Pop. III star with 10M⊙ and an accretion rate of 10-3M⊙ yr-1, and a supermassive star with 105M⊙ and an accretion rate of 10-1M⊙ yr-1. For the 10M⊙ Pop. III star, we find that coherent magnetic fields can be produced on scales of at least 100 AU, which are sufficient to drive a jet with a luminosity of 100L⊙ and a mass outflow rate of 10-3.7M⊙ yr-1. For the supermassive star, the dynamical timescales in its environment are even shorter, implying smaller orbital timescales and an efficient magnetization out to at least 1000 AU. The jet luminosity corresponds to ~106.0L⊙ and a mass outflow rate of 10-2.1M⊙ yr-1. We expect that the feedback from the supermassive star can have a relevant impact on its host galaxy.
3. Accreting planets as dust dams in 'transition' disks
SciTech Connect
Owen, James E.
2014-07-01
We investigate under what circumstances an embedded planet in a protoplanetary disk may sculpt the dust distribution such that it observationally presents as a 'transition' disk. We concern ourselves with 'transition' disks that have large holes (≳ 10 AU) and high accretion rates (∼10{sup –9}-10{sup –8} M {sub ☉} yr{sup –1}), particularly, those disks which photoevaporative models struggle to explain. Adopting the observed accretion rates in 'transition' disks, we find that the accretion luminosity from the forming planet is significant, and can dominate over the stellar luminosity at the gap edge. This planetary accretion luminosity can apply a significant radiation pressure to small (s ≲ 1 μm) dust particles provided they are suitably decoupled from the gas. Secular evolution calculations that account for the evolution of the gas and dust components in a disk with an embedded, accreting planet, show that only with the addition of the radiation pressure can we explain the full observed characteristics of a 'transition' disk (NIR dip in the spectral energy distribution (SED), millimeter cavity, and high accretion rate). At suitably high planet masses (≳ 3-4 M{sub J} ), radiation pressure from the accreting planet is able to hold back the small dust particles, producing a heavily dust-depleted inner disk that is optically thin to infrared radiation. The planet-disk system will present as a 'transition' disk with a dip in the SED only when the planet mass and planetary accretion rate are high enough. At other times, it will present as a disk with a primordial SED, but with a cavity in the millimeter, as observed in a handful of protoplanetary disks.
4. Nonaxisymmetric secular instabilities driven by star/disk coupling
NASA Technical Reports Server (NTRS)
Imamura, James, N.; Toman, Joseph; Durisen, Richard H.; Pickett, Brian K.; Yang, Shelby
1995-01-01
instability can grow to moderate amplitude, then the coupling can transport significant amounts of angular momentum from the star into the circumstellar disk. We find, for the particular case of rotating protostars during the accretion phase, that the instability growth time can be shorter than the accretion time. Further, if the instability can grow to amplitudes on the order of several percent, the star/disk coupling can remove angular momentum from the forming star faster than it is added by accretion.
5. Evolution of Pre-Main Sequence Accretion Disks
NASA Technical Reports Server (NTRS)
Hartmann, Lee W.
2005-01-01
The aim of this project was to develop a comprehensive global picture of the physical conditions in, and evolutionary timescales of, premain sequence accretion disks. The results of this work will help constrain the initial conditions for planet formation. To this end we developed much larger samples of 3-10 Myr-old stars to provide better empirical constraints on protoplanetary disk evolution; measured disk accretion rates in these systems; and constructed detailed model disk structures consistent with observations to infer physical conditions such as grain growth in protoplanetary disks.
6. Evolution of Pre-Main Sequence Accretion Disks
NASA Technical Reports Server (NTRS)
Hartmann, Lee W.
2003-01-01
The aim of this project is to develop a comprehensive global picture of the physical conditions in, and evolutionary timescales of, pre-main sequence accretion disks. The results of this work will help constrain the initial conditions for planet formation. To this end we are developing much larger samples of 3-10 Myr-old stars to provide better empirical constraints on protoplanetary disk evolution; measuring disk accretion rates in these systems; and constructing detailed model disk structures consistent with observations to infer physical conditions such as grain growth in protoplanetary disks.
7. Evolution of Pre-Main Sequence Accretion Disks
NASA Technical Reports Server (NTRS)
Hartmann, Lee W.
2004-01-01
The aim of this project is to develop a comprehensive global picture of the physical conditions in, and evolutionary timescales of, pre-main sequence accretion disks. The results of this work will help constrain the initial conditions for planet formation. To this end we are developing much larger samples of 3-10 Myr-old stars to provide better empirical constraints on protoplanetary disk evolution; measuring disk accretion rates in these systems; and constructing detailed model disk structures consistent with observations to infer physical conditions such as grain growth in protoplanetary disks.
8. An Accretion Model for the Growth of the Central Black Holes Associated with Ionization Instability in Quasars
NASA Technical Reports Server (NTRS)
Lu, Y.; Cheng, K. S.; Zhang, S. N.
2003-01-01
A possible accretion model associated with the ionization instability of quasar disks is proposed to address the growth of the central black hole (BH) harbored in the host galaxy. The evolution of quasars in cosmic time is assumed to change from a highly active state to a quiescent state triggered by the S-shaped ionization instability of the quasar accretion disk. For a given external mass transfer rate supplied by the quasar host galaxy, ionization instability can modify the accretion rate in the disk and separate the accretion flows of the disk into three different phases, like an S-shape. We suggest that the bright quasars observed today are those quasars with disks in the upper branch of the S-shaped instability, and the faint or 'dormant' quasars are simply these systems in the lower branch. The middle branch is the transition state, which is unstable. We assume the quasar disk evolves according to the advection-dominated inflow-outflow solution (ADIOS) configuration in the stable lower branch of the S-shaped instability, and the Eddington accretion rate is used to constrain the accretion rate in the highly active phase. The mass ratio between a BH and its host galactic bulge is a natural consequence of an ADIOS. Our model also demonstrates that a seed BH approx. 2 x 10(exp 6) solar masses similar to those found in spiral galaxies today is needed to produce a BH with a final mass of approx. 2 x 10(exp 8) solar masses.
9. A pure hydrodynamic origin of accretion disk turbulence
2016-07-01
Accretion disks consist of flows for which angular velocity decreases but specific angular momentum increases with increasing radial coordinate. Such flows are Rayleigh stable, but must be turbulent in order to explain observed data. Since molecular viscosity is negligible in these systems, scientists have argued for turbulent viscosity for energy dissipation and hence to explain infall of matter. However, so far, the success to explain the origin of turbulence in accretion disks is done with caveats. Here we investigate the evolution of pure hydrodynamic perturbations in stochastically driven accretion disks. We show that the accretion flows, which are inevitably driven by stochastic noise, are hydrodynamically unstable under linear perturbations. We also argue that in accretion disks, stochastic forcing appears generically due to the presence of shear between different annuli of the disk. This work resolves the turbulence problem of accretion disks from pure hydrodynamics and explains the infall of matter for both hot and cold disks. This would help in explaining the origin of timing and spectral features in the disk flows generically.
10. Temperature fluctuations driven by magnetorotational instability in protoplanetary disks
SciTech Connect
McNally, Colin P.; Hubbard, Alexander; Low, Mordecai-Mark Mac; Yang, Chao-Chin E-mail: [email protected] E-mail: [email protected]
2014-08-10
The magnetorotational instability (MRI) drives magnetized turbulence in sufficiently ionized regions of protoplanetary disks, leading to mass accretion. The dissipation of the potential energy associated with this accretion determines the thermal structure of accreting regions. Until recently, the heating from the turbulence has only been treated in an azimuthally averaged sense, neglecting local fluctuations. However, magnetized turbulence dissipates its energy intermittently in current sheet structures. We study this intermittent energy dissipation using high resolution numerical models including a treatment of radiative thermal diffusion in an optically thick regime. Our models predict that these turbulent current sheets drive order-unity temperature variations even where the MRI is damped strongly by Ohmic resistivity. This implies that the current sheet structures where energy dissipation occurs must be well-resolved to correctly capture the flow structure in numerical models. Higher resolutions are required to resolve energy dissipation than to resolve the magnetic field strength or accretion stresses. The temperature variations are large enough to have major consequences for mineral formation in disks, including melting chondrules, remelting calcium-aluminum-rich inclusions, and annealing silicates; and may drive hysteresis: current sheets in MRI active regions could be significantly more conductive than the remainder of the disk.
11. Magnetohydrodynamic simulations of global accretion disks with vertical magnetic fields
SciTech Connect
Suzuki, Takeru K.; Inutsuka, Shu-ichiro
2014-04-01
We report results of three-dimensional magnetohydrodynamical (MHD) simulations of global accretion disks threaded with weak vertical magnetic fields. We perform the simulations in the spherical coordinates with different temperature profiles and accordingly different rotation profiles. In the cases with a spatially constant temperature, because the rotation frequency is vertically constant in the equilibrium condition, general properties of the turbulence excited by magnetorotational instability are quantitatively similar to those obtained in local shearing box simulations. On the other hand, in the cases with a radially variable temperature profile, the vertical differential rotation, which is inevitable in the equilibrium condition, winds up the magnetic field lines in addition to the usual radial differential rotation. As a result, the coherent wound magnetic fields contribute to the Maxwell stress in the surface regions. We obtain nondimensional density and velocity fluctuations ∼0.1-0.2 at the midplane. The azimuthal power spectra of the magnetic fields show shallower slopes, ∼m {sup 0} – m {sup –1}, than those of velocity and density. The Poynting flux associated with the MHD turbulence drives intermittent and structured disk winds as well as sound-like waves toward the midplane. The mass accretion mainly occurs near the surfaces, and the gas near the midplane slowly moves outward in the time domain of the present simulations. The vertical magnetic fields are also dragged inward in the surface regions, while they stochastically move outward and inward around the midplane. We also discuss an observational implication of induced spiral structure in the simulated turbulent disks.
12. LUNAR ACCRETION FROM A ROCHE-INTERIOR FLUID DISK
SciTech Connect
Salmon, Julien; Canup, Robin M. E-mail: [email protected]
2012-11-20
We use a hybrid numerical approach to simulate the formation of the Moon from an impact-generated disk, consisting of a fluid model for the disk inside the Roche limit and an N-body code to describe accretion outside the Roche limit. As the inner disk spreads due to a thermally regulated viscosity, material is delivered across the Roche limit and accretes into moonlets that are added to the N-body simulation. Contrary to an accretion timescale of a few months obtained with prior pure N-body codes, here the final stage of the Moon's growth is controlled by the slow spreading of the inner disk, resulting in a total lunar accretion timescale of {approx}10{sup 2} years. It has been proposed that the inner disk may compositionally equilibrate with the Earth through diffusive mixing, which offers a potential explanation for the identical oxygen isotope compositions of the Earth and Moon. However, the mass fraction of the final Moon that is derived from the inner disk is limited by resonant torques between the disk and exterior growing moons. For initial disks containing <2.5 lunar masses (M{sub Last-Quarter-Moon }), we find that a final Moon with mass > 0.8 M{sub Last-Quarter-Moon} contains {<=}60% material derived from the inner disk, with this material preferentially delivered to the Moon at the end of its accretion.
13. DISTRIBUTION OF ACCRETING GAS AND ANGULAR MOMENTUM ONTO CIRCUMPLANETARY DISKS
SciTech Connect
Tanigawa, Takayuki; Ohtsuki, Keiji; Machida, Masahiro N.
2012-03-01
We investigate gas accretion flow onto a circumplanetary disk from a protoplanetary disk in detail by using high-resolution three-dimensional nested-grid hydrodynamic simulations, in order to provide a basis of formation processes of satellites around giant planets. Based on detailed analyses of gas accretion flow, we find that most of gas accretion onto circumplanetary disks occurs nearly vertically toward the disk surface from high altitude, which generates a shock surface at several scale heights of the circumplanetary disk. The gas that has passed through the shock surface moves inward because its specific angular momentum is smaller than that of the local Keplerian rotation, while gas near the midplane in the protoplanetary disk cannot accrete to the circumplanetary disk. Gas near the midplane within the planet's Hill sphere spirals outward and escapes from the Hill sphere through the two Lagrangian points L{sub 1} and L{sub 2}. We also analyze fluxes of accreting mass and angular momentum in detail and find that the distributions of the fluxes onto the disk surface are well described by power-law functions and that a large fraction of gas accretion occurs at the outer region of the disk, i.e., at about 0.1 times the Hill radius. The nature of power-law functions indicates that, other than the outer edge, there is no specific radius where gas accretion is concentrated. These source functions of mass and angular momentum in the circumplanetary disk would provide us with useful constraints on the structure and evolution of the circumplanetary disk, which is important for satellite formation.
14. Dynamo magnetic-field generation in turbulent accretion disks
NASA Technical Reports Server (NTRS)
Stepinski, T. F.
1991-01-01
Magnetic fields can play important roles in the dynamics and evolution of accretion disks. The presence of strong differential rotation and vertical density gradients in turbulent disks allows the alpha-omega dynamo mechanism to offset the turbulent dissipation and maintain strong magnetic fields. It is found that MHD dynamo magnetic-field normal modes in an accretion disk are highly localized to restricted regions of a disk. Implications for the character of real, dynamically constrained magnetic fields in accretion disks are discussed. The magnetic stress due to the mean magnetic field is found to be of the order of a viscous stress. The dominant stress, however, is likely to come from small-scale fluctuating magnetic fields. These fields may also give rise to energetic flares above the disk surface, providing a possible explanation for the highly variable hard X-ray emission from objects like Cyg X-l.
15. The Origin of Warped, Precessing Accretion Disks in X-ray Binaries
NASA Technical Reports Server (NTRS)
Maloney, Philip R.; Begelman, Mitchell C.
1997-01-01
The radiation-driven warping instability discovered by Pringle holds considerable promise as the mechanism responsible for producing warped, precessing accretion disks in X-ray binaries. This instability is an inherently global mode of the disk, thereby avoiding the difficulties with earlier models for the precession. Here we follow up on earlier work to study the linear behavior of the instability in the specific context of a binary system. We treat the influence of the companion as an orbit-averaged quadrupole torque on the disk. The presence of this external torque allows the existence of solutions in which the direction of precession of the warp is retrograde with respect to disk rotation, in addition to the prograde solutions that exist in the absence of external torques.
16. Freddi: Fast Rise Exponential Decay accretion Disk model Implementation
Malanchev, K. L.; Lipunova, G. V.
2016-10-01
Freddi (Fast Rise Exponential Decay: accretion Disk model Implementation) solves 1-D evolution equations of the Shakura-Sunyaev accretion disk. It simulates fast rise exponential decay (FRED) light curves of low mass X-ray binaries (LMXBs). The basic equation of the viscous evolution relates the surface density and viscous stresses and is of diffusion type; evolution of the accretion rate can be found on solving the equation. The distribution of viscous stresses defines the emission from the source. The standard model for the accretion disk is implied; the inner boundary of the disk is at the ISCO or can be explicitely set. The boundary conditions in the disk are the zero stress at the inner boundary and the zero accretion rate at the outer boundary. The conditions are suitable during the outbursts in X-ray binary transients with black holes. In a binary system, the accretion disk is radially confined. In Freddi, the outer radius of the disk can be set explicitely or calculated as the position of the tidal truncation radius.
17. TEARING UP THE DISK: HOW BLACK HOLES ACCRETE
SciTech Connect
Nixon, Chris; King, Andrew; Price, Daniel; Frank, Juhan
2012-10-01
We show that in realistic cases of accretion in active galactic nuclei or stellar-mass X-ray binaries, the Lense-Thirring effect breaks the central regions of tilted accretion disks around spinning black holes into a set of distinct planes with only tenuous flows connecting them. If the original misalignment of the outer disk to the spin axis of the hole is 45 Degree-Sign {approx}< {theta} {approx}< 135 Degree-Sign , as in {approx}70% of randomly oriented accretion events, the continued precession of these disks sets up partially counterrotating gas flows. This drives rapid infall as angular momentum is canceled and gas attempts to circularize at smaller radii. Disk breaking close to the black hole leads to direct dynamical accretion, while breaking further out can drive gas down to scales where it can accrete rapidly. For smaller tilt angles breaking can still occur and may lead to other observable phenomena such as quasi-periodic oscillations. For such effects not to appear, the black hole spin must in practice be negligibly small, or be almost precisely aligned with the disk. Qualitatively similar results hold for any accretion disk subject to a forced differential precession, such as an external disk around a misaligned black hole binary.
18. On the role of disks in the formation of stellar systems: A numerical parameter study of rapid accretion
SciTech Connect
Kratter, Kaitlin M.; Matzner, Christopher D.; Krumholz, Mark R.; Klein, Richard I.
2009-12-23
We study rapidly accreting, gravitationally unstable disks with a series of idealized global, numerical experiments using the code ORION. Our numerical parameter study focuses on protostellar disks, showing that one can predict disk behavior and the multiplicity of the accreting star system as a function of two dimensionless parameters which compare the infall rate to the disk sound speed and orbital period. Although gravitational instabilities become strong, we find that fragmentation into binary or multiple systems occurs only when material falls in several times more rapidly than the canonical isothermal limit. The disk-to-star accretion rate is proportional to the infall rate and governed by gravitational torques generated by low-m spiral modes. Furthermore, we also confirm the existence of a maximum stable disk mass: disks that exceed ~50% of the total system mass are subject to fragmentation and the subsequent formation of binary companions.
19. On the role of disks in the formation of stellar systems: A numerical parameter study of rapid accretion
DOE PAGES
Kratter, Kaitlin M.; Matzner, Christopher D.; Krumholz, Mark R.; ...
2009-12-23
We study rapidly accreting, gravitationally unstable disks with a series of idealized global, numerical experiments using the code ORION. Our numerical parameter study focuses on protostellar disks, showing that one can predict disk behavior and the multiplicity of the accreting star system as a function of two dimensionless parameters which compare the infall rate to the disk sound speed and orbital period. Although gravitational instabilities become strong, we find that fragmentation into binary or multiple systems occurs only when material falls in several times more rapidly than the canonical isothermal limit. The disk-to-star accretion rate is proportional to the infallmore » rate and governed by gravitational torques generated by low-m spiral modes. Furthermore, we also confirm the existence of a maximum stable disk mass: disks that exceed ~50% of the total system mass are subject to fragmentation and the subsequent formation of binary companions.« less
20. Accretion Disks in Supersoft X-ray Sources
NASA Technical Reports Server (NTRS)
Popham, Robert; DiStefano, Rosanne
1996-01-01
We examine the role of the accretion disk in the steady-burning white dwarf model for supersoft sources. The accretion luminosity of the disk is quite small compared to the nuclear burning luminosity of the central source. Thus, in contrast to standard accretion disks, the main role of the disk is to reprocess the radiation from the white dwarf. We calculate models of accretion disks around luminous white dwarfs and compare the resulting disk fluxes to optical and UV observations of the LMC supersoft sources CAL 83, CAL 87, and RX J0513.9-6951. We find that if the white dwarf luminosity is near the upper end of the steady-burning region, and the flaring of the disk is included, then reprocessing by the disk can account for the UV fluxes and a substantial fraction of the optical fluxes of these systems. Reprocessing by the companion star can provide additional optical flux, and here too the disk plays an important role: since the disk is fairly thick, it shadows a significant fraction of the companion's surface.
1. Structure Formation through Magnetohydrodynamical Instabilities in Protoplanetary Disks
Noguchi, K.; Tajima, T.; Horton, W.
2000-12-01
The shear flow instabilities under the presence of magnetic fields in the protoplanetary disk can greatly facilitate the formation of density structures that serve as seeds prior to the onset of the gravitational Jeans instability. Such a seeding process may explain several outstanding puzzles in the planetary genesis that are further compounded by the new discoveries of extrasolar planets and a new insight into the equation of state of dense matter. This puzzle also includes the apparent narrow window of the age difference of the Sun and the Earth. We evaluate the effects of the Parker, magnetorotational(Balbus-Hawley), and kinematic dynamo instabilities by comparing the properties of these instabilities. We calculate the mass spectra of aggregated density structures by the above mechanism in the radial direction for an axisymmetric magnetohydrodynamic(MHD) torus equiblium and power-law density profile models. The mass spectrum of the magnetorotational instability may describe the origin of giant planets away from the central star such as Jupiter. Our local three-dimentional MHD simulation indicates that the coupling of the Parker and magnetorotational instabilities creates spiral arms and gas blobs in the accretion disk, reinforcing the theory and model.
2. Jet production in super-Eddington accretion disks
NASA Technical Reports Server (NTRS)
Eggum, G. E.; Coroniti, F. V.; Katz, J. I.
1985-01-01
A two-dimensional, radiation-coupled, Newtonian hydrodynamic simulation is reported for a super-Eddington, mass accretion rate, M = 4 M(E) disk accretion flow onto a 3-solar mass pseudoblack hole. Near the disk midplane, convection cells effectively block the accretion flow, even though viscous heating maximizes there. Accretion predominantly occurs in a supersonic inflow which follows streamlines of approximately constant angular momentum. The optically thick inflow traps radiation so that 80 percent of the luminosity is absorbed by the black hole; the emergent power is sub-Eddington. An axial jet self consistently forms just outside a conical photosphere which bounds the accretion zone; radiation pressure accelerates the jet to about 10 to the 10th cm/s. The jet's mass efflux is only 0.4 percent of the total mass accretion rate.
3. Dynamics of accretion disks in a constant curvature f(R)-gravity
Alipour, N.; Khesali, A. R.; Nozari, K.
2016-07-01
So far the basic physical properties of matter forming a thin accretion disc in the static and spherically symmetric space-time metric of the vacuum f(R) modified gravity models (Pun et al. in Phys. Rev. D 78:024043, 2008) and building radiative models of thin accretion disks for both Schwarzschild and Kerr black holes in f(R) gravity (Perez et al. in Astron. Astrophys. 551:4, 2013) were addressed properly. Also von Zeipel surfaces and convective instabilities in f(R)-Schwarzschild(Kerr) background have been investigated recently (Alipour et al. in Mon. Not. R. Astron. Soc. 454:1992, 2015). In this streamline, here we study the effects of radial and angular pressure gradients on thick accretion disks in Schwarzschild geometries in a constant curvature f(R) modified gravity. Since thick accretion disks have high accretion rate, we study configuration and structure of thick disks by focusing on the effect of pressure gradient on formation of the disks. We clarify our study by assuming two types of equation of state: polytropic and Clapeyron equation of states.
4. Accretion disk emission from a BL Lacertae object
NASA Technical Reports Server (NTRS)
Urry, C. Megan; Wandel, Amri
1990-01-01
The accretion disk is an attractive model for BL Lac objects because of its preferred axis and high efficiency. While the smooth continuum spectra of BL Lacs do not show large UV bumps, in marked contrast to quasars, high quality simultaneous data do reveal deviations from smoothness. Using detailed calculations of cool accretion disk spectra, the best measured ultraviolet and soft x ray spectra of the BL Lac object PKS 2155-304 are fitted. The mass and accretion rate required are determined. A hot disk or corona could comptonize soft photons from the cool disk and produce the observed power law spectrum in the 1 to 10 keV range. The dynamic time scales in the disk regions that contribute most of the observed ultraviolet and soft x ray photons are consistent with the respective time scales for intensity variations. The mass derived from fitting the continuum spectrum is consistent with the limit derived from the fastest hard x ray variability.
5. Vertical Structure of Magnetized Accretion Disks around Young Stars
Lizano, S.; Tapia, C.; Boehler, Y.; D'Alessio, P.
2016-01-01
We model the vertical structure of the magnetized accretion disks that are subject to viscous and resistive heating and irradiation by the central star. We apply our formalism to the radial structure of the magnetized accretion disks that are threaded by the poloidal magnetic field dragged during the process of star formation, which was developed by Shu and coworkers. We consider disks around low-mass protostars, T Tauri, and FU Orionis stars, as well as two levels of disk magnetization: {λ }{sys}=4 (strongly magnetized disks) and {λ }{sys}=12 (weakly magnetized disks). The rotation rates of strongly magnetized disks have large deviations from Keplerian rotation. In these models, resistive heating dominates the thermal structure for the FU Ori disk, and the T Tauri disk is very thin and cold because it is strongly compressed by magnetic pressure; it may be too thin compared with observations. Instead, in the weakly magnetized disks, rotation velocities are close to Keplerian, and resistive heating is always less than 7% of the viscous heating. In these models, the T Tauri disk has a larger aspect ratio, which is consistent with that inferred from observations. All the disks have spatially extended hot atmospheres where the irradiation flux is absorbed, although most of the mass (˜90%-95%) is in the disk midplane. With the advent of ALMA one expects direct measurements of magnetic fields and their morphology at disk scales. It will then be possible to determine the mass-to-flux ratio of magnetized accretion disks around young stars, an essential parameter for their structure and evolution. Our models contribute to the understanding of the vertical structure and emission of these disks.
6. A Hot and Massive Accretion Disk around the High-mass Protostar IRAS 20126+4104
Chen, Huei-Ru Vivien; Keto, Eric; Zhang, Qizhou; Sridharan, T. K.; Liu, Sheng-Yuan; Su, Yu-Nung
2016-06-01
We present new spectral line observations of the CH3CN molecule in the accretion disk around the massive protostar IRAS 20126+4104 with the Submillimeter Array, which, for the first time, measure the disk density, temperature, and rotational velocity with sufficient resolution (0.″37, equivalent to ˜600 au) to assess the gravitational stability of the disk through the Toomre-Q parameter. Our observations resolve the central 2000 au region that shows steeper velocity gradients with increasing upper state energy, indicating an increase in the rotational velocity of the hotter gas nearer the star. Such spin-up motions are characteristics of an accretion flow in a rotationally supported disk. We compare the observed data with synthetic image cubes produced by three-dimensional radiative transfer models describing a thin flared disk in Keplerian motion enveloped within the centrifugal radius of an angular-momentum-conserving accretion flow. Given a luminosity of 1.3 × 104 L ⊙, the optimized model gives a disk mass of 1.5 M ⊙ and a radius of 858 au rotating about a 12.0 M ⊙ protostar with a disk mass accretion rate of 3.9 × 10-5 M ⊙ yr-1. Our study finds that, in contrast to some theoretical expectations, the disk is hot and stable to fragmentation with Q > 2.8 at all radii which permits a smooth accretion flow. These results put forward the first constraints on gravitational instabilities in massive protostellar disks, which are closely connected to the formation of companion stars and planetary systems by fragmentation.
7. Electromagnetic signatures of thin accretion disks in wormhole geometries
SciTech Connect
Harko, Tiberiu; Kovacs, Zoltan; Lobo, Francisco S. N.
2008-10-15
In this paper, we study the physical properties and characteristics of matter forming thin accretion disks in static and spherically symmetric wormhole spacetimes. In particular, the time averaged energy flux, the disk temperature, and the emission spectra of the accretion disks are obtained for these exotic geometries and are compared with the Schwarzschild solution. It is shown that more energy is emitted from the disk in a wormhole geometry than in the case of the Schwarzschild potential and the conversion efficiency of the accreted mass into radiation is more than a factor of 2 higher for the wormholes than for static black holes. These effects in the disk radiation are confirmed in the radial profiles of temperature corresponding to theses flux distributions, and in the emission spectrum {omega}L({omega}) of the accretion disks. We conclude that specific signatures appear in the electromagnetic spectrum, thus leading to the possibility of distinguishing wormhole geometries by using astrophysical observations of the emission spectra from accretion disks.
8. Vertical Structure of Magnetized Accretion Disks Around Young Stars
Tapia, Carlos; Lizano, Susana
2016-01-01
We model the vertical structure of magnetized accretion disks subject to viscous and resistive heating, and irradiation by the central star. We apply our formalism to the radial structure of magnetized accretion disks threaded by a poloidal magnetic field dragged during the process of star formation developed by Shu and coworkers. We consider disks around low mass protostars, T Tauri, and FU Orionis stars. We consider two levels of disk magnetization, λsys = 4 (strongly magnetized disks), and λsys = 12 (weakly magnetized disks). The rotation rates of strongly magnetized disks have large deviations from Keplerian rotation. In these models, resistive heating dominates the thermal structure for the FU Ori disk. The T Tauri disk is very thin and cold because it is strongly compressed by magnetic pressure; it may be too thin compared with observations. Instead, in the weakly magnetized disks, rotation velocities are close to Keplerian, and resistive heating is always less than 7% of the viscous heating. In these models, the T Tauri disk has a larger aspect ratio, consistent with that inferred from observations. All the disks have spatially extended hot atmospheres where the irradiation flux is absorbed, although most of the mass (~ 90 - 95 %) is in the disk midplane.
9. White Dwarf Pollution by Disk Accretion of Tidally Disrupted Rocky Bodies
Feng, Wanda; Desch, Steven
2017-01-01
Approximately 30% of cool white dwarfs (WDs) show heavy elements which should otherwise sediment out of their atmospheres (Koester et al. 2014; Zuckerman et al. 2010). The prevailing model for the pollution of white dwarf photospheres invokes the formation of a solid disk upon a rocky body falling within the WD Roche radius, which is then transported inward by Poynting-Robertson drag (e.g., Metzger et al. 2012, Rafikov 2011). At high temperatures close to the WD, solid particles sublimate to gas that accretes onto the WD and viscously spreads outward. This concept is supported by observations of Ca II emission from WD disks (e.g., Manser et al. 2016). The model by Metzger et al. (2012) successfully explains the range in inferred mass accretion rates (10^10 g/s, Farihi et al. 2010), provided the gaseous disks viscously spread at rates consistent with a partially suppressed magnetorotational instability (MRI). However, Metzger et al. (2012) do not consider disk chemistry or dust-to-gas mixing in their model, and do not calculate the degree of ionization to explore the extent of MRI in WD disks.We present a 1-D model of a gaseous WD disk accretion, to assess the extent of the magnetorotational instability in WD disks. The disk composition is considered with changes in sublimation rate by pressure. The degree of ionization is determined by considering UV, X-ray, and high-temperature ionization. We calculate the rate of viscous spreading and accretion rates of metals onto WDs.
10. ON THE STRUCTURE OF ACCRETION DISKS WITH OUTFLOWS
SciTech Connect
Jiao Chengliang; Wu Xuebing E-mail: [email protected]
2011-06-01
To study the outflows from accretion disks, we solve the set of hydrodynamic equations for accretion disks in spherical coordinates (r{theta}{phi}) to obtain the explicit structure along the {theta}-direction. Using self-similar assumptions in the radial direction, we change the equations to a set of ordinary differential equations about the {theta}-coordinate, which are then solved with symmetrical boundary conditions in the equatorial plane; the velocity field is then obtained. The {alpha} viscosity prescription is applied and an advective factor f is used to simplify the energy equation. The results display thinner, quasi-Keplerian disks for Shakura-Sunyaev disks; thicker, sub-Keplerian disks for advection-dominated accretion flows; and slim disks which are consistent with previous popular analytical models. However, an inflow region and an outflow region always exist, except when the viscosity parameter {alpha} is too large, which supports the results of some recent numerical simulation works. Our results indicate that the outflows should be common in various accretion disks and may be stronger in slim disks, where both advection and radiation pressure are dominant. We also present the structure's dependence on the input parameters and discuss their physical meanings. The caveats of this work and possible improvements for the future are discussed.
11. Evolution of the luminosity function of quasar accretion disks
NASA Technical Reports Server (NTRS)
Caditz, David M.; Petrosian, Vahe; Wandel, Amri
1991-01-01
Using an accretion-disk model, accretion disk luminosities are calculated for a grid of black hole masses and accretion rates. It is shown that, as the black-hole mass increases with time, the monochromatic luminosity at a given frequency first increases and then decreases rapidly as this frequency is crossed by the Wien cutoff. The upper limit on the monochromatic luminosity, which is characteristic for a given epoch, constrains the evolution of quasar luminosities and determines the evolultion of the quasar luminosity function.
12. MAGNETICALLY LEVITATING ACCRETION DISKS AROUND SUPERMASSIVE BLACK HOLES
SciTech Connect
Gaburov, Evghenii; Johansen, Anders; Levin, Yuri
2012-10-20
In this paper, we report on the formation of magnetically levitating accretion disks around supermassive black holes (SMBHs). The structure of these disks is calculated by numerically modeling tidal disruption of magnetized interstellar gas clouds. We find that the resulting disks are entirely supported by the pressure of the magnetic fields against the component of gravitational force directed perpendicular to the disks. The magnetic field shows ordered large-scale geometry that remains stable for the duration of our numerical experiments extending over 10% of the disk lifetime. Strong magnetic pressure allows high accretion rate and inhibits disk fragmentation. This in combination with the repeated feeding of magnetized molecular clouds to an SMBH yields a possible solution to the long-standing puzzle of black hole growth in the centers of galaxies.
13. Propagation of tidal disturbance in gaseous accretion disks
NASA Technical Reports Server (NTRS)
Lin, D. N. C.; Papaloizou, J. C. B.; Savonije, G. J.
1990-01-01
Linear wave propagation is studied in geometrically thin accretion disks where the equilibrium variables, such as density and temperature, are stratified in the direction normal to the plane of the disk; i.e., the vertical direction. It is shown, due to refraction effects, that waves excited by tidal disturbances induced by a satellite or a companion of the central object are not expected to reach the interior regions of the disk with a significant amplitude.
14. Accretion disk emission from a BL Lacertae object
NASA Technical Reports Server (NTRS)
Wandel, Amri; Urry, C. Megan
1991-01-01
It is suggested here that the UV and X-ray emission of BL Lac objects may originate in an accretion disk. Using detailed calculations of accretion disk spectra, the best-measured ultraviolet and soft X-ray spectra of the BL Lac object PKS 2155-304 are fitted, and the mass and accretion rate required is determined. The ultraviolet through soft X-ray continuum is well fitted by the spectrum of an accretion disk, but near-Eddington accretion rates are required to produce the soft X-ray excess. A hot disk or corona could Comptonize soft photons from the cool disk and produce the observed power-law spectrum in the 1-10 keV range. The dynamic time scale in the disk regions that contribute most of the observed ultraviolet and soft X-ray photons are consistent with the respective time scales for intensity variations observed in these two wave bands; the mass derived from fitting the continuum spectrum is consistent with the limit derived from the fastest hard X-ray variability.
15. Young Stellar Objects in Lynds 1641: Disks and Accretion
Fang, Min; Kim, Jinyoung Serena; van Boekel, Roy; Sicilia-Aguilar, Aurora; Henning, Thomas; Flaherty, Kevin
2013-07-01
We investigate the young stellar objects (YSOs) in the Lynds 1641 (L1641) cloud using multi-wavelength data including Spitzer, WISE, 2MASS, and XMM covering 1390 YSOs across a range of evolutionary stages. In addition, we targeted a sub-sample of YSOs for optical spectroscopy with the MMT/Hectospec and the MMT/Hectochelle. We use this data, along with archival photometric data, to derive spectral types, masses, ages and extinction values. We also use the H_alpha and H_beta lines to derive accretion rates. We calculate the disk fraction as N(II)/N(II+III), where N(II) and N(III) are numbers of Class\\ II and Class\\ III sources, respectively, and obtain a disk fraction of 50% in L1641. We find that the disk frequency is almost constant as a function of stellar mass with a slight peak at log(M_*/M_sun) -0.25. The analysis of multi-epoch data indicates that the accretion variability of YSOs cannot explain the two orders of magnitude of scatter for YSOs with similar masses in the M_acc vs. M_* plot. Forty-six new transition disk objects are confirmed in our spectroscopic survey and we find that the fraction of transition disks that are actively accreting is lower than for optically thick disks (40-45% vs. 77-79% respectively). We confirm our previous result that the accreting YSOs with transition disks have a similar median accretion rate to normal optically thick disks. Analyzing the age distributions of various populations, we find that the diskless YSOs are statistically older than the YSOs with optically-thick disks and the transition disk objects have a median age which is intermediate between the two populations.
16. Nucleosynthesis in the accretion disks of Type II collapsars
2013-09-01
We investigate nucleosynthesis inside the gamma-ray burst (GRB) accretion disks formed by the Type II collapsars. In these collapsars, the core collapse of massive stars first leads to the formation of a proto-neutron star. After that, an outward moving shock triggers a successful supernova. However, the supernova ejecta lacks momentum and within a few seconds the newly formed neutron star gets transformed to a stellar mass black hole via massive fallback. The hydrodynamics of such an accretion disk formed from the fallback material of the supernova ejecta has been studied extensively in the past. We use these well-established hydrodynamic models for our accretion disk in order to understand nucleosynthesis, which is mainly advection dominated in the outer regions. Neutrino cooling becomes important in the inner disk where the temperature and density are higher. The higher the accretion rate (dot M) is, the higher the density and temperature are in the disks. We deal with accretion disks with relatively low accretion rates: 0.001 Msolar s-1 ≲ dot M ≲ 0.01 Msolar s-1 and hence these disks are predominantly advection dominated. We use He-rich and Sirich abundances as the initial condition of nucleosynthesis at the outer disk, and being equipped with the disk hydrodynamics and the nuclear network code, we study the abundance evolution as matter inflows and falls into the central object. We investigate the variation in the nucleosynthesis products in the disk with the change in the initial abundance at the outer disk and also with the change in the mass accretion rate. We report the synthesis of several unusual nuclei like 31P, 39K, 43Sc, 35Cl and various isotopes of titanium, vanadium, chromium, manganese and copper. We also confirm that isotopes of iron, cobalt, nickel, argon, calcium, sulphur and silicon get synthesized in the disk, as shown by previous authors. Much of these heavy elements thus synthesized are ejected from the disk via outflows and hence they
17. Flux distributions and colors of accretion disks
NASA Technical Reports Server (NTRS)
Pacharintanakul, P.; Katz, J. I.
1980-01-01
The disk model of Shakura and Sunyaev (1973) and Novikov and Thorne (1973) is used to calculate temperature distributions and integrated spectral fluxes for disks around a typical white dwarf and a typical neutron star, under the assumption that each element of the disk locally radiates as a blackbody. In addition, the disks' integrated UBV colors are calculated using the grid colors for real model atmospheres calculated by Buser and Kurucz (1978) and the observed colors given by Allen (1973). In all the calculations the effect of radiation from one part of the disk on all the other parts is included.
18. Evolution of Pre-Main Sequence Accretion Disks
NASA Technical Reports Server (NTRS)
Hartmann, Lee W.
2000-01-01
The aim of this project was to develop a comprehensive global picture of the physical conditions in, and evolutionary timescales of, pre-main sequence accretion disks. The results of this work will help constrain the initial conditions for planet formation. To this end we: (1) Developed detailed calculations of disk structure to study physical conditions and investigate the observational effects of grain growth in T Tauri disks; (2) Studied the dusty emission and accretion rates in older disk systems, with ages closer to the expected epoch of (giant) planet formation at 3-10 Myr, and (3) Began a project to develop much larger samples of 3-10 Myr-old stars to provide better empirical constraints on protoplanetary disk evolution.
19. Time-dependent X-ray emission from unstable accretion disks around black holes
NASA Technical Reports Server (NTRS)
Mineshige, Shin; Kim, Soon-Wook; Wheeler, J. Craig
1990-01-01
The spectral evolution of accretion disks in X-ray binaries containing black holes is studied, based on the disk instability model. The thermal transition of the outer portions of the disk controls the mass flow rate into the inner portions of the disk, thus modulating the soft X-ray flux which is thought to arise from the inner disk. Calculated soft X-ray spectra are consistent with the observations of the X-ray transient A0620 - 00 and especially ASM 2000 + 25, the soft X-ray spectra of which are well fitted by blackbody radiation with a fixed inner edge of the disk, Rin, and with monotonically decreasing temperature at Rin with time. Since the gas pressure is always dominant over the radiation pressure during the decay in these models, a two-temperature region is difficult to create. Instead, it is suggested that hard X-rays are generated in a hot (kT greater than 10 keV) accretion disk corona above the cool (kT less than 1 keV) disk.
20. [Predicting Spectra of Accretion Disks Around Galactic Black Holes
NASA Technical Reports Server (NTRS)
Krolik, Julian H.
2004-01-01
The purpose of this grant was to construct detailed atmosphere solutions in order to predict the spectra of accretion disks around Galactic black holes. Our plan of action was to take an existing disk atmosphere code (TLUSTY, created by Ivan Hubeny) and introduce those additional physical processes necessary to make it applicable to disks of this variety. These modifications include: treating Comptonization; introducing continuous opacity due to heavy elements; incorporating line opacity due to heavy elements; adopting a disk structure that reflects readjustments due to radiation pressure effects; and injecting heat via a physically-plausible vertical distribution.
1. Viscous pulsational instability of the transonic region of isothermal geometrically thin accretion discs. I - Analytical results
NASA Technical Reports Server (NTRS)
Kato, Shoji; Honma, Fumio; Matsumoto, Ryoji
1988-01-01
Viscous instability of the transonic region of the conventional geometrically thin alpha-type accretion disks is examined analytically. For simplicity, isothermal disks and isothermal perturbations are assumed. It is found that when the value of alpha is larger than a critical value the disk is unstable against two types of perturbations. One is local propagating perturbations of inertial acoustic waves. Results suggest the possibility that unstable perturbations develop to overstable global oscillations which are restricted only in the innermost region of the disk. The other is standing growing perturbations localized just at the transonic point. The cause of these instabilities is that the azimuthal component of the Lagrangian velocity variation associated with the perturbations becomes in phase with the variation of the viscous stress force. Because of this phase matching work is done on perturbations, and they are amplified.
2. ACCRETION DISKS AROUND KICKED BLACK HOLES: POST-KICK DYNAMICS
SciTech Connect
Ponce, Marcelo; Faber, Joshua A.; Lombardi, James C. E-mail: [email protected]
2012-01-20
Numerical calculations of merging black hole binaries indicate that asymmetric emission of gravitational radiation can kick the merged black hole at up to thousands of km s{sup -1}, and a number of systems have been observed recently whose properties are consistent with an active galactic nucleus containing a supermassive black hole moving with substantial velocity with respect to its broader accretion disk. We study here the effect of an impulsive kick delivered to a black hole on the dynamical evolution of its accretion disk using a smoothed particle hydrodynamics code, focusing attention on the role played by the kick angle with respect to the orbital angular momentum vector of the pre-kicked disk. We find that for more vertical kicks, for which the angle between the kick and the normal vector to the disk {theta} {approx}< 30 Degree-Sign , a gap remains present in the inner disk, in accordance with the prediction from an analytic collisionless Keplerian disk model, while for more oblique kicks with {theta} {approx}> 45 Degree-Sign , matter rapidly accretes toward the black hole. There is a systematic trend for higher potential luminosities for more oblique kick angles for a given black hole mass, disk mass, and kick velocity, and we find large amplitude oscillations in time in the case of a kick oriented 60 Degree-Sign from the vertical.
3. STANDING SHOCK INSTABILITY IN ADVECTION-DOMINATED ACCRETION FLOWS
SciTech Connect
Le, Truong; Wood, Kent S.; Wolff, Michael T.; Becker, Peter A.; Putney, Joy
2016-03-10
Depending on the values of the energy and angular momentum per unit mass in the gas supplied at large radii, inviscid advection-dominated accretion flows can display velocity profiles with either preshock deceleration or preshock acceleration. Nakayama has shown that these two types of flow configurations are expected to have different stability properties. By employing the Chevalier and Imamura linearization method and the Nakayama instability boundary conditions, we discover that there are regions of parameter space where disks/shocks with outflows can be stable or unstable. In regions of instability, we find that preshock deceleration is always unstable to the zeroth mode with zero frequency of oscillation, but is always stable to the fundamental mode and overtones. Furthermore, we also find that preshock acceleration is always unstable to the zeroth mode and that the fundamental mode and overtones become increasingly less stable as the shock location moves away from the horizon when the disk half-height expands above ∼12 gravitational radii at the shock radius. In regions of stability, we demonstrate the zeroth mode to be stable for the velocity profiles that exhibit preshock acceleration and deceleration. Moreover, for models that are linearly unstable, our model suggests the possible existence of quasi-periodic oscillations (QPOs) with ratios 2:3 and 3:5. These ratios are believed to occur in stellar and supermassive black hole candidates, for example, in GRS 1915+105 and Sgr A*, respectively. We expect that similar QPO ratios also exist in regions of stable shocks.
4. A nonlinear investigation of corrugation instabilities in magnetic accretion shocks
Ernst, Scott
2011-05-01
Accretion shock waves are present in many important astrophysical systems and have been a focus of research for decades. These investigations provide a large body of understanding as to the nature, characteristics, and evolutionary behaviors of accretion shock waves over a wide range of conditions. However, largely absent are investigations into the properties of accretion shock waves in the presence of strong magnetic fields. In such cases these strong magnetic fields can significantly alter the stability behaviors and evolution of the accretion shock wave through the production and propagation of magnetic waves as well as magnetically constrained advection. With strong magnetic fields likely found in a number of accretion shock systems, such as compact binary and protostellar systems, a better understanding of the behaviors of magnetic accretion shock waves is needed. A new magnetohydrodynamics simulation tool, IMOGEN, was developed to carry out an investigation of instabilities in strong, slow magnetic accretion shocks by modelling their long-term, nonlinear evolution. IMOGEN implements a relaxed, second-order, total variation diminishing, monotonic upwind scheme for conservation laws and incorporates a staggered-grid constrained transport scheme for magnetic advection. Through the simulated evolution of magnetic accretion shocks over a wide range of initial conditions, it has been shown, for sufficiently high magnetic field strengths, that magnetic accretion shocks are generally susceptible to corrugation instabilities, which arise in the presence of perturbations of the initial shock front. As these corrugation instabilities grow, they manifestas magnetic wave propagation in the upstream region of the accretion column, which propagate away from the accretion shock front, and as density columns, or fingers, that grow into the higher density downstream flow, defined and constrained by current loops created during the early evolution of the instability.
5. Plasma (Accretion) Disks with High Magnetic Energy Densities
Rousseau, F.; Coppi, B.
2006-04-01
Corrugated'' plasma disks can form in the dominant gravity of a central object when the peak plasma pressure in the disk is of the same order as that of the pressure of the external'' magnetic field, while the magnetic field resulting from internal plasma currents is of the same order as the external field. The corrugation refers to a periodic variation of the plasma density in a region around the equatorial plane. The considered structure represents a transition between a classical'' accretion disk and a rings sequence'' configuration^2. The common feature of the corrugated'' and the rings sequence'' configurations is the crystal'' structure of the magnetic surfaces that consist of a sequence of pairs of oppositely directed toroidal current density filaments. The connection between the characteristics of these configurations and those of the marginally stable ballooning modes that can be found in a model accretion disk is pointed out and analyzed.
6. EFFECTIVE INNER RADIUS OF TILTED BLACK HOLE ACCRETION DISKS
SciTech Connect
Fragile, P. Chris
2009-12-01
One of the primary means of determining the spin a of an astrophysical black hole is by actually measuring the inner radius r {sub in} of a surrounding accretion disk and using that to infer a. By comparing a number of different estimates of r {sub in} from simulations of tilted accretion disks with differing black hole spins, we show that such a procedure can give quite wrong answers. Over the range 0 <= a/M <= 0.9, we find that, for moderately thick disks (H/r approx 0.2) with modest tilt (15 deg.), r {sub in} is nearly independent of spin. This result is likely dependent on tilt, such that for larger tilts, it may even be that r {sub in} would increase with increasing spin. In the opposite limit, we confirm through numerical simulations of untilted disks that, in the limit of zero tilt, r {sub in} recovers approximately the expected dependence on a.
7. Evolution of Pre-Main Sequence Accretion Disks
NASA Technical Reports Server (NTRS)
Hartmann, Lee W.
2002-01-01
The aim of this project is to develop a comprehensive global picture of the physical conditions in, and evolutionary timescales of, pre-main sequence accretion disks. The results of this work will help constrain the initial conditions for planet formation. To this end we plan to: (1) Develop much larger samples of 3-10 Myr-old stars to provide better empirical constraints on protoplanetary disk evolution; (2) Study the dusty emission and accretion rates in these systems, with ages closer to the expected epoch of (giant) planet formation at 3-10 Myr; and (3) Develop detailed model disk structures consistent with observations to infer physical conditions in protoplanetary disks and to constrain possible grain growth as the first stage of planetesimal formation.
8. Orbital Evolution of Moons in Weakly Accreting Circumplanetary Disks
Fujii, Yuri I.; Kobayashi, Hiroshi; Takahashi, Sanemichi Z.; Gressel, Oliver
2017-04-01
We investigate the formation of hot and massive circumplanetary disks (CPDs) and the orbital evolution of satellites formed in these disks. Because of the comparatively small size-scale of the sub-disk, quick magnetic diffusion prevents the magnetorotational instability (MRI) from being well developed at ionization levels that would allow MRI in the parent protoplanetary disk. In the absence of significant angular momentum transport, continuous mass supply from the parental protoplanetary disk leads to the formation of a massive CPD. We have developed an evolutionary model for this scenario and have estimated the orbital evolution of satellites within the disk. We find, in a certain temperature range, that inward migration of a satellite can be stopped by a change in the structure due to the opacity transitions. Moreover, by capturing second and third migrating satellites in mean motion resonances, a compact system in Laplace resonance can be formed in our disk models.
9. The growth of supermassive black holes fed by accretion disks
Montesinos Armijo, M. A.; de Freitas Pacheco, J. A.
2011-02-01
Context. Supermassive black holes are probably present in the centre of the majority of the galaxies. There is consensus that these exotic objects are formed by the growth of seeds either by mass accretion from a circumnuclear disk and/or by coalescences during merger episodes. Aims: The mass fraction of the disk captured by the central object and the related timescale are still open questions, as is how these quantities depend on parameters, such as the initial mass of the disk or the seed, or on the angular momentum transport mechanism. This paper addresses these particular aspects of the accretion disk evolution and the growth of seeds. Methods: The time-dependent hydrodynamic equations were solved numerically for an axisymmetric disk in which the gravitational potential includes contributions from both the central object and the disk itself. The numerical code is based on a Eulerian formalism, using a finite difference method of second-order, according to the Van Leer upwind algorithm on a staggered mesh. Results: The present simulations indicate that seeds capture about a half of the initial disk mass, a result weakly dependent on model parameters. The timescales required for accreting 50% of the disk mass are in the range 130-540 Myr, depending on the adopted parameters. These timescales can explain the presence of bright quasars at z ~ 6.5. Moreover, at the end of the disk evolution, a "torus-like" geometry develops, offering a natural explanation for the presence of these structures in the central regions of AGNs, representing an additional support to the unified model.
10. Structure and Spectroscopy of Black Hole Accretion Disks
SciTech Connect
Liedahl, D; Mauche, C
2005-02-14
The warped spacetime near black holes is one of the most exotic observable environments in the Universe. X-ray spectra from active galaxies obtained with the current generation of X-ray observatories reveal line emission that is modified by both special relativistic and general relativistic effects. The interpretation is that we are witnessing X-ray irradiated matter orbiting in an accretion disk around a supermassive black hole, as it prepares to cross the event horizon. This interpretation, however, is based upon highly schematized models of accretion disk structure. This report describes a project to design a detailed computer model of accretion disk atmospheres, with the goal of elucidating the high radiation density environments associated with mass flows in the curved spacetime near gravitationally collapsed objects. We have evolved the capability to generate realistic theoretical X-ray line spectra of accretion disks, thereby providing the means for a workable exploration of the behavior of matter in the strong-field limit of gravitation.
11. Durability of the accretion disk of millisecond pulsars.
PubMed
Michel, F C; Dessler, A J
1985-05-24
Pulsars with pulsation periods in the millisecond range are thought to be neutron stars that have acquired an extraordinarily short spin period through the accretion of stellar material spiraling down onto the neutron star from a nearby companion. Nearly all the angular momentum and most of the mass of the companion star is transferred to the neutron star. During this process, wherein the neutron star consumes its companion, it is required that a disk of stellar material be formed around the neutron star. In conventional models it is supposed that the disk is somehow lost when the accretion phase is finished, so that only the rapidly spinning neutron star remains. However, it is possible that, after the accretion phase, a residual disk remains in stable orbit around the neutron star. The end result of such an accretion process is an object that looks much like a miniature (about 100 kilometers), heavy version of Saturn: a central object (the neutron star) surrounded by a durable disk.
12. Accretion Disk Structure in Various Spectral States of GRS 1915+105
Remillard, Ronald
2000-09-01
GRS 1915+105 displays 9 types of light curves that fall in 3 categories. In the steady-hard states, the Fe line is strongest, and there is a steady type of jet. In the soft states, the accretion disk dominates the X-ray spectrum, and we often detect the 67 Hz QPO thought to arise from GR effects in the inner disk. The remaining states show a variety of instability oscillations, some producing violent mass ejections. Progress on all fronts requires high resolution spectra to help interpret the disk structure. We have particular interest in the profiles of broad Fe emission, intending to gain physical insights using theoretical models of Nayakshin et al. With monitoring timescales selected to randomize the 9 states, we request 3 obs likely to sample different conditions in the disk.
13. Accretion Disk Emission Around Kerr Black Holes
Campitiello, Samuele; Sbarrato, T.; Ghisellini, G.
2016-10-01
Measuring the spin of supermassive Black holes in Active Galactic Nuclei is a further step towards a better understanding of the evolution of their physics. We proposed a new method to estimate the Black hole spin, based on data-fitting. We consider a numerical model called KERRBB, including all relativistic effects (i.e. light-bending, gravitational redshift and Doppler beaming). We found that the same spectrum can be produced by different masses, accretion rates and spins, but that these three quantities are related. In other words, having a robust indipendent estimate on one of these three quantities fixes the other two. By using the Black hole mass, estimated by the virial method, we can pinpoint a narrow range of possible spins and accretion rates for the 32 blazars we have studied. For these objects, we found a lower limit of the spin, that must be a/M > 0.6-0.7
14. Accretion Disk Illumination in Schwarzschild and Kerr Geometries: Fitting Formulae
Fukumura, Keigo; Kazanas, Demosthenes
2007-07-01
We describe the methodology and compute the illumination of geometrically thin accretion disks around black holes of arbitrary spin parameter a exposed to the radiation of a pointlike isotropic source at arbitrary height above the disk on its symmetry axis. We then provide analytic fitting formulae for the illumination as a function of the source height h and the black hole angular momentum a. We find that for a source on the disk symmetry axis and with h/M>3, the main effect of the parameter a is allowing the disk to extend to smaller radii (approaching r/M-->1 as a/M-->1) and thus allowing the illumination of regions of much higher rotational velocity and redshift. We also compute the illumination profiles for anisotropic emission associated with the motion of the source relative to the accretion disk and present the fractions of photons absorbed by the black hole, intercepted by the disk, or escaping to infinity for both isotropic and anisotropic emission for a/M=0 and 0.99. As the anisotropy (of a source approaching the disk) increases, the illumination profile reduces (approximately) to a single power law, whose index q, because of absorption of the beamed photons by the black hole, saturates to a value no higher than q>~3. Finally, we compute the fluorescent Fe line profiles associated with the specific illumination and compare them among various cases.
15. Tilted Thick-Disk Accretion onto a Kerr Black Hole
SciTech Connect
Fragile, P C; Anninos, P
2003-12-12
We present the first results from fully general relativistic numerical studies of thick-disk accretion onto a rapidly-rotating (Kerr) black hole with a spin axis that is tilted (not aligned) with the angular momentum vector of the disk. We initialize the problem with the solution for an aligned, constant angular momentum, accreting thick disk around a black hole with spin a/M = J/M{sup 2} = +0.9 (prograde disk). The black hole is then instantaneously tilted, through a change in the metric, by an angle {beta}{sub 0}. In this Letter we report results with {beta}{sub 0} = 0, 15, and 30{sup o}. The disk is allowed to respond to the Lense-Thirring precession of the tilted black hole. We find that the disk settles into a quasi-static, twisted, warped configuration with Lense-Thirring precession dominating out to a radius analogous to the Bardeen-Petterson transition in tilted Keplerian disks.
16. Radial Transport and Meridional Circulation in Accretion Disks
Philippov, Alexander A.; Rafikov, Roman R.
2017-03-01
Radial transport of particles, elements and fluid driven by internal stresses in three-dimensional (3D) astrophysical accretion disks is an important phenomenon, potentially relevant for the outward dust transport in protoplanetary disks, origin of the refractory particles in comets, isotopic equilibration in the Earth–Moon system, etc. To gain better insight into these processes, we explore the dependence of meridional circulation in 3D disks with shear viscosity on their thermal stratification, and demonstrate a strong effect of the latter on the radial flow. Previous locally isothermal studies have normally found a pattern of the radial outflow near the midplane, switching to inflow higher up. Here we show, both analytically and numerically, that a flow that is inward at all altitudes is possible in disks with entropy and temperature steeply increasing with height. Such thermodynamic conditions may be typical in the optically thin, viscously heated accretion disks. Disks in which these conditions do not hold should feature radial outflow near the midplane, as long as their internal stress is provided by the shear viscosity. Our results can also be used for designing hydrodynamical disk simulations with a prescribed pattern of the meridional circulation.
17. X-Ray Binary Phenomenology and Their Accretion Disk Structure
Kazanas, Demosthenes
We propose a scheme that accounts for the broader spectral and temporal properties of galactic black hole X-ray transients. The fundamental notion behind this proposal is that the mass accretion rate, dot{M}, of the disks of these systems depends on the radius, as it has been proposed for ADIOS. We propose that, because of this dependence of dot{M} on radius, an accretion disk which is geometrically thin and cool at large radii converts into a geometrically thick, advection dominated, hot disk interior to a transition radius at which the local accretion rate drops below the square of the viscosity parameter, a condition for the existence of advection dominated flows. We argue also that such a transition requires in addition that the vertical disk support be provided by magnetic fields. As discussed in other chapters of this book, the origin of these fields is local to the disk by the Poynting Robertson battery, thereby providing a complete self-contained picture for the spectra and evolution of these systems.
18. Accreting protoplanets in the LkCa 15 transition disk.
PubMed
Sallum, S; Follette, K B; Eisner, J A; Close, L M; Hinz, P; Kratter, K; Males, J; Skemer, A; Macintosh, B; Tuthill, P; Bailey, V; Defrère, D; Morzinski, K; Rodigas, T; Spalding, E; Vaz, A; Weinberger, A J
2015-11-19
Exoplanet detections have revolutionized astronomy, offering new insights into solar system architecture and planet demographics. While nearly 1,900 exoplanets have now been discovered and confirmed, none are still in the process of formation. Transition disks, protoplanetary disks with inner clearings best explained by the influence of accreting planets, are natural laboratories for the study of planet formation. Some transition disks show evidence for the presence of young planets in the form of disk asymmetries or infrared sources detected within their clearings, as in the case of LkCa 15 (refs 8, 9). Attempts to observe directly signatures of accretion onto protoplanets have hitherto proven unsuccessful. Here we report adaptive optics observations of LkCa 15 that probe within the disk clearing. With accurate source positions over multiple epochs spanning 2009-2015, we infer the presence of multiple companions on Keplerian orbits. We directly detect Hα emission from the innermost companion, LkCa 15 b, evincing hot (about 10,000 kelvin) gas falling deep into the potential well of an accreting protoplanet.
19. Ultraviolet line diagnostics of accretion disk winds in cataclysmic variables
NASA Technical Reports Server (NTRS)
Vitello, Peter; Shlosman, Isaac
1993-01-01
The IUE data base is used to analyze the UV line shapes of the cataclysmic variables RW Sex, RW Tri, and V Sge. Observed lines are compared to synthetic line profiles computed using a model of rotating biconical winds from accretion disks. The wind model calculates the wind ionization structure self-consistently including photoionization from the disk and boundary layer and treats 3D line radiation transfer in the Sobolev approximation. It is found that winds from accretion disks provide a good fit for reasonable parameters to the observed UV lines which include the P Cygni profiles for low-inclination systems and pure emission at large inclination. Disk winds are preferable to spherical winds which originate on the white dwarf because they: (1) require a much lower ratio of mass-loss rate to accretion rate and are therefore more plausible energetically; (2) provide a natural source for a biconical distribution of mass outflow which produces strong scattering far above the disk leading to P Cygni profiles for low-inclination systems and pure line emission profiles at high inclination with the absence of eclipses in UV lines; and (3) produce rotation-broadened pure emission lines at high inclination.
20. UV line diagnostics of accretion disk winds in cataclysmic variables
NASA Technical Reports Server (NTRS)
Vitello, Peter; Shlosman, Isaac
1992-01-01
The IUE data base is used to analyze the UV line shapes of cataclysmic variables RW Sex, RW Tri, and V Sge. Observed lines are compared to synthetic line profiles computed using a model of rotating bi-conical winds from accretion disks. The wind model calculates the wind ionization structure self-consistently including photoionization from the disk and boundary layer and treats 3-D line radiation transfer in the Sobolev approximation. It is found that winds from accretion disks provide a good fit for reasonable parameters to the observed UV lines which include the P Cygni profiles for low inclination systems and pure emission at large inclination. Disk winds are preferable to spherical winds which originate on the white dwarf because they (1) require a much lower ratio of mass loss rate to accretion rate and are therefore more plausible energetically, (2) provide a natural source for a bi-conical distribution of mass outflow which produces strong scattering far above the disk leading to P Cygni profiles for low inclination systems, and pure line emission profiles at high inclination with the absence of eclipses in UV lines, and (3) produce rotation broadened pure emission lines at high inclination.
1. Where do Accretion Disks Around Black Holes End?
Asmus, D.; Duschl, W. J.
2010-10-01
Accretion disks around (supermassive) black holes act as "machines" which extract gravitational energy. In fact, the observed radiation allows to sample the physical conditions very close to the event horizon. For a test particle, the innermost stable circular orbit (ISCO) is located at 3 rS for a non-rotating hole (Schwarzschild metrics; at smaller radii for a rotating black hole). This ISCO is usually identified with the inner edge of the accretion disk. For a given black hole mass, it allows, in principle, to determine the Kerr parameter. In "real life," however, we deal not with test particles but with a viscous flow, which introduces additional forces. We have calculated the location of the inner edge in a more realistic environment. The results show that the true inner edge of the disk is no longer located at the ISCO, when radial advection of energy is taken into account with a careful treatment of the transonic nature of the flow.
2. Isothermal, Compton-heated coronae above accretion disks
NASA Technical Reports Server (NTRS)
Ostriker, Eve C.; Mckee, Christopher F.; Klein, Richard I.
1991-01-01
The structure of Compton-heated coronae above accretion disks is studied here by using analytic and numerical approaches are used here to determine the direct and scattered radiation reaching the base of the corona for a range of central source luminosities. It is found that the outer region of the corona is unaffected by multiple scattering in the interior, provided that the luminosity of the central source is sufficient below the Eddington limit. How attenuation and scattering by the corona affects the strength of chromospheric emission lines is determined, as is the condition for which the irradiation due to the central source exceeds the locally generated flux from the disk. Finally, it is shown that the stability analysis for irradiated accretion disks of Tuchman et al. is not substantially altered by the corona.
3. Gas accretion from halos to disks: observations, curiosities, and problems
Elmegreen, Bruce G.
2016-08-01
Accretion of gas from the cosmic web to galaxy halos and ultimately their disks is a prediction of modern cosmological models but is rarely observed directly or at the full rate expected from star formation. Here we illustrate possible large-scale cosmic HI accretion onto the nearby dwarf starburst galaxy IC10, observed with the VLA and GBT. We also suggest that cosmic accretion is the origin of sharp metallicity drops in the starburst regions of other dwarf galaxies, as observed with the 10-m GTC. Finally, we question the importance of cosmic accretion in normal dwarf irregulars, for which a recent study of their far-outer regions sees no need for, or evidence of, continuing gas buildup.
4. Magnetic reconnection process in accretion disk systems
Piovezan, P.; de Gouveia Dal Pino, E. M.
2009-08-01
At the present study, we investigate the role of magnetic reconnection in three different astrophysical systems, namely young stellar objects (YSO's), microquasars, and active galactic nuclei (AGN's). In the case of microquasars and AGN's, violent reconnection episodes between the magnetic field lines of the inner disk region (which are established by a turbulent dynamo) and those anchored into the black hole are able to heat the coronal/disk gas and accelerate particles to relativistic velocities through a diffusive first-order Fermi-like process within the reconnection site that will produce relativistic blobs. The heating of the coronal/disk gas is able to produce a steep X-ray spectrum with a luminosity that is consistent with the observations and we argue that it is being produced mainly at the foot of the reconnection zone, while the Fermi-like acceleration process within the reconnection site results a power-law electron distribution with N(E) ∝ E-α, with α=5/2, and a corresponding synchrotron radio power-law spectrum with a spectral index that is compatible with that observed during the radio flares in microquasars (Sν ∝ ν-0.75). The scaling laws that we derive for AGN's indicate that the same mechanism may be occurring there. Finally, in the case of the YSO's, a similar magnetic configuration can be reached. The amount of magnetic energy that can be extracted from the inner disk region can heat the coronal gas to temperatures of the order of 10^8 K and could explain the observed X-ray flaring emission.
5. THE PARKER INSTABILITY IN DISK GALAXIES
SciTech Connect
Rodrigues, L. F. S.; Sarson, G. R.; Shukurov, A.; Bushby, P. J.; Fletcher, A. E-mail: [email protected] E-mail: [email protected]
2016-01-01
We examine the evolution of the Parker instability in galactic disks using 3D numerical simulations. We consider a local Cartesian box section of a galactic disk, where gas, magnetic fields, and cosmic rays are all initially in a magnetohydrostatic equilibrium. This is done for different choices of initial cosmic-ray density and magnetic field. The growth rates and characteristic scales obtained from the models, as well as their dependences on the density of cosmic rays and magnetic fields, are in broad agreement with previous (linearized, ideal) analytical work. However, this nonideal instability develops a multimodal 3D structure, which cannot be quantitatively predicted from the earlier linearized studies. This 3D signature of the instability will be of importance in interpreting observations. As a preliminary step toward such interpretations, we calculate synthetic polarized intensity and Faraday rotation measure (RM) maps, and the associated structure functions of the latter, from our simulations; these suggest that the correlation scales inferred from RM maps are a possible probe for the cosmic-ray content of a given galaxy. Our calculations highlight the importance of cosmic rays in these measures, making them an essential ingredient of realistic models of the interstellar medium.
6. Accretion by rotating magnetic neutron stars. II - Radial and vertical structure of the transition zone in disk accretion
NASA Technical Reports Server (NTRS)
Ghosh, P.; Lamb, F. K.
1979-01-01
The radial and vertical structure of the transition zone at the magnetospheric boundary of an aligned rotating neutron star accreting matter from a Keplerian disk are calculated. The results obtained indicate that: (1) the inner edge of the disk is located where the integrated magnetic stress acting on the disk plasma becomes comparable to the integrated material stress associated with its inward radial drift and orbital motion; (2) the stellar magnetic field threads the disk near its inner edge via the Kelvin-Helmholtz instability, turbulent diffusion, and reconnection, producing a broad transition zone between the unperturbed disk flow and corotating magnetosphere; (3) the transition zone consists of two qualitatively different regions, viz., a broad outer transition zone where the motion is Keplerian and a narrow inner zone, or boundary layer, where the departure from Keplerian motion is substantial; (4) the stellar magnetic field is largely but not entirely screened by currents flowing in the boundary layer; and (5) there are no steady-flow solutions for sufficiently fast stellar rotation.
7. Wind-driven Accretion in Transitional Protostellar Disks
Wang, Lile; Goodman, Jeremy J.
2017-01-01
Transitional protostellar disks have inner cavities that are heavily depleted in dust and gas, yet most of them show signs of ongoing accretion, often at rates comparable to full disks. We show that recent constraints on the gas surface density in a few well-studied disk cavities suggest that the accretion speed is at least transsonic. We propose that this is the natural result of accretion driven by magnetized winds. Typical physical conditions of the gas inside these cavities are estimated for plausible X-ray and FUV radiation fields. The gas near the midplane is molecular and predominantly neutral, with a dimensionless ambipolar parameter in the right general range for wind solutions of the type developed by Königl, Wardle, and others. That is to say, the density of ions and electrons is sufficient for moderately good coupling to the magnetic field, but it is not so good that the magnetic flux needs to be dragged inward by the accreting neutrals.
8. Simulations of Accretion Disk Wind Models
Brooks, Craig L.; Yong, Suk Yee; O'Dowd, Matthew; Webster, Rachel L.; Bate, Nicholas
2016-01-01
The kinematics of the broad emission line region (BELR) in quasars is largely unknown, however there is strong evidence that outflows may be a key component. For example, in approximately 15% of quasars we observe broad, blue-shifted absorption features which may be ubiquitous based on line-of-sight arguments. We use a new mathematical description of an outflowing disk-wind with an initial rotational component to predict surface brightness distributions of this wind at different orientations. These surface brightness distributions will allow us to simulate gravitational microlensing of BELR light, with a view to mapping the structure and better understanding the kinematics of these flows.
9. Constraints on r-process nucleosynthesis in accretion disks
NASA Technical Reports Server (NTRS)
Jin, Liping
1991-01-01
Systems in which accretion drives an outflow from a region near a compact object may enrich the interstellar medium in r-process elements. A detailed assessment of the efficacy of this mechanism for the r-process is presented here, taking into account the constraints imposed by typical accretion-disk conditions. It is concluded that r-process elements are unlikely to have been made in this way, largely because the total production is too low, by a factor of about 100,000, to explain the observed abundances.
10. ACCRETION DISK DYNAMO AS THE TRIGGER FOR X-RAY BINARY STATE TRANSITIONS
SciTech Connect
Begelman, Mitchell C.; Armitage, Philip J.; Reynolds, Christopher S.
2015-08-20
Magnetohydrodynamic accretion disk simulations suggest that much of the energy liberated by the magnetorotational instability (MRI) can be channeled into large-scale toroidal magnetic fields through dynamo action. Under certain conditions, this field can dominate over gas and radiation pressure in providing vertical support against gravity, even close to the midplane. Using a simple model for the creation of this field, its buoyant rise, and its coupling to the gas, we show how disks could be driven into this magnetically dominated state and deduce the resulting vertical pressure and density profiles. Applying an established criterion for MRI to operate in the presence of a toroidal field, we show that magnetically supported disks can have two distinct MRI-active regions, separated by a “dead zone” where local MRI is suppressed, but where magnetic energy continues to flow upward from the dynamo region below. We suggest that the relative strengths of the MRI zones, and the local poloidal flux, determine the spectral states of X-ray binaries. Specifically, “intermediate” and “hard” accretion states occur when MRI is triggered in the hot, upper zone of the corona, while disks in “soft” states do not develop the upper MRI zone. We discuss the conditions under which various transitions should take place and speculate on the relationship of dynamo activity to the various types of quasi-periodic oscillations that sometimes appear in the hard spectral components. The model also explains why luminous accretion disks in the “soft” state show no signs of the thermal/viscous instability predicted by standard α-models.
11. Accretion Disk Dynamo as the Trigger for X-Ray Binary State Transitions
Begelman, Mitchell C.; Armitage, Philip J.; Reynolds, Christopher S.
2015-08-01
Magnetohydrodynamic accretion disk simulations suggest that much of the energy liberated by the magnetorotational instability (MRI) can be channeled into large-scale toroidal magnetic fields through dynamo action. Under certain conditions, this field can dominate over gas and radiation pressure in providing vertical support against gravity, even close to the midplane. Using a simple model for the creation of this field, its buoyant rise, and its coupling to the gas, we show how disks could be driven into this magnetically dominated state and deduce the resulting vertical pressure and density profiles. Applying an established criterion for MRI to operate in the presence of a toroidal field, we show that magnetically supported disks can have two distinct MRI-active regions, separated by a “dead zone” where local MRI is suppressed, but where magnetic energy continues to flow upward from the dynamo region below. We suggest that the relative strengths of the MRI zones, and the local poloidal flux, determine the spectral states of X-ray binaries. Specifically, “intermediate” and “hard” accretion states occur when MRI is triggered in the hot, upper zone of the corona, while disks in “soft” states do not develop the upper MRI zone. We discuss the conditions under which various transitions should take place and speculate on the relationship of dynamo activity to the various types of quasi-periodic oscillations that sometimes appear in the hard spectral components. The model also explains why luminous accretion disks in the “soft” state show no signs of the thermal/viscous instability predicted by standard α-models.
12. Bulk viscosity of accretion disks around non rotating black holes
2017-01-01
In this paper, we study the Keplerian, relativistic accretion disks around the non rotating black holes with the bulk viscosity. Many of authors studied the relativistic accretion disks around the black holes, but they ignored the bulk viscosity. We introduce a simple method to calculate the bulk in these disks. We use the simple form for the radial component of the four velocity in the Schwarzschild metric, then the other components of the four velocity and the components of the shear and the bulk tensor are calculated. Also all components of the bulk viscosity, the shear viscosity and stress tensor are calculated. It is seen that some components of the bulk tensor are comparable with the shear tensor. We calculate some of the thermodynamic quantities of the relativistic disks. Comparison of thermodynamic quantities shows that in some states influences of the bulk viscosity are important, especially in the inner radiuses. All calculations are done analytically and we do not use the boundary conditions. Finally, we find that in the relativistic disks around the black holes, the bulk viscosity is non-negligible in all the states.
13. The average size and temperature profile of quasar accretion disks
SciTech Connect
Jiménez-Vicente, J.; Mediavilla, E.; Muñoz, J. A.; Motta, V.; Falco, E.
2014-03-01
We use multi-wavelength microlensing measurements of a sample of 10 image pairs from 8 lensed quasars to study the structure of their accretion disks. By using spectroscopy or narrowband photometry, we have been able to remove contamination from the weakly microlensed broad emission lines, extinction, and any uncertainties in the large-scale macro magnification of the lens model. We determine a maximum likelihood estimate for the exponent of the size versus wavelength scaling (r{sub s} ∝λ {sup p}, corresponding to a disk temperature profile of T∝r {sup –1/p}) of p=0.75{sub −0.2}{sup +0.2} and a Bayesian estimate of p = 0.8 ± 0.2, which are significantly smaller than the prediction of the thin disk theory (p = 4/3). We have also obtained a maximum likelihood estimate for the average quasar accretion disk size of r{sub s}=4.5{sub −1.2}{sup +1.5} lt-day at a rest frame wavelength of λ = 1026 Å for microlenses with a mean mass of M = 1 M {sub ☉}, in agreement with previous results, and larger than expected from thin disk theory.
14. The SEDs of Gapped Accretion Disks surrounding Binary Black Holes
Gultekin, Kayhan; Miller, J. M.
2014-01-01
We calculate the observability of a black hole (BH) accretion disk with a gap or a hole created by a secondary BH embedded in the disk. We find that for an interesting range of parameters of BH masses 10^6-10^9 M⊙), orbital separation 1 AU to ~0.1 pc), and gap width (10-190 disk scale heights), the missing thermal emission from a gap manifests itself in an observable decrement in the spectral energy distribution (SED). The change in slope in the broken power law is strongly dependent on the width of the gap in the accretion disk, which in turn is uniquely determined by the mass ratio of the BHs (under our assumptions), such that it scales roughly as q^(5/12). Thus, one can use spectral observations of the continuum of bright AGNs to infer not only the presence of a closely separated BH binary, but also the mass ratio. When the BH merger opens an entire hole (or cavity) in the inner disk, the broadband SED of the AGNs or quasar may serve as a diagnostic. We note future directions for this research.
15. Magnetized Accretion and Dead Zones in Protoplanetary disks
Dzyurkevich, Natalia; Turner, Neal J.; Henning, Thomas; Kley, Wilhelm
2013-07-01
16. Super-spinning compact objects generated by thick accretion disks
SciTech Connect
Li, Zilong; Bambi, Cosimo E-mail: [email protected]
2013-03-01
If astrophysical black hole candidates are the Kerr black holes predicted by General Relativity, the value of their spin parameter must be subject to the theoretical bound |a{sub *}| ≤ 1. In this work, we consider the possibility that these objects are either non-Kerr black holes in an alternative theory of gravity or exotic compact objects in General Relativity. We study the accretion process when their accretion disk is geometrically thick with a simple version of the Polish doughnut model. The picture of the accretion process may be qualitatively different from the one around a Kerr black hole. The inner edge of the disk may not have the typical cusp on the equatorial plane any more, but there may be two cusps, respectively above and below the equatorial plane. We extend previous work on the evolution of the spin parameter and we estimate the maximum value of a{sub *} for the super-massive black hole candidates in galactic nuclei. Since measurements of the mean radiative efficiency of AGNs require η > 0.15, we infer the ''observational'' bound |a{sub *}|∼<1.3, which seems to be quite independent of the exact nature of these objects. Such a bound is only slightly weaker than |a{sub *}|∼<1.2 found in previous work for thin disks.
17. Circumstellar disks of the most vigorously accreting young stars
PubMed Central
Liu, Hauyu Baobab; Takami, Michihiro; Kudo, Tomoyuki; Hashimoto, Jun; Dong, Ruobing; Vorobyov, Eduard I.; Pyo, Tae-Soo; Fukagawa, Misato; Tamura, Motohide; Henning, Thomas; Dunham, Michael M.; Karr, Jennifer L.; Kusakabe, Nobuhiko; Tsuribe, Toru
2016-01-01
Stars may not accumulate their mass steadily, as was previously thought, but in a series of violent events manifesting themselves as sharp stellar brightening. These events can be caused by fragmentation due to gravitational instabilities in massive gaseous disks surrounding young stars, followed by migration of dense gaseous clumps onto the star. Our high-resolution near-infrared imaging has verified the presence of the key associated features, large-scale arms and arcs surrounding four young stellar objects undergoing luminous outbursts. Our hydrodynamics simulations and radiative transfer models show that these observed structures can indeed be explained by strong gravitational instabilities occurring at the beginning of the disk formation phase. The effect of those tempestuous episodes of disk evolution on star and planet formation remains to be understood. PMID:26989772
18. Circumstellar disks of the most vigorously accreting young stars.
PubMed
Liu, Hauyu Baobab; Takami, Michihiro; Kudo, Tomoyuki; Hashimoto, Jun; Dong, Ruobing; Vorobyov, Eduard I; Pyo, Tae-Soo; Fukagawa, Misato; Tamura, Motohide; Henning, Thomas; Dunham, Michael M; Karr, Jennifer L; Kusakabe, Nobuhiko; Tsuribe, Toru
2016-02-01
Stars may not accumulate their mass steadily, as was previously thought, but in a series of violent events manifesting themselves as sharp stellar brightening. These events can be caused by fragmentation due to gravitational instabilities in massive gaseous disks surrounding young stars, followed by migration of dense gaseous clumps onto the star. Our high-resolution near-infrared imaging has verified the presence of the key associated features, large-scale arms and arcs surrounding four young stellar objects undergoing luminous outbursts. Our hydrodynamics simulations and radiative transfer models show that these observed structures can indeed be explained by strong gravitational instabilities occurring at the beginning of the disk formation phase. The effect of those tempestuous episodes of disk evolution on star and planet formation remains to be understood.
19. Magnetized Accretion and Dead Zones in Protostellar Disks
Dzyurkevich, Natalia; Turner, Neal J.; Henning, Thomas; Kley, Wilhelm
2013-03-01
20. MAGNETIZED ACCRETION AND DEAD ZONES IN PROTOSTELLAR DISKS
SciTech Connect
Dzyurkevich, Natalia; Henning, Thomas; Turner, Neal J.; Kley, Wilhelm
2013-03-10
1. The frequency of accretion disks around single stars: Chamaeleon I
Daemgen, Sebastian; Elliot Meyer, R.; Jayawardhana, Ray; Petr-Gotzens, Monika G.
2016-02-01
Context. It is well known that stellar companions can influence the evolution of a protoplanetary disk. Nevertheless, previous disk surveys did not - and could not - consistently exclude binaries from their samples. Aims: We present a study dedicated to investigating the frequency of ongoing disk accretion around single stars in a star-forming region. Methods: We obtained near-infrared spectroscopy of 54 low-mass stars selected from a high-angular resolution survey in the 2-3 Myr-old Chamaeleon I region to determine the presence of Brackett-γ emission, taking the residual chance of undetected multiplicity into account, which we estimate to be on the order of 30%. The result is compared with previous surveys of the same feature in binary stars of the same region to provide a robust estimate of the difference between the accretor fractions of single stars and individual components of binary systems. Results: We find Brγ emission among 39.5+ 14.0-9.9% of single stars, which is a significantly higher fraction than for binary stars in Chamaeleon I. In particular, close binary systems with separations <100 AU show emission in only 6.5+ 16.5-3.0% of the cases according to the same analysis. The emitter frequency of wider binaries appears consistent with the single star value. Interpreting Brγ emission as a sign of ongoing accretion and correcting for sensitivity bias, we infer an accretor fraction of single stars of Facc = 47.8+ 14.0-9.9%. This is slightly higher but consistent with previous estimates that do not clearly exclude binaries from their samples. Conclusions: Through our robust and consistent analysis, we confirm that the fraction of young single stars harboring accretion disks is much larger than that of close binaries at the same age. Our findings have important implications for the timescales of disk evolution and planet formation.
2. Magneto-rotational instability in the protolunar disk
Carballido, Augusto; Desch, Steven J.; Taylor, G. Jeffrey
2016-04-01
We perform the first study of magnetohydrodynamic processes in the protolunar disk (PLD). With the use of published data on the chemical composition of the PLD, along with existing analytical models of the disk structure, we show that the high temperatures that were prevalent in the disk would have led to ionization of Na, K, SiO, Zn and, to a lesser extent, O2. For simplicity, we assume that the disk has a vapor structure. The resulting ionization fractions, together with a relatively weak magnetic field, possibly of planetary origin, would have been sufficient to trigger the magneto-rotational instability, or MRI, as demonstrated by the fact that the Elsasser criterion was met in the PLD: a magnetic field embedded in the flow would have diffused more slowly than the growth rate of the linear perturbations. We calculate the intensity of the resulting magnetohydrodynamic turbulence, as parameterized by the dimensionless ratio α of turbulent stresses to gas pressure, and obtain maximum values α ∼10-2 along most of the vertical extent of the disk, and at different orbital radii. This indicates that, under these conditions, turbulent mixing within the PLD due to the MRI was likely capable of transporting isotopic and chemical species efficiently. To test these results in a conservative manner, we carry out a numerical magnetohydrodynamic simulation of a small, rectangular patch of the PLD, located at 4 Earth radii (rE) from the center of the Earth, and assuming once again that the disk is completely gaseous. We use a polytrope-like equation of state. The rectangular patch is threaded initially by a vertical magnetic field with zero net magnetic flux. This field configuration is known to produce relatively weak MRI turbulence in studies of astrophysical accretion disks. We accordingly obtain turbulence with an average intensity α ∼ 7 ×10-6 over the course of 280 orbital periods (133 days at 4rE). Despite this relatively low value of α , the effective turbulent
3. Accretion Disks, Magnetospheres, and Disk Winds as Emitters of the Hydrogen Lines in Herbig Ae/Be Stars
Tambovtseva, L. V.; Grinin, V. P.; Weigelt, G.; Schertl, D.; Hofmann, K.-H.; Caratti o Garatti, A.; Garcia Lopez, R.
2017-02-01
Various disk and outflow components of the circumstellar environment of young Herbig Ae/Be stars may contribute to the hydrogen line emission. These are a magnetosphere, a disk wind, and a gaseous accretion disk. Non-LTE modeling was performed to show the influence of the model parameters on the intensity and the line profiles for each emitting region to present the spatial distribution of the brightness for each component and to compare their contributions to the total line emission. The modeling shows that the disk wind is the dominant contributor to the Brγ and Hα lines rather than the magnetospheric accretion and gaseous accretion disk.
4. Accretion Effects on Disks Around Non-Magnetic Compact Objects
Montgomery, Michele M.
2013-02-01
Accretion disks in compact binaries are thought to sometimes tilt and precess in the retrograde direction as indicated by modulations in light curves and/or signals. Using 3D Smoothed Particle Hydrodynamics and a low mass transfer rate, Montgomery (2012) shows the disk in non-magnetic Cataclysmic Variables tilts naturally after enough time has passed. In that work, twice the fundamental negative superhump signal 2ν_ is associated with disk tilt around the line of nodes, gas stream overflow approximately twice per orbital period, and retrograde precession. In this work, we show that after enough additional time has passed in the same simulation, the 4ν_ harmonic appears. The decrease in the 2ν_ amplitude approximately equals the amplitude of the 4ν_ harmonic. We discuss the implications.
5. The dim inner accretion disk of the quiescent black hole A0620-00
NASA Technical Reports Server (NTRS)
Mcclintock, Jeffrey E.; Horne, Keith; Remillard, Ronald A.
1995-01-01
We observed the X-ray nova A0620-00 with the Hubble Space Telescope (HST) Faint object Spectrograph 16 yr after its 1975 outburst. We present a single spectrum (1250-4750 A), which is approximately an average over a full 7.8 hr orbital cycle of the source. The continuum can be fitted approximately by a blackbody model with T = 9000 K and a small projected source area, which is approximately 1 % of the expected area of the accretion disk. AS0620-00 is faint in the far-UV band; its luminosity is comparable to the luminosity of the quiescent dwarf-nova accretion disk (i.e., excluding the white dwarf). By analogy with dwarf novae, the optical luminosity of the disk (M(sub nu) approximately = 7) and the orbital period of A0620-00 imply that the rate of mass transfer onto the outer disk in M(sub d) approximately 10(exp -10) solar mass/yr. We also observed A0620-00 with the ROSAT PSPC X-ray detector for 3 x 10(exp 4) s and detected a faint source (5 sigma) at the location of the X-ray nova. For an assumed blackbody spectrum the source temperature and luminosity are approximately 0.16 keV and 6 x 10(exp 30) ergs/s, respectively (d = 1 kpc). This luminosity implies that the rate of mass transfer into the black hole is extraordinarily small: M(sub BH) less than 5 x 10(exp -15) solar mass/yr. The much larger mass transfer rate onto the outer disk, and the UV/X-ray faintness of the inner disk confirm key predictions of the disk instability model for the nova outburst of A0620-00 published by Huang and Wheeler and by Mineshige and Wheeler.
6. Explosive magnetorotational instability in Keplerian disks
Shtemler, Yu.; Liverts, E.; Mond, M.
2016-06-01
Differentially rotating disks under the effect of axial magnetic field are prone to a nonlinear explosive magnetorotational instability (EMRI). The dynamic equations that govern the temporal evolution of the amplitudes of three weakly detuned resonantly interacting modes are derived. As distinct from exponential growth in the strict resonance triads, EMRI occurs due to the resonant interactions of an MRI mode with stable Alfvén-Coriolis and magnetosonic modes. Numerical solutions of the dynamic equations for amplitudes of a triad indicate that two types of perturbations behavior can be excited for resonance conditions: (i) EMRI which leads to infinite values of the three amplitudes within a finite time, and (ii) bounded irregular oscillations of all three amplitudes. Asymptotic explicit solutions of the dynamic equations are obtained for EMRI regimes and are shown to match the numerical solutions near the explosion time.
7. An Eccentric Accretion Disk In V691 Cra?
Peris, Charith; Vrtilek, S. D.
2011-09-01
We present phase-resolved spectroscopic observations over a full orbital period (5.6 hr) of the low-mass X-ray binary, V691 CrA, obtained with IMACS on the 6.5-m Walter Baade telescope at Las Campanas Observatory in June 2010. This is part of an on-going program to construct modulated tomograms in selected optical lines that enable us to study the geometry of the accretion flow and to examine spectral signatures that differentiate between neutron stars and black holes. The images constructed via tomography provide one of the few paths toward detailed insight into the structure of spatially unresolved accretion processes and the dependence of that structure on the nature of the central condensed object. Apparent in the V691 CrA spectrum are emission lines from H, He, and Fe with Hα and HeII 4686 showing clear double peaks varying with phase. Using K1= 94.5 km/s (Casares et al., 2010) and K2 = 324 km/s (Jonker et al 2003) we confirm a systemic velocity γ = -43 km/s (Casares et al 2003). Using these values to generate Modulation maps in Hα we find strong disk emission and a bright spot at the point where the accreting stream hits the disk. The center of the disk appears significantly offset from the center-of-mass of the system indicating an eccentric disk that may be associated with precession. We will present these results in the context of both black hole and neutron star systems observed by our project. SDV has been supported in part by NSF grant AST-0507637 awarded to the Smithsonian Astrophysical Observatory and a Smithsonian Institution Scholarly Studies Grant.
8. Search for and follow-up imaging of subparsec accretion disks in AGN
Kondratko, Paul Thomas
We report results of several large surveys for water maser emission among Active Galactic Nuclei with the 100-m Green Bank Telescope and the two NASA Deep Space Network 70-m antennas at Tidbinbilla, Australia and at Robledo, Spain. We detected 23 new sources, which resulted in a 60% increase in the number of then known nuclear water maser sources. Eight new detections show the characteristic spectral signature of emission from an edge-on accretion disk and therefore constitute good candidates for the determination of black hole mass and geometric distance. This increase in the number of known sources has enabled us to reconsider statistical properties of the resulting sample. For the 30 water maser sources with available hard X-ray data, we found a possible correlation between unabsorbed X-ray luminosity (2-10 keV) and total isotropic water maser luminosity of the form L 2-10 0([Special characters omitted.] , consistent with the model proposed by Neufeld et al. (1994) in which X-ray irradiation of molecular accretion disk gas by the central engine excites the maser emission. We mapped for the first time with Very Long Baseline Interferomatey (VLBI) the full extent of the pc-scale accretion disk in NGC 3079 as traced by water maser emission. Positions and line-of-sight velocities of maser emission are consistent with a nearly edge-on pc-scale disk and a central mass of ~ 2 x 10^6 [Special characters omitted.] enclosed within ~ 0.4 pc. Based on the kinematics of the system, we propose that the disk is geometrically-thick, massive, subject to gravitational instabilities, and hence most likely clumpy and star- forming. The accretion disk in NGC 3079 is thus markedly different from the compact, thin, warped, differentially rotating disk in the archetypal maser galaxy NGC 4258. We also detect maser emission at high latitudes above the disk and suggest that it traces an inward extension of the kpc-scale bipolar wide- angle outflow previously observed along the galactic
9. Black Hole Accretion and Feedback Driven by Thermal Instability
Gaspari, M.; Ruszkowski, M.; Oh, S. P.; Churazov, E.; Brighenti, F.; Ettori, S.; Sharma, P.; Temi, P.
2013-03-01
Multiwavelength data indicate that the cores of several galaxy clusters are moderately cooling, though not catastrophically, showing signs of filamentary extended multiphase gas. Through 3D AMR hydrodynamic simulations, we study the impact of thermal instability in the evolution of the intracluster medium. Common moderate turbulence of just over 100 km/s leads to the growth of nonlinear thermal instability within the central few tens kpc. In the presence of a global counterbalancing heating, the condensation of extended filamentary cold gas is violent, occurring when the cooling time falls below 10 times the free-fall time. The frequent stochastic collisions, fragmentations and shearing motions between the cold clouds, filaments and the central torus, efficiently reduce angular momentum. Tracking the accreting gas with a dynamical range of 10 million, we find that the accretion rate is boosted up to 100 times with respect to the Bondi rate. In a commonly turbulent and quasi-stable atmosphere, the mode of black accretion is cold and chaotic, substantially different from the classic idealized scenario. Only in the transonic regime, turbulent dissipation starts to inhibit thermal instability. On sub-parsec scales the cold phase is channeled via a funnel, triggering the black hole feedback likely linked to mechanical jets/outflows. As shown by long-term self-regulated simulations, the interplay of chaotic cold accretion and AGN feedback is crucial in order to avoid the cooling catastrophe and to reproduce the key thermodynamical features of observed clusters.
10. Iron Opacity Bump Changes the Stability and Structure of Accretion Disks in Active Galactic Nuclei
Jiang, Yan-Fei; Davis, Shane W.; Stone, James M.
2016-08-01
Accretion disks around supermassive black holes have regions where the Rosseland mean opacity can be larger than the electron scattering opacity due to the large number of bound-bound transitions in iron. We study the effects of this iron opacity “bump” on the thermal stability and vertical structure of radiation-pressure-dominated accretion disks, utilizing three-dimensional radiation magnetohydrodynamic (MHD) simulations in the local shearing box approximation. The simulations self-consistently calculate the heating due to MHD turbulence caused by magneto-rotational instability and radiative cooling by using the radiative transfer module based on a variable Eddington tensor in Athena. For a 5 × 108 solar mass black hole with ˜3% of the Eddington luminosity, a model including the iron opacity bump maintains its structure for more than 10 thermal times without showing significant signs of thermal runaway. In contrast, if only electron scattering and free-free opacity are included as in the standard thin disk model, the disk collapses on the thermal timescale. The difference is caused by a combination of (1) an anti-correlation between the total optical depth and the midplane pressure, and (2) enhanced vertical advective energy transport. These results suggest that the iron opacity bump may have a strong impact on the stability and structure of active galactic nucleus (AGN) accretion disks, and may contribute to a dependence of AGN properties on metallicity. Since this opacity is relevant primarily in UV emitting regions of the flow, it may help to explain discrepancies between observation and theory that are unique to AGNs.
11. Testing the Star-Disk Connection: CIV and MGII Maps of Accretion Disks CYC3-MED
Horne, Keith
1992-06-01
Empirical scaling laws among magnetic activity indicators are well established for the sun and other cool stars. Ground-based studies of Balmer and CaII emission suggest that similar relationships may hold for the accretion disks and tidally-locked secondary stars in cataclysmic variables. We propose to test this star-disk connection by using HST to make Doppler maps of MgII and CIV emission in three quiescent dwarf novae. These lines sensitive to chromospheric and transition region temperature regimes are predicted to scale as radius to the -3/2 and -3 respectively in the Keplerian accretion disk. Our experiment tests the hypothesis that dynamo action powers emission lines from accretion disk chromospheres. The disk and secondary star rotate much faster than the stars for which magnetic activity relations have been previously determined. By expanding the study of magnetic activity to higher rotation rates and different geometries, we expect to gain insights into the basic physics that will advance our understanding of dynamos and magnetic activity in a broad context. NOTE: THE TAC CUT THIS PROPOSAL FROM 3 TO 1 OBJECT.
12. The Standing Accretion Shock Instability: Enhanced Growth in Rotating Progenitors
Blondin, John M.; Gipson, Emily; Harris, Sawyer; Mezzacappa, Anthony
2017-02-01
We investigate the effect of progenitor rotation on the standing accretion shock instability (SASI) using two- and three-dimensional hydrodynamic simulations. We find that the growth rate of the SASI is a near-linearly increasing function of the specific angular momentum in the accreting gas. Both the growth rate and the angular frequency in the two-dimensional model with cylindrical geometry agree well with previous linear stability analyses. When excited by very small random perturbations, a one-armed spiral mode dominates the small rotation rates predicted by current stellar evolution models, while progressively higher-order modes are seen as the specific angular momentum increases.
13. Hydraulic jumps in 'viscous' accretion disks. [in astronomical models
NASA Technical Reports Server (NTRS)
Michel, F. C.
1984-01-01
It is proposed that the dissipative process necessary for rapid accretion disk evolution is driven by hydraulic jump waves on the surface of the disk. These waves are excited by the asymmetric nature of the central rotator (e.g., neutron star magnetosphere) and spiral out into the disk to form a pattern corotating with the central object. Disk matter in turn is slowed slightly at each encounter with the jump and spirals inward. In this process, the disk is heated by true turbulence produced in the jumps. Additional effects, such as a systematic misalignment of the magnetic moment of the neutron star until it is nearly orthogonal, and systematic distortion of the magnetosphere in such a way as to form an even more asymmetric central 'paddle wheel', may enhance the interaction with inflowing matter. The application to X-ray sources corresponds to the 'slow' solutions of Ghosh and Lamb, and therefore to rms magnetic fields of about 4 x 10 to the 10th gauss. Analogous phenomena have been proposed to act in the formation of galactic spiral structure.
14. ACCRETION RATES OF MOONLETS EMBEDDED IN CIRCUMPLANETARY PARTICLE DISKS
SciTech Connect
Ohtsuki, Keiji; Yasui, Yuki; Daisaka, Hiroshi
2013-08-01
We examine the gravitational capture probability of colliding particles in circumplanetary particle disks and accretion rates of small particles onto an embedded moonlet, using analytic calculation, three-body orbital integrations, and N-body simulations. Expanding our previous work, we take into account the Rayleigh distribution of particles' orbital eccentricities and inclinations in our analytic calculation and orbital integration and confirm agreement between them when the particle velocity dispersion is comparable to or larger than their mutual escape velocity and the ratio of the sum of the physical radii of colliding particles to their mutual Hill radius (r-tilde{sub p}) is much smaller than unity. As shown by our previous work, the capture probability decreases significantly when the velocity dispersion is larger than the escape velocity and/or r-tilde{sub p}{approx}>0.7. Rough surfaces of particles can enhance the capture probability. We compare the results of three-body calculations with N-body simulations for accretion of small particles by an embedded moonlet and find agreement at the initial stage of accretion. However, when particles forming an aggregate on the moonlet surface nearly fill the Hill sphere, the aggregate reaches a quasi-steady state with a nearly constant number of particles covering the moonlet, and the accretion rate is significantly reduced compared to the three-body results.
15. The Instability in Accretion Flows: GvMRI
Yardimci, Melis; Ebru Devlen, Doç.
2016-07-01
In this study, we discuss the physical instability defining the expected turbulence in Radiatively Inefficient Accretion Flows (RIAFs) around the supermassive black holes (e.g., Sagittarius A* in the center of our Galaxy). These flows, with a high probability, include weakly collisional hot, optically thin and dilute plasmas. Within these flows, gravitational potential energy brought about by turbulent stresses is trapped as heat energy. Thus, in order accretion to be realized, outward transport of heat as well as angular momentum is required. This outward heat transport may reduce the mass inflow rate on black hole. We solve MHD equations including variation of viscosity coefficients with pressure in the momentum conservation equation. We plot the wave number-frequency diagrams for the wave modes. We show that one of the most probable candidates for definition of mass accretion and the source of excess heat energy in RIAFs is the gyroviscous modified magnetorotational instabilitiy (GvMRI).
16. Luminosity limit for alpha-viscosity accretion disks
NASA Technical Reports Server (NTRS)
Liang, Edison P.; Wandel, Amri
1991-01-01
The existence of a luminosity limit for alpha-viscosity physically thin accretion disks around black holes is established, using a new formulation of the radiation equation bridging optically thick and thin regimes. For alpha close to unity, this limit can be lower than the Eddington limit. Physically, this limit is due to the combined effects of gas and radiation pressure which become too large to satisfy vertical hydrostatic balance at intermediate optical depths for sufficiently high luminosities. This effect was overlooked in previous treatments using only the optically thin or thick limits of the radiative equation.
17. On the thermal stability of radiation-dominated accretion disks
SciTech Connect
Jiang, Yan-Fei; Stone, James M.; Davis, Shane W.
2013-11-20
We study the long-term thermal stability of radiation-dominated disks in which the vertical structure is determined self-consistently by the balance of heating due to the dissipation of MHD turbulence driven by magneto-rotational instability (MRI) and cooling due to radiation emitted at the photosphere. The calculations adopt the local shearing box approximation and utilize the recently developed radiation transfer module in the Athena MHD code based on a variable Eddington tensor rather than an assumed local closure. After saturation of the MRI, in many cases the disk maintains a steady vertical structure for many thermal times. However, in every case in which the box size in the horizontal directions are at least one pressure scale height, fluctuations associated with MRI turbulence and dynamo action in the disk eventually trigger a thermal runaway that causes the disk to either expand or contract until the calculation must be terminated. During runaway, the dependence of the heating and cooling rates on total pressure satisfy the simplest criterion for classical thermal instability. We identify several physical reasons why the thermal runaway observed in our simulations differ from the standard α disk model; for example, the advection of radiation contributes a non-negligible fraction to the vertical energy flux at the largest radiation pressure, most of the dissipation does not happen in the disk mid-plane, and the change of dissipation scale height with mid-plane pressure is slower than the change of density scale height. We discuss how and why our results differ from those published previously. Such thermal runaway behavior might have important implications for interpreting temporal variability in observed systems, but fully global simulations are required to study the saturated state before detailed predictions can be made.
18. Disk-accreting magnetic neutron stars as high-energy particle accelerators
NASA Technical Reports Server (NTRS)
Hamilton, Russell J.; Lamb, Frederick K.; Miller, M. Coleman
1994-01-01
Interaction of an accretion disk with the magnetic field of a neutron star produces large electromotive forces, which drive large conduction currents in the disk-magnetosphere-star circuit. Here we argue that such large conduction currents will cause microscopic and macroscopic instabilities in the magnetosphere. If the minimum plasma density in the magnetosphere is relatively low is less than or aproximately 10(exp 9)/cu cm, current-driven micro-instabilities may cause relativistic double layers to form, producing voltage differences in excess of 10(exp 12) V and accelerating charged particles to very high energies. If instead the plasma density is higher (is greater than or approximately = 10(exp 9)/cu cm, twisting of the stellar magnetic field is likely to cause magnetic field reconnection. This reconnection will be relativistic, accelerating plasma in the magnetosphere to relativistic speeds and a small fraction of particles to very high energies. Interaction of these high-energy particles with X-rays, gamma-rays, and accreting plasma may produce detectable high-energy radiation.
19. Quiescent accretion disks in black hole X-ray novae
NASA Technical Reports Server (NTRS)
Orosz, Jerome A.; Bailyn, Charles D.; Remillard, Ronald A.; Mcclintock, Jeffrey E.; Foltz, Craig B.
1994-01-01
We present detailed time-resolved spectroscopy of the Balmer emission lines from two black hole binary systems in quiescence, A0620-00 and Nova Muscae 1991. We find extraordinary similarities between the two systems. There are 30-40 km/s velocity variations of the emission lines over the orbital period, the phases of which are not aligned with the expected phase of the motion of the compact primary. Detailed modeling of both systems is complicated by variable hot spot components, regions of optical thickness, and intermittent excess emission in the blue line wings of the H-alpha lines. Both sources also display low velocities at the outer edge of the accretion disk, implying a large primary Roche lobe and extreme mass ratios. These complications suggest that although simple optically thin, Keplerian alpha-disk models provide a useful parameterization of emission lines from these systems, the straightforward physical models they imply should be treated with great caution.
20. Crystalline structure of accretion disks: Features of a global model
Montani, Giovanni; Benini, Riccardo
2011-08-01
In this paper, we develop the analysis of a two-dimensional magnetohydrodynamical configuration for an axially symmetric and rotating plasma (embedded in a dipolelike magnetic field), modeling the structure of a thin accretion disk around a compact astrophysical object. Our study investigates the global profile of the disk plasma, in order to fix the conditions for the existence of a crystalline morphology and ring sequence, as outlined by the local analysis pursued in Coppi [Phys. PlasmasPHPAEN1070-664X10.1063/1.1883667 12, 7302 (2005)] and Coppi and Rousseau [Astrophys. J.AJLEEY0004-637X10.1086/500315 641, 458 (2006)]. In the linear regime, when the electromagnetic back-reaction of the plasma is small enough, we show the existence of an oscillating radial behavior for the flux surface function, which very closely resembles the one outlined in the local model, apart from a radial modulation of the amplitude. In the opposite limit, corresponding to a dominant back-reaction in the magnetic structure over the field of central object, we can recognize the existence of a ringlike decomposition of the disk, according to the same modulation of the magnetic flux surface, and a smoother radial decay of the disk density, with respect to the linear case. In this extreme nonlinear regime, the global model seems to predict a configuration very close to that of the local analysis, but here the thermostatic pressure, crucial for the equilibrium setting, is also radially modulated. Among the conditions requested for the validity of such a global model, the confinement of the radial coordinate within a given value sensitive to the disk temperature and to the mass of the central objet, stands; however, this condition corresponds to dealing with a thin disk configuration.
1. Chemistry in a Forming Protoplanetary Disk: Main Accretion Phase
Yoneda, Haruaki; Tsukamoto, Yusuke; Furuya, Kenji; Aikawa, Yuri
2016-12-01
We investigate the chemistry in a radiation-hydrodynamics model of a star-forming core that evolves from a cold (∼10 K) prestellar core to the main accretion phase in ∼105 years. A rotationally supported gravitationally unstable disk is formed around a protostar. We extract the temporal variation of physical parameters in ∼1.5 × 103 SPH particles that end up in the disk, and perform post-processing calculations of the gas-grain chemistry adopting a three-phase model. Inside the disk, the SPH particles migrate both inward and outward. Since a significant fraction of volatiles such as CO can be trapped in the water-dominant ice in the three-phase model, the ice mantle composition depends not only on the current position in the disk, but also on whether the dust grain has ever experienced higher temperatures than the water sublimation temperature. Stable molecules such as H2O, CH4, NH3, and CH3OH are already abundant at the onset of gravitational collapse and are simply sublimated as the fluid parcels migrate inside the water snow line. On the other hand, various molecules such as carbon chains and complex organic molecules (COMs) are formed in the disk. The COMs abundance sensitively depends on the outcomes of photodissociation and diffusion rates of photofragments in bulk ice mantle. As for S-bearing species, H2S ice is abundant in the collapse phase. In the warm regions in the disk, H2S is sublimated to be destroyed, while SO, H2CS, OCS, and SO2 become abundant.
2. Crystalline structure of accretion disks: features of a global model.
PubMed
Montani, Giovanni; Benini, Riccardo
2011-08-01
In this paper, we develop the analysis of a two-dimensional magnetohydrodynamical configuration for an axially symmetric and rotating plasma (embedded in a dipolelike magnetic field), modeling the structure of a thin accretion disk around a compact astrophysical object. Our study investigates the global profile of the disk plasma, in order to fix the conditions for the existence of a crystalline morphology and ring sequence, as outlined by the local analysis pursued in Coppi [Phys. Plasmas 12, 7302 (2005)] and Coppi and Rousseau [Astrophys. J. 641, 458 (2006)]. In the linear regime, when the electromagnetic back-reaction of the plasma is small enough, we show the existence of an oscillating radial behavior for the flux surface function, which very closely resembles the one outlined in the local model, apart from a radial modulation of the amplitude. In the opposite limit, corresponding to a dominant back-reaction in the magnetic structure over the field of central object, we can recognize the existence of a ringlike decomposition of the disk, according to the same modulation of the magnetic flux surface, and a smoother radial decay of the disk density, with respect to the linear case. In this extreme nonlinear regime, the global model seems to predict a configuration very close to that of the local analysis, but here the thermostatic pressure, crucial for the equilibrium setting, is also radially modulated. Among the conditions requested for the validity of such a global model, the confinement of the radial coordinate within a given value sensitive to the disk temperature and to the mass of the central objet, stands; however, this condition corresponds to dealing with a thin disk configuration.
3. The spectra of relativistic accretion disks - Application to A0620-00
NASA Technical Reports Server (NTRS)
Fu, Albert; Taam, Ronald E.
1990-01-01
The X-ray flux emitted from a geometrically thin, relativistic accretion disk in the steady state approximation is investigated in order to place limits on the quiescent state mass flow rate in the soft X-ray transient black hole candidate source A0620-00. Specific attention is focused on the effects associated with gravitational redshifts, Doppler shifts, and on the enhancement of the apparent accretion disk area due to gravitational light bending on the continuum spectrum. It is found that the upper limit to the mass flow rate within the inner regions of the disk, constrained by the lack of soft X-rays in the quiescent state, is about 2.8 x 10 to the -11th solar mass/yr for black hole masses greater than about 5.4 solar mass. The optical data are consistent with these upper limits provided that the inclination angle of the binary system is less than about 65 deg. The upper limits and the lack of a hard X-ray flux, together, suggest that the soft X-ray transient model based upon a mass transfer instability situated in the stellar envelope of the companion is inapplicable to A0620-00.
4. Testing Horava-Lifshitz gravity using thin accretion disk properties
SciTech Connect
Harko, Tiberiu; Kovacs, Zoltan; Lobo, Francisco S. N.
2009-08-15
Recently, a renormalizable gravity theory with higher spatial derivatives in four dimensions was proposed by Horava. The theory reduces to Einstein gravity with a nonvanishing cosmological constant in IR, but it has improved UV behaviors. The spherically symmetric black hole solutions for an arbitrary cosmological constant, which represent the generalization of the standard Schwarzschild-(anti) de Sitter solution, have also been obtained for the Horava-Lifshitz theory. The exact asymptotically flat Schwarzschild-type solution of the gravitational field equations in Horava gravity contains a quadratic increasing term, as well as the square root of a fourth order polynomial in the radial coordinate, and it depends on one arbitrary integration constant. The IR-modified Horava gravity seems to be consistent with the current observational data, but in order to test its viability more observational constraints are necessary. In the present paper we consider the possibility of observationally testing Horava gravity by using the accretion disk properties around black holes. The energy flux, the temperature distribution, the emission spectrum, as well as the energy conversion efficiency are obtained, and compared to the standard general relativistic case. Particular signatures can appear in the electromagnetic spectrum, thus leading to the possibility of directly testing Horava gravity models by using astrophysical observations of the emission spectra from accretion disks.
5. Broad band variability of SS433: accretion disk at work?
Revnivtsev, M.; Fabrika, S.; Abolmasov, P.; Postnov, K.; Bikmaev, I.; Burenin, R.; Pavlinsky, M.; Sunyaev, R.; Khamitov, I.; Sakhibullin, N.
2006-02-01
6. A High-mass Protobinary System with Spatially Resolved Circumstellar Accretion Disks and Circumbinary Disk
Kraus, S.; Kluska, J.; Kreplin, A.; Bate, M.; Harries, T. J.; Hofmann, K.-H.; Hone, E.; Monnier, J. D.; Weigelt, G.; Anugu, A.; de Wit, W. J.; Wittkowski, M.
2017-01-01
High-mass multiples might form via fragmentation of self-gravitational disks or alternative scenarios such as disk-assisted capture. However, only a few observational constraints exist on the architecture and disk structure of high-mass protobinaries and their accretion properties. Here, we report the discovery of a close (57.9 ± 0.2 mas = 170 au) high-mass protobinary, IRAS17216-3801, where our VLTI/GRAVITY+AMBER near-infrared interferometry allows us to image the circumstellar disks around the individual components with ∼3 mas resolution. We estimate the component masses to ∼20 and ∼18 M⊙ and find that the radial intensity profiles can be reproduced with an irradiated disk model, where the inner regions are excavated of dust, likely tracing the dust sublimation region in these disks. The circumstellar disks are strongly misaligned with respect to the binary separation vector, which indicates that the tidal forces did not have time to realign the disks, pointing toward a young dynamical age of the system. We constrain the distribution of the Brγ and CO-emitting gas using VLTI/GRAVITY spectro-interferometry and VLT/CRIRES spectro-astrometry and find that the secondary is accreting at a higher rate than the primary. VLT/NACO imaging shows L‧-band emission on (3–4)× larger scales than the binary separation, matching the expected dynamical truncation radius for the circumbinary disk. The IRAS17216-3801 system is ∼3× more massive and ∼5× more compact than other high-mass multiplies imaged at infrared wavelength and the first high-mass protobinary system where circumstellar and circumbinary dust disks could be spatially resolved. This opens exciting new opportunities for studying star–disk interactions and the role of multiplicity in high-mass star formation. Based on observations made with ESO telescopes at Paranal Observatory under program IDs 60.A-9174(A), 089.C-0819(A,C), 089.C-0959(D,E), 094.C-0153(A), 096.C-0652(A).
7. GLOBAL PROPERTIES OF FULLY CONVECTIVE ACCRETION DISKS FROM LOCAL SIMULATIONS
SciTech Connect
Bodo, G.; Ponzo, F.; Rossi, P.; Cattaneo, F.; Mignone, A.
2015-08-01
We present an approach to deriving global properties of accretion disks from the knowledge of local solutions derived from numerical simulations based on the shearing box approximation. The approach consists of a two-step procedure. First, a local solution valid for all values of the disk height is constructed by piecing together an interior solution obtained numerically with an analytical exterior radiative solution. The matching is obtained by assuming hydrostatic balance and radiative equilibrium. Although in principle the procedure can be carried out in general, it simplifies considerably when the interior solution is fully convective. In these cases, the construction is analogous to the derivation of the Hayashi tracks for protostars. The second step consists of piecing together the local solutions at different radii to obtain a global solution. Here we use the symmetry of the solutions with respect to the defining dimensionless numbers—in a way similar to the use of homology relations in stellar structure theory—to obtain the scaling properties of the various disk quantities with radius.
8. Reverberation Mapping of Accretion Disk Winds in Active Galactic Nuclei
Mangham, S.
2015-09-01
Reverberation mapping is commonly used for determining black holes masses in AGN from the delayed response of the Broad Line Region (BLR) to fluctuations in the intensity of the AGN continuum source. However, it can also be an effective tool for investigating the structure and kinematics of the BLR itself. Much prior work has been performed to simulate the transfer functions associated with a range of basic geometries (e.g. Keplerian disks, Hubble-like outflows, etc). One promising model for the BLR is that the emission lines are formed in an equatorial accretion disk wind. Here, we predict the reverberation signatures expected from such a model, by modifying the radiative transfer and ionisation code Python that has previously been used to model broad absorption line quasars. This allows to account self-consistently for ionization and radiative transfer effects in the predicted BLR response, which are normally ignored in such calculations. We discuss the agreement between our results and prior work and consider the possibility of detecting the signature of rotating equatorial disk winds in observations obtained by velocity-resolved reverberation mapping campaigns.
9. Accretion disk coronae in high-luminosity systems
NASA Technical Reports Server (NTRS)
Murray, Stephen D.; Castor, John I.; Klein, Richard I.; Mckee, Christopher F.
1994-01-01
We present the results of self-consistent models of Compton-heated accretion disk coronae. The models are calculated using a new method for computing monochromatic radiative transfer n two dimensions. The method splits the radiation into direct and scattered components. The direct radiation is computed by calculating the optical depth along rays, while transfer of the scattered radiation is approximated by flux-limited diffusion. The resulting code agrees with more accurate treatments to within 50%, and is highly efficient, making it practical for use in large hydrodynamic simulations. The coronal models are used to confirm the results of earlier work, and to extend it to higher luminosities. In contrast to earlier work, which found the outer disks to be shadowed by the inner corona at high luminosities, we find our results to form an almost continuous extension of the models at lower luminosities. This is due to the presence of multiply scattered radiation, which acts to partially offset the loss of direct radiation from the central source. Although the analytic methods derived at lower luminosities cannot be used to derive the coronal structure for L/L(sub Edd) approx. greater than 0.1, the results of the models are amenable to semiempirical fits. We also discuss possible observational consequences of the results for coronal veiling and line fluorescence from the disk.
10. PATCHY ACCRETION DISKS IN ULTRA-LUMINOUS X-RAY SOURCES
SciTech Connect
Miller, J. M.; Bachetti, M.; Barret, D.; Webb, N. A.; Harrison, F. A.; Walton, D. J.; Rana, V.; Fabian, A. C.
2014-04-10
The X-ray spectra of the most extreme ultra-luminous X-ray sources—those with L ≥ 10{sup 40} erg s{sup –1}—remain something of a mystery. Spectral roll-over in the 5-10 keV band was originally detected in the deepest XMM-Newton observations of the brightest sources; this is confirmed in subsequent NuSTAR spectra. This emission can be modeled via Comptonization, but with low electron temperatures (kT{sub e} ≅ 2 keV) and high optical depths (τ ≅ 10) that pose numerous difficulties. Moreover, evidence of cooler thermal emission that can be fit with thin disk models persists, even in fits to joint XMM-Newton and NuSTAR observations. Using NGC 1313 X-1 as a test case, we show that a patchy disk with a multiple temperature profile may provide an excellent description of such spectra. In principle, a number of patches within a cool disk might emit over a range of temperatures, but the data only require a two-temperature profile plus standard Comptonization, or three distinct blackbody components. A mechanism such as the photon bubble instability may naturally give rise to a patchy disk profile, and could give rise to super-Eddington luminosities. It is possible, then, that a patchy disk (rather than a disk with a standard single-temperature profile) might be a hallmark of accretion disks close to or above the Eddington limit. We discuss further tests of this picture and potential implications for sources such as narrow-line Seyfert-1 galaxies and other low-mass active galactic nuclei.
11. ON THE ROLE OF THE ACCRETION DISK IN BLACK HOLE DISK-JET CONNECTIONS
SciTech Connect
Miller, J. M.; Reis, R. C.; Pooley, G. G.; Fabian, A. C.; Cackett, E. M.; Nowak, M. A.; Pottschmidt, K.; Wilms, J.
2012-09-20
Models of jet production in black hole systems suggest that the properties of the accretion disk-such as its mass accretion rate, inner radius, and emergent magnetic field-should drive and modulate the production of relativistic jets. Stellar-mass black holes in the 'low/hard' state are an excellent laboratory in which to study disk-jet connections, but few coordinated observations are made using spectrometers that can incisively probe the inner disk. We report on a series of 20 Suzaku observations of Cygnus X-1 made in the jet-producing low/hard state. Contemporaneous radio monitoring was done using the Arcminute MicroKelvin Array radio telescope. Two important and simple results are obtained: (1) the jet (as traced by radio flux) does not appear to be modulated by changes in the inner radius of the accretion disk and (2) the jet is sensitive to disk properties, including its flux, temperature, and ionization. Some more complex results may reveal aspects of a coupled disk-corona-jet system. A positive correlation between the reflected X-ray flux and radio flux may represent specific support for a plasma ejection model of the corona, wherein the base of a jet produces hard X-ray emission. Within the framework of the plasma ejection model, the spectra suggest a jet base with v/c {approx_equal} 0.3 or the escape velocity for a vertical height of z {approx_equal} 20 GM/c {sup 2} above the black hole. The detailed results of X-ray disk continuum and reflection modeling also suggest a height of z {approx_equal} 20 GM/c {sup 2} for hard X-ray production above a black hole, with a spin in the range 0.6 {<=} a {<=} 0.99. This height agrees with X-ray time lags recently found in Cygnus X-1. The overall picture that emerges from this study is broadly consistent with some jet-focused models for black hole spectral energy distributions in which a relativistic plasma is accelerated at z = 10-100 GM/c {sup 2}. We discuss these results in the context of disk-jet connections
12. A global three-dimensional radiation magneto-hydrodynamic simulation of super-eddington accretion disks
SciTech Connect
Jiang, Yan-Fei; Stone, James M.; Davis, Shane W.
2014-12-01
13. Using High Speed Rotating Gas to Study Angular Momentum in Accretion Disks
Berrios, William; Greess, Samuel; Merino, Enrique; Ji, Hantao
2013-10-01
Accretion disks are a sheet of gas and dust which surrounds black holes and quasars. The angular momentum in accretion disks is one of the biggest mysteries in astrophysics. A machine was recently built to create accretion disks in a closed chamber. In order to study this, there are several important instruments that are used: a fog machine to see the accretion disks form within the chamber, a high speed camera to observe and record the formation of the accretion disks, and Particle Image Velocimetry (PIV) to analyze velocity profile of the rotating gas and better understand this phenomenon. By collecting relevant data and subsequent computational analysis, results from a previous experiment are reproduced, expanded and the new properties observed with this experiment are characterized. A discussion of any modifications done to the machine, technical challenges and preliminary results will be presented.
14. The vertical structure and stability of accretion disks surrounding black holes and neutron stars
NASA Technical Reports Server (NTRS)
Milsom, J. A.; Chen, Xingming; Taam, Ronald E.
1994-01-01
The structure and stability of the inner regions of accretion disks surrounding neutron stars and black holes have been investigated. Within the framework of the alpha viscosity prescription for optically thick disks, we assume the viscous stress scales with gas pressure only, and the alpha parameter, which is less than or equal to unity, is formulated as alpha(sub 0)(h/r)(exp n), where h is the local scale height and n and alpha(sub 0) are constants. We neglect advective energy transport associated with radial motions and construct the vertical structure of the disks by assuming a Keplerian rotation law and local hydrostatic and thermal equilibrium. The vertical structures have been calculated with and without convective energy transport, and it has been demonstrated that convection is important especially for mass accretion rates, M-dot, greater than about 0.1 times the Eddington value, M-dot(sub Edd). Although the efficiency of convection is not high, convection significantly modifies the vertical structure of the disk (as compared with a purely radiative model) and leads to lower temperatures at a given M-dot. The results show that the disk can be locally unstable and that for n greater than or = 0.75, an S-shaped relation can exist between M-dot and the column density, sigma, at a given radius. While the lower stable branch (derivative of M-dot/derivative of sigma greater than 0) and middle unstable branch (derivative of M-dot/derivative of sigma less than 0) represent structures for which the gas and radiation pressure dominate respectively, the stable upper branch (derivative of M-dot/derivative of sigma greater than 0) is a consequence of the saturation of alpha. This saturation of alpha can occur for large alpha(sub 0) and at M-dot less than or = M-dot(sub Edd). The instability is found to occur at higher mass accretion rates for neutron stars than for black holes. In particular, the disk is locally unstable for M-dot greater than or = 0.5 M-dot(sub Edd
15. Spectral energy distributions of T Tauri stars - Disk flaring and limits on accretion
NASA Technical Reports Server (NTRS)
Kenyon, S. J.; Hartmann, L.
1987-01-01
The Adams et al. (1987) conclusion that much of the IR excess emission in the spectral energy distribution of T Tauri stars arises from reprocessing of stellar radiation by a dusty circumstellar disk is presently supported by analyses conducted in light of various models of these stars' spectra. A low mass reprocessing disk can, however, produce these spectra as well as a massive accretion disk. The detection of possible boundary layer radiation in the optical and near-UV regions poses the strongest limits on accretion rates. Disk accretion in the T Tauri phase does not significantly modify stellar evolution.
16. You’re Cut Off: HD and MHD Simulations of Truncated Accretion Disks
Hogg, J. Drew; Reynolds, Christopher S.
2017-01-01
Truncated accretion disks are commonly invoked to explain the spectro-temporal variability from accreting black holes in both small systems, i.e. state transitions in galactic black hole binaries (GBHBs), and large systems, i.e. low-luminosity active galactic nuclei (LLAGNs). In the canonical truncated disk model of moderately low accretion rate systems, gas in the inner region of the accretion disk occupies a hot, radiatively inefficient phase, which leads to a geometrically thick disk, while the gas in the outer region occupies a cooler, radiatively efficient phase that resides in the standard geometrically thin disk. Observationally, there is strong empirical evidence to support this phenomenological model, but a detailed understanding of the disk behavior is lacking. We present well-resolved hydrodynamic (HD) and magnetohydrodynamic (MHD) numerical models that use a toy cooling prescription to produce the first sustained truncated accretion disks. Using these simulations, we study the dynamics, angular momentum transport, and energetics of a truncated disk in the two different regimes. We compare the behaviors of the HD and MHD disks and emphasize the need to incorporate a full MHD treatment in any discussion of truncated accretion disk evolution.
17. Effects of Accretion Disks on Spins and Eccentricities of Binaries, and Implications for Gravitational Waves
NASA Technical Reports Server (NTRS)
Baker, John
2012-01-01
Effects of accretion disks on spins and eccentricities of binaries, and implications for gravitational waves. John Baker Space-based gravitational wave observations will allow exquisitely precise measurements of massive black hole binary properties. Through several recently suggested processes, these properties may depend on interactions with accretion disks through the merger process. I will discuss ways that accretion may influence those binary properties which may be probed by gravitational-wave observations.
18. Evidence for accretion disks in highly polarized quasars
NASA Technical Reports Server (NTRS)
Smith, Paul S.; Elston, Richard; Berriman, Graham; Allen, Richard G.; Balonek, Thomas J.
1988-01-01
The results of a search for thermal components in 11 highly polarized quasars (HPQs) using UVBRI polarimetry and photometry are reported. The 2000-2500 A luminosities of the thermal components are calculated and the estimated luminosities of the broad-line region (BLR) are given in the same wavelength for comparison. The observed optical continua are modeled as a combination of polarized synchrotron emission, unpolarized emission from the BLR, and an unpolarized flat spectral component that may be optically thick thermal emission from an accretion disk. Evidence for thermal emission components is found in three HPQs: PKS 0420-014, B2 1156+295, and 3C 454.3, with marginal evidence in another two, PKS 1510-089 and PKS 2345-167.
19. Laboratory Study of Angular Momentum Transport in Astrophysical Accretion Disks
Ji, Hantao
2014-10-01
Studying astrophysical processes in the lab becomes increasingly possible and exciting, as one of Stirling's favorite subjects throughout his scientific career. In this talk, I will describe experimental efforts to study mechanisms of rapid angular momentum transport required to occur in accretion disks to explain a wide range of phenomena from star formation, energetic activity of cataclysmic variables, to powering quasars, the most luminous steady sources in the Universe. By carefully isolating effects due to artificial boundaries, which are inherent to terrestrial experiments, certain astrophysical questions regarding hydrodynamic and magnetohydrodynamic stabilities are being addressed in the laboratory. Inspirations from Stirling as well as scientific exchanges with him will be mentioned during this talk as part of my scientific journey on this subject.
20. Magnetic Field Generation by the Stationary Accretion Shock Instability
SciTech Connect
Endeve, Eirik; Cardall, Christian Y; Budiardja, R. D.; Mezzacappa, Anthony
2008-01-01
By adding a weak magnetic field to a spherically symmetric fluid configuration that caricatures a stalled shock in the post-bounce supernova environment, we explore the capacity of the stationary accretion shock instability (SASI) to generate magnetic fields. The SASI develops upon perturbation of the initial condition, and the ensuing flow generates - in the absence of rotation - dynamically significant magnetic fields (~ 10^{15} G) on a time scale that is relevant for the explosion mechanism of core collapse supernovae. We describe our model, present some recent results, and discuss their potential relevance for supernova models.
1. ON THE DOUBLE NATURED SOLUTIONS OF THE TWO-TEMPERATURE EXTERNAL SOFT PHOTON COMPTONIZED ACCRETION DISKS
SciTech Connect
Meirelles Filho, Cesar
2009-08-01
We have analyzed pair production in the innermost region of a two-temperature external soft photon Comptonized accretion disk. We have shown that, if the viscosity parameter is greater than a critical value {alpha}{sub c}, the solution to the disk equation is double valued: one, advection dominated, and the other, radiation dominated. When {alpha} {<=} {alpha}{sub c}, the accretion rate has to satisfy m-dot{sub 1}{<=}m-dot{<=}m-dot{sub c} in order to have two steady-state solutions. It is shown that these critical parameters m-dot{sub 1}, m-dot{sub c} are functions of r, {alpha}, and {theta}{sub e}, and {alpha}{sub c} is a function of r and {theta}{sub e}. Depending on the combination of the parameters, the advection-dominated solution may not be physically consistent. It is also shown that the electronic temperature is maximum at the onset of the thermal instability, from which results this inner region. These solutions are stable against perturbations in the electron temperature and in the density of pairs.
2. Numerical simulation of the disk dynamics around the black hole: Bondi-Hoyle accretion
Koyuncu, Fahrettin; Dönmez, Orhan
2014-06-01
We have solved the General Relativistic Hydrodynamic (GRH) equations using the high resolution shock capturing scheme (HRSCS) to find out the dependency of the disk dynamics to the Mach number, adiabatic index, the black hole rotation parameter and the outer boundary of the computational domain around the non-rotating and rotating black holes. We inject the gas to computational domain at upstream and downstream regions at the same time with different initial conditions. It is found that variety of the mass accretion rates and shock cone structures strongly depend on Mach number and adiabatic index of the gas. The shock cones on the accretion disk are important physical mechanisms to trap existing oscillation modes, thereupon these trapped modes may generate strong X-rays observed by different X-ray satellites. Besides, our numerical approach also show that the shock cones produces the flip-flop oscillation around the black holes. The flip-flop instabilities which are monitored in our simulations may explain the erratic spin behavior of the compact objects (the black holes and neutron stars) seen from observed data.
3. Black hole accretion disks - Electrodynamic coupling of accretion-disk coronae and the partitioning of soft and hard X-ray emission
NASA Technical Reports Server (NTRS)
Kuperus, M.; Ionson, J. A.
1985-01-01
It is demonstrated that the observed large ratio of hard to soft X-ray emission and the bimodel behavior of black hole accreting X-ray sources such as Cygnus X-1 can be described in terms of a magnetically structured accretion disk corona which is electrodynamically coupled to the disk turbulent motions while the disk is thermodynamically coupled to the corona as described by a feedback parameter delta. The observed ratio of hard to soft X-ray emission is independent of the disk thickness, and weakly dependent of the disk parameter alpha relating the disk viscous stresses to the total pressure. Observed values of the luminosity ratio point towards strong differences of the feedback of the low state compared to the high state, in the sense that low state means small feedback (delta less than 0.2) and high state means strong feedback delta of about 0.5.
4. NUMERICAL SIMULATIONS OF NATURALLY TILTED, RETROGRADELY PRECESSING, NODAL SUPERHUMPING ACCRETION DISKS
SciTech Connect
Montgomery, M. M.
2012-02-15
Accretion disks around black hole, neutron star, and white dwarf systems are thought to sometimes tilt, retrogradely precess, and produce hump-shaped modulations in light curves that have a period shorter than the orbital period. Although artificially rotating numerically simulated accretion disks out of the orbital plane and around the line of nodes generate these short-period superhumps and retrograde precession of the disk, no numerical code to date has been shown to produce a disk tilt naturally. In this work, we report the first naturally tilted disk in non-magnetic cataclysmic variables using three-dimensional smoothed particle hydrodynamics. Our simulations show that after many hundreds of orbital periods, the disk has tilted on its own and this disk tilt is without the aid of radiation sources or magnetic fields. As the system orbits, the accretion stream strikes the bright spot (which is on the rim of the tilted disk) and flows over and under the disk on different flow paths. These different flow paths suggest the lift force as a source to disk tilt. Our results confirm the disk shape, disk structure, and negative superhump period and support the source to disk tilt, source to retrograde precession, and location associated with X-ray and He II emission from the disk as suggested in previous works. Our results identify the fundamental negative superhump frequency as the indicator of disk tilt around the line of nodes.
5. Do Accretion Disks Exist in High Energy Astrophysics?
Coppi, B.
2006-10-01
The familiar concept of an accretion disk is based on its gas dynamic description where, in particular, the vertical equilibrium is maintained by the (weak) vertical component of the gravitational force due to the central object. When a plasma structure differentially rotating around the same kind of object is considered in which the magnetic field diffusion due to finite resistivity is realistically weak, a radially periodic sequence of pairs of opposite current channels is found. Moreover, the vertical confinement of the structure is maintained by the resulting Lorentz force rather than by gravity. Thus, a Lorentz compression'' occurs. In addition, sequences of plasma rings^2 rather than disks emerge. (Note that H. Alfvén had proposed that planetary rings may be fossils'' of pre- existing envisioned plasma rings. Moreover, a large ring is the most prominent feature emerging from the high resolution X- ray image of the Crab). The seed'' magnetic field in which the structure is immersed is considerably smaller than that produced by the internal toroidal currents. The magnetic pressure is of the order of the plasma pressure. Thus, ring sequence configurations can be suitable for the emergence of a jet from their center. Two coupled non-linear equations have been solved, representing the vertical and the horizontal equilibrium conditions for the structure.*Sponsored in part by the U.S. D.O.E. B. Coppi, Phys. Plasmas 12, 057301, (2005) B. Coppi and F. Rousseau, Ap. J. 641 (1), 458 (2006)
6. The intrinsic quasar luminosity function: Accounting for accretion disk anisotropy
SciTech Connect
DiPompeo, M. A.; Myers, A. D.; Brotherton, M. S.; Runnoe, J. C.; Green, R. F.
2014-05-20
Quasar luminosity functions are a fundamental probe of the growth and evolution of supermassive black holes. Measuring the intrinsic luminosity function is difficult in practice, due to a multitude of observational and systematic effects. As sample sizes increase and measurement errors drop, characterizing the systematic effects is becoming more important. It is well known that the continuum emission from the accretion disk of quasars is anisotropic—in part due to its disk-like structure—but current luminosity function calculations effectively assume isotropy over the range of unobscured lines of sight. Here, we provide the first steps in characterizing the effect of random quasar orientations and simple models of anisotropy on observed luminosity functions. We find that the effect of orientation is not insignificant and exceeds other potential corrections such as those from gravitational lensing of foreground structures. We argue that current observational constraints may overestimate the intrinsic luminosity function by as much as a factor of ∼2 on the bright end. This has implications for models of quasars and their role in the universe, such as quasars' contribution to cosmological backgrounds.
7. Smearing of mass accretion rate variation by viscous processes in accretion disks in compact binary systems
Ghosh, A.; Chakrabarti, Sandip K.
2016-09-01
Variation of mass supply rate from the companion can be smeared out by viscous processes inside an accretion disk. Hence, by the time the flow reaches the inner edge, the variation in X-rays need not reflect the true variation of the mass supply rate at the outer edge. However, if the viscosity fluctuates around a mean value, one would expect the viscous time scale t_{{visc}} also to spread around a mean value. In high mass X-ray binaries, which are thought to be primarily wind-fed, the size of the viscous Keplerian disk is smaller and thus such a spread could be lower as compared to the low mass X-ray binaries which are primarily fed by Roche lobe overflow. If there is an increasing or decreasing trend in viscosity, the interval between enhanced emission would be modified systematically. In the absence of a detailed knowledge about the variation of mass supply rates at the outer edge, we study ideal circumstances where modulation must take place exactly in orbital time scales, such as when there is an ellipticity in the orbit. We study a few compact binaries using long term All Sky monitor (ASM) data (1.5-12 keV) of Rossi X-ray Timing Explorer (RXTE) and all sky survey data (15-50 keV) of Swift satellites by different methods to look for such smearing effects and to infer what these results can tell us about the viscous processes inside the respective disks. We employ three different methods to seek imprints of periodicity on the X-ray variation and found that in all the cases, the location of the peak in the power density spectra is consistent with the orbital frequencies. Interestingly, in high mass X-ray binaries the peaks are sharp with high rms values, consistent with a small Keplerian disk in a wind fed system. However, in low mass X-ray binaries with larger Keplerian disk component, the peaks are spreaded out with much lower rms values. X-ray reflections, or superhump phenomena which may also cause such X-ray modulations would not be affected by the size of
8. Accretion Disk Lifetimes and Stellar Rotation Periods for Young Stars in NGC 2264
Makidon, R. B.; Strom, S. E.; Tingley, B.; Adams, M. T.; Hillenbrand, L.; Hartmann, L.; Calvet, N.; Jones, B. F.
1997-12-01
We present the initial results of a study aimed at: (1) determining the lifetime of the disk accretion phase among low mass pre-main sequence stars; (2) establishing the time dependence of disk mass accretion rates; and (3) further exploring the role played by accretion disks in regulating stellar rotation. Our laboratory for this study is NGC 2264, a young cluster which contains more than 300 proper motion members with ages ranging from 0.1 to 10 Myr and masses ranging from 0.1 to 10 Msun. We diagnose the presence of circumstellar accretion disks from observed ultraviolet excesses, estimate accretion rates from the magnitude of those excesses, and determine stellar rotation periods for more than 200 stars from the analysis of spot-modulated I-band light curves. We find for PMS stars with masses M <= 0.4 Msun: (1) that accretion disk lifetimes can exceed 10 Myr; (2) that accretion rates decay with time (dM/dt ~ M(-n) ; 0.9 < n < 2); and (3) that disks appear to play a critical role in regulating stellar rotation periods. In particular, PMS stars stars surrounded by accretion disks on average rotate more slowly than their counterparts which show no evidence of such disks: the median rotation period for stars surrounded by disks is 7.91 days, while for stars which lack disks the median period is 3.97 days. However, our results suggest the range of periods (0.5 < P < 30 days) among stars surrounded by disks is considerably larger than reported in previous studies. The authors would like to thank Dr. Brian Patten for his many contributions to this project. This work was supported by a grant awarded under the NASA Origins of Solar Systems Program.
9. TURBULENCE AND STEADY FLOWS IN THREE-DIMENSIONAL GLOBAL STRATIFIED MAGNETOHYDRODYNAMIC SIMULATIONS OF ACCRETION DISKS
SciTech Connect
Flock, M.; Dzyurkevich, N.; Klahr, H.; Turner, N. J.; Henning, Th.
2011-07-10
We present full 2{pi} global three-dimensional stratified magnetohydrodynamic (MHD) simulations of accretion disks. We interpret our results in the context of protoplanetary disks. We investigate the turbulence driven by the magnetorotational instability (MRI) using the PLUTO Godunov code in spherical coordinates with the accurate and robust HLLD Riemann solver. We follow the turbulence for more than 1500 orbits at the innermost radius of the domain to measure the overall strength of turbulent motions and the detailed accretion flow pattern. We find that regions within two scale heights of the midplane have a turbulent Mach number of about 0.1 and a magnetic pressure two to three orders of magnitude less than the gas pressure, while in those outside three scale heights the magnetic pressure equals or exceeds the gas pressure and the turbulence is transonic, leading to large density fluctuations. The strongest large-scale density disturbances are spiral density waves, and the strongest of these waves has m = 5. No clear meridional circulation appears in the calculations because fluctuating radial pressure gradients lead to changes in the orbital frequency, comparable in importance to the stress gradients that drive the meridional flows in viscous models. The net mass flow rate is well reproduced by a viscous model using the mean stress distribution taken from the MHD calculation. The strength of the mean turbulent magnetic field is inversely proportional to the radius, so the fields are approximately force-free on the largest scales. Consequently, the accretion stress falls off as the inverse square of the radius.
10. Modeling X-ray Absorbers in AGNs with MHD-Driven Accretion-Disk Winds
Fukumura, Keigo; Kazanas, D.; Shrader, C. R.; Tombesi, F.; Contopoulos, J.; Behar, E.
2013-04-01
We have proposed a systematic view of the observed X-ray absorbers, namely warm absorbers (WAs) in soft X-ray and highly-ionized ultra-fast outflows (UFOs), in the context of magnetically-driven accretion-disk wind models. While potentially complicated by variability and thermal instability in these energetic outflows, in this simplistic model we have calculated 2D kinematic field as well as density and ionization structure of the wind with density profile of 1/r corresponding to a constant column distribution per decade of ionization parameter. In particular we show semi-analytically that the inner layer of the disk-wind manifests itself as the strongly-ionized fast outflows while the outer layer is identified as the moderately-ionized absorbers. The computed characteristics of these two apparently distinct absorbers are consistent with X-ray data (i.e. a factor of ~100 difference in column and ionization parameters as well as low wind velocity vs. near-relativistic flow). With the predicted contour curves for these wind parameters one can constrain allowed regions for the presence of WAs and UFOs.The model further implies that the UFO's gas pressure is comparable to that of the observed radio jet in 3C111 suggesting that the magnetized disk-wind with density profile of 1/r is a viable agent to help sustain such a self-collimated jet at small radii.
11. Magnetorotational dynamo chimeras. The missing link to turbulent accretion disk dynamo models?
Riols, A.; Rincon, F.; Cossu, C.; Lesur, G.; Ogilvie, G. I.; Longaretti, P.-Y.
2017-02-01
In Keplerian accretion disks, turbulence and magnetic fields may be jointly excited through a subcritical dynamo mechanisminvolving magnetorotational instability (MRI). This dynamo may notably contribute to explaining the time-variability of various accreting systems, as high-resolution simulations of MRI dynamo turbulence exhibit statistical self-organization into large-scale cyclic dynamics. However, understanding the physics underlying these statistical states and assessing their exact astrophysical relevance is theoretically challenging. The study of simple periodic nonlinear MRI dynamo solutions has recently proven useful in this respect, and has highlighted the role of turbulent magnetic diffusion in the seeming impossibility of a dynamo at low magnetic Prandtl number (Pm), a common regime in disks. Arguably though, these simple laminar structures may not be fully representative of the complex, statistically self-organized states expected in astrophysical regimes. Here, we aim at closing this seeming discrepancy by reporting the numerical discovery of exactly periodic, yet semi-statistical "chimeral MRI dynamo states" which are the organized outcome of a succession of MRI-unstable, non-axisymmetric dynamical stages of different forms and amplitudes. Interestingly, these states, while reminiscent of the statistical complexity of turbulent simulations, involve the same physical principles as simpler laminar cycles, and their analysis further confirms the theory that subcritical turbulent magnetic diffusion impedes the sustainment of an MRI dynamo at low Pm. Overall, chimera dynamo cycles therefore offer an unprecedented dual physical and statistical perspective on dynamos in rotating shear flows, which may prove useful in devising more accurate, yet intuitive mean-field models of time-dependent turbulent disk dynamos. Movies associated to Fig. 1 are available at http://www.aanda.org
12. STRONG FIELD EFFECTS ON EMISSION LINE PROFILES: KERR BLACK HOLES AND WARPED ACCRETION DISKS
SciTech Connect
Wang Yan; Li Xiangdong
2012-01-10
If an accretion disk around a black hole is illuminated by hard X-rays from non-thermal coronae, fluorescent iron lines will be emitted from the inner region of the accretion disk. The emission line profiles will show a variety of strong field effects, which may be used as a probe of the spin parameter of the black hole and the structure of the accretion disk. In this paper, we generalize the previous relativistic line profile models by including both the black hole spinning effects and the non-axisymmetries of warped accretion disks. Our results show different features from the conventional calculations for either a flat disk around a Kerr black hole or a warped disk around a Schwarzschild black hole by presenting, at the same time, multiple peaks, rather long red tails, and time variations of line profiles with the precession of the disk. We show disk images as seen by a distant observer, which are distorted by the strong gravity. Although we are primarily concerned with the iron K-shell lines in this paper, the calculation is general and is valid for any emission lines produced from a warped accretion disk around a black hole.
13. Convection causes enhanced magnetic turbulence in accretion disks in outburst
SciTech Connect
Hirose, Shigenobu; Blaes, Omer; Coleman, Matthew S. B.; Krolik, Julian H.; Sano, Takayoshi
2014-05-20
We present the results of local, vertically stratified, radiation magnetohydrodynamic (MHD) shearing box simulations of magneto-rotational instability (MRI) turbulence appropriate for the hydrogen ionizing regime of dwarf nova and soft X-ray transient outbursts. We incorporate the frequency-integrated opacities and equation of state for this regime, but neglect non-ideal MHD effects and surface irradiation, and do not impose net vertical magnetic flux. We find two stable thermal equilibrium tracks in the effective temperature versus surface mass density plane, in qualitative agreement with the S-curve picture of the standard disk instability model. We find that the large opacity at temperatures near 10{sup 4} K, a corollary of the hydrogen ionization transition, triggers strong, intermittent thermal convection on the upper stable branch. This convection strengthens the magnetic turbulent dynamo and greatly enhances the time-averaged value of the stress to thermal pressure ratio α, possibly by generating vertical magnetic field that may seed the axisymmetric MRI, and by increasing cooling so that the pressure does not rise in proportion to the turbulent dissipation. These enhanced stress to pressure ratios may alleviate the order of magnitude discrepancy between the α-values observationally inferred in the outburst state and those that have been measured from previous local numerical simulations of magnetorotational turbulence that lack net vertical magnetic flux.
14. Disk-mediated accretion burst in a high-mass young stellar object
Caratti O Garatti, A.; Stecklum, B.; Garcia Lopez, R.; Eislöffel, J.; Ray, T. P.; Sanna, A.; Cesaroni, R.; Walmsley, C. M.; Oudmaijer, R. D.; de Wit, W. J.; Moscadelli, L.; Greiner, J.; Krabbe, A.; Fischer, C.; Klein, R.; Ibañez, J. M.
2016-11-01
Solar-mass stars form via disk-mediated accretion. Recent findings indicate that this process is probably episodic in the form of accretion bursts, possibly caused by disk fragmentation. Although it cannot be ruled out that high-mass young stellar objects arise from the coalescence of their low-mass brethren, the latest results suggest that they more likely form via disks. It follows that disk-mediated accretion bursts should occur. Here we report on the discovery of the first disk-mediated accretion burst from a roughly twenty-solar-mass high-mass young stellar object. Our near-infrared images show the brightening of the central source and its outflow cavities. Near-infrared spectroscopy reveals emission lines typical for accretion bursts in low-mass protostars, but orders of magnitude more luminous. Moreover, the released energy and the inferred mass-accretion rate are also orders of magnitude larger. Our results identify disk-accretion as the common mechanism of star formation across the entire stellar mass spectrum.
15. AN OBSERVED LINK BETWEEN ACTIVE GALACTIC NUCLEI AND VIOLENT DISK INSTABILITIES IN HIGH-REDSHIFT GALAXIES
SciTech Connect
Bournaud, Frederic; Juneau, Stephanie; Le Floc'h, Emeric; Mullaney, James; Daddi, Emanuele; Duc, Pierre-Alain; Elbaz, David; Salmi, Fadia; Dekel, Avishai; Dickinson, Mark
2012-09-20
We provide evidence for a correlation between the presence of giant clumps and the occurrence of active galactic nuclei (AGNs) in disk galaxies. Giant clumps of 10{sup 8}-10{sup 9} M{sub Sun} arise from violent gravitational instability in gas-rich galaxies, and it has been proposed that this instability could feed supermassive black holes (BHs). We use emission line diagnostics to compare a sample of 14 clumpy (unstable) disks and a sample of 13 smoother (stable) disks at redshift z {approx} 0.7. The majority of clumpy disks in our sample have a high probability of containing AGNs. Their [O III] {lambda}5007 emission line is strongly excited, inconsistent with low-metallicity star formation (SF) alone. [Ne III] {lambda}3869 excitation is also higher. Stable disks rarely have such properties. Stacking ultra sensitive Chandra observations (4 Ms) reveals an X-ray excess in clumpy galaxies, which confirms the presence of AGNs. The clumpy galaxies in our intermediate-redshift sample have properties typical of gas-rich disk galaxies rather than mergers, being in particular on the main sequence of SF. This suggests that our findings apply to the physically similar and numerous gas-rich unstable disks at z > 1. Using the observed [O III] and X-ray luminosities, we conservatively estimate that AGNs hosted by clumpy disks have typical bolometric luminosities of the order of a few 10{sup 43} erg s{sup -1}, BH growth rates m-dot{sub BH}{approx}10{sup -2} M{sub Sun} yr{sup -1}, and that these AGNs are substantially obscured in X-rays. This moderate-luminosity mode could provide a large fraction of today's BH mass with a high duty cycle (>10%), accretion bursts with higher luminosities being possible over shorter phases. Violent instabilities at high redshift (giant clumps) are a much more efficient driver of BH growth than the weak instabilities in nearby spirals (bars), and the evolution of disk instabilities with mass and redshift could explain the simultaneous downsizing of
16. The Hall Instability of Weakly Ionized, Radially Stratified, Rotating Disks
Liverts, Edward; Mond, Michael; Chernin, Arthur D.
2007-09-01
Cool weakly ionized gaseous rotating disks are considered by many models to be the origin of the evolution of protoplanetary clouds. Instabilities against perturbations in such disks play an important role in the theory of the formation of stars and planets. Thus, a hierarchy of successive fragmentations into smaller and smaller pieces as a part of the Kant-Laplace theory of formation of the planetary system remains valid also for contemporary cosmogony. Traditionally, axisymmetric magnetohydrodynamic (MHD) and, recently, Hall-MHD instabilities have been thoroughly studied as providers of an efficient mechanism for radial transfer of angular momentum and of radial density stratification. In the current work, the Hall instability against nonaxisymmetric perturbations in compressible rotating fluid in external magnetic field is proposed as a viable mechanism for the azimuthal fragmentation of the protoplanetary disk and, thus, perhaps initiates the road to planet formation. The Hall instability is excited due to the combined effect of the radial stratification of the disk and the Hall electric field, and its growth rate is of the order of the rotation period. This family of instabilities is introduced here for the first time in an astrophysical context.
17. Connecting Clump Sizes in Turbulent Disk Galaxies to Instability Theory
Fisher, David B.; Glazebrook, Karl; Abraham, Roberto G.; Damjanov, Ivana; White, Heidi A.; Obreschkow, Danail; Basset, Robert; Bekiaris, Georgios; Wisnioski, Emily; Green, Andy; Bolatto, Alberto D.
2017-04-01
In this letter we study the mean sizes of Hα clumps in turbulent disk galaxies relative to kinematics, gas fractions, and Toomre Q. We use ∼100 pc resolution HST images, IFU kinematics, and gas fractions of a sample of rare, nearby turbulent disks with properties closely matched to z∼ 1.5{--}2 main-sequence galaxies (the DYNAMO sample). We find linear correlations of normalized mean clump sizes with both the gas fraction and the velocity dispersion-to-rotation velocity ratio of the host galaxy. We show that these correlations are consistent with predictions derived from a model of instabilities in a self-gravitating disk (the so-called “violent disk instability model”). We also observe, using a two-fluid model for Q, a correlation between the size of clumps and self-gravity-driven unstable regions. These results are most consistent with the hypothesis that massive star-forming clumps in turbulent disks are the result of instabilities in self-gravitating gas-rich disks, and therefore provide a direct connection between resolved clump sizes and this in situ mechanism.
18. The Destruction of Thin Stellar Disks Via Cosmologically Common Satellite Accretion Events
Purcell, Chris W.; Kazantzidis, Stelios; Bullock, James S.
2009-04-01
Most Galaxy-sized systems (M host sime 1012 M sun) in the ΛCDM cosmology are expected to have interacted with at least one satellite with a total mass M sat sime 1011 M sun sime 3M disk in the past 8 Gyr. Analytic and numerical investigations suggest that this is the most precarious type of accretion for the survival of thin galactic disks because more massive accretion events are relatively rare and less massive ones preserve thin disk components. We use high-resolution, dissipationless N-body simulations to study the response of an initially thin, fully formed Milky Way-type stellar disk to these cosmologically common satellite accretion events, and show that the thin disk does not survive. Regardless of orbital configuration, the impacts transform the disks into structures that are roughly three times as thick and more than twice as kinematically hot as the observed dominant thin disk component of the Milky Way. We conclude that if the Galactic thin disk is a representative case, then the presence of a stabilizing gas component is the only recourse for explaining the preponderance of disk galaxies in a ΛCDM universe; otherwise, the disk of the Milky Way must be uncommonly cold and thin for its luminosity, perhaps as a consequence of an unusually quiescent accretion history.
19. NEUTRINO SPECTRA FROM ACCRETION DISKS: NEUTRINO GENERAL RELATIVISTIC EFFECTS AND THE CONSEQUENCES FOR NUCLEOSYNTHESIS
SciTech Connect
Caballero, O. L.; McLaughlin, G. C.; Surman, R. E-mail: [email protected] E-mail: [email protected]
2012-02-01
Black hole (BH) accretion disks have been proposed as good candidates for a range of interesting nucleosynthesis, including the r-process. The presence of the BH influences the neutrino fluxes and affects the nucleosynthesis resulting from the interaction of the emitted neutrinos and hot outflowing material ejected from the disk. We study the impact of general relativistic effects on the neutrinos emitted from BH accretion disks. We present abundances obtained by considering null geodesics and energy shifts for two different disk models. We find that both the bending of the neutrino trajectories and the energy shifts have important consequences for the nucleosynthetic outcome.
20. Workshop on Physics of Accretion Disks Around Compact and Young Stars
NASA Technical Reports Server (NTRS)
Liang, E (Editor); Stepinski, T. F. (Editor)
1995-01-01
The purpose of the two-day Workshop on Physics of Accretion Disks Around Compact and Young Stars was to bring together workers on accretion disks in the western Gulf region (Texas and Louisiana). Part 2 presents the workshop program, a list of poster presentations, and a list of workshop participants. Accretion disks are believed to surround many stars. Some of these disks form around compact stars, such as white dwarfs, neutron stars, or black holes that are members of binary systems and reveal themselves as a power source, especially in the x-ray and gamma regions of the spectrum. On the other hand, protostellar disks are believed to be accretion disks associated with young, pre-main-sequence stars and manifest themselves mostly in infrared and radio observations. These disks are considered to be a natural outcome of the star formation process. The focus of this workshop included theory and observations relevant to accretion disks around compact objects and newly forming stars, with the primary purpose of bringing the two communities together for intellectual cross-fertilization. The nature of the workshop was exploratory, to see how much interaction is possible between distinct communities and to better realize the local potential in this subject. A critical workshop activity was identification and documentation of key issues that are of mutual interest to both communities.
1. Instability and transition in rotating disk flow
NASA Technical Reports Server (NTRS)
Malik, M. R.
1981-01-01
The stability of three dimensional rotating disk flow and the effects of Coriolis forces and streamline curvature were investigated. It was shown that this analysis gives better growth rates than Orr-Sommerfeld equation. Results support the numerical prediction that the number of stationary vortices varies directly with the Reynolds number.
2. Numerical simulation of the Hall effect in magnetized accretion disks with the Pluto code
Nakhaei, Mohammad; Safaei, Ghasem; Abbassi, Shahram
2014-01-01
We investigate the Hall effect in a standard magnetized accretion disk which is accompanied by dissipation due to viscosity and magnetic resistivity. By considering an initial magnetic field, using the PLUTO code, we perform a numerical magnetohydrodynamic simulation in order to study the effect of Hall diffusion on the physical structure of the disk. Current density and temperature of the disk are significantly modified by Hall diffusion, but the global structure of the disk is not substantially affected. The changes in the current densities and temperature of the disk lead to a modification in the disk luminosity and radiation.
3. The structure and appearance of winds from supercritical accretion disks. I - Numerical models
NASA Technical Reports Server (NTRS)
Meier, D. L.
1979-01-01
Equations for the structure and appearance of supercritical accretion disks and the radiation-driven winds which emanate from them are derived and solved by a steady-state hydrodynamic computer code with a relaxation technique used in stellar structure problems. The present model takes into account the mass of the accreting star, the total accretion rate, a generalization of the disk alpha parameter which accounts for heating by processes in addition to viscosity, and the ratio of the total luminosity to the Eddington luminosity. Solutions indicate that for accretion onto a hard-surfaced star, steady, optically thick winds result for even slightly supercritical accretion, and the object will appear as a supergiant star with a high mass loss rate and a nonblackbody spectrum. Winds from black hole accretion disks are expected to depend on the form of the accretion interior to the critical radius, possibly consisting of no ejection at all, a wind similar to that of a hard-surfaced star, or a column of material ejected from a hole in the accretion disk.
4. The connection of standard thin disk with advection-dominated accretion flow
Lin, Yi-qing; Lu, Ju-fu; G. U., Wei-min
2005-04-01
Using the standard Runge-Kutta method, a global solution of the basic equations describing black hole accretion flows is derived. It is proved that transition from a standard thin disk to an advection-dominated accretion flow is realizable in case of high viscosity, without introducing any additional mechanism of energy transfer or specifying any ad hoc outer boundary condition.
5. ACCRETION OF GAS ONTO GAP-OPENING PLANETS AND CIRCUMPLANETARY FLOW STRUCTURE IN MAGNETIZED TURBULENT DISKS
SciTech Connect
Uribe, A. L.; Klahr, H.; Henning, Th.
2013-06-01
We have performed three-dimensional magnetohydrodynamical simulations of stellar accretion disks, using the PLUTO code, and studied the accretion of gas onto a Jupiter-mass planet and the structure of the circumplanetary gas flow after opening a gap in the disk. We compare our results with simulations of laminar, yet viscous disks with different levels of an {alpha}-type viscosity. In all cases, we find that the accretion flow across the surface of the Hill sphere of the planet is not spherically or azimuthally symmetric, and is predominantly restricted to the mid-plane region of the disk. Even in the turbulent case, we find no significant vertical flow of mass into the Hill sphere. The outer parts of the circumplanetary disk are shown to rotate significantly below Keplerian speed, independent of viscosity, while the circumplanetary disk density (therefore the angular momentum) increases with viscosity. For a simulation of a magnetized turbulent disk, where the global averaged alpha stress is {alpha}{sub MHD} = 10{sup -3}, we find the accretion rate onto the planet to be M-dot {approx}2 Multiplication-Sign 10{sup -6}M{sub J} yr{sup -1} for a gap surface density of 12 g cm{sup -2}. This is about a third of the accretion rate obtained in a laminar viscous simulation with equivalent {alpha} parameter.
6. The applicability of the viscous α-parameterization of gravitational instability in circumstellar disks
Vorobyov, E. I.
2010-01-01
We study numerically the applicability of the effective-viscosity approach for simulating the effect of gravitational instability (GI) in disks of young stellar objects with different disk-to-star mass ratios ξ . We adopt two α-parameterizations for the effective viscosity based on Lin and Pringle [Lin, D.N.C., Pringle, J.E., 1990. ApJ 358, 515] and Kratter et al. [Kratter, K.M., Matzner, Ch.D., Krumholz, M.R., 2008. ApJ 681, 375] and compare the resultant disk structure, disk and stellar masses, and mass accretion rates with those obtained directly from numerical simulations of self-gravitating disks around low-mass (M∗ ∼ 1.0M⊙) protostars. We find that the effective viscosity can, in principle, simulate the effect of GI in stellar systems with ξ≲ 0.2- 0.3 , thus corroborating a similar conclusion by Lodato and Rice [Lodato, G., Rice, W.K.M., 2004. MNRAS 351, 630] that was based on a different α-parameterization. In particular, the Kratter et al.'s α-parameterization has proven superior to that of Lin and Pringle's, because the success of the latter depends crucially on the proper choice of the α-parameter. However, the α-parameterization generally fails in stellar systems with ξ≳ 0.3 , particularly in the Classes 0 and I phases of stellar evolution, yielding too small stellar masses and too large disk-to-star mass ratios. In addition, the time-averaged mass accretion rates onto the star are underestimated in the early disk evolution and greatly overestimated in the late evolution. The failure of the α-parameterization in the case of large ξ is caused by a growing strength of low-order spiral modes in massive disks. Only in the late Class II phase, when the magnitude of spiral modes diminishes and the mode-to-mode interaction ensues, may the effective viscosity be used to simulate the effect of GI in stellar systems with ξ≳ 0.3 . A simple modification of the effective viscosity that takes into account disk fragmentation can somewhat improve
7. Evolution of dynamo-generated magnetic fields in accretion disks around compact and young stars
NASA Technical Reports Server (NTRS)
Stepinski, Tomasz F.
1994-01-01
Geometrically thin, optically thick, turbulent accretion disks are believed to surround many stars. Some of them are the compact components of close binaries, while the others are throught to be T Tauri stars. These accretion disks must be magnetized objects because the accreted matter, whether it comes from the companion star (binaries) or from a collapsing molecular cloud core (single young stars), carries an embedded magnetic field. In addition, most accretion disks are hot and turbulent, thus meeting the condition for the MHD turbulent dynamo to maintain and amplify any seed field magnetic field. In fact, for a disk's magnetic field to persist long enough in comparison with the disk viscous time it must be contemporaneously regenerated because the characteristic diffusion time of a magnetic field is typically much shorter than a disk's viscous time. This is true for most thin accretion disks. Consequently, studying magentic fields in thin disks is usually synonymous with studying magnetic dynamos, a fact that is not commonly recognized in the literature. Progress in studying the structure of many accretion disks was achieved mainly because most disks can be regarded as two-dimensional flows in which vertical and radial structures are largely decoupled. By analogy, in a thin disk, one may expect that vertical and radial structures of the magnetic field are decoupled because the magnetic field diffuses more rapidly to the vertical boundary of the disk than along the radius. Thus, an asymptotic method, called an adiabatic approximation, can be applied to accretion disk dynamo. We can represent the solution to the dynamo equation in the form B = Q(r)b(r,z), where Q(r) describes the field distribution along the radius, while the field distribution across the disk is included in the vector function b, which parametrically depends on r and is normalized by the condition max (b(z)) = 1. The field distribution across the disk is established rapidly, while the radial
8. THE STRUCTURE OF THE ACCRETION DISK IN THE ACCRETION DISK CORONA X-RAY BINARY 4U 1822-371 AT OPTICAL AND ULTRAVIOLET WAVELENGTHS
SciTech Connect
Bayless, Amanda J.; Robinson, Edward L.; Cornell, Mark E.; Hynes, Robert I.; Ashcraft, Teresa A.
2010-01-20
The eclipsing low-mass X-ray binary 4U 1822-371 is the prototypical accretion disk corona (ADC) system. We have obtained new time-resolved UV spectroscopy of 4U 1822-371 with the Advanced Camera for Surveys/Solar Blind Channel on the Hubble Space Telescope and new V- and J-band photometry with the 1.3 m SMARTS telescope at Cerro Tololo Inter-American Observatory. We use the new data to construct its UV/optical spectral energy distribution and its orbital light curve in the UV, V, and J bands. We derive an improved ephemeris for the optical eclipses and confirm that the orbital period is changing rapidly, indicating extremely high rates of mass flow in the system, and we show that the accretion disk in the system has a strong wind with projected velocities up to 4000 km s{sup -1}. We show that the disk has a vertically extended, optically thick component at optical wavelengths. This component extends almost to the edge of the disk and has a height equal to approx0.5 of the disk radius. As it has a low brightness temperature, we identify it as the optically thick base of a disk wind, not as the optical counterpart of the ADC. Like previous models of 4U 1822-371, ours needs a tall obscuring wall near the edge of the accretion disk, but we interpret the wall as a layer of cooler material at the base of the disk wind, not as a tall, luminous disk rim.
9. Angular momentum regulation in low-mass young stars surrounded by accretion disks
NASA Technical Reports Server (NTRS)
Edwards, Suzan; Strom, Stephen E.; Hartigan, Patrick; Strom, Karen M.; Hillenbrand, Lynne A.; Herbst, William; Attridge, Joanne; Merrill, K. M.; Probst, Ron; Gatley, Ian
1993-01-01
From study of a sample of 34 T Tauri stars with photometrically derived rotation periods and spectral types later than KS, we find that the observed periods appear to be related to the presence or absence of an accretion disk. Those stars which we infer to be surrounded by accretion disks have rotation periods P(rot) over 4 days with a most probable P(rot) of about 8.5 days, while those stars which lack accretion disk signatures cover a wide range of P(rot) from 1.5 to 16 days, including a significant number of objects with P(rot) less than 4 days. This suggests the possibility that the 'initial' angular momentum of a star is not established until it dissipates its circumstellar accretion disk. During the disk accretion phase, the stellar angular velocity appears to be regulated at a low value, countering the tendency of the star to spin up both from contraction toward the main sequence and from the accretion of inner disk material of high specific angular momentum. When the accretion disk is dissipated, this regulation mechanism will cease to function. At this point, the star is no longer maintained at a low angular velocity, but is 'free' to conserve its angular momentum, and thus to increase its angular velocity in response to contraction and changes in moment of inertia. This hypothesis, combined with a spread in disk dispersal time scales, provides a context for explaining the observed distribution of stellar rotational velocities for stars on the ZAMS in young clusters.
10. Lunar volatile depletion due to incomplete accretion within an impact-generated disk
Canup, Robin M.; Visscher, Channon; Salmon, Julien; Fegley, Bruce, Jr.
2015-12-01
The Moon may have formed from an Earth-orbiting disk of vapour and melt produced by a giant impact. The mantles of the Moon and Earth have similar compositions. However, it is unclear why lunar samples are more depleted in volatile elements than terrestrial mantle rocks, given that an evaporative escape mechanism seems inconsistent with expected disk conditions. Dynamical models suggest that the Moon initially accreted from the outermost disk, but later acquired up to 60% of its mass from melt originating from the inner disk. Here we combine dynamical, thermal and chemical models to show that volatile depletion in the Moon can be explained by preferential accretion of volatile-rich melt in the inner disk to the Earth, rather than to the growing Moon. Melt in the inner disk is initially hot and volatile poor, but volatiles condense as the disk cools. In our simulations, the delivery of inner disk melt to the Moon effectively ceases when gravitational interactions cause the Moon’s orbit to expand away from the disk, and this termination of lunar accretion occurs before condensation of potassium and more volatile elements. Thus, the portion of the Moon derived from the inner disk is expected to be volatile depleted. We suggest that this mechanism may explain part or all of the Moon’s volatile depletion, depending on the degree of mixing within the lunar interior.
11. Self-collimated electromagnetic jets from magnetized accretion disks - The even-symmetry case
NASA Technical Reports Server (NTRS)
Wang, J. C. L.; Sulkanen, M. E.; Lovelace, R. V. E.
1990-01-01
This paper extends the previous treatment (Lovelace et al., 1987) of the origin of self-collimated EM jets to the case of even field symmetry, where the magnetic flux function Psi(r, z) is an even function of z. A viscous resistive accretion disk is assumed to surround a black hole with a force-free plasma outside of the disk. Inside the disk, the induction equation is solved for Psi(r, z) and the toroidal magnetic field. Outside the disk, previous results are used to study the formation of self-collimated EM jets. In contrast with the odd-symmetry case, for even symmetry the toroidal magnetic field acts to vertically compress the disk; a comparatively large toroidal magnetic field can exist inside the disk; and an appreciable fraction (possibly all) of the available accretion power can go into the jets.
12. Disk instability and the time-dependent X-ray emission from the intermediate polar GK Persei
NASA Technical Reports Server (NTRS)
Yi, Insu; Kim, Soon-Wook; Vishniac, Ethan T.; Wheeler, J. C.
1992-01-01
The correlation between the disk instability model for the 1981-1989 optical outbursts of the intermediate polar GK Per and the accompanying X-ray emission is examined, and the self-consistency of the combined optical-X-ray model is investigated. Special attention is given to the nature of the transition in the X-ray emission due to the time-dependent accretion rates in the simple column accretion model. The large variation in the efficiency of hard X-ray production is explained.
13. A search for the lasts gasps of disk accretion in Orion T Tauri stars
Clark, Catherine; Briceno, Cesar; Calvet, Nuria; Hernandez, Jesus
2017-01-01
Using the echelle mode of the Michigan/Magellan Fiber System (M2FS) on the Magellan/Clay telescope at Las Campanas Observatory, we obtained high resolution spectra (R~35000) of a sample of ~4 - 10 Myr old T Tauri stars distributed in ten 0.5 deg diameter fields in the Orion OB1 association.We present here a search for accretion signatures among a sample of weak-line T Tauri stars (WTTS). These are young stars that on the basis of their classification in low-resolution spectra, are assumed to lack a primordial disk and therefore should not be actively accreting. We look for signatures of disk accretion at modest or low levels by measuring the width at 10% height of the H-alpha profile, and looking for a redshifted absorption feature. In parallel, we determine which WTTS among the M2FS sample have infrared excesses indicating a circumstellar disk, to see which disk-bearing WTTS also show indications of accretion. We propose that such WTTS accreting at low levels are T Tauri stars at or nearing the end of their accretion phase. Our goal is to build a large sample of these objects so that we can place statistical contraints on how long the accretion phase lasts in solar-like and low-mass stars.
14. On the effects of tidal interaction on thin accretion disks: An analytic study
NASA Technical Reports Server (NTRS)
Dgani, R.; Livio, M.; Regev, O.
1994-01-01
We calculate tidal effects on two-dimensional thin accretion disks in binary systems. We apply a perturbation expansion to obtain an analytic solution of the tidally induced waves. We obtain spiral waves that are stronger at the inner parts of the disks, in addition to a local disturbance which scales like the strength of the local tidal force. Our results agree with recent calculations of the linear response of the disk to tidal interaction.
15. CONSTRAINTS ON COMPTON-THICK WINDS FROM BLACK HOLE ACCRETION DISKS: CAN WE SEE THE INNER DISK?
SciTech Connect
Reynolds, Christopher S.
2012-11-01
Strong evidence is emerging that winds can be driven from the central regions of accretion disks in both active galactic nuclei and Galactic black hole binaries. Direct evidence for highly ionized, Compton-thin inner-disk winds comes from observations of blueshifted (v {approx} 0.05-0.1c) iron-K X-ray absorption lines. However, it has been suggested that the inner regions of black hole accretion disks can also drive Compton-thick winds-such winds would enshroud the inner disk, preventing us from seeing direct signatures of the accretion disk (i.e., the photospheric thermal emission, or the Doppler/gravitationally broadened iron K{alpha} line). Here, we show that, provided the source is sub-Eddington, the well-established wind-driving mechanisms fail to launch a Compton-thick wind from the inner disk. For the accelerated region of the wind to be Compton-thick, the momentum carried in the wind must exceed the available photon momentum by a factor of at least 2/{lambda}, where {lambda} is the Eddington ratio of the source, ruling out radiative acceleration unless the source is very close to the Eddington limit. Compton-thick winds also carry large mass fluxes, and a consideration of the connections between the wind and the disk shows this to be incompatible with magneto-centrifugal driving. Finally, thermal driving of the wind is ruled out on the basis of the large Compton radii that typify black hole systems. In the absence of some new acceleration mechanisms, we conclude that the inner regions of sub-Eddington accretion disks around black holes are indeed naked.
16. A New Paradigm for Gamma Ray Bursts: Long Term Accretion Rate Modulation by an External Accretion Disk
NASA Technical Reports Server (NTRS)
Cannizzo, John; Gehrels, Neil
2009-01-01
We present a new way of looking at the very long term evolution of GRBs in which the disk of material surrounding the putative black hole powering the GRB jet modulates the mass flow, and hence the efficacy of the process that extracts rotational energy from the black hole and inner accretion disk. The pre-Swift paradigm of achromatic, shallow-to-steep "breaks" in the long term GRB light curves has not been borne out by detailed Swift data amassed in the past several years. We argue that, given the initial existence of a fall-back disk near the progenitor, an unavoidable consequence will be the formation of an "external disk" whose outer edge continually moves to larger radii due to angular momentum transport and lack of a confining torque. The mass reservoir at large radii moves outward with time and gives a natural power law decay to the GRB light curves. In this model, the different canonical power law decay segments in the GRB identified by Zhang et al. and Nousek et al. represent different physical states of the accretion disk. We identify a physical disk state with each power law segment.
17. Global Simulations of Dynamo and Magnetorotational Instability in Madison Plasma Experiments and Astrophysical Disks
SciTech Connect
Ebrahimi, Fatima
2014-07-31
Large-scale magnetic fields have been observed in widely different types of astrophysical objects. These magnetic fields are believed to be caused by the so-called dynamo effect. Could a large-scale magnetic field grow out of turbulence (i.e. the alpha dynamo effect)? How could the topological properties and the complexity of magnetic field as a global quantity, the so called magnetic helicity, be important in the dynamo effect? In addition to understanding the dynamo mechanism in astrophysical accretion disks, anomalous angular momentum transport has also been a longstanding problem in accretion disks and laboratory plasmas. To investigate both dynamo and momentum transport, we have performed both numerical modeling of laboratory experiments that are intended to simulate nature and modeling of configurations with direct relevance to astrophysical disks. Our simulations use fluid approximations (Magnetohydrodynamics - MHD model), where plasma is treated as a single fluid, or two fluids, in the presence of electromagnetic forces. Our major physics objective is to study the possibility of magnetic field generation (so called MRI small-scale and large-scale dynamos) and its role in Magneto-rotational Instability (MRI) saturation through nonlinear simulations in both MHD and Hall regimes.
18. WAVE-VORTEX MODE COUPLING IN ASTROPHYSICAL ACCRETION DISKS UNDER COMBINED RADIAL AND VERTICAL STRATIFICATION
SciTech Connect
Salhi, A.; Lehner, T.; Godeferd, F.; Cambon, C.
2013-07-10
We examine accretion disk flow under combined radial and vertical stratification utilizing a local Cartesian (or ''shearing box'') approximation. We investigate both axisymmetric and nonaxisymmetric disturbances with the Boussinesq approximation. Under axisymmetric disturbances, a new dispersion relation is derived. It reduces to the Solberg-Hoieland criterion in the case without vertical stratification. It shows that, asymptotically, stable radial and vertical stratification cannot induce any linear instability; Keplerian flow is accordingly stable. Previous investigations strongly suggest that the so-called bypass concept of turbulence (i.e., that fine-tuned disturbances of any inviscid smooth shear flow can reach arbitrarily large transient growth) can also be applied to Keplerian disks. We present an analysis of this process for three-dimensional plane-wave disturbances comoving with the shear flow of a general rotating shear flow under combined stable radial and vertical rotation. We demonstrate that large transient growth occurs for K{sub 2}/k{sub 1} >> 1 and k{sub 3} = 0 or k{sub 1} {approx} k{sub 3}, where k{sub 1}, K{sub 2}, and k{sub 3} are the azimuthal, radial, and vertical components of the initial wave vector, respectively. By using a generalized ''wave-vortex'' decomposition of the disturbance, we show that the large transient energy growth in a Keplerian disk is mainly generated by the transient dynamics of the vortex mode. The analysis of the power spectrum of total (kinetic+potential) energy in the azimuthal or vertical directions shows that the contribution coming from the vortex mode is dominant at large scales, while the contribution coming from the wave mode is important at small scales. These findings may be confirmed by appropriate numerical simulations in the high Reynolds number regime.
19. Integrated accretion disk angular momentum removal and astrophysical jet acceleration mechanism
Bellan, Paul
2015-11-01
A model has been developed for how accretion disks discard angular momentum while powering astrophysical jets. The model depends on the extremely weak ionization of disks. This causes disk ions to be collisionally locked to adjacent disk neutrals so a clump of disk ions and neutrals has an effective cyclotron frequency αωci where α is the fractional ionization. When αωci is approximately twice the Kepler orbital frequency, conservation of canonical momentum shows that the clump spirals radially inwards producing a radially inward disk electric current as electrons cannot move radially in the disk. Upon reaching the jet radius, this current then flows axially away from the disk plane along the jet, producing a toroidal magnetic field that drives the jet. Electrons remain frozen to poloidal flux surfaces everywhere and electron motion on flux surfaces in the ideal MHD region outside the disk completes the current path. Angular momentum absorbed from accreting material in the disk by magnetic counter-torque -JrBz is transported by the electric circuit and ejected at near infinite radius in the disk plane. This is like an electric generator absorbing angular momentum and wired to a distant electric motor that emits angular momentum. Supported by USDOE/NSF Partnership in Plasma Science.
20. Generation of Magnetic Fields by the Stationary Accretion Shock Instability
SciTech Connect
Endeve, Eirik; Cardall, Christian Y; Budiardja, R. D.; Mezzacappa, Anthony
2010-01-01
We begin an exploration of the capacity of the stationary accretion shock instability (SASI) to generate magnetic fields by adding a weak, stationary, and radial (but bipolar) magnetic field, and in some cases rotation, to an initially spherically symmetric fluid configuration that models a stalled shock in the post-bounce supernova environment. In axisymmetric simulations we find that cycles of latitudinal flows into and radial flows out of the polar regions amplify the field parallel to the symmetry axis, typically increasing the total magnetic energy by about two orders of magnitude. Nonaxisymmetric calculations result in fundamentally different flows and a larger magnetic energy increase: shearing associated with the SASI spiral mode contributes to a widespread and turbulent field amplification mechanism, boosting the magnetic energy by almost four orders of magnitude (a result which remains very sensitive to the spatial resolution of the numerical simulations). While the SASI may contribute to neutron star magnetization, these simulations do not show qualitatively new features in the global evolution of the shock as a result of SASI-induced magnetic field amplification.
1. Magnetic Field Roles in Black-Holes Accretion Disk's Structure
2016-09-01
We study several factors which play remarkable roles in vertical structure and dynamics of hot accretion flows around black holes. These factors are large-scale magnetic field, thermal conduction, outflow and self-gravity. We consider an axisymmetric, rotating, steady viscous-resistive hot accretion flows.
2. Binary Black Hole Accretion from a Circumbinary Disk: Gas Dynamics inside the Central Cavity
Farris, Brian D.; Duffell, Paul; MacFadyen, Andrew I.; Haiman, Zoltan
2014-03-01
We present the results of two-dimensional (2D) hydrodynamical simulations of circumbinary disk accretion using the finite-volume code DISCO. This code solves the 2D viscous Navier-Stokes equations on a high-resolution moving mesh which shears with the fluid flow, greatly reducing advection errors in comparison with a fixed grid. We perform a series of simulations for binary mass ratios in the range 0.026 <= q <= 1.0, each lasting longer than a viscous time so that we reach a quasi-steady accretion state. In each case, we find that gas is efficiently stripped from the inner edge of the circumbinary disk and enters the cavity along accretion streams, which feed persistent "mini disks" surrounding each black hole. We find that for q >~ 0.1, the binary excites eccentricity in the inner region of the circumbinary disk, creating an overdense lump which gives rise to enhanced periodicity in the accretion rate. The dependence of the periodicity on mass ratio may provide a method for observationally inferring mass ratios from measurements of the accretion rate. We also find that for all mass ratios studied, the magnitude of the accretion onto the secondary is sufficient to drive the binary toward larger mass ratio. This suggests a mechanism for biasing mass-ratio distributions toward equal mass.
3. Binary black hole accretion from a circumbinary disk: Gas dynamics inside the central cavity
SciTech Connect
Farris, Brian D.; Duffell, Paul; MacFadyen, Andrew I.; Haiman, Zoltan
2014-03-10
We present the results of two-dimensional (2D) hydrodynamical simulations of circumbinary disk accretion using the finite-volume code DISCO. This code solves the 2D viscous Navier-Stokes equations on a high-resolution moving mesh which shears with the fluid flow, greatly reducing advection errors in comparison with a fixed grid. We perform a series of simulations for binary mass ratios in the range 0.026 ≤ q ≤ 1.0, each lasting longer than a viscous time so that we reach a quasi-steady accretion state. In each case, we find that gas is efficiently stripped from the inner edge of the circumbinary disk and enters the cavity along accretion streams, which feed persistent 'mini disks' surrounding each black hole. We find that for q ≳ 0.1, the binary excites eccentricity in the inner region of the circumbinary disk, creating an overdense lump which gives rise to enhanced periodicity in the accretion rate. The dependence of the periodicity on mass ratio may provide a method for observationally inferring mass ratios from measurements of the accretion rate. We also find that for all mass ratios studied, the magnitude of the accretion onto the secondary is sufficient to drive the binary toward larger mass ratio. This suggests a mechanism for biasing mass-ratio distributions toward equal mass.
4. Can neutron stars have auroras ? : electromagnetic coupling process between neutron star and magnetized accretion disk
Kimura, T.; Iwakiri, W. B.; Enoto, T.; Wada, T.; Tao, C.
2015-12-01
In the binary neutron star system, angular momentum transfer from accretion disk to a star is essential process for spin-up/down of stars. The angular momentum transfer has been well formulated for the accretion disk strongly magnetized by the neutron star [e.g., Ghosh and Lamb, 1978, 1979a, b]. However, the electromagnetic (EM) coupling between the neutron star and accretion disk has not been self-consistently solved in the previous studies although the magnetic field lines from the star are strongly tied with the accretion disk. In this study, we applied the planet-magnetosphere coupling process established for Jupiter [Hill, 1979] to the binary neutron star system. Angular momentum distribution is solved based on the torque balance between the neutron star's surface and accretion disk coupled by the magnetic field tensions. We found the EM coupling can transfer significantly larger fraction of the angular momentum from the magnetized accretion disk to the star than the unmagnetized case. The resultant spin-up rate is estimated to ~10^-14 [sec/sec] for the nominal binary system parameters, which is comparable with or larger than the other common spin-down/up processes: e.g., the magnetic dipole radiation spin-down. The Joule heating energy dissipated in the EM coupling is estimated to be up to ~10^36 [erg/sec] for the nominal binary system parameters. The release is comparable to that of gravitation energy directly caused by the matters accreting onto the neutron star. This suggests the EM coupling at the neutron star can accompany the observable radiation as auroras with a similar manner to those at the rotating planetary magnetospheres like Jupiter, Saturn, and other gas giants.
5. ALIGNMENTS OF BLACK HOLES WITH THEIR WARPED ACCRETION DISKS AND EPISODIC LIFETIMES OF ACTIVE GALACTIC NUCLEI
SciTech Connect
Li, Yan-Rong; Wang, Jian-Min; Qiu, Jie; Cheng, Cheng
2015-05-01
Warped accretion disks have attracted intense attention because of their critical role in shaping the spin of supermassive massive black holes (SMBHs) through the Bardeen–Petterson effect, a general relativistic effect that leads to final alignments or anti-alignments between black holes and warped accretion disks. We study such alignment processes by explicitly taking into account the finite sizes of accretion disks and the episodic lifetimes of active galactic nuclei (AGNs) that delineate the duration of gas fueling onto accretion disks. We employ an approximate global model to simulate the evolution of accretion disks, allowing us to determine the gravitomagnetic torque that drives the alignments in a simple way. We then track down the evolutionary paths for mass and spin of black holes both in a single activity episode and over a series of episodes. Given with randomly and isotropically oriented gas fueling over episodes, we calculate the spin evolution with different episodic lifetimes and find that it is quite sensitive to the lifetimes. We therefore propose that the spin distribution of SMBHs can place constraints on the episodic lifetimes of AGNs and vice versa. The applications of our results on the observed spin distributions of SMBHs and the observed episodic lifetimes of AGNs are discussed, although both measurements at present are too ambiguous for us to draw a firm conclusion. Our prescription can be easily incorporated into semi-analytic models for black hole growth and spin evolution.
6. Alignments Of Black Holes with Their Warped Accretion Disks and Episodic Lifetimes of Active Galactic Nuclei
Li, Yan-Rong; Wang, Jian-Min; Cheng, Cheng; Qiu, Jie
2015-05-01
Warped accretion disks have attracted intense attention because of their critical role in shaping the spin of supermassive massive black holes (SMBHs) through the Bardeen-Petterson effect, a general relativistic effect that leads to final alignments or anti-alignments between black holes and warped accretion disks. We study such alignment processes by explicitly taking into account the finite sizes of accretion disks and the episodic lifetimes of active galactic nuclei (AGNs) that delineate the duration of gas fueling onto accretion disks. We employ an approximate global model to simulate the evolution of accretion disks, allowing us to determine the gravitomagnetic torque that drives the alignments in a simple way. We then track down the evolutionary paths for mass and spin of black holes both in a single activity episode and over a series of episodes. Given with randomly and isotropically oriented gas fueling over episodes, we calculate the spin evolution with different episodic lifetimes and find that it is quite sensitive to the lifetimes. We therefore propose that the spin distribution of SMBHs can place constraints on the episodic lifetimes of AGNs and vice versa. The applications of our results on the observed spin distributions of SMBHs and the observed episodic lifetimes of AGNs are discussed, although both measurements at present are too ambiguous for us to draw a firm conclusion. Our prescription can be easily incorporated into semi-analytic models for black hole growth and spin evolution.
7. TRUNCATION OF THE INNER ACCRETION DISK AROUND A BLACK HOLE AT LOW LUMINOSITY
SciTech Connect
Tomsick, John A.; Yamaoka, Kazutaka; Corbel, Stephane; Kaaret, Philip; Kalemci, Emrah; Migliari, Simone
2009-12-10
Most black hole binaries show large changes in X-ray luminosity caused primarily by variations in mass accretion rate. An important question for understanding black hole accretion and jet production is whether the inner edge of the accretion disk recedes at low accretion rate. Measurements of the location of the inner edge (R {sub in}) can be made using iron emission lines that arise due to fluorescence of iron in the disk, and these indicate that R {sub in} is very close to the black hole at high and moderate luminosities (approx>1% of the Eddington luminosity, L {sub Edd}). Here, we report on X-ray observations of the black hole GX 339 - 4 in the hard state by Suzaku and the Rossi X-ray Timing Explorer that extend iron line studies to 0.14% L {sub Edd} and show that R {sub in} increases by a factor of >27 over the value found when GX 339 - 4 was bright. The exact value of R {sub in} depends on the inclination of the inner disk (i), and we derive 90% confidence limits of R {sub in} > 35R{sub g} at i = 0{sup 0} and R {sub in} > 175R{sub g} at i = 30{sup 0}. This provides direct evidence that the inner portion of the disk is not present at low luminosity, allowing for the possibility that the inner disk is replaced by advection- or magnetically dominated accretion flows.
8. On the viability of the magnetorotational instability in circumplanetary disks
SciTech Connect
Fujii, Yuri I.; Okuzumi, Satoshi; Inutsuka, Shu-ichiro; Tanigawa, Takayuki
2014-04-20
We examine whether the magnetorotational instability (MRI) can serve as a mechanism of angular momentum transport in circumplanetary disks. For the MRI to operate the ionization degree must be sufficiently high and the magnetic pressure must be sufficiently lower than the gas pressure. We calculate the spatial distribution of the ionization degree and search for the MRI-active region where the two criteria are met. We find that there can be thin active layers at the disk surface depending on the model parameters, however, we find hardly any region which can sustain well-developed MRI turbulence; when the magnetic field is enhanced by MRI turbulence at the disk surface layer, a magnetically dominated atmosphere encroaches on a lower altitude and a region of well-developed MRI turbulence becomes smaller. We conclude that if there are no angular momentum transfer mechanisms other than MRI in gravitationally stable circumplanetary disks, gas is likely to pile up until disks become gravitationally unstable, and massive disks may survive for a long time.
9. RADIATION PRESSURE-SUPPORTED ACCRETION DISKS: VERTICAL STRUCTURE, ENERGY ADVECTION, AND CONVECTIVE STABILITY
SciTech Connect
Gu Weimin
2012-07-10
By taking into account the local energy balance per unit volume between the viscous heating and the advective cooling plus the radiative cooling, we investigate the vertical structure of radiation pressure-supported accretion disks in spherical coordinates. Our solutions show that the photosphere of the disk is close to the polar axis and therefore the disk seems to be extremely thick. However, the density profile implies that most of the accreted matter exists in a moderate range around the equatorial plane. We show that the well-known polytropic relation between the pressure and the density is unsuitable for describing the vertical structure of radiation pressure-supported disks. More importantly, we find that the energy advection is significant even for slightly sub-Eddington accretion disks. We argue that the non-negligible advection may help us understand why the standard thin disk model is likely to be inaccurate above {approx}0.3 Eddington luminosity, which was found by some works on black hole spin measurement. Furthermore, the solutions satisfy the Solberg-Hoiland conditions, which indicate the disk to be convectively stable. In addition, we discuss the possible link between our disk model and ultraluminous X-ray sources.
10. DYNAMO ACTIVITIES DRIVEN BY MAGNETOROTATIONAL INSTABILITY AND THE PARKER INSTABILITY IN GALACTIC GASEOUS DISKS
SciTech Connect
Machida, Mami; Nakamura, Kenji E.; Kudoh, Takahiro; Akahori, Takuya; Sofue, Yoshiaki; Matsumoto, Ryoji
2013-02-10
We carried out global three-dimensional magnetohydrodynamic simulations of dynamo activities in galactic gaseous disks without assuming equatorial symmetry. Numerical results indicate the growth of azimuthal magnetic fields non-symmetric to the equatorial plane. As the magnetorotational instability (MRI) grows, the mean strength of magnetic fields is amplified until the magnetic pressure becomes as large as 10% of the gas pressure. When the local plasma {beta} (=p {sub gas}/p {sub mag}) becomes less than 5 near the disk surface, magnetic flux escapes from the disk by the Parker instability within one rotation period of the disk. The buoyant escape of coherent magnetic fields drives dynamo activities by generating disk magnetic fields with opposite polarity to satisfy the magnetic flux conservation. The flotation of the azimuthal magnetic flux from the disk and the subsequent amplification of disk magnetic field by the MRI drive quasi-periodic reversal of azimuthal magnetic fields on a timescale of 10 rotation periods. Since the rotation speed decreases with radius, the interval between the reversal of azimuthal magnetic fields increases with radius. The rotation measure computed from the numerical results shows symmetry corresponding to a dipole field.
11. Testing the Propagating Fluctuations Model with a Long, Global Accretion Disk Simulation
Hogg, J. Drew; Reynolds, Christopher S.
2016-07-01
The broadband variability of many accreting systems displays characteristic structures; log-normal flux distributions, root-mean square (rms)-flux relations, and long inter-band lags. These characteristics are usually interpreted as inward propagating fluctuations of the mass accretion rate in an accretion disk driven by stochasticity of the angular momentum transport mechanism. We present the first analysis of propagating fluctuations in a long-duration, high-resolution, global three-dimensional magnetohydrodynamic (MHD) simulation of a geometrically thin (h/r ≈ 0.1) accretion disk around a black hole. While the dynamical-timescale turbulent fluctuations in the Maxwell stresses are too rapid to drive radially coherent fluctuations in the accretion rate, we find that the low-frequency quasi-periodic dynamo action introduces low-frequency fluctuations in the Maxwell stresses, which then drive the propagating fluctuations. Examining both the mass accretion rate and emission proxies, we recover log-normality, linear rms-flux relations, and radial coherence that would produce inter-band lags. Hence, we successfully relate and connect the phenomenology of propagating fluctuations to modern MHD accretion disk theory.
12. STUDIES OF THERMALLY UNSTABLE ACCRETION DISKS AROUND BLACK HOLES WITH ADAPTIVE PSEUDOSPECTRAL DOMAIN DECOMPOSITION METHOD. II. LIMIT-CYCLE BEHAVIOR IN ACCRETION DISKS AROUND KERR BLACK HOLES
SciTech Connect
Xue Li; Lu Jufu; Sadowski, Aleksander; Abramowicz, Marek A. E-mail: [email protected]
2011-07-01
For the first time ever, we derive equations governing the time evolution of fully relativistic slim accretion disks in the Kerr metric and numerically construct their detailed non-stationary models. We discuss applications of these general results to a possible limit-cycle behavior of thermally unstable disks. Our equations and numerical method are applicable in a wide class of possible viscosity prescriptions, but in this paper we use a diffusive form of the 'standard alpha prescription' that assumes that the viscous torque is proportional to the total pressure. In this particular case, we find that the parameters that dominate the limit-cycle properties are the mass-supply rate and the value of the alpha-viscosity parameter. Although the duration of the cycle (or the outburst) does not exhibit any clear dependence on the black hole spin, the maximal outburst luminosity (in the Eddington units) is positively correlated with the spin value. We suggest a simple method for a rough estimate of the black hole spin based on the maximal luminosity and the ratio of outburst to cycle durations. We also discuss a temperature-luminosity relation for the Kerr black hole accretion disk limit cycle. Based on these results, we discuss the limit-cycle behavior observed in microquasar GRS 1915+105. We also extend this study to several non-standard viscosity prescriptions, including a 'delayed heating' prescription recently addressed by the MHD simulations of accretion disks.
13. ANISOTROPY OF X-RAY BURSTS FROM NEUTRON STARS WITH CONCAVE ACCRETION DISKS
SciTech Connect
He, C.-C.; Keek, L.
2016-03-01
Emission from neutron stars and accretion disks in low-mass X-ray binaries is anisotropic. The non-spherical shape of the disk as well as blocking of the neutron star by the disk make the observed flux dependent on the inclination angle of the disk with respect to the line of sight. This is of importance for the interpretation of thermonuclear X-ray bursts from neutron stars. Because part of the X-ray burst is reflected off the disk, the observed burst flux depends on the anisotropies for both direct emission from the neutron star and reflection off the disk. This influences measurements of source distance, mass accretion rate, and constraints on the neutron star’s equation of state. Previous predictions of the anisotropy factors assumed a geometrically flat disk. Detailed observations of two so-called superbursts allowed for the direct and the reflected burst fluxes to each be measured separately. The reflection fraction was much higher than what the anisotropies of a flat disk can account for. We create numerical models to calculate the anisotropy factors for different disk shapes, including concave disks. We present the anisotropy factors of the direct and reflected burst fluxes separately, as well as the anisotropy of the persistent flux. Reflection fractions substantially larger than unity are produced in the case where the inner accretion disk increases steeply in height, such that part of the star is blocked from view. Such a geometry could possibly be induced by the X-ray burst if X-ray heating causes the inner disk to puff up.
14. GLOBAL MODELING OF RADIATIVELY DRIVEN ACCRETION OF METALS FROM COMPACT DEBRIS DISKS ONTO WHITE DWARFS
SciTech Connect
Bochkarev, Konstantin V.; Rafikov, Roman R. E-mail: [email protected]
2011-11-01
Recent infrared observations have revealed the presence of compact (radii {approx}< R{sub sun}) debris disks around more than a dozen metal-rich white dwarfs (WDs), likely produced by a tidal disruption of asteroids. Accretion of high-Z material from these disks may account for the metal contamination of these WDs. It was previously shown using local calculations that the Poynting-Robertson (PR) drag acting on the dense, optically thick disk naturally drives metal accretion onto the WD at the typical rate M-dot{sub PR}{approx}10{sup 8} g s{sup -1}. Here we extend this local analysis by exploring the global evolution of the debris disk under the action of the PR drag for a variety of assumptions about the disk properties. We find that massive disks (mass {approx}> 10{sup 20} g), which are optically thick to incident stellar radiation, inevitably give rise to metal accretion at rates M-dot {approx}>0.2 M-dot{sub PR}. The magnitude of M-dot and its time evolution are determined predominantly by the initial pattern of the radial distribution of the debris (i.e., ring-like versus disk-like) but not by the total mass of the disk. The latter determines only the disk lifetime, which can be several Myr or longer. The evolution of an optically thick disk generically results in the development of a sharp outer edge of the disk. We also find that the low-mass ({approx}< 10{sup 20} g), optically thin disks exhibit M-dot << M-dot{sub PR} and evolve on a characteristic timescale {approx}10{sup 5}-10{sup 6} yr, independent of their total mass.
15. RELATIVISTIC LINES AND REFLECTION FROM THE INNER ACCRETION DISKS AROUND NEUTRON STARS
SciTech Connect
Cackett, Edward M.; Miller, Jon M.; Ballantyne, David R.; Barret, Didier; Boutelier, Martin; Miller, M. Coleman; Strohmayer, Tod E.
2010-09-01
A number of neutron star low-mass X-ray binaries (LMXBs) have recently been discovered to show broad, asymmetric Fe K emission lines in their X-ray spectra. These lines are generally thought to be the most prominent part of a reflection spectrum, originating in the inner part of the accretion disk where strong relativistic effects can broaden emission lines. We present a comprehensive, systematic analysis of Suzaku and XMM-Newton spectra of 10 neutron star LMXBs, all of which display broad Fe K emission lines. Of the 10 sources, 4 are Z sources, 4 are atolls, and 2 are accreting millisecond X-ray pulsars (also atolls). The Fe K lines are fit well by a relativistic line model for a Schwarzschild metric, and imply a narrow range of inner disk radii (6-15 GM/c {sup 2}) in most cases. This implies that the accretion disk extends close to the neutron star surface over a range of luminosities. Continuum modeling shows that for the majority of observations, a blackbody component (plausibly associated with the boundary layer) dominates the X-ray emission from 8 to 20 keV. Thus it appears likely that this spectral component produces the majority of the ionizing flux that illuminates the accretion disk. Therefore, we also fit the spectra with a blurred reflection model, wherein a blackbody component illuminates the disk. This model fits well in most cases, supporting the idea that the boundary layer illuminates a geometrically thin disk.
16. Relativistic Lines and Reflection from the Inner Accretion Disks Around Neutron Stars
Cackett, Edward M.; Miller, Jon M.; Ballantyne, David R.; Barret, Didier; Bhattacharyya, Sudip; Boutelier, Martin; Miller, M. Coleman; Strohmayer, Tod E.; Wijnands, Rudy
2010-09-01
A number of neutron star low-mass X-ray binaries (LMXBs) have recently been discovered to show broad, asymmetric Fe K emission lines in their X-ray spectra. These lines are generally thought to be the most prominent part of a reflection spectrum, originating in the inner part of the accretion disk where strong relativistic effects can broaden emission lines. We present a comprehensive, systematic analysis of Suzaku and XMM-Newton spectra of 10 neutron star LMXBs, all of which display broad Fe K emission lines. Of the 10 sources, 4 are Z sources, 4 are atolls, and 2 are accreting millisecond X-ray pulsars (also atolls). The Fe K lines are fit well by a relativistic line model for a Schwarzschild metric, and imply a narrow range of inner disk radii (6-15 GM/c 2) in most cases. This implies that the accretion disk extends close to the neutron star surface over a range of luminosities. Continuum modeling shows that for the majority of observations, a blackbody component (plausibly associated with the boundary layer) dominates the X-ray emission from 8 to 20 keV. Thus it appears likely that this spectral component produces the majority of the ionizing flux that illuminates the accretion disk. Therefore, we also fit the spectra with a blurred reflection model, wherein a blackbody component illuminates the disk. This model fits well in most cases, supporting the idea that the boundary layer illuminates a geometrically thin disk.
17. Towards Bayesian Machine Learning for Estimating Parameters of Accretion Disk Models for SPH Simulations
Goel, Amit; Montgomery, Michele; Wiegand, Paul
2016-01-01
Accretion disks are ubiquitous in Active Galactic Nuclei, in protostellar systems forming protoplanets, and in close binary star systems such as X-ray binaries, Cataclysmic Variables, and Algols, for example. Observations such as disk tilt are found in all of these different accreting system types, suggesting a common physics must be present. To understand the common connections between these different system types, which can help us understand their unique evolutions, we need to better understand the physics of accretion. For example, viscosity is typically a constant value in the disk of a system that is in a specific state such as a quiescent state. However, viscosity can't be constant throughout the disk, especially at the boundaries. To learn more about viscosity and other common parameters in these disk, we use Bayesian Inference and Markov Chain Monte Carlo techniques to make predictions of events to come in the numerical simulations of these accreting disks. In this work, we present our techniques and initial findings.
18. The Spiral Wave Instability Induced by a Giant Planet. I. Particle Stirring in the Inner Regions of Protoplanetary Disks
Bae, Jaehan; Nelson, Richard P.; Hartmann, Lee
2016-12-01
We have recently shown that spiral density waves propagating in accretion disks can undergo a parametric instability by resonantly coupling with and transferring energy into pairs of inertial waves (or inertial-gravity waves when buoyancy is important). In this paper, we perform inviscid three-dimensional global hydrodynamic simulations to examine the growth and consequence of this instability operating on the spiral waves driven by a Jupiter-mass planet in a protoplanetary disk. We find that the spiral waves are destabilized via the spiral wave instability (SWI), generating hydrodynamic turbulence and sustained radially alternating vertical flows that appear to be associated with long wavelength inertial modes. In the interval 0.3 {R}{{p}}≤slant R≤slant 0.7{R}{{p}}, where R p denotes the semimajor axis of the planetary orbit (assumed to be 5 au), the estimated vertical diffusion rate associated with the turbulence is characterized by {α }{diff}∼ (0.2{--}1.2)× {10}-2. For the disk model considered here, the diffusion rate is such that particles with sizes up to several centimeters are vertically mixed within the first pressure scale height. This suggests that the instability of spiral waves launched by a giant planet can significantly disperse solid particles and trace chemical species from the midplane. In planet formation models where the continuous local production of chondrules/pebbles occurs over Myr timescales to provide a feedstock for pebble accretion onto these bodies, this stirring of solid particles may add a time constraint: planetary embryos and large asteroids have to form before a gas giant forms in the outer disk, otherwise the SWI will significantly decrease the chondrule/pebble accretion efficiency.
19. Diffusive Particle Acceleration in Shocked, Viscous Accretion Disks: Green's Function Energy Distribution
Becker, Peter A.; Das, Santabrata; Le, Truong
2011-12-01
The acceleration of relativistic particles in a viscous accretion disk containing a standing shock is investigated as a possible explanation for the energetic outflows observed around radio-loud black holes. The energy/space distribution of the accelerated particles is computed by solving a transport equation that includes the effects of first-order Fermi acceleration, bulk advection, spatial diffusion, and particle escape. The velocity profile of the accreting gas is described using a model for shocked viscous disks recently developed by the authors, and the corresponding Green's function distribution for the accelerated particles in the disk and the outflow is obtained using a classical method based on eigenfunction analysis. The accretion-driven, diffusive shock acceleration scenario explored here is conceptually similar to the standard model for the acceleration of cosmic rays at supernova-driven shocks. However, in the disk application, the distribution of the accelerated particles is much harder than would be expected for a plane-parallel shock with the same compression ratio. Hence the disk environment plays a key role in enhancing the efficiency of the shock acceleration process. The presence of the shock helps to stabilize the disk by reducing the Bernoulli parameter, while channeling the excess binding energy into the escaping relativistic particles. In applications to M87 and Sgr A*, we find that the kinetic power in the jet is {\\sim}0.01\\,\\dot{M} c^2, and the outflowing relativistic particles have a mean energy ~300 times larger than that of the thermal gas in the disk at the shock radius. Our results suggest that a standing shock may be an essential ingredient in accretion onto underfed black holes, helping to resolve the long-standing problem of the stability of advection-dominated accretion disks.
20. Exploring non-normality in magnetohydrodynamic rotating shear flows: Application to astrophysical accretion disks
2016-10-01
shear flows are ubiquitous in astrophysics, especially accretion disks, where molecular viscosity is too low to account for observed data. The primary accepted cause of energy-momentum transport therein is turbulent viscosity. Hence, these results would have important implications in astrophysics.
1. Critical condition for the propeller effect in systems with magnetized neutron stars accreting from geometrically thin accretion disks
Ertan, Unal
2016-07-01
2. Evidence for large temperature fluctuations in quasar accretion disks from spectral variability
SciTech Connect
Ruan, John J.; Anderson, Scott F.; Agol, Eric; Dexter, Jason
2014-03-10
The well-known bluer-when-brighter trend observed in quasar variability is a signature of the complex processes in the accretion disk and can be a probe of the quasar variability mechanism. Using a sample of 604 variable quasars with repeat spectra in the Sloan Digital Sky Survey-I/II (SDSS), we construct difference spectra to investigate the physical causes of this bluer-when-brighter trend. The continuum of our composite difference spectrum is well fit by a power law, with a spectral index in excellent agreement with previous results. We measure the spectral variability relative to the underlying spectra of the quasars, which is independent of any extinction, and compare to model predictions. We show that our SDSS spectral variability results cannot be produced by global accretion rate fluctuations in a thin disk alone. However, we find that a simple model of an inhomogeneous disk with localized temperature fluctuations will produce power-law spectral variability over optical wavelengths. We show that the inhomogeneous disk will provide good fits to our observed spectral variability if the disk has large temperature fluctuations in many independently varying zones, in excellent agreement with independent constraints from quasar microlensing disk sizes, their strong UV spectral continuum, and single-band variability amplitudes. Our results provide an independent constraint on quasar variability models and add to the mounting evidence that quasar accretion disks have large localized temperature fluctuations.
3. The Evolution of the Accretion Disk Around 4U 1820-30 During a Superburst
NASA Technical Reports Server (NTRS)
Ballantyne, D. R.; Strohmayer, T. E.
2004-01-01
Accretion from a disk onto a collapsed, relativistic star - a neutron star or black hole - is the mechanism widely believed to be responsible for the emission from compact X-ray binaries. Because of the extreme spatial resolution required, it is not yet possible to directly observe the evolution or dynamics of the inner parts of the accretion disk where general relativistic effects are dominant. Here, we use the bright X-ray emission from a superburst on the surface of the neutron star 4U 1820-30 as a spotlight to illuminate the disk surface. The X-rays cause iron atoms in the disk t o fluoresce, allowing a determination of the ionization state, covering factor and inner radius of the disk over the course of the burst. The time-resolved spectral fitting shows that the inner region of the disk is disrupted by the burst, possibly being heated into a thicker, more tenuous flow, before recovering its previous form in approximately 1000 s. This marks the first instance that the evolution of the inner regions of an accretion disk has been observed in real-time.
4. The Effect of Protoplanetary Disk Cooling Times on the Formation of Gas Giant Planets by Gravitational Instability
Boss, Alan P.
2017-02-01
Observational evidence exists for the formation of gas giant planets on wide orbits around young stars by disk gravitational instability, but the roles of disk instability and core accretion for forming gas giants on shorter period orbits are less clear. The controversy extends to population synthesis models of exoplanet demographics and to hydrodynamical models of the fragmentation process. The latter refers largely to the handling of radiative transfer in three-dimensional (3D) hydrodynamical models, which controls heating and cooling processes in gravitationally unstable disks, and hence dense clump formation. A suite of models using the β cooling approximation is presented here. The initial disks have masses of 0.091 M ⊙ and extend from 4 to 20 au around a 1 M ⊙ protostar. The initial minimum Toomre Q i values range from 1.3 to 2.7, while β ranges from 1 to 100. We show that the choice of Q i is equal in importance to the β value assumed: high Q i disks can be stable for small β, when the initial disk temperature is taken as a lower bound, while low Q i disks can fragment for high β. These results imply that the evolution of disks toward low Q i must be taken into account in assessing disk fragmentation possibilities, at least in the inner disk, i.e., inside about 20 au. The models suggest that if low Q i disks can form, there should be an as yet largely undetected population of gas giants orbiting G dwarfs between about 6 au and 16 au.
5. A Spectrum Synthesis Program for Binary Stars with Optically Thick Accretion Disks
Linnell, A. P.; Hubeny, I.
1994-12-01
We recently reported a spectrum synthesis program for binary stars (Linnell & Hubeny, 1994, ApJ, 434, Oct.20). That program now has been extended to include the case of an optically thick accretion disk associated with either of the two stellar components. Our model of the accretion disk uses the Pringle expression (Pringle, 1981, ARA&A, 19, 137) for T_eff values on the accretion disk face, and the results of Hubeny and Plavec (1981, ApJ, 102, 1156) for rim T_eff values. The treatment of the stellar components is the same as in our 1994 paper. The program divides the rim into NRIM latitude values, typically 9, and divides the visible accretion disk face into NRING concentric ring boundaries, typically 31. The individual rings (for both the rim and the face) subdivide into NSEG pixels, typically 101. An individual synthetic spectrum, appropriate to the local T_eff value, is attached to each pixel. For illustration purposes we have used synthetic spectra prepared from Kurucz atmospheres. The extended program constructs a synthetic spectrum for the accretion disk face, rim, the separate stellar components, and the entire system by producing a sum of contributions, at each wavelength (with due allowance for Doppler shift), from each visible pixel on the accretion disk or the separate stellar components, weighted by the projected area of the pixel. A separate program establishes a visibility key for each pixel and cosine of the zenith angle of the observer as seen from each pixel, for the current value of orbital inclination and orbital longitude. These data combine with synthetic spectra in the spectrum synthesis program to determine line of sight light intensities at each wavelength, i.e., the contributions needed for the sum. Separate data from related programs permit a plot of the system projected on the plane of the sky. This project received partial support from NSF grant AST9020459.
6. Application of the Cubed-Sphere Grid to Tilted Black-Hole Accretion Disks
SciTech Connect
Fragile, P C; Lindner, C C; Anninos, P; Salmonson, J D
2008-09-24
In recent work we presented the first results of global general relativistic magnetohydrodynamic (GRMHD) simulations of tilted (or misaligned) accretion disks around rotating black holes. The simulated tilted disks showed dramatic differences from comparable untilted disks, such as asymmetrical accretion onto the hole through opposing 'plunging streams' and global precession of the disk powered by a torque provided by the black hole. However, those simulations used a traditional spherical-polar grid that was purposefully underresolved along the pole, which prevented us from assessing the behavior of any jets that may have been associated with the tilted disks. To address this shortcoming we have added a block-structured 'cubed-sphere' grid option to the Cosmos++ GRMHD code, which will allow us to simultaneously resolve the disk and polar regions. Here we present our implementation of this grid and the results of a small suite of validation tests intended to demonstrate that the new grid performs as expected. The most important test in this work is a comparison of identical tilted disks, one evolved using our spherical-polar grid and the other with the cubed-sphere grid. We also demonstrate an interesting dependence of the early-time evolution of our disks on their orientation with respect to the grid alignment. This dependence arises from the differing treatment of current sheets within the disks, especially whether they are aligned with symmetry planes of the grid or not.
7. Signatures of Gravitational Instability in Resolved Images of Protostellar Disks
Dong, Ruobing; Vorobyov, Eduard; Pavlyuchenkov, Yaroslav; Chiang, Eugene; Liu, Hauyu Baobab
2016-06-01
Protostellar (class 0/I) disks, which have masses comparable to those of their nascent host stars and are fed continuously from their natal infalling envelopes, are prone to gravitational instability (GI). Motivated by advances in near-infrared (NIR) adaptive optics imaging and millimeter-wave interferometry, we explore the observational signatures of GI in disks using hydrodynamical and Monte Carlo radiative transfer simulations to synthesize NIR scattered light images and millimeter dust continuum maps. Spiral arms induced by GI, located at disk radii of hundreds of astronomical units, are local overdensities and have their photospheres displaced to higher altitudes above the disk midplane; therefore, arms scatter more NIR light from their central stars than inter-arm regions, and are detectable at distances up to 1 kpc by Gemini/GPI, VLT/SPHERE, and Subaru/HiCIAO/SCExAO. In contrast, collapsed clumps formed by disk fragmentation have such strong local gravitational fields that their scattering photospheres are at lower altitudes; such fragments appear fainter than their surroundings in NIR scattered light. Spiral arms and streamers recently imaged in four FU Ori systems at NIR wavelengths resemble GI-induced structures and support the interpretation that FUors are gravitationally unstable protostellar disks. At millimeter wavelengths, both spirals and clumps appear brighter in thermal emission than the ambient disk and can be detected by ALMA at distances up to 0.4 kpc with one hour integration times at ˜0.″1 resolution. Collapsed fragments having masses ≳1 M J can be detected by ALMA within ˜10 minutes.
8. Optical Microlensing and Accretion Disk Structure in the Lensed Quasar SDSS 1520+530
Manickam, Vigneshwar; Grinaski, Ian; MacLeod, Chelsea; Morgan, Christopher W.; Harris, Hugh C.; Kennington, James
2015-01-01
We analyze uncorrelated variability in seven seasons of SDSS r-band monitoring data from the doubly-imaged gravitationally lensed quasar SBS 1520+530 to yield a measurement of the size of the near-UV continuum emission region in this quasar. Photometry in the SBS 1520+530 system is complicated significantly by the proximity of a very bright star whose diffraction spike blends with the the lens, so we employed a mirror-flip subtraction technique to correct for this contamination. We conclude by testing our accretion disk measurement against the Quasar Accretion Disk Size - Black Hole Mass Relation.
9. Accretion to magnetized stars through the Rayleigh-Taylor instability: global 3D simulations
Kulkarni, A. K.; Romanova, M. M.
2008-05-01
We present results of 3D simulations of magnetohydrodynamics (MHD) instabilities at the accretion disc-magnetosphere boundary. The instability is Rayleigh-Taylor, and develops for a fairly broad range of accretion rates and stellar rotation rates and magnetic fields. It manifests itself in the form of tall, thin tongues of plasma that penetrate the magnetosphere in the equatorial plane. The shape and number of the tongues changes with time on the inner disc dynamical time-scale. In contrast with funnel flows, which deposit matter mainly in the polar region, the tongues deposit matter much closer to the stellar equator. The instability appears for relatively small misalignment angles, Θ <~ 30°, between the star's rotation and magnetic axes, and is associated with higher accretion rates. The hotspots and light curves during accretion through instability are generally much more chaotic than during stable accretion. The unstable state of accretion has possible implications for quasi-periodic oscillations and intermittent pulsations from accreting systems, as well as planet migration.
10. Simulating the Formation of Massive Protostars. I. Radiative Feedback and Accretion Disks
Klassen, Mikhail; Pudritz, Ralph E.; Kuiper, Rolf; Peters, Thomas; Banerjee, Robi
2016-05-01
We present radiation hydrodynamic simulations of collapsing protostellar cores with initial masses of 30, 100, and 200 M ⊙. We follow their gravitational collapse and the formation of a massive protostar and protostellar accretion disk. We employ a new hybrid radiative feedback method blending raytracing techniques with flux-limited diffusion for a more accurate treatment of the temperature and radiative force. In each case, the disk that forms becomes Toomre-unstable and develops spiral arms. This occurs between 0.35 and 0.55 freefall times and is accompanied by an increase in the accretion rate by a factor of 2-10. Although the disk becomes unstable, no other stars are formed. In the case of our 100 and 200 M ⊙ simulations, the star becomes highly super-Eddington and begins to drive bipolar outflow cavities that expand outwards. These radiatively driven bubbles appear stable, and appear to be channeling gas back onto the protostellar accretion disk. Accretion proceeds strongly through the disk. After 81.4 kyr of evolution, our 30 M ⊙ simulation shows a star with a mass of 5.48 M ⊙ and a disk of mass 3.3 M ⊙, while our 100 M ⊙ simulation forms a 28.8 M ⊙ mass star with a 15.8 M ⊙ disk over the course of 41.6 kyr, and our 200 M ⊙ simulation forms a 43.7 M ⊙ star with an 18 M ⊙ disk in 21.9 kyr. In the absence of magnetic fields or other forms of feedback, the masses of the stars in our simulation do not appear to be limited by their own luminosities.
11. Cold Dark Matter Substructure and Galactic Disks I: Morphological Signatures of Hierarchical SatelliteAccretion
SciTech Connect
Kazantzidis, Stelios; Bullock, James S.; Zentner, Andrew R.; Kravtsov, Andrey V.; Moustakas, Leonidas A.
2007-12-03
We conduct a series of high-resolution, fully self-consistent dissipation less N-body simulations to investigate the cumulative effect of substructure mergers onto thin disk galaxies in the context of the {Lambda}CDM paradigm of structure formation. Our simulation campaign is based on a hybrid approach combining cosmological simulations and controlled numerical experiments. Substructure mass functions, orbital distributions, internal structures, and accretion times are culled directly from cosmological simulations of galaxy-sized cold dark matter (CDM) halos. We demonstrate that accretions of massive subhalos onto the central regions of host halos, where the galactic disk resides, since z {approx} 1 should be common occurrences. In contrast, extremely few satellites in present-day CDM halos are likely to have a significant impact on the disk structure. This is due to the fact that massive subhalos with small orbital pericenters that are most capable of strongly perturbing the disk become either tidally disrupted or suffer substantial mass loss prior to z = 0. One host halo merger history is subsequently used to seed controlled N-body experiments of repeated satellite impacts on an initially-thin Milky Way-type disk galaxy. These simulations track the effects of six dark matter substructures, with initial masses in the range {approx} (0.7-2) x 10{sup 10} M{sub {circle_dot}} ({approx} 20-60% of the disk mass), crossing the disk in the past {approx} 8 Gyr. We show that these accretion events produce several distinctive observational signatures in the stellar disk including: a long-lived, low-surface brightness, ring-like feature in the outskirts; a significant flare; a central bar; and faint filamentary structures that (spuriously) resemble tidal streams in configuration space. The final distribution of disk stars exhibits a complex vertical structure that is well-described by a standard 'thin-thick' disk decomposition, where the 'thick' disk component has emerged
12. A CORRELATION BETWEEN THE IONIZATION STATE OF THE INNER ACCRETION DISK AND THE EDDINGTON RATIO OF ACTIVE GALACTIC NUCLEI
SciTech Connect
Ballantyne, D. R.; McDuffie, J. R.; Rusin, J. S.
2011-06-20
X-ray reflection features observed from the innermost regions of accretion disks in active galactic nuclei (AGNs) allow important tests of accretion theory. In recent years, it has been possible to use the Fe K{alpha} line and reflection continuum to parameterize the ionization state of the irradiated inner accretion disk. Here, we collect 10 measurements of {xi}, the disk ionization parameter, from eight AGNs with strong evidence for reflection from the inner accretion disk and good black hole mass estimates. We find strong statistical evidence (98.56% confidence) for a nearly linear correlation between {xi} and the AGN Eddington ratio. Moreover, such a correlation is predicted by a simple application of {alpha}-disk accretion theory, albeit with a stronger dependence on the Eddington ratio. The theory shows that there will be intrinsic scatter to any correlation as a result of different black hole spins and radii of reflection. There are several possibilities to soften the predicted dependence on the Eddington ratio to allow a closer agreement with the observed correlation, but the current data do not allow for a unique explanation. The correlation can be used to estimate that MCG-6-30-15 should have a highly ionized inner accretion disk, which would imply a black hole spin of {approx}0.8. Additional measurements of {xi} from a larger sample of AGNs are needed to confirm the existence of this correlation, and will allow investigation of the accretion disk/corona interaction in the inner regions of accretion disks.
13. Direct detection of a magnetic field in the innermost regions of an accretion disk
Donati, Jean-François; Paletou, Fréderic; Bouvier, Jérome; Ferreira, Jonathan
2005-11-01
Models predict that magnetic fields play a crucial role in the physics of astrophysical accretion disks and their associated winds and jets. For example, the rotation of the disk twists around the rotation axis the initially vertical magnetic field, which responds by slowing down the plasma in the disk and by causing it to fall towards the central star. The magnetic energy flux produced in this process points away from the disk, pushing the surface plasma outwards, leading to a wind from the disk and sometimes a collimated jet. But these predictions have hitherto not been supported by observations. Here we report the direct detection of the magnetic field in the core of the protostellar accretion disk FU Orionis. The surface field reaches strengths of about 1kG close to the centre of the disk, and it includes a significant azimuthal component, in good agreement with recent models. But we find that the field is very filamentary and slows down the disk plasma much more than models predict, which may explain why FU Ori fails to collimate its wind into a jet.
14. MIGRATION OF EXTRASOLAR PLANETS: EFFECTS FROM X-WIND ACCRETION DISKS
SciTech Connect
Adams, Fred C.; Cai, Mike J.; Lizano, Susana
2009-09-10
Magnetic fields are dragged in from the interstellar medium during the gravitational collapse that forms star/disk systems. Consideration of mean field magnetohydrodynamics in these disks shows that magnetic effects produce sub-Keplerian rotation curves and truncate the inner disk. This Letter explores the ramifications of these predicted disk properties for the migration of extrasolar planets. Sub-Keplerian flow in gaseous disks drives a new migration mechanism for embedded planets and modifies the gap-opening processes for larger planets. This sub-Keplerian migration mechanism dominates over Type I migration for sufficiently small planets (m{sub P} {approx}< 1 M {sub +}) and/or close orbits (r {approx}< 1 AU). Although the inclusion of sub-Keplerian torques shortens the total migration time by only a moderate amount, the mass accreted by migrating planetary cores is significantly reduced. Truncation of the inner disk edge (for typical system parameters) naturally explains final planetary orbits with periods P {approx} 4 days. Planets with shorter periods, P {approx} 2 days, can be explained by migration during FU-Orionis outbursts, when the mass accretion rate is high and the disk edge moves inward. Finally, the midplane density is greatly increased at the inner truncation point of the disk (the X-point); this enhancement, in conjunction with continuing flow of gas and solids through the region, supports the in situ formation of giant planets.
15. The characteristic blue spectra of accretion disks in quasars as uncovered in the infrared.
PubMed
Kishimoto, Makoto; Antonucci, Robert; Blaes, Omer; Lawrence, Andy; Boisson, Catherine; Albrecht, Marcus; Leipski, Christian
2008-07-24
Quasars are thought to be powered by supermassive black holes accreting surrounding gas. Central to this picture is a putative accretion disk which is believed to be the source of the majority of the radiative output. It is well known, however, that the most extensively studied disk model-an optically thick disk which is heated locally by the dissipation of gravitational binding energy-is apparently contradicted by observations in a few major respects. In particular, the model predicts a specific blue spectral shape asymptotically from the visible to the near-infrared, but this is not generally seen in the visible wavelength region where the disk spectrum is observable. A crucial difficulty has been that, towards the infrared, the disk spectrum starts to be hidden under strong, hot dust emission from much larger but hitherto unresolved scales, and thus has essentially been impossible to observe. Here we report observations of polarized light interior to the dust-emitting region that enable us to uncover this near-infrared disk spectrum in several quasars. The revealed spectra show that the near-infrared disk spectrum is indeed as blue as predicted. This indicates that, at least for the outer near-infrared-emitting radii, the standard picture of the locally heated disk is approximately correct.
16. Quasi-static model of collimated jets and radio lobes. I. Accretion disk and jets
SciTech Connect
Colgate, Stirling A.; Li, Hui; Fowler, T. Kenneth; Pino, Jesse
2014-07-10
This is the first of a series of papers showing that when an efficient dynamo can be maintained by accretion disks around supermassive black holes in active galactic nuclei, it can lead to the formation of a powerful, magnetic helix that could explain both the observed radio jet/lobe structures on very large scales and ultimately the enormous power inferred from the observed ultra-high-energy cosmic rays. In this work, we solve a set of one-dimensional equations similar to the steady-state standard accretion disk model, but now including the large-scale magnetic fields giving rises to jets. We find that the frequently made assumption that large-scale fields are frozen into the disk is fundamentally incorrect, due to the necessity for current and the accreting mass to flow perpendicular to magnetic flux surfaces. A correct treatment greatly simplifies the calculations, yielding fields that leave the disk nearly vertically with magnetic profiles uniquely determined by disk angular momentum conservation. Representative solutions of the magnetic fields in different radial regions of the disk surface are given, and they determine the overall key features in the jet structure and its dissipation, which will be the subjects of later papers.
17. Hydrodynamic Models of Line-Driven Accretion Disk Winds III: Local Ionization Equilibrium
NASA Technical Reports Server (NTRS)
Pereyra, Nicolas Antonio; Kallman, Timothy R.; White, Nicholas E. (Technical Monitor)
2002-01-01
We present time-dependent numerical hydrodynamic models of line-driven accretion disk winds in cataclysmic variable systems and calculate wind mass-loss rates and terminal velocities. The models are 2.5-dimensional, include an energy balance condition with radiative heating and cooling processes, and includes local ionization equilibrium introducing time dependence and spatial dependence on the line radiation force parameters. The radiation field is assumed to originate in an optically thick accretion disk. Wind ion populations are calculated under the assumption that local ionization equilibrium is determined by photoionization and radiative recombination, similar to a photoionized nebula. We find a steady wind flowing from the accretion disk. Radiative heating tends to maintain the temperature in the higher density wind regions near the disk surface, rather than cooling adiabatically. For a disk luminosity L (sub disk) = solar luminosity, white dwarf mass M(sub wd) = 0.6 solar mass, and white dwarf radii R(sub wd) = 0.01 solar radius, we obtain a wind mass-loss rate of M(sub wind) = 4 x 10(exp -12) solar mass yr(exp -1) and a terminal velocity of approximately 3000 km per second. These results confirm the general velocity and density structures found in our earlier constant ionization equilibrium adiabatic CV wind models. Further we establish here 2.5D numerical models that can be extended to QSO/AGN winds where the local ionization equilibrium will play a crucial role in the overall dynamics.
18. Axisymmetric Two-Dimensional Computation of Magnetic Field Dragging in Accretion Disks
NASA Technical Reports Server (NTRS)
Reyes-Ruiz, Mauricio; Stepinski, Tomasz F.
1996-01-01
In this paper we model a geometrically thin accretion disk interacting with an externally imposed, uniform, vertical magnetic field. The accretion flow in the disk drags and distorts field lines, amplifying the magnetic field in the process. Inside the disk the radial component of the field is sheared into a toroidal component. The aim of this work is to establish the character of the resultant magnetic field and its dependence on the disk's parameters. We concentrate on alpha-disks driven by turbulent viscosity. Axisymmetric, two-dimensional solutions are obtained without taking into account the back-reaction of the magnetic field on the structure of the disk. The character of the magnetic field depends strongly on the magnitude of the magnetic Prandtl number, P . We present two illustrative examples of viscous disks: a so-called 'standard' steady state model of a disk around a compact star (e.g., cataclysmic variable), and a steady state model of a proto-planetary disk. In both cases, P = 1, P = 10(sup -1), and P = 10(sup -2) scenarios are calculated. Significant bending and magnification of the magnetic field is possible only for disks characterized by P of the order of 10(sup -2). In such a case, the field lines are bent sufficiently to allow the development of a centrifugally driven wind. Inside the disk the field is dominated by its toroidal component. We also investigate the dragging of the magnetic field by a nonviscous protoplanetary disk described by a phenomenological model. This scenario leads to large distortion and magnification of the magnetic field.
19. Circumbinary ring, circumstellar disks, and accretion in the binary system UY Aurigae
SciTech Connect
Tang, Ya-Wen; Ho, Paul T. P.; Dutrey, Anne; Guilloteau, Stéphane; Di Folco, Emmanuel; Piétu, Vincent; Gueth, Fréderic; Beck, Tracy; Boehler, Yann; Bary, Jeff; Simon, Michal
2014-09-20
Recent exo-planetary surveys reveal that planets can orbit and survive around binary stars. This suggests that some fraction of young binary systems which possess massive circumbinary (CB) disks may be in the midst of planet formation. However, there are very few CB disks detected. We revisit one of the known CB disks, the UY Aurigae system, and probe {sup 13}CO 2-1, C{sup 18}O 2-1, SO 5(6)-4(5) and {sup 12}CO 3-2 line emission and the thermal dust continuum. Our new results confirm the existence of the CB disk. In addition, the circumstellar (CS) disks are clearly resolved in dust continuum at 1.4 mm. The spectral indices between the wavelengths of 0.85 mm and 6 cm are found to be surprisingly low, being 1.6 for both CS disks. The deprojected separation of the binary is 1.''26 based on our 1.4 mm continuum data. This is 0.''07 (10 AU) larger than in earlier studies. Combining the fact of the variation of UY Aur B in R band, we propose that the CS disk of an undetected companion UY Aur Bb obscures UY Aur Ba. A very complex kinematical pattern inside the CB disk is observed due to a mixing of Keplerian rotation of the CB disk, the infall and outflow gas. The streaming gas accreting from the CB ring toward the CS disks and possible outflows are also identified and resolved. The SO emission is found to be at the bases of the streaming shocks. Our results suggest that the UY Aur system is undergoing an active accretion phase from the CB disk to the CS disks. The UY Aur B might also be a binary system, making the UY Aur a triple system.
20. THE CENTRAL ENGINE STRUCTURE OF 3C120: EVIDENCE FOR A RETROGRADE BLACK HOLE OR A REFILLING ACCRETION DISK
SciTech Connect
Cowperthwaite, Philip S.; Reynolds, Christopher S.
2012-06-20
The broad-line radio galaxy 3C120 is a powerful source of both X-ray and radio emission including superluminal jet outflows. We report on our reanalysis of 160 ks of Suzaku data taken in 2006, previously examined by Kataoka et al. Spectral fits to the X-ray Imaging Spectrometer and Hard X-ray Detector/positive intrinsic negative data over a range of 0.7-45 keV reveal a well-defined iron K line complex with a narrow K{alpha} core and relativistically broadened features consistent with emission from the inner regions of the accretion disk. Furthermore, the inner region of the disk appears to be truncated, with an inner radius of r{sub in} = 11.7{sup +3.5}{sub -5.2} r{sub g} . If we assume that fluorescent iron line features terminate at the inner-most stable circular orbit (ISCO), then we measure a black hole spin of a-hat < -0.1 at a 90% confidence level. A rapidly spinning prograde black hole ( a-hat > 0.8) can be ruled out at the 99% confidence level. Alternatively, the disk may be truncated well outside of the ISCO of a rapid prograde hole. The most compelling scenario is the possibility that the inner regions of the disk were destroyed/ejected by catastrophic instabilities just prior to the time these observations were made.
1. The Behavior of Accretion Disks in Low Mass X-ray Binaries: Disk Winds and Alpha Model
Bayless, Amanda J.
2010-01-01
This dissertation presents research on two low mass X-ray binaries. The eclipsing low-mass X-ray binary 4U 1822-371 is the prototypical accretion disk corona (ADC) system. We have obtained new time-resolved UV spectroscopy with the ACS/SBC on the Hubble Space Telescope and new V- and J-band photometry with the 1.3-m SMARTS telescope at CTIO. We show that the accretion disk in the system has a strong wind with projected velocities up to 4000 km/s as determined from the Doppler width of the C IV emission line. The broad and shallow eclipse indicates that the disk has a vertically-extended, optically-thick component at optical wavelengths. This component extends almost to the edge of the disk and has a height equal to 50% of the disk radius. As it has a low brightness temperature, we identify it as the optically-thick base of the disk wind. V1408 Aql (=4U 1957+115) is a low mass X-ray binary which continues to be a black hole candidate. We have new photometric data of this system from the Otto Struve 2.1-m telescope's high speed CCD photometer at McDonald Observatory. The light curve is largely sinusoidal which we model with two components: a constant light source from the disk and a sinusoidal modulation at the orbital period from the irradiated face of the companion star. This is a radical re-interpretation of the orbital light curve. We do not require a large or asymmetric disk rim to account for the modulation in the light curve. Thus, the orbital inclination is unconstrained in our new model, removing the foundation for any claims of the compact object being a black hole.
2. Evidence for a correlation between mass accretion rates onto young stars and the mass of their protoplanetary disks
Manara, C. F.; Rosotti, G.; Testi, L.; Natta, A.; Alcalá, J. M.; Williams, J. P.; Ansdell, M.; Miotello, A.; van der Marel, N.; Tazzari, M.; Carpenter, J.; Guidi, G.; Mathews, G. S.; Oliveira, I.; Prusti, T.; van Dishoeck, E. F.
2016-06-01
A relation between the mass accretion rate onto the central young star and the mass of the surrounding protoplanetary disk has long been theoretically predicted and observationally sought. For the first time, we have accurately and homogeneously determined the photospheric parameters, mass accretion rate, and disk mass for an essentially complete sample of young stars with disks in the Lupus clouds. Our work combines the results of surveys conducted with VLT/X-Shooter and ALMA. With this dataset we are able to test a basic prediction of viscous accretion theory, the existence of a linear relation between the mass accretion rate onto the central star and the total disk mass. We find a correlation between the mass accretion rate and the disk dust mass, with a ratio that is roughly consistent with the expected viscous timescale when assuming an interstellar medium gas-to-dust ratio. This confirms that mass accretion rates are related to the properties of the outer disk. We find no correlation between mass accretion rates and the disk mass measured by CO isotopologues emission lines, possibly owing to the small number of measured disk gas masses. This suggests that the mm-sized dust mass better traces the total disk mass and that masses derived from CO may be underestimated, at least in some cases.
3. Efficiency of the Keplerian accretion in braneworld Kerr-Newman spacetimes and mining instability of some naked singularity spacetimes
Blaschke, Martin; Stuchlík, Zdeněk
2016-10-01
We show that the braneworld rotating Kerr-Newman black hole and naked singularity spacetimes with both positive and negative braneworld tidal charge parameters can be separated into 14 classes according to the properties of circular geodesics governing the Keplerian accretion. We determine the efficiency of the Keplerian accretion disks for all braneworld Kerr-Newman spacetimes. We demonstrate the occurrence of an infinitely deep gravitational potential in Kerr-Newman naked singularity spacetimes having the braneworld dimensionless tidal charge b ∈(1 /4 ,1 ) and the dimensionless spin a ∈(2 √{b }-√{b (4 b -1 ) } , 2 √{b }+√{b (4 b -1 ) }) , implying unbound efficiency of the Keplerian accretion and the possibility of extracting the whole naked singularity mass. Therefore, we call them braneworld "mining-unstable" Kerr-Newman naked singularity spacetimes. Fundamental restriction on the relevance of the extraordinary—but fully classical—phenomenon of the mining instability is given by validity of the assumption of geodesic motion of the accreting matter.
4. Cooling Requirements for the Vertical Shear Instability in Protoplanetary Disks
Lin, Min-Kai; Youdin, Andrew N.
2015-09-01
The vertical shear instability (VSI) offers a potential hydrodynamic mechanism for angular momentum transport in protoplanetary disks (PPDs). The VSI is driven by a weak vertical gradient in the disk’s orbital motion, but must overcome vertical buoyancy, a strongly stabilizing influence in cold disks, where heating is dominated by external irradiation. Rapid radiative cooling reduces the effective buoyancy and allows the VSI to operate. We quantify the cooling timescale tc needed for efficient VSI growth, through a linear analysis of the VSI with cooling in vertically global, radially local disk models. We find the VSI is most vigorous for rapid cooling with {t}{{c}}\\lt {{{Ω }}}{{K}}-1h| q| /(γ -1) in terms of the Keplerian orbital frequency, {{{Ω }}}{{K}}, the disk’s aspect-ratio, h\\ll 1, the radial power-law temperature gradient, q, and the adiabatic index, γ. For longer tc, the VSI is much less effective because growth slows and shifts to smaller length scales, which are more prone to viscous or turbulent decay. We apply our results to PPD models where tc is determined by the opacity of dust grains. We find that the VSI is most effective at intermediate radii, from ∼5 to ∼50 AU with a characteristic growth time of ∼30 local orbital periods. Growth is suppressed by long cooling times both in the opaque inner disk and the optically thin outer disk. Reducing the dust opacity by a factor of 10 increases cooling times enough to quench the VSI at all disk radii. Thus the formation of solid protoplanets, a sink for dust grains, can impede the VSI.
5. Young Stellar Objects in Lynds 1641: Disks, Accretion, and Star Formation History
Fang, Min; Kim, Jinyoung Serena; van Boekel, Roy; Sicilia-Aguilar, Aurora; Henning, Thomas; Flaherty, Kevin
2013-07-01
We investigate the young stellar objects (YSOs) in the Lynds 1641 (L1641) cloud using multi-wavelength data including Spitzer, WISE, the Two Micron All Sky Survey, and XMM covering ~1390 YSOs across a range of evolutionary stages. In addition, we targeted a sub-sample of YSOs for optical spectroscopy with the MMT/Hectospec and the MMT/Hectochelle. We use these data, along with archival photometric data, to derive spectral types, extinction values, masses, ages, and accretion rates. We obtain a disk fraction of ~50% in L1641. The disk frequency is almost constant as a function of stellar mass with a slight peak at log (M */M ⊙) ≈ -0.25. The analysis of multi-epoch spectroscopic data indicates that the accretion variability of YSOs cannot explain the two orders of magnitude of scatter for YSOs with similar masses. Forty-six new transition disk (TD) objects are confirmed in this work, and we find that the fraction of accreting TDs is lower than for optically thick disks (40%-45% versus 77%-79%, respectively). We confirm our previous result that the accreting TDs have a median accretion rate similar to normal optically thick disks. We confirm that two star formation modes (isolated versus clustered) exist in L1641. We find that the diskless YSOs are statistically older than the YSOs with optically thick disks and the TD objects have a median age that is intermediate between those of the other two populations. We tentatively study the star formation history in L1641 based on the age distribution and find that star formation started to be active 2-3 Myr ago.
6. General relativistic hydrodynamics with Adaptive-Mesh Refinement (AMR) and modeling of accretion disks
Donmez, Orhan
We present a general procedure to solve the General Relativistic Hydrodynamical (GRH) equations with Adaptive-Mesh Refinement (AMR) and model of an accretion disk around a black hole. To do this, the GRH equations are written in a conservative form to exploit their hyperbolic character. The numerical solutions of the general relativistic hydrodynamic equations is done by High Resolution Shock Capturing schemes (HRSC), specifically designed to solve non-linear hyperbolic systems of conservation laws. These schemes depend on the characteristic information of the system. We use Marquina fluxes with MUSCL left and right states to solve GRH equations. First, we carry out different test problems with uniform and AMR grids on the special relativistic hydrodynamics equations to verify the second order convergence of the code in 1D, 2 D and 3D. Second, we solve the GRH equations and use the general relativistic test problems to compare the numerical solutions with analytic ones. In order to this, we couple the flux part of general relativistic hydrodynamic equation with a source part using Strang splitting. The coupling of the GRH equations is carried out in a treatment which gives second order accurate solutions in space and time. The test problems examined include shock tubes, geodesic flows, and circular motion of particle around the black hole. Finally, we apply this code to the accretion disk problems around the black hole using the Schwarzschild metric at the background of the computational domain. We find spiral shocks on the accretion disk. They are observationally expected results. We also examine the star-disk interaction near a massive black hole. We find that when stars are grounded down or a hole is punched on the accretion disk, they create shock waves which destroy the accretion disk.
7. Shock-driven Accretion in Circumplanetary Disks: Observables and Satellite Formation
Zhu, Zhaohuan; Ju, Wenhua; Stone, James M.
2016-12-01
Circumplanetary disks (CPDs) control the growth of planets, supply material for satellites to form, and provide observational signatures of young forming planets. We have carried out two-dimensional hydrodynamical simulations with radiative cooling to study CPDs and suggested a new mechanism to drive the disk accretion. Two spiral shocks are present in CPDs, excited by the central star. We find that spiral shocks can at least contribute to, if not dominate, the angular momentum transport and energy dissipation in CPDs. Meanwhile, dissipation and heating by spiral shocks have a positive feedback on shock-driven accretion itself. As the disk is heated up by spiral shocks, the shocks become more open, leading to more efficient angular momentum transport. This shock-driven accretion is, on the other hand, unsteady due to production and destruction of vortices in disks. After being averaged over time, a quasi-steady accretion is reached from the planet’s Hill radius all the way to the planet surface, and the disk α coefficient characterizing angular momentum transport is ˜0.001-0.02. The disk surface density ranges from 10 to 1000 g cm-2 in our simulations, which is at least three orders of magnitude smaller than the “minimum-mass subnebula” model used to study satellite formation; instead it is more consistent with the “gas-starved” satellite formation model. Finally, we calculate the millimeter flux emitted by CPDs at ALMA and EVLA wavelength bands and predict the flux for several recently discovered CPD candidates, which suggests that ALMA is capable of discovering these accreting CPDs.
8. THE TORQUING OF CIRCUMNUCLEAR ACCRETION DISKS BY STARS AND THE EVOLUTION OF MASSIVE BLACK HOLES
SciTech Connect
Bregman, Michal; Alexander, Tal
2012-03-20
An accreting massive black hole (MBH) in a galactic nucleus is surrounded by a dense stellar cluster. We analyze and simulate numerically the evolution of a thin accretion disk due to its internal viscous torques, due to the frame-dragging torques of a spinning MBH (the Bardeen-Petterson effect), and due to the orbit-averaged gravitational torques by the stars (resonant relaxation). We show that the evolution of the MBH mass accretion rate, the MBH spin growth rate, and the covering fraction of the disk relative to the central ionizing continuum source, are all strongly coupled to the stochastic fluctuations of the stellar potential via the warps that the stellar torques excite in the disk. These lead to fluctuations by factors of up to a few in these quantities over a wide range of timescales, with most of the power on timescales {approx}> (M{sub .}/M{sub d} )P(R{sub d} ), where M{sub .} and M{sub d} are the masses of the MBH and disk, and P is the orbital period at the disk's mass-weighted mean radius R{sub d}. The response of the disk is stronger the lighter it is and the more centrally concentrated the stellar cusp. As proof of concept, we simulate the evolution of the low-mass maser disk in NGC 4258 and show that its observed O(10 Degree-Sign ) warp can be driven by the stellar torques. We also show that the frame dragging of a massive active galactic nucleus disk couples the stochastic stellar torques to the MBH spin and can excite a jitter of a few degrees in its direction relative to that of the disk's outer regions.
9. The Torquing of Circumnuclear Accretion Disks by Stars and the Evolution of Massive Black Holes
Bregman, Michal; Alexander, Tal
2012-03-01
An accreting massive black hole (MBH) in a galactic nucleus is surrounded by a dense stellar cluster. We analyze and simulate numerically the evolution of a thin accretion disk due to its internal viscous torques, due to the frame-dragging torques of a spinning MBH (the Bardeen-Petterson effect), and due to the orbit-averaged gravitational torques by the stars (resonant relaxation). We show that the evolution of the MBH mass accretion rate, the MBH spin growth rate, and the covering fraction of the disk relative to the central ionizing continuum source, are all strongly coupled to the stochastic fluctuations of the stellar potential via the warps that the stellar torques excite in the disk. These lead to fluctuations by factors of up to a few in these quantities over a wide range of timescales, with most of the power on timescales >~ (M •/Md )P(Rd ), where M • and Md are the masses of the MBH and disk, and P is the orbital period at the disk's mass-weighted mean radius Rd . The response of the disk is stronger the lighter it is and the more centrally concentrated the stellar cusp. As proof of concept, we simulate the evolution of the low-mass maser disk in NGC 4258 and show that its observed O(10°) warp can be driven by the stellar torques. We also show that the frame dragging of a massive active galactic nucleus disk couples the stochastic stellar torques to the MBH spin and can excite a jitter of a few degrees in its direction relative to that of the disk's outer regions.
10. Thermo-galvanometric instabilities in magnetized plasma disks
Franco, Alessio; Montani, Giovanni; Carlevaro, Nakia
2014-11-01
In this work, we present a linear stability analysis of fully-ionized rotating plasma disks with a temperature gradient and a sub-thermal background magnetic field (oriented towards the axial direction). We describe how the plasma reacts when galvanometric and thermo-magnetic phenomena, such as Hall and Nernst-Ettingshausen effects, are taken into account, and meridian perturbations of the plasma are considered. It is shown how, in the ideal case, this leads to a significant overlap of the Magneto-rotational Instability and the Thermo-magnetic one. Considering dissipative effects, an overall damping of the unstable modes, although not sufficient to fully suppress the instability, appears especially in the thermo-magnetic related branch of the curve.
11. Super-Eddington accretion disks in Ultraluminous X-ray sources
Fabrika, S.; Vinokurov, A.; Atapin, K.; Sholukhova, O.
2016-06-01
The origin of Ultraluminous X-ray sources (ULXs) in external galaxies whose X-ray luminosities exceed those of the brightest black holes in our Galaxy hundreds and thousands times is mysterious. The most popular models for the ULXs involve either intermediate mass black holes (IMBHs) or stellar-mass black holes accreting at super-Eddington rates. Here we review the ULX properties, their X-ray spectra indicate the presence of hot winds in their accretion disks supposing the supercritical accretion. However, the strongest evidences come from optical spectroscopy. The spectra of the ULX counterparts are very similar to that of SS433, the only known supercritical accretor in our Galaxy. The spectra are apparently of WNL type (late nitrogen Wolf-Rayet stars) or LBV (luminous blue variables) in their hot state, which are very scarce stellar objects. We find that the spectra do not originate from WNL/LBV type donors but from very hot winds from the accretion disks, whose physical conditions are similar to those in stellar winds from these stars. The results suggest that bona-fide ULXs must constitute a homogeneous class of objects, which most likely have supercritical accretion disks.
12. Crossing the Eddington Limit: Examining Disk Spectra at High Accretion Rates
Sutton, Andrew D.; Swartz, Douglas A.; Roberts, Timothy P.; Middleton, Matthew J.; Soria, Roberto; Done, Chris
2017-02-01
The faintest ultraluminous X-ray sources (ULXs), those with 0.3–10 keV luminosities 1< {L}{{X}}/{10}39< 3 {erg} {{{s}}}-1, tend to have X-ray spectra that are disk-like but broader than expected for thin accretion disks. These “broadened disk (BD)” spectra are thought to indicate near- or mildly super-Eddington accretion onto stellar remnant black holes. Here we report that a sample of bright thermal-dominant black hole binaries, which have Eddington ratios constrained to moderate values, also show BD spectra in the 0.3–10 keV band at an order of magnitude lower luminosities. This broadening would be missed in studies that only look above ∼ 2 {keV}. While this may suggest that BD ULXs could be powered by accretion onto massive stellar remnant black holes with close to maximal spin, we argue in favor of a scenario where they are at close to the Eddington luminosity, such that radiation pressure would be expected to result in geometrically slim, advective accretion disks. However, this implies that an additional physical mechanism is required to produce the observed broad spectra at low Eddington ratios.
13. Accretion disk winds as the jet suppression mechanism in the microquasar GRS 1915+105.
PubMed
Neilsen, Joseph; Lee, Julia C
2009-03-26
Stellar-mass black holes with relativistic jets, also known as microquasars, mimic the behaviour of quasars and active galactic nuclei. Because timescales around stellar-mass black holes are orders of magnitude smaller than those around more distant supermassive black holes, microquasars are ideal nearby 'laboratories' for studying the evolution of accretion disks and jet formation in black-hole systems. Whereas studies of black holes have revealed a complex array of accretion activity, the mechanisms that trigger and suppress jet formation remain a mystery. Here we report the presence of a broad emission line in the faint, hard states and narrow absorption lines in the bright, soft states of the microquasar GRS 1915+105. ('Hard' and 'soft' denote the character of the emitted X-rays.) Because the hard states exhibit prominent radio jets, we argue that the broad emission line arises when the jet illuminates the inner accretion disk. The jet is weak or absent during the soft states, and we show that the absorption lines originate when the powerful radiation field around the black hole drives a hot wind off the accretion disk. Our analysis shows that this wind carries enough mass away from the disk to halt the flow of matter into the radio jet.
14. Instabilities of interacting vortex rings generated by an oscillating disk
Deng, Jian; Teng, Lubao; Caulfield, C. P.; Mao, Xuerui
2016-09-01
We propose a natural model to probe in a controlled fashion the instability of interacting vortex rings shed from the edge of an oblate spheroid disk of major diameter c , undergoing oscillations of frequency f0 and amplitude A . We perform a Floquet stability analysis to determine the characteristics of the instability modes, which depend strongly on the azimuthal (integer) wave number m . We vary two key control parameters, the Keulegan-Carpenter number KC=2 π A /c and the Stokes number β =f0c2/ν , where ν is the kinematic viscosity of the fluid. We observe two distinct flow regimes. First, for sufficiently small β , and hence low frequency of oscillation corresponding to relatively weak interaction between sequentially shedding vortex rings, symmetry breaking occurs directly to a single unstable mode with m =1 . Second, for sufficiently large yet fixed values of β , corresponding to a higher oscillation frequency and hence stronger ring-ring interaction, the onset of asymmetry is predicted to occur due to two branches of high m instabilities as the amplitude is increased, with m =1 structures being dominant only for sufficiently large values of KC. These two branches can be distinguished by the phase properties of the vortical structures above and below the disk. The region in (KC,β ) parameter space where these two high m instability branches arise can be described accurately in terms of naturally defined Reynolds numbers, using appropriately chosen characteristic length scales. We subsequently carry out direct numerical simulations of the fully three-dimensional flow to verify the principal characteristics of the Floquet analysis, in particular demonstrating that high wave-number symmetry-breaking generically occurs when vortex rings sequentially interact sufficiently strongly.
15. From Dust to Dust: Protoplanetary Disk Accretion, Hot Jupiter Climates, and the Evaporation of Rocky Planets
Perez-Becker, Daniel Alonso
2013-12-01
This dissertation is composed of three independent projects in astrophysics concerning phenomena that are concurrent with the birth, life, and death of planets. In Chapters 1 & 2, we study surface layer accretion in protoplanetary disks driven stellar X-ray and far-ultraviolet (FUV) radiation. In Chapter 3, we identify the dynamical mechanisms that control atmospheric heat redistribution on hot Jupiters. Finally, in Chapter 4, we characterize the death of low-mass, short-period rocky planets by their evaporation into a dusty wind. Chapters 1 & 2: Whether protoplanetary disks accrete at observationally significant rates by the magnetorotational instability (MRI) depends on how well ionized they are. We find that disk surface layers ionized by stellar X-rays are susceptible to charge neutralization by condensates---ranging from mum-sized dust to angstrom-sized polycyclic aromatic hydrocarbons (PAHs). Ion densities in X-ray-irradiated surfaces are so low that ambipolar diffusion weakens the MRI. In contrast, ionization by stellar FUV radiation enables full-blown MRI turbulence in disk surface layers. Far-UV ionization of atomic carbon and sulfur produces a plasma so dense that it is immune to ion recombination on grains and PAHs. Even though the FUV-ionized layer is ˜10--100 times more turbulent than the X-ray-ionized layer, mass accretion rates of both layers are comparable because FUV photons penetrate to lower surface densities than do X-rays. We conclude that surface layer accretion occurs at observationally significant rates at radii ≳ 1--10 AU. At smaller radii, both X-ray- and FUV-ionized surface layers cannot sustain the accretion rates generated at larger distance and an additional means of transport is needed. In the case of transitional disks, it could be provided by planets. Chapter 3: Infrared light curves of transiting hot Jupiters present a trend in which the atmospheres of the hottest planets are less efficient at redistributing the stellar energy
16. Regulation of black-hole accretion by a disk wind during a violent outburst of V404 Cygni.
PubMed
Muñoz-Darias, T; Casares, J; Mata Sánchez, D; Fender, R P; Armas Padilla, M; Linares, M; Ponti, G; Charles, P A; Mooley, K P; Rodriguez, J
2016-06-02
Accretion of matter onto black holes is universally associated with strong radiative feedback and powerful outflows. In particular, black-hole transients have outflows whose properties are strongly coupled to those of the accretion flow. This includes X-ray winds of ionized material, expelled from the accretion disk encircling the black hole, and collimated radio jets. Very recently, a distinct optical variability pattern has been reported in the transient stellar-mass black hole V404 Cygni, and interpreted as disrupted mass flow into the inner regions of its large accretion disk. Here we report observations of a sustained outer accretion disk wind in V404 Cyg, which is unlike any seen hitherto. We find that the outflowing wind is neutral, has a large covering factor, expands at one per cent of the speed of light and triggers a nebular phase once accretion drops sharply and the ejecta become optically thin. The large expelled mass (>10(-8) solar masses) indicates that the outburst was prematurely ended when a sizeable fraction of the outer disk was depleted by the wind, detaching the inner regions from the rest of the disk. The luminous, but brief, accretion phases shown by transients with large accretion disks imply that this outflow is probably a fundamental ingredient in regulating mass accretion onto black holes.
17. Accretion Disks Around Binary Black Holes of Unequal Mass: GRMHD Simulations Near Decoupling
NASA Technical Reports Server (NTRS)
Gold, Roman; Paschalidis, Vasileios; Etienne, Zachariah B.; Shapiro, Stuart L.; Pfeiffer, Harald, P.
2013-01-01
We report on simulations in general relativity of magnetized disks onto black hole binaries. We vary the binary mass ratio from 1:1 to 1:10 and evolve the systems when they orbit near the binary disk decoupling radius. We compare (surface) density profiles, accretion rates (relative to a single, non-spinning black hole), variability, effective alpha-stress levels and luminosities as functions of the mass ratio. We treat the disks in two limiting regimes: rapid radiative cooling and no radiative cooling. The magnetic field lines clearly reveal jets emerging from both black hole horizons and merging into one common jet at large distances. The magnetic fields give rise to much stronger shock heating than the pure hydrodynamic flows, completely alter the disk structure, and boost accretion rates and luminosities. Accretion streams near the horizons are among the densest structures; in fact, the 1:10 no-cooling evolution results in a refilling of the cavity. The typical effective temperature in the bulk of the disk is approx. 10(exp5) (M / 10(exp 8)M solar mass (exp -1/4(L/L(sub edd) (exp 1/4K) yielding characteristic thermal frequencies approx. 10 (exp 15) (M /10(exp 8)M solar mass) (exp -1/4(L/L (sub edd) (1+z) (exp -1)Hz. These systems are thus promising targets for many extragalactic optical surveys, such as LSST, WFIRST, and PanSTARRS.
18. Integrated mechanism that both removes accretion disk angular momentum and drives astrophysical jets
Bellan, Paul
2016-10-01
Using concepts from laboratory experiments, Hamiltonian mechanics, Hall MHD, and weakly ionized plasmas, I propose a mechanism that simultaneously drives astrophysical jets and removes accretion disk angular momentum. The mechanism depends on the extreme stratification of ionization between the weakly ionized accretion disk and the highly ionized exterior region. In the exterior region, axisymmetric Hamiltonian mechanics constrain charged particles to move on nested poloidal flux surfaces. In contrast, fluid elements in the weakly ionized, highly collisional accretion disk behave like collisionless meta-particles with effective q / m reduced from than that of an ion by the nominal disk 10-15 - 10-8 fractional ionization; this means that the meta-particle effective cyclotron frequency ωc can be of order of the Kepler frequency ωK =(MG /r3) 1 / 2 . Meta-particles with ωc = - 2ωK have zero canonical angular momentum, experience no centrifugal force and spiral in towards the central body. Because these inward spiraling meta-particles are positive, their accumulation near the central body produces radially and axially outward electric fields. The axial outward electric field drives an out-of-plane poloidal electric current along poloidal flux surfaces in the external region. As in lab experiments, this current and its associated toroidal magnetic field drive astrophysical jets flowing normal to and away from the disk. Supported by NSF/DOE Partnership in Plasma Physics.
19. Shrinking galaxy disks with fountain-driven accretion from the halo
SciTech Connect
Elmegreen, Bruce G.; Struck, Curtis; Hunter, Deidre A. E-mail: [email protected]
2014-12-01
Star formation in most galaxies requires cosmic gas accretion because the gas consumption time is short compared to the Hubble time. This accretion presumably comes from a combination of infalling satellite debris, cold flows, and condensation of hot halo gas at the cool disk interface, perhaps aided by a galactic fountain. In general, the accretion will have a different specific angular momentum than the part of the disk that receives it, even if the gas comes from the nearby halo. The gas disk then expands or shrinks over time. Here we show that condensation of halo gas at a rate proportional to the star formation rate in the fountain model will preserve an initial shape, such as an exponential, with a shrinking scale length, leaving behind a stellar disk with a slightly steeper profile of younger stars near the center. This process is slow for most galaxies, producing imperceptible radial speeds, and it may be dominated by other torques, but it could be important for blue compact dwarfs, which tend to have large, irregular gas reservoirs and steep blue profiles in their inner stellar disks.
20. Truncation of the Inner Accretion Disk Around a Black Hole at Low Luminosity
NASA Technical Reports Server (NTRS)
Tomsick, John A.; Yamoka, Kazutaka; Corbel, Stephane; Kaaret, Philip; Kalemci, Emrah; Migliari, Simone
2011-01-01
Most black hole binaries show large changes in X-ray luminosity caused primarily by variations in mass accretion rate. An important question for understanding black hole accretion and jet production is whether the inner edge of the accretion disk recedes at low accretion rate. Measurements of the location of the inner edge (R(sub in)) can be made using iron emission lines that arise due to fluorescence of iron in the disk, and these indicate that R(sub in) is very close to the black hole at high and moderate luminosities (greater than or equal to 1% of the Eddington luminosity, L(sub Edd). Here, we report on X-ray observations of the black hole GX 339-4 in the hard state by Suzaku and the Rossi X-ray Timing Explorer that extend iron line studies to 0.14% L(sub Edd) and show that R(sub in) increases by a factor of greater than 27 over the value found when GX 339-4 was bright. The exact value of R(sub in) depends on the inclination of the inner disk (i), and we derive 90% confidence limits of R(sub in) greater than 35 R(sub g) at i = 0 degrees and R(sub in) greater than 175 R(sub g) at i = 30 degrees. This provides direct evidence that the inner portion of the disk is not present at low luminosity, allowing for the possibility that the inner disk is replaced by advection- or magnetically dominated accretion flows.
1. Truncation of the Inner Accretion Disk Around a Black Hole at Low Luminosity
NASA Technical Reports Server (NTRS)
Tomsick, John A.; Yamaoka, Kazutaka; Corbel, Stephane; Kaaret, Philip; Kalemci, Emrah; Migliari, Simone
2009-01-01
Most black hole binaries show large changes in X-ray luminosity caused primarily by variations in mass accretion rate. An important question for understanding black hole accretion and jet production is whether the inner edge of the accretion disk recedes at low accretion rate. Measurements of the location of the inner edge (R(sub in)) can be made using iron emission lines that arise due to fluorescence of iron in the disk, and these indicate that R(sub in) is very close to the black hole at high and moderate luminosities (greater than approximately equal to 1% of the Eddington luminosity, L(sub Edd). Here, we report on X-ray observation of the black hole GX 339-4 in the hard state by Suzaku and the Rossi X-ray Timing Explorer (RXTE) that extend iron line studies to 0.14% L(sub Edd) and show that R(sub in) increases by a factor of greater than 27 over the value found when GX 339-4 was bright. The exact value of R(sub in) depends on the inclination of the inner disk (i), and we derive 90% confidence limits of R(sub in) greater than 35R(sub g) at i = 0 degrees and R(sub in) greater than 175R(sub g) at i = 30 degrees. This provides direct evidence that the inner portion of the disk is not present at low luminosity, allowing for the possibility that the inner disk is replaced by advection- or magnetically-dominated accretion flows.
2. Wind from the black-hole accretion disk driving a molecular outflow in an active galaxy.
PubMed
Tombesi, F; Meléndez, M; Veilleux, S; Reeves, J N; González-Alfonso, E; Reynolds, C S
2015-03-26
Powerful winds driven by active galactic nuclei are often thought to affect the evolution of both supermassive black holes and their host galaxies, quenching star formation and explaining the close relationship between black holes and galaxies. Recent observations of large-scale molecular outflows in ultraluminous infrared galaxies support this quasar-feedback idea, because they directly trace the gas from which stars form. Theoretical models suggest that these outflows originate as energy-conserving flows driven by fast accretion-disk winds. Proposed connections between large-scale molecular outflows and accretion-disk activity in ultraluminous galaxies were incomplete because no accretion-disk wind had been detected. Conversely, studies of powerful accretion-disk winds have until now focused only on X-ray observations of local Seyfert galaxies and a few higher-redshift quasars. Here we report observations of a powerful accretion-disk wind with a mildly relativistic velocity (a quarter that of light) in the X-ray spectrum of IRAS F11119+3257, a nearby (redshift 0.189) optically classified type 1 ultraluminous infrared galaxy hosting a powerful molecular outflow. The active galactic nucleus is responsible for about 80 per cent of the emission, with a quasar-like luminosity of 1.5 × 10(46) ergs per second. The energetics of these two types of wide-angle outflows is consistent with the energy-conserving mechanism that is the basis of the quasar feedback in active galactic nuclei that lack powerful radio jets (such jets are an alternative way to drive molecular outflows).
3. A GENERAL RELATIVISTIC MODEL OF ACCRETION DISKS WITH CORONAE SURROUNDING KERR BLACK HOLES
SciTech Connect
You Bei; Cao Xinwu; Yuan Yefei E-mail: [email protected]
2012-12-20
We calculate the structure of a standard accretion disk with a corona surrounding a massive Kerr black hole in the general relativistic frame, in which the corona is assumed to be heated by the reconnection of the strongly buoyant magnetic fields generated in the cold accretion disk. The emergent spectra of accretion disk-corona systems are calculated by using the relativistic ray-tracing method. We propose a new method to calculate the emergent Comptonized spectra from the coronae. The spectra of disk-corona systems with a modified {alpha}-magnetic stress show that both the hard X-ray spectral index and the hard X-ray bolometric correction factor L{sub bol}/L{sub X,2-10keV} increase with the dimensionless mass accretion rate, which is qualitatively consistent with the observations of active galactic nuclei. The fraction of the power dissipated in the corona decreases with increasing black hole spin parameter a, which leads to lower electron temperatures of the coronae for rapidly spinning black holes. The X-ray emission from the coronae surrounding rapidly spinning black holes becomes weak and soft. The ratio of the X-ray luminosity to the optical/UV luminosity increases with the viewing angle, while the spectral shape in the X-ray band is insensitive to the viewing angle. We find that the spectral index in the infrared waveband depends on the mass accretion rate and the black hole spin a, which deviates from the f{sub {nu}}{proportional_to}{nu}{sup 1/3} relation expected by the standard thin disk model.
4. How to Determine The Precession of the Inner Accretion Disk in Cygnus X-1
SciTech Connect
Torres, D F; Romero, G E; Barcons, X; Lu, Y
2005-01-05
We show that changes in the orientation of the inner accretion disk of Cygnus X-1 affect the shape of the broad Fe K{alpha} emission line emitted from this object, in such a way that eV-level spectral resolution observations (such as those that will be carried out by the ASTRO-E2 satellite) can be used to analyze the dynamics of the disk. We here present a new diagnosis tool, supported by numerical simulations, by which short observations of Cygnus X-1, separated in time, can determine whether its accretion disk actually processes, and if so, determine its period and precession angle. Knowing the precession parameters of Cygnus X-1 would result in a clarification of the origin of such precession, distinguishing between tidal and spin-spin coupling. This approach could also be used for similar studies in other microquasar systems.
5. High-energy particle acceleration by explosive electromagnetic interaction in an accretion disk
NASA Technical Reports Server (NTRS)
Haswell, C. A.; Tajima, T.; Sakai, J.-I.
1992-01-01
By examining electromagnetic field evolution occurring in an accretion disk around a compact object, we arrive at an explosive mechanism of particle acceleration. Flux-freezing in the differentially rotating disk causes the seed and/or generated magnetic field to wrap up tightly, becoming highly sheared and locally predominantly azimuthal in orientation. We show how asymptotically nonlinear solutions for the electromagnetic fields may arise in isolated plasma blobs as a result of the driving of the fluid equations by the accretion flow. These fields are capable of rapidly accelerating charged particles from the disk. Acceleration through the present mechanism from AGN can give rise to energies beyond 10 exp 20 eV. Such a mechanism may present an explanation for the extragalactic origin of the most energetic observed cosmic rays.
6. The Production of Jets From Magnetic Accretion Disks: Simulation of the Blandford-Payne Mechanism
NASA Technical Reports Server (NTRS)
Meier, David L.
1995-01-01
We have performed magnetohydrodynamic (MRD) simulations of the production of jets from magnetized accretion disks with a factor of 5 greater extent in space and time, and with more models, than any study published so far. We find that jets are produced by such disks in a broad range of parameter space, and by at least two different mechanisms. We also are able to follow the propagation of the jet well beyond the accretion disk into the region of hydrodynamic collimation. The code used is our MHD simulation code FLOW (K. Lind, D. Payne, D. Meier, and R. Blandford, 1989), converted to run on Caltech's massively parallel Intel Touchstone Delta supercomputer. Some of these models may be directly applicable to observed radio sources.
7. GLOBAL DRAG-INDUCED INSTABILITIES IN PROTOPLANETARY DISKS
SciTech Connect
Jalali, Mir Abbas
2013-07-20
We use the Fokker-Planck equation and model the dispersive dynamics of solid particles in annular protoplanetary disks whose gas component is more massive than the particle phase. We model particle-gas interactions as hard sphere collisions, determine the functional form of diffusion coefficients, and show the existence of two global unstable modes in the particle phase. These modes have spiral patterns with the azimuthal wavenumber m = 1 and rotate slowly. We show that in ring-shaped disks, the phase-space density of solid particles increases linearly in time toward an accumulation point near the location of pressure maximum, while instabilities grow exponentially. Therefore, planetesimals and planetary cores can be efficiently produced near the peaks of unstable density waves. In this mechanism, particles migrating toward the accumulation point will not participate in the formation of planets, and should eventually form a debris ring like the main asteroid belt or classical Kuiper Belt objects. We present the implications of global instabilities to the formation of ice giants and terrestrial planets in the solar system.
8. Pair-density transitions in accretion disk coronae
NASA Technical Reports Server (NTRS)
Kusunose, Masaaki; Mineshige, Shin
1991-01-01
The thermal and e(+)e(-)-pair equilibrium structure of two-temperature disk coronae above a cool (about 10 exp 6 K) disk around a black hole of 10 solar masses are investigated. Soft photons are assumed to be amply supplied from the cool disk. Two-pair thermal equilibrium points are found for a given proton column density: the low state with very small pair density and the high state dominated by pairs. Both states are thermally unstable, while for perturbations in pair density the high state is unstable and the low state is stable. Two possible scenarios are discussed for the fate of a two-temperature corona. When the proton optical depth is relatively small (e.g., less than 1) and the temperature of input soft photons is low (e.g., less than 10 exp 6 K), the corona will undergo a limit cycle between the high state and the low state on a time scale of milliseconds. As a consequence of Compton scattering of the soft photons, the emergent spectrum in the high state is rather flat with a big Wien bump at about 100 keV, whereas it is composed of a power-law component in the low state. Some observational consequences are briefly discussed in connection with the high-low spectral transition in Cyg X-1.
9. Gravitational Influences on Magnetic Field Structure in Accretion Disks*
Schneck, K.; Coppi, B.
2009-11-01
The structure of the magnetic fields associated with plasma disks surrounding black holes is identified when the effects of gravitational and Lorentz forces on the dynamics of the disk are comparable. The effects of corrections to the radial gravitational force% ρGM*R(R^2+z^2)^3/2 are explored within the geometry of a thin disk. A significant external magnetic field component is considered, along with an internal component due to the azimuthal current configuration. The relation of the resulting configuration to the field structure when the gravitational force can be neglectedfootnotetextB. Coppi, Phys. Plasmas 12, 057302 (2005)^,footnotetextCoppi, B. and Rousseau, F. Astrophysical Journal, 641: 458-470 (2006) is discussed. The relevant equations for the pseudo-Newtonian potentialfootnotetextPaczy'nski, B. and Wiita, P. J. Astron. Astrophys. 88: 23 (1980) describing the physics near the event horizon of the black hole are also derived and the physical consequences are explored. *Sponsored in part by the U.S. Department of Energy and the MIT Undergraduate Research Opportunities Program.
10. BIPOLAR JETS LAUNCHED FROM MAGNETICALLY DIFFUSIVE ACCRETION DISKS. I. EJECTION EFFICIENCY VERSUS FIELD STRENGTH AND DIFFUSIVITY
SciTech Connect
Sheikhnezami, Somayeh; Fendt, Christian; Porth, Oliver; Vaidya, Bhargav; Ghanbari, Jamshid E-mail: [email protected]
2012-09-20
We investigate the launching of jets and outflows from magnetically diffusive accretion disks. Using the PLUTO code, we solve the time-dependent resistive magnetohydrodynamic equations taking into account the disk and jet evolution simultaneously. The main question we address is which kind of disks launch jets and which kind of disks do not? In particular, we study how the magnitude and distribution of the (turbulent) magnetic diffusivity affect mass loading and jet acceleration. We apply a turbulent magnetic diffusivity based on {alpha}-prescription, but also investigate examples where the scale height of diffusivity is larger than that of the disk gas pressure. We further investigate how the ejection efficiency is governed by the magnetic field strength. Our simulations last for up to 5000 dynamical timescales corresponding to 900 orbital periods of the inner disk. As a general result, we observe a continuous and robust outflow launched from the inner part of the disk, expanding into a collimated jet of superfast-magnetosonic speed. For long timescales, the disk's internal dynamics change, as due to outflow ejection and disk accretion the disk mass decreases. For magnetocentrifugally driven jets, we find that for (1) less diffusive disks, (2) a stronger magnetic field, (3) a low poloidal diffusivity, or (4) a lower numerical diffusivity (resolution), the mass loading of the outflow is increased-resulting in more powerful jets with high-mass flux. For weak magnetization, the (weak) outflow is driven by the magnetic pressure gradient. We consider in detail the advection and diffusion of magnetic flux within the disk and we find that the disk and outflow magnetization may substantially change in time. This may have severe impact on the launching and formation process-an initially highly magnetized disk may evolve into a disk of weak magnetization which cannot drive strong outflows. We further investigate the jet asymptotic velocity and the jet rotational velocity in
11. Is the Blazar Sequence related to accretion disk winds?
Boula, Stella; Mastichiadis, Apostolos; Kazanas, Demosthenes
2016-08-01
Adopting the hypothesis that the nonthermal emission of blazars is primarily due to the acceleration of electrons, we construct a simple leptonic model in order to explain the Blazar Sequence. The acceleration process is assumed to be of the first order Fermi type and the injected electrons and photons in the emitting region of the blazar are described by spatially averaged kinetic equations. According to the leptonic scenario, the spectral energy distributions of blazars have two basic components: a low frequency component, peaking in the optical through X-rays, from synchrotron emission; and a high frequency one, peaking in the γ rays, probably originating from Compton scattering of some seed photon source, either internal (synchrotron self-Compton) and/or external to the jet (external Compton). We find an adequate description of the Blazar Sequence by assuming a wind density profile of the form n(r) 1/r. Higher luminosity objects have higher accretion rates, higher optical thicknesses of the wind to Compton scattering and thus higher external photon fields than the lower luminosity ones. Therefore, we present indicative Blazar Sequence models which reproduce the basic observational trends just by varying one parameter, namely the mass accretion rate dot{m}.
12. Detailed Mid- and Far- Ultraviolet Model Spectra for Accretion Disks in Cataclysmic Binaries
NASA Technical Reports Server (NTRS)
1998-01-01
We present a large grid of computed far- and mid-ultraviolet spectra (850-2000 A) of the integrated light from steady-state accretion disks in luminous cataclysmic variables. The spectra are tabulated at 0.25 A intervals with an adopted FWHM resolution of 1.0 A, so they are suitable for use with observed spectra from a variety of modern space-borne observatories. Twenty-six different combinations of white dwarf mass M(sub wd) and mass accretion rate dot-m are considered, and spectra are presented for six different disk inclinations i. The disk models are computed self-consistently in the plane-parallel approximation, assuming LTE and vertical hydrostatic equilibrium, by solving simultaneously the radiative transfer, hydrostatic equilibrium, and energy balance equations. Irradiation from external sources is neglected. Local spectra of disk annuli are computed taking into account line transitions from elements 1-28 (H through Ni). Limb darkening as well as Doppler broadening and blending of lines are taken into account in computing the integrated disk spectra. The radiative properties of the models are discussed, including the dependence of ultraviolet fluxes and colors on M(sub wd), dot-m, and i. The appearance of the disk spectra is illustrated, with regard to changes in the same three parameters. Finally, possible future improvements to the present models and spectra are discussed.
13. How surface density of galaxy disks affects metallicity? Outflow and Accretion
Wu, Po-Feng; Kudritzki, Rolf-Peter; Tully, R. Brent; Neill, J. D.
2015-08-01
The surface density of disk is considered as a second parameter affecting the evolution of disk galaxies other than mass. Several physical and chemical properties of galaxies are found to be correlated with surface density of disk galaxies. However, the surface density, or surface brightness, is also strongly correlated with mass. It's not clear whether surface density really plays a role, or those correlations simply reflect the effect from stellar mass. To ask the question properly, one should take away the dependence on mass of galaxies, i.e., compare galaxies with the same mass but different surface densities.In this study, we ask, besides stellar mass, whether the surface density of disks also affects chemical evolution of galaxies. We demonstrate that, after removing the dependence on stellar mass and gas mass, the metallicity of galaxy still correlates with surface density of the galaxy disk. At the same stellar and gas mass, higher surface brightness galaxies on average possess both higher stellar and gas-phase metallicity, inferred from broadband color and spectrosopy of HII regions, respectively.We use an analytical model of chemical evolution involving gas outflow and accretion to explore possible reasons causing the difference in metallicity. Accroding to the model, at the same mass, lower metallicity galaxies should have experienced severer mass loss during star-formation events, and/or be inert to gas accretion. Both scenarios are consistent with general expections from properties of low surface density disks of shallow potential wells and dynamical stability.
14. Escape conditions of radiative-driven strati from luminous accretion disks
Nakai, Takuya; Fukue, Jun
2015-10-01
We examine the dynamical motion and escape conditions of continuum-driven strati (flat cloud) with finite optical depth from luminous accretion disks around a black hole. We adopt the near-disk approximation, and treat the problem in the framework of special relativity, where the radiation drag force as well as the radiation pressure are included. We find that the optically thin strati are easy to accelerate, compared with the particles, and the escape condition of the stratus is reduced. That is, when the disk luminosity is around the Eddington luminosity, the optically thin strati can escape from the inner disk (≲ 20 rg; rg being the Schwarzschild radius). When the disk luminosity is about half the Eddington luminosity, it can escape at around 5 rg. This is due to the translucent effect. In addition, the trajectories of the strati are not vertical, but a funnel-like shape due to the centrifugal force. Stratus outflow could easily blow out from usual accretion disks with sub-Eddington luminosities, and this may explain outflows observed in broad absorption line quasars and ultra-fast outflow objects.
15. The formation of a massive protostar through the disk accretion of gas.
PubMed
Chini, Rolf; Hoffmeister, Vera; Kimeswenger, Stefan; Nielbock, Markus; Nürnberger, Dieter; Schmidtobreick, Linda; Sterzik, Michael
2004-05-13
The formation of low-mass stars like our Sun can be explained by the gravitational collapse of a molecular cloud fragment into a protostellar core and the subsequent accretion of gas and dust from the surrounding interstellar medium. Theoretical considerations suggest that the radiation pressure from the protostar on the in-falling material may prevent the formation of stars above ten solar masses through this mechanism, although some calculations have claimed that stars up to 40 solar masses can in principle be formed via accretion through a disk. Given this uncertainty and the fact that most massive stars are born in dense clusters, it was suggested that high-mass stars are the result of the runaway merging of intermediate-mass stars. Here we report observations that clearly show a massive star being born from a large rotating accretion disk. The protostar has already assembled about 20 solar masses, and the accretion process is still going on. The gas reservoir of the circumstellar disk contains at least 100 solar masses of additional gas, providing sufficient fuel for substantial further growth of the forming star.
16. NUCLEOSYNTHESIS IN THE OUTFLOWS ASSOCIATED WITH ACCRETION DISKS OF TYPE II COLLAPSARS
SciTech Connect
Banerjee, Indrani; Mukhopadhyay, Banibrata E-mail: [email protected]
2013-11-20
We investigate nucleosynthesis inside the outflows from gamma-ray burst (GRB) accretion disks formed by the Type II collapsars. In these collapsars, massive stars undergo core collapse to form a proto-neutron star initially, and a mild supernova (SN) explosion is driven. The SN ejecta lack momentum, and subsequently this newly formed neutron star gets transformed to a stellar mass black hole via massive fallback. The hydrodynamics and the nucleosynthesis in these accretion disks have been studied extensively in the past. Several heavy elements are synthesized in the disk, and much of these heavy elements are ejected from the disk via winds and outflows. We study nucleosynthesis in the outflows launched from these disks by using an adiabatic, spherically expanding outflow model, to understand which of these elements thus synthesized in the disk survive in the outflow. While studying this, we find that many new elements like isotopes of titanium, copper, zinc, etc., are present in the outflows. {sup 56}Ni is abundantly synthesized in most of the cases in the outflow, which implies that the outflows from these disks in a majority of cases will lead to an observable SN explosion. It is mainly present when outflow is considered from the He-rich, {sup 56}Ni/{sup 54}Fe-rich zones of the disks. However, outflow from the Si-rich zone of the disk remains rich in silicon. Although emission lines of many of these heavy elements have been observed in the X-ray afterglows of several GRBs by Chandra, BeppoSAX, XMM-Newton, etc., Swift seems to have not yet detected these lines.
17. Supermassive star formation via episodic accretion: protostellar disc instability and radiative feedback efficiency
Sakurai, Y.; Vorobyov, E. I.; Hosokawa, T.; Yoshida, N.; Omukai, K.; Yorke, H. W.
2016-06-01
The formation of supermassive stars (SMSs) is a potential pathway to seed supermassive black holes in the early universe. A critical issue for forming SMSs is stellar UV feedback, which may limit the stellar mass growth via accretion. In this paper, we study the evolution of an accreting SMS and its UV emissivity with realistic variable accretion from a circumstellar disc. First we conduct a 2D hydrodynamical simulation to follow the protostellar accretion until the stellar mass exceeds 104 M⊙. The disc fragments by gravitational instability, creating many clumps that migrate inward to fall on to the star. The resulting accretion history is highly time-dependent: short episodic accretion bursts are followed by longer quiescent phases. We show that the disc for the direct collapse model is more unstable and generates greater variability than normal Pop III cases. Next, we conduct a stellar evolution calculation using the obtained accretion history. Our results show that, regardless of the variable accretion, the stellar radius monotonically increases with almost constant effective temperature at Teff ≃ 5000 K as the stellar mass increases. The resulting UV feedback is too weak to hinder accretion due to the low flux of stellar UV photons. The insensitivity of stellar evolution to variable accretion is attributed to the fact that time-scales of variability, ≲103 yr, are too short to affect the stellar structure. We argue that this evolution will continue until the SMS collapses to produce a black hole by the general relativistic instability after the mass reaches ≳105 M⊙.
18. The impact of non-thermal electrons on resolved black hole accretion disk images
Mao, Shengkai; Dexter, Jason; Quataert, Eliot
2015-01-01
Recent developments in radio astronomy (in particular, the Event Horizon Telescope) allow us for the first time to resolve length scales around the Milky Way's Sgr A* comparable to the event horizon radius. These observations are opening up new opportunities to study strong gravity and accretion physics in the vicinity of a supermassive black hole. However, the processes governing black hole accretion are not well understood. In particular, the electron thermodynamics in black hole accretion disks remain mysterious, and current models vary significantly from each other. The impact of these differences between current electron thermodynamics models on results obtained from EHT images is not well understood. Thus, in this work, we explore the effects of non-thermal electrons on black hole images and radio spectra in the context of both semi-analytic and numerical models of accretion flows. Using general relativistic ray-tracing and radiative transfer code, we simulate images of the accretion disk around Sgr A* and compare our simulations to observed radio data. We estimate the range of electron energy distribution functions permissible by the data. In so doing, we also explore the range and variety of black hole images obtained by varying the distribution function.
19. Binary Black Holes, Accretion Disks and Relativistic Jets: Photocenters of Nearby AGN and Quasars
NASA Technical Reports Server (NTRS)
Wehrle, Ann E.; Jones, Dayton L.; Meier, David L.; Piner, B. Glenn; Unwin, Stephen C.
2004-01-01
One of the most challenging questions in astronomy today is to understand the origin, structure, and evolution of the central engines in the nuclei of quasars and active galaxies (AGNs). The favoured theory involves the activation of relativistic jets from the fueling of a supermassive black hole through an accretion disk. In some AGN an outer optically thick, dusty torus is seen orbiting the black hole system. This torus is probably related to an inner accretion disk - black hole system that forms the actual powerhouse of the AGN. In radio-loud AGN two oppositely-directed radio jets are ejected perpendicular to the torus/disk system. Although there is a wealth of observational data on AGN, some very basic questions have not been definitively answered. The Space Interferometry Mission (SIM) will address the following three key questions about AGN. 1) Does the most compact optical emission from an AGN come from an accretion disk or from a relativistic jet? 2) Does the separation of the radio core and optical photocenter of the quasars used for the reference frame tie, change on the timescales of their photometric variability, or is the separation stable at the level of a few microarcseconds? 3) Do the cores of galaxies harbor binary supermassive black holes remaining from galaxy mergers? It is not known whether such mergers are common, and whether binaries would persist for a significant time.
20. Lunar volatile depletion due to incomplete accretion within an impact-generated disk
Canup, Robin M.; Visscher, Channon; Salmon, Julien; Fegley, Bruce
2015-11-01
The Moon likely formed from a disk produced by a giant impact with the Earth. The Moon and the bulk silicate Earth (BSE) share many compositional similarities (e.g., Ringwood 1979; Dauphas et al. 2014). However compared with the BSE, the Moon is more depleted in volatile elements, including moderately volatile K and Na, as well as more highly volatile elements, e.g., Zn (e.g., O’Neill 1991; Taylor & Wieczorek 2014). The origin of this depletion is poorly understood. Prior results suggest escape (e.g., Paniello et al. 2012), but at least hydrodynamic escape appears minimal for expected disk conditions (Nakajima & Stevenson 2014).In the limit of no escape and a closed system, a depletion could instead result if disk volatiles were preferentially accreted by the Earth rather than by the Moon. Taylor et al. (2006) advocated that the lunar depletion pattern is most consistent with incomplete condensation from an initially high temperature vapor, with the accretion of condensates by the Moon “cut-off” at a temperature allowing incorporation of a small component of alkalis (e.g., K and Na) but only a tiny fraction of more volatile elements (e.g., Zn). Neither the mechanism that would produce the cut-off, nor what the relevant cut-off temperature would be in an oxygen-rich protolunar disk (e.g., Visscher & Fegley 2013), were known.We identify a mechanism wherein a depletion results because disk volatiles are preferentially accreted by the Earth rather than by the Moon. The Moon may acquire the final tens to 60% of its mass from melt originating from the inner portions of the disk (Salmon & Canup 2012). Initially the inner disk melt is hot and volatile-poor, but as the disk cools, volatiles condense. We combine dynamical, thermal and chemical models to show that delivery of inner disk material to the Moon effectively ends as gravitational interactions cause the Moon’s orbit to expand away from the disk, with this cut-off occurring prior to condensation of key
1. Manifestations of dynamo driven large-scale magnetic field in accretion disks of compact objects
NASA Technical Reports Server (NTRS)
Chagelishvili, G. D.; Chanishvili, R. G.; Lominadze, J. G.; Sokhadze, Z. A.
1991-01-01
A turbulent dynamo nonlinear theory of turbulence was developed that shows that in the compact objects of accretion disks, the generated large-scale magnetic field (when the generation takes place) has a practically toroidal configuration. Its energy density can be much higher than turbulent pulsations energy density, and it becomes comparable with the thermal energy density of the medium. On this basis, the manifestations to which the large-scale magnetic field can lead at the accretion onto black holes and gravimagnetic rotators, respectively, are presented.
2. Growth of an ice disk: dependence of critical thickness for disk instability on supercooling of water.
PubMed
Yokoyama, Etsuro; Sekerka, Robert F; Furukawa, Yoshinori
2009-04-09
The appearance of an asymmetrical pattern that occurs when a disk crystal of ice grows from supercooled water was studied by using an analysis of growth rates for radius and thickness. The growth of the radius is controlled by transport of latent heat and is calculated by solving the diffusion equation for the temperature field surrounding the disk. The growth of the thickness is governed by the generation and lateral motion of steps and is expressed as a power function of the supercooling at the center of a basal face. Symmetry breaking with respect to the basal face of an ice disk crystal is observed when the thickness reaches a critical value; then one basal face becomes larger than the other and the disk loses its cylindrical shape. Subsequently, morphological instability occurs at the edge of the larger basal face of the asymmetrical shape (Shimada, W.; Furukawa, Y. J. Phys. Chem. 1997, B101, 6171-6173). We show that the critical thickness is related to the critical condition for the stable growth of a basal face. A difference of growth rates between two basal faces is a possible mechanism for the appearance of the asymmetrical shape.
3. Mass loss from pre-main-sequence accretion disks. I - The accelerating wind of FU Orionis
NASA Technical Reports Server (NTRS)
Calvet, Nuria; Hartmann, Lee; Kenyon, Scott J.
1993-01-01
We present evidence that the wind of the pre-main-sequence object FU Orionis arises from the surface of the luminous accretion disk. A disk wind model calculated assuming radiative equilibrium explains the differential behavior of the observed asymmetric absorption-line profiles. The model predicts that strong lines should be asymmetric and blueshifted, while weak lines should be symmetric and double-peaked due to disk rotation, in agreement with observations. We propose that many blueshifted 'shell' absorption features are not produced in a true shell of material, but rather form in a differentially expanding wind that is rapidly rotating. The inference of rapid rotation supports the proposal that pre-main-sequence disk winds are rotationally driven.
4. Accretion Disks around Black Holes: Dynamical Evolution, Meridional Circulations, and Gamma-Ray Bursts
Lee, William H.; Ramirez-Ruiz, Enrico
2002-10-01
We study the hydrodynamic evolution of massive accretion disks around black holes, formed when a neutron star is disrupted by a black hole in a binary system. The initial conditions are taken from three-dimensional calculations of coalescing binaries. By assuming azimuthal symmetry we are able to follow the time dependence of the disk structure for 0.2 s in cylindrical coordinates (r,z). We use an ideal gas equation of state and assume that all the dissipated energy is radiated away. The disks evolve because of viscous stresses, modeled with an α law. We study the disk structure and, in particular, the strong meridional circulations that are established and persist throughout our calculations. These consist of strong outflows along the equatorial plane that reverse direction close to the surface of the disk and converge on the accretor. In the context of gamma-ray bursts (GRBs), we estimate the energy released from the system in neutrinos and through magnetic-dominated mechanisms and find it can be as high as Eν~1052 ergs and EBZ~1051 ergs, respectively, during an estimated accretion timescale of 0.1-0.2 s. The νν annihilation is likely to produce bursts from only a short, impulsive energy input Lνν~t-5/2 and so would be unable to account for a large fraction of bursts that show complicated light curves. On the other hand, a gas mass ~0.1-0.25 Msolar survives in the orbiting debris, which enables strong magnetic fields ~1016 G to be anchored in the dense matter long enough to power short duration GRBs. We highlight the effects that the initial disk and black holes masses, viscosity, and binary mass ratio have on the evolution of the disk structure. Finally, we investigate the continuous energy injection that arises as the black hole slowly swallows the rest of the disk and discuss its consequences on the GRB afterglow emission.
5. Star formation and accretion in the circumnuclear disks of active galaxies
Wutschik, Stephanie; Schleicher, Dominik R. G.; Palmer, Thomas S.
2013-12-01
Aims: We explore the evolution of supermassive black holes (SMBH) centered in a circumnuclear disk (CND) as a function of the mass supply from the host galaxy and considering different star formation laws, which may give rise to a self-regulation via the injection of supernova-driven turbulence. Methods: A system of equations describing star formation, black hole accretion and angular momentum transport in the disk was solved self-consistently for an axisymmetric disk in which the gravitational potential includes contributions from the black hole, the disk and the hosting galaxy. Our model extends the framework provided by Kawakatu & Wada (2008, ApJ, 681, 73), by separately considering the inner and outer part of the disk, and by introducing a potentially non-linear dependence of the star formation rate on the gas surface density and the turbulent velocity. The star formation recipes are calibrated using observational data for NGC 1097, while the accretion model is based on turbulent viscosity as a source of angular momentum transport in a thin viscous accretion disk. Results: We find that current data provide no strong constraint on the star formation recipe, and can in particular not distinguish between models entirely regulated by the surface density, and models including a dependence on the turbulent velocity. The evolution of the black hole mass, on the other hand, strongly depends on the applied star formation law, as well as the mass supply from the host galaxy. We suggest to explore the star formation process in local AGN with high-resolution ALMA observations to break the degeneracy between different star formation models.
6. Accretion disks in the IRAS 23151+5912 region
SciTech Connect
2014-06-20
We present observations of radio continuum emission at 1.3 and 3.6 cm and H{sub 2}O masers toward the high-mass star-forming region IRAS 23151+5912 carried out with the Very Large Array-Expanded Very Large Array (in transition phase) in configuration A. We detected one continuum source at 1.3 cm in the region, but the counterpart of this radio continuum source at 3.6 cm was not detected at a level of 3σ. We also detected 13 water maser spots toward IRAS 23151+5912, which are distributed in three groups aligned along the northeast-southwest direction. Our results suggest that the 1.3 cm emission is consistent with a hypercompact H II region, probably with an embedded zero-age main-sequence star of type B2. In particular, we find that this young star is spatially associated with a maser group, which is tracing a disk-like structure of about 460 AU. We also find that the masers of the second group are probably describing a circumstellar disk of about 86 AU, whose central protostar, still undetected, should have a mass of ∼11 M {sub ☉}. We also suggest that the third water maser group is possibly associated with the SiO outflow and the undetected driving source of the system. Finally, we noted that the 1.3 cm continuum source and the three maser groups are aligned in the northeast-southwest direction, similar to the elongation of the large ionized region, which probably is the result of shock-wave induced star formation on the densest region of the medium.
7. A model of an X-ray-illuminated accretion disk and corona
NASA Technical Reports Server (NTRS)
Raymond, John C.
1993-01-01
The X-ray-illuminated surface of the accretion disk in a low-mass X-ray Binary (LMXRB) and the X-ray-heated corona above the disk produce optical, UV, and soft X-ray emission lines. This paper presents 1D models of the emission line spectra and the vertical temperature and density structures at different radii. The models include a detailed treatment of the important atomic processes and an escape probability treatment of radiative transfer. Soker and Raymond (1993) use the density structure predicted by these models for a 2D Monte Carlo simulation of the photon scattering in the accretion disk corona (ADC) to examine the effects of the ADC on the angular distribution of X-rays and the flux of X-rays incident on the outer disk. This paper concentrates on the emission line fluxes for various elemental abundances and disk parameters. The UV lines of the classic LMXRBs are consistent with the model predictions. Some CNO processing is necessary to account for the nitrogen and helium abundances in Sco X-1 and other LMXRBs. Comparison of the models with observed spectra also points to a soft X-ray component with luminosity comparable to the hard X-rays. The models predict a substantial luminosity in the group of highly ionized iron lines near 100 A.
8. The power of relativistic jets is larger than the luminosity of their accretion disks.
PubMed
Ghisellini, G; Tavecchio, F; Maraschi, L; Celotti, A; Sbarrato, T
2014-11-20
Theoretical models for the production of relativistic jets from active galactic nuclei predict that jet power arises from the spin and mass of the central supermassive black hole, as well as from the magnetic field near the event horizon. The physical mechanism underlying the contribution from the magnetic field is the torque exerted on the rotating black hole by the field amplified by the accreting material. If the squared magnetic field is proportional to the accretion rate, then there will be a correlation between jet power and accretion luminosity. There is evidence for such a correlation, but inadequate knowledge of the accretion luminosity of the limited and inhomogeneous samples used prevented a firm conclusion. Here we report an analysis of archival observations of a sample of blazars (quasars whose jets point towards Earth) that overcomes previous limitations. We find a clear correlation between jet power, as measured through the γ-ray luminosity, and accretion luminosity, as measured by the broad emission lines, with the jet power dominating the disk luminosity, in agreement with numerical simulations. This implies that the magnetic field threading the black hole horizon reaches the maximum value sustainable by the accreting matter.
9. The power of relativistic jets is larger than the luminosity of their accretion disks
Ghisellini, G.; Tavecchio, F.; Maraschi, L.; Celotti, A.; Sbarrato, T.
2014-11-01
Theoretical models for the production of relativistic jets from active galactic nuclei predict that jet power arises from the spin and mass of the central supermassive black hole, as well as from the magnetic field near the event horizon. The physical mechanism underlying the contribution from the magnetic field is the torque exerted on the rotating black hole by the field amplified by the accreting material. If the squared magnetic field is proportional to the accretion rate, then there will be a correlation between jet power and accretion luminosity. There is evidence for such a correlation, but inadequate knowledge of the accretion luminosity of the limited and inhomogeneous samples used prevented a firm conclusion. Here we report an analysis of archival observations of a sample of blazars (quasars whose jets point towards Earth) that overcomes previous limitations. We find a clear correlation between jet power, as measured through the γ-ray luminosity, and accretion luminosity, as measured by the broad emission lines, with the jet power dominating the disk luminosity, in agreement with numerical simulations. This implies that the magnetic field threading the black hole horizon reaches the maximum value sustainable by the accreting matter.
10. Bipolar flows, molecular gas disks, and the collapse and accretion of rotating interstellar clouds
NASA Technical Reports Server (NTRS)
Boss, Alan P.
1987-01-01
Rigorous numerical models of the collapse and accretion of rotating, axisymmetric, isothermal interstellar clouds are studied. The results show that molecular gas disks and evacuated bipolar cavities both appear to be natural consequences of the collapse of rotating interstellar clouds. Dynamically significant magnetic fields may not be necessary for explaining either phenomenon. The models strongly support theoretical models of the type where an isotropic wind from a pre-main sequence star is extrinsically collimated by a rotationally derived molecular gas cloud. The models imply that collimation should be strongest on small scales where rotational effects are most important, i.e., in the dense region of the molecular gas disk.
11. Variations in the accretion rate and luminosity in gravitationally unstable protostellar disks
Elbakyan, V. G.; Vorobyov, E. I.; Glebova, G. M.
2016-10-01
Self-consistent modeling of a protostar and protostellar disk is carried out for early stages of their evolution. The accretion rate at distances of sevral astronomical units from the protostar is appreciably variable, which is reflected in the protostar's luminosity. The amplitude of the variations in the accretion rate and luminosity grows together with the sampling period, as a consequence of the nature of gravitationally unstable protostellar disks. A comparison of model luminosity variations with those derived from observations of nearby sites of star formation shows that the model variations are appreciably lower than the observed values for sampling periods of less than 10 years, indicating the presence of additional sources of variability on small dynamical distances from the protostar.
12. X-ray Reflected Spectra from Accretion Disk Models. I. Constant Density Atmospheres
NASA Technical Reports Server (NTRS)
Garcia, Javier; Kallman, Timothy R.
2009-01-01
We present new models for illuminated accretion disks, their structure and reprocessed emission. We consider the effects of incident X-rays on the surface of an accretion disk by solving simultaneously the equations of radiative transfer, energy balance and ionization equilibrium over a large range of column densities. We assume plane-parallel geometry and azimuthal symmetry, such that each calculation corresponds to a ring at a given distance from the central object. Our models include recent and complete atomic data for K-shell of the iron and oxygen isonuclear sequences. We examine the effect on the spectrum of fluorescent Ka line emission and absorption in the emitted spectrum. We also explore the dependence of the spectrum on the strength of the incident X-rays and other input parameters, and discuss the importance of Comptonization on the emitted spectrum.
13. Azimuthal instability of vortex rings generated by an oscillating disk
Deng, Jian; Caulfield, C. P.
2015-11-01
We report the instabilities of vortex rings generated by an oscillating disk. Assuming sinusoidal variation in the azimuthal direction with mode number, m, a Floquet linear stability analysis is performed. We study the dynamics for a range of the two control parameters, the Keulegan-Carpenter number KC = 2 πA / c and the Stokes number β = fc2 / ν , where A is the amplitude of oscillation, f is the frequency of oscillation, c is the diameter of the disk, and ν is the kinematic viscosity of the fluid. We observe two distinctive flow regions in the (KC , β) parameter space. First, in the low β region, the flow breaks its symmetry with a single wavenumber mode getting a positive growth rate. Second, in the high β region, high-order unstable modes emerge, with the highest mode number m = 9 recorded. Furthermore, we carry out Direct Numerical Simulations (DNS) on the fully three-dimensional Navier-stokes equations. The results reproduce the main features of the high-order unstable modes predicted by the Floquet analysis, exhibiting the highest mode number m = 6 . We conjecture that the inconsistence in the highest mode number between the Floquet linear stability analysis and the DNS implies the non-linear characteristic of the current problem. Supported by the National Natural Science Foundation of China (Grant No: 11272283) and Zhejiang Provincial Natural Science Foundation of China (Grant No: LY12A02006).
14. Photo-Reverberation Mapping of a Protoplanetary Accretion Disk around a T Tauri star
Meng, Huan; Plavchan, Peter; Rieke, George
2015-12-01
Theoretical models and spectroscopic observations of newborn stars suggest that protoplantary disks have an inner "wall", where material is depleted by sublimation and/or magnetospheric accretion. Around T Tauri stars, the size of this disk hole is expected to be on a 0.1-AU scale that is unresolved by current adaptive optics imaging, though some model-dependent constraints have been obtained by near-infrared interferometry. Here we report the first measurement of the inner disk wall around a solar-mass young stellar object, YLW 16B in the ρ Ophiuchi star-forming region, by detecting the light travel time of the variable radiation from the stellar surface to the disk. Consistent time lags were detected on two nights, when the time series in H and K bands were synchronized while the 4.5 μm emission lagged by 74.5 ± 3.2 seconds. Considering the nearly edge-on geometry of the disk, the inner rim should be 0.084 ± 0.004 AU from the protostar on average. This size is likely larger than the range of magnetospheric truncations, but consistent with an optically and geometrically thick disk front at the dust sublimation radius of ~1500 K. The detection of a definite time lag places constraints on the geometry of the disk.
15. Equilibrium configuration and stability of a stratus floating above accretion disks
Nakai, Takuya; Fukue, Jun
2016-04-01
We examine the equilibrium configurations of a stratus floating above an accretion disk, using the radiative force from the luminous disk just below the stratus. For various disk luminosities and optical depths of the stratus, the stratus can stably float on the outer disk, while a stable configuration does not exist on the inner disk. When the disk luminosity normalized by the Eddington luminosity is unity, and the stratus optical depth is around unity, the stable configuration disappears at r ≲ 50rg, rg being the Schwarzschild radius, and the stratus would be blown off as a cloudy wind, which consists of many strati with appropriate conditions. In the outer region of r ≳ 50rg, on the other hand, we find that the stable floating height is z ˜ 20rg, which is approximately two times larger than in the case of the particle. This difference is due to the anisotropic scattering effect; the stratus can get twice the momentum from radiation than it can in the particle case. The present results, that the radiation-driven cloudy wind can be easily blown off from the luminous disk, can explain observed outflows in broad absorption line quasars and ultra-fast outflow objects.
16. HST UV observations of the accretion disk corona X-ray binary X1822-371
NASA Technical Reports Server (NTRS)
Puchnarewicz, E. M.; Mason, K. O.; Cordova, F. A.
1995-01-01
The Faint Object Spectrograph (FOS) on the Hubble Space Telescope (HST) has provided the first ultraviolet orbital light curve of the low-mass X-ray binary X1822-371. The shape of the UV light curve changes with wavelength providing the first direct clues to the temperature of the various system components. The data support the idea that the system contains a thick, structured accretion disk.
17. On the stream-accretion disk interaction - Response to increased mass transfer rate
NASA Technical Reports Server (NTRS)
Dgani, Ruth; Livio, Mario; Soker, Noam
1989-01-01
The time-dependent interaction between the stream of mass from the inner Lagrangian point and the accretion disk, resulting from an increasing mass transfer rate is calculated. The calculation is fully three-dimensional, using a pseudoparticle description of the hydrodynamics. It is demonstrated that the results of such calculations, when combined with specific observations, have the potential of both determining essential parameters, such as the viscosity parameter alpha, and can distinguish between different models of dwarf nova eruptions.
18. ACCRETION DISK WARPING BY RESONANT RELAXATION: THE CASE OF MASER DISK NGC 4258
SciTech Connect
Bregman, Michal; Alexander, Tal
2009-08-01
The maser disk around the massive black hole (MBH) in active galaxy NGC 4258 exhibits an O(10 deg.) warp on the O(0.1 pc) scale. The physics driving the warp is still debated. Suggested mechanisms include torquing by relativistic frame dragging or by radiation pressure. We propose here a new warping mechanism: resonant torquing of the disk by stars in the dense cusp around the MBH. We show that resonant torquing can induce such a warp over a wide range of observed and deduced physical parameters of the maser disk.
19. Cygnus X-1: A Case for a Magnetic Accretion Disk?
NASA Technical Reports Server (NTRS)
Nowak, Michael A.; Vaughan, B. A.; Dove, J.; Wilms, J.
1996-01-01
With the advent of Rossi X-ray Timing Explorer (RXTE), which is capable of broad spectral coverage and fast timing, as well as other instruments which are increasingly being used in multi-wavelength campaigns (via both space-based and ground-based observations), we must demand more of our theoretical models. No current model mimics all facets of a system as complex as an x-ray binary. However, a modern theory should qualitatively reproduce - or at the very least not fundamentally disagree with - all of Cygnus X-l's most basic average properties: energy spectrum (viewed within a broader framework of black hole candidate spectral behavior), power spectrum (PSD), and time delays and coherence between variability in different energy bands. Below we discuss each of these basic properties in turn, and we assess the health of one of the currently popular theories: Comptonization of photons from a cold disk. We find that the data pose substantial challenges for this theory, as well as all other in currently discussed models.
20. Optical spectroscopy of Z Canis Majoris, V1057 Cygni, and FU Orionis - Accretion disks and signatures of disk winds
NASA Technical Reports Server (NTRS)
Welty, Alan D.; Strom, Stephen E.; Edwards, Suzan; Kenyon, Scott J.; Hartmann, Lee W.
1992-01-01
High resolution, high SNR optical spectra have been used to investigate the hypothesis that in outburst, FU Ori objects are self-luminous accretion disks whose light dominates at optical and near-IR wavelengths. Strong evidence has been found for linewidth versus wavelength correlation in good agreement with model predictions for Z CMa and V1057 Cyg, but not for FU Ori itself. Linewidth varies continuously with wavelength at optical wavelengths in the former two objects, In the case of FU Ori, it is argued that a combination of strong wind components to spectral lines, and surface gravity possibly being lower than that of supergiants, conceals the underlying linewidth versus wavelength relationship. A marginal correlation is found between linewidth and lower excitation potential in all three objects. Synthetic disk spectra are subtracted from observed spectral, and remarkably good fits are found for all three objects for wavelengths longer than about 5000 A.
1. Optical spectroscopy of Z Canis Majoris, V1057 Cygni, and FU Orionis - Accretion disks and signatures of disk winds
Welty, Alan D.; Strom, Stephen E.; Edwards, Suzan; Kenyon, Scott J.; Hartmann, Lee W.
1992-09-01
High resolution, high SNR optical spectra have been used to investigate the hypothesis that in outburst, FU Ori objects are self-luminous accretion disks whose light dominates at optical and near-IR wavelengths. Strong evidence has been found for linewidth versus wavelength correlation in good agreement with model predictions for Z CMa and V1057 Cyg, but not for FU Ori itself. Linewidth varies continuously with wavelength at optical wavelengths in the former two objects, In the case of FU Ori, it is argued that a combination of strong wind components to spectral lines, and surface gravity possibly being lower than that of supergiants, conceals the underlying linewidth versus wavelength relationship. A marginal correlation is found between linewidth and lower excitation potential in all three objects. Synthetic disk spectra are subtracted from observed spectral, and remarkably good fits are found for all three objects for wavelengths longer than about 5000 A.
2. Depletion of Molecular Gas by an Accretion Outburst in a Protoplanetary Disk
Banzatti, A.; Pontoppidan, K. M.; Bruderer, S.; Muzerolle, J.; Meyer, M. R.
2015-01-01
We investigate new and archival 3-5 μm high-resolution (~3 km s-1) spectroscopy of molecular gas in the inner disk of the young solar-mass star EX Lupi, taken during and after the strong accretion outburst of 2008. The data were obtained using the CRIRES spectrometer at the European Southern Observatory Very Large Telescope in 2008 and 2014. In 2008, emission lines from CO, H2O, and OH were detected with broad profiles tracing gas near and within the corotation radius (0.02-0.3 AU). In 2014, the spectra display marked differences. The CO lines, while still detected, are much weaker, and the H2O and OH lines have disappeared altogether. At 3 μm a veiled stellar photospheric spectrum is observed. Our analysis finds that the molecular gas mass in the inner disk has decreased by an order of magnitude since the outburst, matching a similar decrease in the accretion rate onto the star. We discuss these findings in the context of a rapid depletion of material accumulated beyond the disk corotation radius during quiescent periods, as proposed by models of episodic accretion in EXor-type young stars.
3. DEPLETION OF MOLECULAR GAS BY AN ACCRETION OUTBURST IN A PROTOPLANETARY DISK
SciTech Connect
Banzatti, A.; Pontoppidan, K. M.; Muzerolle, J.; Bruderer, S.; Meyer, M. R.
2015-01-01
We investigate new and archival 3-5 μm high-resolution (∼3 km s{sup –1}) spectroscopy of molecular gas in the inner disk of the young solar-mass star EX Lupi, taken during and after the strong accretion outburst of 2008. The data were obtained using the CRIRES spectrometer at the European Southern Observatory Very Large Telescope in 2008 and 2014. In 2008, emission lines from CO, H{sub 2}O, and OH were detected with broad profiles tracing gas near and within the corotation radius (0.02-0.3 AU). In 2014, the spectra display marked differences. The CO lines, while still detected, are much weaker, and the H{sub 2}O and OH lines have disappeared altogether. At 3 μm a veiled stellar photospheric spectrum is observed. Our analysis finds that the molecular gas mass in the inner disk has decreased by an order of magnitude since the outburst, matching a similar decrease in the accretion rate onto the star. We discuss these findings in the context of a rapid depletion of material accumulated beyond the disk corotation radius during quiescent periods, as proposed by models of episodic accretion in EXor-type young stars.
4. REVISITING PUTATIVE COOL ACCRETION DISKS IN ULTRALUMINOUS X-RAY SOURCES
SciTech Connect
Miller, J. M.; King, A. L.; Reynolds, M. T.; Reis, R. C.; Walton, D. J.; Fabian, A. C.; Miller, M. C.
2013-10-20
Soft, potentially thermal spectral components observed in some ultra-luminous X-ray sources (ULXs) can be fit with models for emission from cool, optically thick accretion disks. If that description is correct, the low temperatures that are observed imply accretion onto 'intermediate-mass' black holes. Subsequent work has found that these components may follow an inverse relationship between luminosity and temperature, implying a non-blackbody origin for this emission. We have re-analyzed numerous XMM-Newton spectra of extreme ULXs. Crucially, observations wherein the source fell on a chip gap were excluded owing to their uncertain flux calibration, and the neutral column density along the line of sight to a given source was jointly determined by multiple spectra. The luminosity of the soft component is found to be positively correlated with temperature, and to be broadly consistent with L∝T {sup 4} in the measured band pass, as per blackbody emission from a standard thin disk. These results are nominally consistent with accretion onto black holes with masses above the range currently known in Galactic X-ray binaries, though there are important caveats. Emission from inhomogeneous or super-Eddington disks may also be consistent with the data.
5. Electron Heating by the Ion Cyclotron Instability in Collisionless Accretion Flows. I. Compression-driven Instabilities and the Electron Heating Mechanism
Sironi, Lorenzo; Narayan, Ramesh
2015-02-01
In systems accreting well below the Eddington rate, such as the central black hole in the Milky Way (Sgr A*), the plasma in the innermost regions of the disk is believed to be collisionless and have two temperatures, with the ions substantially hotter than the electrons. However, whether a collisionless faster-than-Coulomb energy transfer mechanism exists in two-temperature accretion flows is still an open question. We study the physics of electron heating during the growth of ion velocity-space instabilities by means of multidimensional, fully kinetic, particle-in-cell (PIC) simulations. A background large-scale compression—embedded in a novel form of the PIC equations—continuously amplifies the field. This constantly drives a pressure anisotropy P > P ∥ because of the adiabatic invariance of the particle magnetic moments. We find that, for ion plasma beta values β0i ~ 5-30 appropriate for the midplane of low-luminosity accretion flows (here, β0i is the ratio of ion thermal pressure to magnetic pressure), mirror modes dominate if the electron-to-proton temperature ratio is T 0e /T 0i >~ 0.2, whereas for T 0e /T 0i <~ 0.2 the ion cyclotron instability triggers the growth of strong Alfvén-like waves, which pitch-angle scatter the ions to maintain marginal stability. We develop an analytical model of electron heating during the growth of the ion cyclotron instability, which we validate with PIC simulations. We find that for cold electrons (β0e <~ 2 me /mi , where β0e is the ratio of electron thermal pressure to magnetic pressure), the electron energy gain is controlled by the magnitude of the E-cross-B velocity induced by the ion cyclotron waves. This term is independent of the initial electron temperature, so it provides a solid energy floor even for electrons starting with extremely low temperatures. On the other hand, the electron energy gain for β0e >~ 2 me /mi —governed by the conservation of the particle magnetic moment in the growing fields of
6. ELECTRON HEATING BY THE ION CYCLOTRON INSTABILITY IN COLLISIONLESS ACCRETION FLOWS. I. COMPRESSION-DRIVEN INSTABILITIES AND THE ELECTRON HEATING MECHANISM
SciTech Connect
Sironi, Lorenzo; Narayan, Ramesh E-mail: [email protected]
2015-02-20
In systems accreting well below the Eddington rate, such as the central black hole in the Milky Way (Sgr A*), the plasma in the innermost regions of the disk is believed to be collisionless and have two temperatures, with the ions substantially hotter than the electrons. However, whether a collisionless faster-than-Coulomb energy transfer mechanism exists in two-temperature accretion flows is still an open question. We study the physics of electron heating during the growth of ion velocity-space instabilities by means of multidimensional, fully kinetic, particle-in-cell (PIC) simulations. A background large-scale compression—embedded in a novel form of the PIC equations—continuously amplifies the field. This constantly drives a pressure anisotropy P > P {sub ∥} because of the adiabatic invariance of the particle magnetic moments. We find that, for ion plasma beta values β{sub 0i} ∼ 5-30 appropriate for the midplane of low-luminosity accretion flows (here, β{sub 0i} is the ratio of ion thermal pressure to magnetic pressure), mirror modes dominate if the electron-to-proton temperature ratio is T {sub 0e}/T {sub 0i} ≳ 0.2, whereas for T {sub 0e}/T {sub 0i} ≲ 0.2 the ion cyclotron instability triggers the growth of strong Alfvén-like waves, which pitch-angle scatter the ions to maintain marginal stability. We develop an analytical model of electron heating during the growth of the ion cyclotron instability, which we validate with PIC simulations. We find that for cold electrons (β{sub 0e} ≲ 2 m{sub e} /m{sub i} , where β{sub 0e} is the ratio of electron thermal pressure to magnetic pressure), the electron energy gain is controlled by the magnitude of the E-cross-B velocity induced by the ion cyclotron waves. This term is independent of the initial electron temperature, so it provides a solid energy floor even for electrons starting with extremely low temperatures. On the other hand, the electron energy gain for β{sub 0e} ≳ 2 m{sub e} /m{sub i}
7. New Insights on the Accretion Disk-Winds Connection in Radio-Loud AGNs from Suzaku
NASA Technical Reports Server (NTRS)
Tombesi, F.; Sambruna, R. M.; Reeves, J. N.; Braito, V.; Cappi, M.; Reynolds, S.; Mushotzky, R. F.
2011-01-01
From the spectral analysis of long Suzaku observations of five radio-loud AGNs we have been able to discover the presence of ultra-fast outflows with velocities ,,approx.0.1 c in three of them, namely 3C III, 3C 120 and 3C 390.3. They are consistent with being accretion disk winds/outflows. We also performed a follow-up on 3C III to monitor its outflow on approx.7 days time-scales and detected an anti-correlated variability of a possible relativistic emission line with respect to blue-shifted Fe K features, following a flux increase. This provides the first direct evidence for an accretion disc-wind connection in an AGN. The mass outflow rate of these outflows can be comparable to the accretion rate and their mechanical power can correspond to a significant fraction of the bolometric luminosity and is comparable to their typical jet power. Therefore, they can possibly play a significant role in the expected feedback from AGNs and can give us further clues on the relation between the accretion disk and the formation of winds/jets.
8. Correlation analysis of radio properties and accretion-disk luminosity for low luminosity AGNs
Su, Renzhi; Liu, Xiang; Zhang, Zhen
2017-01-01
The correlation between the jet power and accretion disk luminosity is investigated and analyzed with our model for 7 samples of low luminosity active galactic nuclei (LLAGNs). The main results are: (1) the power-law correlation index (P_{jet} ∝ L_{disk} ^{μ}) typically ranges μ=0.4-0.7 for the LLAGN samples, and there is a hint of steep index for the LLAGN sample which hosted by a high fraction of elliptical galaxies, and there are no significant correlation between the μ and the LLAGN types (Seyfert, LINER); (2) for μ≈1, as noted in Liu et al., the accretion disk dominates the jet power and the black hole (BH) spin is not important, for the LLAGN samples studied in this paper we find that the μ is significantly less than unity, implying that BH spin may play a significant role in the jet power of LLAGNs; (3) the BH spin-jet power is negatively correlated with the BH mass in our model, which means a high spin-jet efficiency in the low' BH-mass LLAGNs; (4) an anti-correlation between radio loudness and disk luminosity is found, which is apparently due to the flatter power-law index in the jet-disk correlation of the LLAGNs, and the radio loudness can be higher in the LLAGNs than in luminous AGNs/quasars when the BH spin-jet power is comparable to or dominate over the accretion-jet power in the LLAGNs. The high radio-core dominance of the LLAGNs is also discussed.
9. THE FATE OF PLANETESIMALS IN TURBULENT DISKS WITH DEAD ZONES. II. LIMITS ON THE VIABILITY OF RUNAWAY ACCRETION
SciTech Connect
Ormel, C. W.; Okuzumi, S. E-mail: [email protected]
2013-07-01
A critical phase in the standard model for planet formation is the runaway growth (RG) phase. During RG bodies in the 0.1-100 km size range (planetesimals) quickly produce a number of much larger seeds. The RG phase is essential for planet formation as the emergent planetary embryos can accrete the leftover planetesimals at large gravitational focusing factors. However, torques resulting from turbulence-induced density fluctuations may violate the criterion for the onset of RG, which is that the magnitude of the planetesimals' random (eccentric) motions is less than their escape velocity. This condition represents a more stringent constraint than the condition that planetesimals survive their mutual collisions. To investigate the effects of magneto-rotational instability turbulence on the viability of the RG scenario, we apply our semi-analytical recipes of Paper I, which we augment by a coagulation/fragmentation model for the dust component. We find that the surface-area-equivalent abundance of 0.1 {mu}m particles is reduced by factors 10{sup 2}-10{sup 3}, which tends to render the dust irrelevant to the turbulence. We express the turbulent activity in the midplane regions in terms of a size s{sub run} above which planetesimals will experience RG. We find that s{sub run} is mainly determined by the strength of the vertical net field that threads the disks and the disk radius. At disk radii beyond 5 AU, s{sub run} becomes larger than {approx}100 km and the collision times among these bodies longer than the duration of the nebula phase. Our findings imply that the classical, planetesimal-dominated model for planet formation is not viable in the outer regions of a turbulent disk.
10. THE EFFECT OF A TIME-VARYING ACCRETION DISK SIZE ON QUASAR MICROLENSING LIGHT CURVES
SciTech Connect
Blackburne, Jeffrey A.; Kochanek, Christopher S. E-mail: [email protected]
2010-08-01
Microlensing perturbations to the magnification of gravitationally lensed quasar images are dependent on the angular size of the quasar. If quasar variability at visible wavelengths is caused by a change in the area of the accretion disk, it will affect the microlensing magnification. We derive the expected signal, assuming that the luminosity scales with some power of the disk area, and estimate its amplitude using simulations. We discuss the prospects for detecting the effect in real-world data and for using it to estimate the logarithmic slope of the luminosity's dependence on disk area. Such an estimate would provide a direct test of the standard thin accretion disk model. We tried fitting six seasons of the light curves of the lensed quasar HE 0435-1223 including this effect as a modification to the Kochanek et al. approach to estimating time delays. We find a dramatic improvement in the goodness of fit and relatively plausible parameters, but a robust estimate will require a full numerical calculation in order to correctly model the strong correlations between the structure of the microlensing magnification patterns and the magnitude of the effect. We also comment briefly on the effect of this phenomenon for the stability of time-delay estimates.
11. Angular Momentum Transport and Variability in Boundary Layers of Accretion Disks Driven by Global Acoustic Modes
Belyaev, Mikhail A.; Rafikov, Roman R.; Stone, James M.
2012-11-01
Disk accretion onto a weakly magnetized central object, e.g., a star, is inevitably accompanied by the formation of a boundary layer near the surface, in which matter slows down from the highly supersonic orbital velocity of the disk to the rotational velocity of the star. We perform high-resolution two-dimensional hydrodynamical simulations in the equatorial plane of an astrophysical boundary layer with the goal of exploring the dynamics of non-axisymmetric structures that form there. We generically find that the supersonic shear in the boundary layer excites non-axisymmetric quasi-stationary acoustic modes that are trapped between the surface of the star and a Lindblad resonance in the disk. These modes rotate in a prograde fashion, are stable for hundreds of orbital periods, and have a pattern speed that is less than and of the order of the rotational velocity at the inner edge of the disk. The origin of these intrinsically global modes is intimately related to the operation of a corotation amplifier in the system. Dissipation of acoustic modes in weak shocks provides a universal mechanism for angular momentum and mass transport even in purely hydrodynamic (i.e., non-magnetized) boundary layers. We discuss the possible implications of these trapped modes for explaining the variability seen in accreting compact objects.
12. Accretion disk modeling of AGN continuum using non-LTE stellar atmospheres. [active galactic nuclei (AGN)
NASA Technical Reports Server (NTRS)
Sun, Wei-Hsin; Malkan, Matthew A.
1988-01-01
Active galactic nuclei (AGN) accretion disk spectra were calculated using non-LTE stellar atmosphere models for Kerr and Schwarzschild geometries. It is found that the Lyman limit absorption edge, probably the most conclusive observational evidence for the accretion disk, would be drastically distorted and displaced by the relativistic effects from the large gravitational field of the central black hole and strong Doppler motion of emitting material on the disk surface. These effects are especially pronounced in the Kerr geometry. The strength of the Lyman limit absorption is very sensitive to the surface gravity in the stellar atmosphere models used. For models at the same temperature but different surface gravities, the strength of the Lyman edge exhibits an almost exponential decrease as the surface gravity approach the Eddington limit, which should approximate the thin disk atmosphere. The relativistic effects as well as the vanishing of the Lyman edge at the Eddington gravity may be the reasons that not many Lyman edges in the rest frames of AGNs and quasars are found.
13. Generation of magnetic field on the accretion disk around a proto-first-star
SciTech Connect
Shiromoto, Yuki; Susa, Hajime; Hosokawa, Takashi
2014-02-20
The generation process of a magnetic field around a proto-first-star is studied. Utilizing the recent numerical results of proto-first-star formation based on radiation hydrodynamics simulations, we assess the magnetic field strength generated by the radiative force and the Biermann battery effect. We find that a magnetic field of ∼10{sup –9} G is generated on the surface of the accretion disk around the proto-first-star. The field strength on the accretion disk is smaller by two orders of magnitude than the critical value, above which the gravitational fragmentation of the disk is suppressed. Thus, the generated seed magnetic field hardly affect the dynamics of on-site first star formation directly, unless an efficient amplification process is taken into consideration. We also find that the generated magnetic field is continuously blown out from the disk on the outflows to the poles, that are driven by the thermal pressure of photoheated gas. The strength of the diffused magnetic field in low-density regions is ∼10{sup –14}-10{sup –13} G at n {sub H} = 10{sup 3} cm{sup –3}, which could play an important role in the next generation star formation, as well as the seeds of the magnetic field in the present-day universe.
14. ANGULAR MOMENTUM TRANSPORT AND VARIABILITY IN BOUNDARY LAYERS OF ACCRETION DISKS DRIVEN BY GLOBAL ACOUSTIC MODES
SciTech Connect
Belyaev, Mikhail A.; Stone, James M.; Rafikov, Roman R.
2012-11-20
Disk accretion onto a weakly magnetized central object, e.g., a star, is inevitably accompanied by the formation of a boundary layer near the surface, in which matter slows down from the highly supersonic orbital velocity of the disk to the rotational velocity of the star. We perform high-resolution two-dimensional hydrodynamical simulations in the equatorial plane of an astrophysical boundary layer with the goal of exploring the dynamics of non-axisymmetric structures that form there. We generically find that the supersonic shear in the boundary layer excites non-axisymmetric quasi-stationary acoustic modes that are trapped between the surface of the star and a Lindblad resonance in the disk. These modes rotate in a prograde fashion, are stable for hundreds of orbital periods, and have a pattern speed that is less than and of the order of the rotational velocity at the inner edge of the disk. The origin of these intrinsically global modes is intimately related to the operation of a corotation amplifier in the system. Dissipation of acoustic modes in weak shocks provides a universal mechanism for angular momentum and mass transport even in purely hydrodynamic (i.e., non-magnetized) boundary layers. We discuss the possible implications of these trapped modes for explaining the variability seen in accreting compact objects.
15. Ion Viscosity Mediated by Tangled Magnetic Fields: An Application to Black Hole Accretion Disks
NASA Technical Reports Server (NTRS)
Subramanian, Prasad; Becker, Peter A.; Kafatos, Menas
1996-01-01
We examine the viscosity associated with the shear stress exerted by ions in the presence of a tangled magnetic field. As an application, we consider the effect of this mechanism on the structure of black hole accretion disks. We do not attempt to include a self-consistent description of the magnetic field. Instead, we assume the existence of a tangled field with coherence length lambda(sub coh), which is the average distance between the magnetic 'kinks' that scatter the particles. For simplicity, we assume that the field is self-similar, and take lambda(sub coh) to be a fixed fraction zeta of the local disk height H. Ion viscosity in the presence of magnetic fields is generally taken to be the cross-field viscosity, wherein the effective mean free path is the ion Larmor radius lambda(sub L), which is much less than the ion-ion Coulomb mean free path A(sub ii) in hot accretion disks. However, we arrive at a formulation for a 'hybrid' viscosity in which the tangled magnetic field acts as an intermediary in the transfer of momentum between different layers in the shear flow. The hybrid viscosity greatly exceeds the standard cross-field viscosity when (lambda/lambda(sub L)) much greater than (lambda(sub L)/lambda(sub ii)), where lambda = ((lambda(sub ii)(sup -1) + lambda(sub (coh)(sup -1))(sup -1) is the effective mean free path for the ions. This inequality is well satisfied in hot accretion disks, which suggests that the ions may play a much larger role in the momentum transfer process in the presence of magnetic fields than was previously thought. The effect of the hybrid viscosity on the structure of a steady-state, two-temperature, quasi-Keplerian accretion disk is analyzed. The hybrid viscosity is influenced by the degree to which the magnetic field is tangled (represented by zeta = lambda(sub coh)), and also by the relative accretion rate M/M(sub E), where M(sub E) = L(sub E)/c(sup 2) and L(sub E) is the Eddington luminosity. We find that ion viscosity in the
16. CHEMICAL ABUNDANCES IN THE POLAR DISK OF NGC 4650A: IMPLICATIONS FOR COLD ACCRETION SCENARIO
SciTech Connect
Spavone, M.; Longo, G.; Iodice, E.; Arnaboldi, M.; Gerhard, O.; Saglia, R.
2010-05-10
The aim of the present study is to test whether the cold accretion of gas through a 'cosmic filament' is a possible formation scenario for the polar disk galaxy NGC 4650A. If polar disks form from cold accretion of gas, the abundances of the H II regions may be similar to those of very late-type spiral galaxies, regardless of the presence of a bright central stellar spheroid, with total luminosity of few 10{sup 9} L{sub sun}. We use deep long-slit spectra obtained with the FORS2 spectrograph at the Very Large Telescope in the optical and near-infrared wavelength ranges for the brightest H II regions in the polar disk of NGC 4650A. The strongest emission lines ([O II] H{sub {beta}}, [O III], H{sub {alpha}}) were used to derive oxygen abundances, metallicities, and the global star formation rates for the disk. The available deep spectra allowed us to measure the oxygen abundances (12 + log(O/H)) using the empirical method based on intensities of the strongest emission lines and the direct method based on the determination of electron temperature from the detection of weak auroral lines, as the [O III] at 4363 A. The oxygen abundance measured for the polar disk is then compared with those measured for different galaxy types of similar total luminosities and then compared against the predictions of different polar ring formation scenarios. The average metallicity values for the polar disk in NGC 4650A is Z = 0.2 Z{sub sun}, and it is lower than the values measured for ordinary spirals of similar luminosity. Moreover, the gradient of the metallicity is flat along the polar disk major axis, which implies none or negligible metal enrichment from the stars in the older central spheroid. The low-metallicity value in the polar disk NGC 4650A and the flat metallicity gradient are both consistent with a latter infall of metal-poor gas, as expected in the cold accretion processes.
17. Hot accretion disks with pairs: Effects of magnetic field and thermal cyclocsynchrotron radiation
NASA Technical Reports Server (NTRS)
Kusunose, Masaaki; Zdziarski, Andrzej A.
1994-01-01
We show the effects of thermal cyclosynchrotron radiation and magnetic viscosity on the structure of hot, two-temperature accretion disks. Magnetic field, B, is assumed to be randomly oriented and the ratio of magnetic pressure to either gas pressure, alpha = P(sub mag)/P(sub gas), or the sum of the gas and radiation pressures, alpha = (P(sub mag)/P(sub gas) + P(sub rad)), is fixed. We find those effects do not change the qualitative properties of the disks, i.e., there are still two critical accretion rates related to production of e(sup +/-) pairs, (M dot)((sup U)(sub cr)) and (M dot)((sup L)(sub cr)), that affect the number of local and global disk solutions, as recently found by Bjoernsson and Svensson for the case with B = 0. However, a critical value of the alpha-viscosity parameter above which those critical accretion rates disappear becomes smaller than alpha(sub cr) = 1 found in the case of B = 0, for P(sub mag) = alpha(P(sub gas) + P(sub rad)). If P(sub mag) = alpha P(sub gas), on the other hand, alpha(sub cr) is still about unity. Moreover, when Comptonized cyclosynchrotron radiation dominates Comptonized bremsstrahlung, radiation from the disk obeys a power law with the energy spectral index of approximately 0.5, in a qualitative agreement with X-ray observations of active galactic nuclei (AGNS) and Galactic black hole candidates. We also extend the hot disk solutions for P(sub mag) = alpha(P(sub gas) + P(sub rad)) to the effectively optically thick region, where they merge with the standard cold disk solutions. We find that the mapping method by Bjoernsson and Svensson gives a good approximation to the disk structure in the hot region and show where it breaks in the transition region. Finally, we find a region in the disk parameter space with no solutions due to the inability of Coulomb heating to supply enough energy to electrons.
18. Sustained Accretion on Gas Giants Surrounded by Low-Turbulence Circumplanetary Disks
D'Angelo, Gennaro; Marzari, Francesco
2015-11-01
Gas giants more massive than Saturn acquire most of their envelope while surrounded by a circumplanetary disk (CPD), which extends over a fraction of the planet’s Hill radius. Akin to circumstellar disks, CPDs may be subject to MRI-driven turbulence and contain low-turbulence regions, i.e., dead zones. It was suggested that CPDs may inhibit sustained gas accretion, thus limiting planet growth, because gas transport through a CPD may be severely reduced by a dead zone, a consequence at odds with the presence of Jupiter-mass (and larger) planets. We studied how an extended dead zone influences gas accretion on a Jupiter-mass planet, using global 3D hydrodynamics calculations with mesh refinements. The accretion flow from the circumstellar disk to the CPD is resolved locally at the length scale Rj, Jupiter's radius. The gas kinematic viscosity is assumed to be constant and the dead zone around the planet is modeled as a region of much lower viscosity, extending from ~Rj out to ~60Rj and off the mid-plane for a few CPD scale heights. We obtain accretion rates only marginally smaller than those reported by, e.g., D'Angelo et al. (2003), Bate et al. (2003), Bodenheimer et al. (2013), who applied the same constant kinematic viscosity everywhere, including in the CPD. As found by several previous studies (e.g., D’Angelo et al. 2003; Bate et al. 2003; Tanigawa et al. 2012; Ayliffe and Bate 2012; Gressel et al. 2013; Szulágyi et al. 2014), the accretion flow does not proceed through the CPD mid-plane but rather at and above the CPD surface, hence involving MRI-active regions (Turner et al. 2014). We conclude that the presence of a dead zone in a CPD does not inhibit gas accretion on a giant planet. Sustained accretion in the presence of a CPD is consistent not only with the formation of Jupiter but also with observed extrasolar planets more massive than Jupiter. We place these results in the context of the growth and migration of a pair of giant planets locked in the 2
19. The Black-Hole Accretion Disk in NGC 4258: One of Nature's Most Beautiful Dynamical Systems
Moran, J. M.
2008-08-01
In this talk I will summarize some of the work that the CfA group has done to study the structure of the water masers in the accretion disk of NGC 4258. A series of 18 epochs of VLBA data taken from 1997.3 to 2000.8 were used for this study. The vertical distribution of maser features in the systemic group was found to be Gaussian, as expected for hydrostatic equilibrium, with a σ-width of 5.1 microarcsec (μas). If the disk is in hydrostatic equilibrium, its temperature is about 600 K. The systemic features exhibit a small, but persistent, gradient in acceleration versus impact parameter. This characteristic may indicate the presence of a spiral density wave rotating at sub-Keplerian speed. A more precise understanding of the dynamical properties of the disk is expected to lead to a more refined estimate of the distance to the galaxy.
20. Probing the connection between the accretion disk, outflows and the jet in 3C111
Tombesi, Francesco
2011-10-01
Recent XMM-Newton and Suzaku observations of 3C111 demonstrated the presence of ultra-fast outflows (UFOs) with v~0.1c and their relation with the accretion disk. Independent studies found that X-ray dips are followed by ejection of superluminal radio knots, therefore providing a proof of the disk-jet connection. We acquired evidence that UFOs are preferentially present between X-ray dips and new knots, possibly indicating also a link between disk outflows and the jet. The goal of this XMM-Newton proposal is to confirm this evidence. Given the strong correlation with X-rays, we will use an ongoing optical monitoring campaign to trigger a 90ks observation within two days of a dip to detect a UFO and we request a possible additional 60ks >15 days after to compare with the non-dipped state.
1. A test of synthetic accretion disk spectra using ultraviolet flux distributions of novalike variables
NASA Technical Reports Server (NTRS)
1988-01-01
Ultraviolet (UV) fluxes and other data for a sample of nine novalike cataclysmic variables are assembled from the literature. The UV fluxes and colors are compared to spectra of steady state model accretion disks, constructed using either Planck functions or model stellar atmosphere spectra as elementary radiators. Deficiencies are found with both sets of models. The restriction to steady state radial temperature profiles is relaxed for the 'stellar atmosphere' disks, but these generalized models still fail to account simultaneously for the absolute UV flux and the UV color of the observed spectra. The conclusion is that stellar atmosphere models do not reflect the physics of disks. More appropriate models probably can and definitely should be made.
2. Non-LTE effects on the strength of the Lyman edge in quasar accretion disks
NASA Technical Reports Server (NTRS)
Stoerzer, H.; Hauschildt, P. H.; Allard, F.
1994-01-01
We have calculated UV/EUV (300 A which is less than or equal to lambda which is less than or equal to 1500 A) continuous energy distributions of accretion disks in the centers of active galactic nuclei (AGNs) for disk luminosities in the range 0.1 L(sub Edd) less than or equal to L(sub acc) less than 1.0 L(sub Edd) and central masses ranging from 10(exp 8) solar mass to 10(exp 9) solar mass. The vertical gas pressure structure of the disk and the disk height are obtained analytically; the temperature stratification and the resulting continuum radiation fields are calculated numerically. We have included non-Local Thermodynamic Equilibrium (LTE) effects of both the ionization equilibrium and the level populations of hydrogen and helium. We show that these non-LTE effects reduce the strength of the Lyman edge when comapred to the LTE case. In non-LTE we find that the edge can be weakly in emission or absorption for disks seen face-on, depending on the disk parameters.
3. The role of magnetic reconnection on jet/accretion disk systems
de Gouveia Dal Pino, E. M.; Piovezan, P. P.; Kadowaki, L. H. S.
2010-07-01
Context. It was proposed earlier that the relativistic ejections observed in microquasars could be produced by violent magnetic reconnection episodes at the inner disk coronal region (de Gouveia Dal Pino & Lazarian 2005). Aims: Here we revisit this model, which employs a standard accretion disk description and fast magnetic reconnection theory, and discuss the role of magnetic reconnection and associated heating and particle acceleration in different jet/disk accretion systems, namely young stellar objects (YSOs), microquasars, and active galactic nuclei (AGNs). Methods: In microquasars and AGNs, violent reconnection episodes between the magnetic field lines of the inner disk region and those that are anchored in the black hole are able to heat the coronal/disk gas and accelerate the plasma to relativistic velocities through a diffusive first-order Fermi-like process within the reconnection site that will produce intermittent relativistic ejections or plasmons. Results: The resulting power-law electron distribution is compatible with the synchrotron radio spectrum observed during the outbursts of these sources. A diagram of the magnetic energy rate released by violent reconnection as a function of the black hole (BH) mass spanning 109 orders of magnitude shows that the magnetic reconnection power is more than sufficient to explain the observed radio luminosities of the outbursts from microquasars to low luminous AGNs. In addition, the magnetic reconnection events cause the heating of the coronal gas, which can be conducted back to the disk to enhance its thermal soft X-ray emission as observed during outbursts in microquasars. The decay of the hard X-ray emission right after a radio flare could also be explained in this model due to the escape of relativistic electrons with the evolving jet outburst. In the case of YSOs a similar magnetic configuration can be reached that could possibly produce observed X-ray flares in some sources and provide the heating at the
4. Links between the Shock Instability in Core-collapse Supernovae and Asymmetric Accretions of Envelopes
Takahashi, Kazuya; Iwakami, Wakana; Yamamoto, Yu; Yamada, Shoichi
2016-11-01
The explosion mechanism of core-collapse supernovae (CCSNe) has not been fully understood yet, but multidimensional fluid instabilities such as standing accretion shock instability and convection are now believed to be crucial for shock revival. Another multidimensional effect that has been recently argued is the asymmetric structures in progenitors, which are induced by violent convections in silicon/oxygen layers that occur before the onset of collapse, as revealed by recent numerical simulations of the last stage of massive star evolutions. Furthermore, it has been also demonstrated numerically that accretions of such nonspherical envelopes could facilitate shock revival. These two multidimensional effects may hence hold a key to successful explosions. In this paper, we performed a linear stability analysis of the standing accretion shock in CCSNe, taking into account nonspherical, unsteady accretion flows onto the shock to clarify the possible links between the two effects. We found that such preshock perturbations can excite the fluid instabilities efficiently and hence help the shock revive in CCSNe.
5. The Accretion Disk Wind in the Black Hole GRS 1915 + 105
NASA Technical Reports Server (NTRS)
Miller, J.M.; Raymond, J.; Fabian, A. C.; Gallo, E.; Kaastra, J.; Kallman, T.; King, A. L.; Proga, D.; Reynolds, C. S.; Zoghbi, A.
2016-01-01
We report on a 120 kiloseconds Chandra/HETG spectrum of the black hole GRS 1915+105. The observation was made during an extended and bright soft state in 2015 June. An extremely rich disk wind absorption spectrum is detected, similar to that observed at lower sensitivity in 2007. The very high resolution of the third-order spectrum reveals four components to the disk wind in the Fe K band alone; the fastest has a blueshift of v = 0.03 c (velocity equals 0.03 the speed of light). Broadened reemission from the wind is also detected in the first-order spectrum, giving rise to clear accretion disk P Cygni profiles. Dynamical modeling of the re-emission spectrum gives wind launching radii of r approximately equal to 10 (sup 2-4) GM (Gravitational constant times Mass) divided by c (sup 2) (the speed of light squared). Wind density values of n approximately equal to 10 (sup 13-16) per cubic centimeter are then required by the ionization parameter formalism. The small launching radii, high density values, and inferred high mass outflow rates signal a role for magnetic driving. With simple, reasonable assumptions, the wind properties constrain the magnitude of the emergent magnetic field to be B approximately equal to 10 (sup 3-4) G (Gravitational constant) if the wind is driven via magnetohydrodynamic (MHD) pressure from within the disk and B approximately equal to 10 (sup 4-5) G (Gravitational constant) if the wind is driven by magnetocentrifugal acceleration. The MHD estimates are below upper limits predicted by the canonical alpha-disk model. We discuss these results in terms of fundamental disk physics and black hole accretion modes.
6. Improved reflection models of black hole accretion disks: Treating the angular distribution of X-rays
SciTech Connect
García, J.; Steiner, J. F.; McClintock, J. E.; Brenneman, L. E-mail: [email protected] E-mail: [email protected]; and others
2014-02-20
X-ray reflection models are used to constrain the properties of the accretion disk, such as the degree of ionization of the gas and the elemental abundances. In combination with general relativistic ray tracing codes, additional parameters like the spin of the black hole and the inclination to the system can be determined. However, current reflection models used for such studies only provide angle-averaged solutions for the flux reflected at the surface of the disk. Moreover, the emission angle of the photons changes over the disk due to relativistic light bending. To overcome this simplification, we have constructed an angle-dependent reflection model with the XILLVER code and self-consistently connected it with the relativistic blurring code RELLINE. The new model, relxill, calculates the proper emission angle of the radiation at each point on the accretion disk and then takes the corresponding reflection spectrum into account. We show that the reflected spectra from illuminated disks follow a limb-brightening law highly dependent on the ionization of disk and yet different from the commonly assumed form I∝ln (1 + 1/μ). A detailed comparison with the angle-averaged model is carried out in order to determine the bias in the parameters obtained by fitting a typical relativistic reflection spectrum. These simulations reveal that although the spin and inclination are mildly affected, the Fe abundance can be overestimated by up to a factor of two when derived from angle-averaged models. The fit of the new model to the Suzaku observation of the Seyfert galaxy Ark 120 clearly shows a significant improvement in the constraint of the physical parameters, in particular by enhancing the accuracy in the inclination angle and the spin determinations.
7. A VLT/X-Shooter study of accretion and photoevaporation in Transitional Disks
Manara, Carlo Felice; Testi, Leonardo; Natta, Antonella; Ricci, Luca; Benisty, Myriam; Rosotti, Giovanni; Ercolano, Barbara
2013-07-01
Transitional Disks (TDs) are considered to be a late evolutionary stage of optically thick massive disks whose inner regions are being evacuated, leaving behind large holes that can be detected both by modeling the infrared spectral energy distribution (SED) or, in some cases, by mm-interferometry. These holes could be produced by processes of photoevaporation, grain growth, or planet formation. Still, none of these processes alone has been shown to be sufficient to explain all observations. In this context, the combination of inner hole size, mass accretion rate and wind properties is a powerful observational diagnostic of disk evolution models, but the current measurements of mass accretion rates for TDs are mostly based on secondary indicators (such as the 10% Ha width), and very few data on the wind properties for these objects are available. Here we present a detailed study of the accretion and wind properties of TDs carried out with the VLT/X-Shooter spectrograph. Combining new and archival X-Shooter observations, we collected a sample of more than 20 TDs from different nearby star-forming regions. Our sample includes objects with both small (<5-15 AU) and large (>20-30 AU) known inner hole size from the literature (either from mm-observations or infrared SED fitting). We check their stellar parameters (Teff, Lstar, Av, Mstar) and derive their accretion properties (Lacc, Macc) in a self-consistent way, which makes use of the wide wavelength coverage of X-Shooter, and study their wind properties by mean of different forbidden emission lines analysis. Here we present some preliminary results.
8. Orbital circularization of a planet accreting disk gas: the formation of distant jupiters in circular orbits based on a core accretion model
SciTech Connect
Kikuchi, Akihiro; Higuchi, Arika; Ida, Shigeru E-mail: [email protected]
2014-12-10
Recently, gas giant planets in nearly circular orbits with large semimajor axes (a ∼ 30-1000 AU) have been detected by direct imaging. We have investigated orbital evolution in a formation scenario for such planets, based on a core accretion model. (1) Icy cores accrete from planetesimals at ≲ 30 AU, (2) they are scattered outward by an emerging nearby gas giant to acquire highly eccentric orbits, and (3) their orbits are circularized through the accretion of disk gas in outer regions, where they spend most of their time. We analytically derived equations to describe the orbital circularization through gas accretion. Numerical integrations of these equations show that the eccentricity decreases by a factor of more than 5 while the planetary mass increases by a factor of 10. Because runaway gas accretion increases planetary mass by ∼10-300, the orbits are sufficiently circularized. On the other hand, a is reduced at most only by a factor of two, leaving the planets in the outer regions. If the relative velocity damping by shock is considered, the circularization slows down, but is still efficient enough. Therefore, this scenario potentially accounts for the formation of observed distant jupiters in nearly circular orbits. If the apocenter distances of the scattered cores are larger than the disk sizes, their a shrink to a quarter of the disk sizes; the a-distribution of distant giants could reflect the outer edges of the disks in a similar way that those of hot jupiters may reflect inner edges.
9. Light Curves from an MHD Simulation of a Black Hole Accretion Disk
Schnittman, Jeremy D.; Krolik, Julian H.; Hawley, John F.
2006-11-01
We use a relativistic ray-tracing code to calculate the light curves observed from a global, general relativistic, magnetohydrodynamic simulation of an accretion flow onto a Schwarzschild black hole. We apply three basic emission models to sample different properties of the time-dependent accretion disk. With one of these models, which assumes thermal blackbody emission and free-free absorption, we can predict qualitative features of the high-frequency power spectrum from stellar-mass black holes in the thermal dominant'' state. The simulated power spectrum is characterized by a power law of index Γ~3 and total rms fractional variance of <~2% above 10 Hz. For each emission model, we find that the variability amplitude should increase with increasing inclination angle. On the basis of a newly developed formalism for quantifying the significance of quasi-periodic oscillations (QPOs) in simulation data, we find that these simulations are able to identify any such features with (rms/mean) amplitudes >~1% near the orbital frequency at the innermost stable orbit. Initial results indicate the existence of transient QPO peaks with frequency ratios of nearly 2:3 at a 99.9% confidence limit, but they are not generic features, because at any given time they are seen only from certain observer directions. In addition, we present detailed analysis of the azimuthal structure of the accretion disk and the evolution of density perturbations in the inner disk. These hot-spot'' structures appear to be roughly self-similar over a range of disk radii, with a single characteristic size δφ=25deg and δr/r=0.3, and typical lifetimes Tl~0.3Torb.
10. Circumplanetary discs around young giant planets: a comparison between core-accretion and disc instability
Szulágyi, J.; Mayer, L.; Quinn, T.
2017-01-01
Circumplanetary discs can be found around forming giant planets, regardless of whether core accretion or gravitational instability built the planet. We carried out state-of-the-art hydrodynamical simulations of the circumplanetary discs for both formation scenarios, using as similar initial conditions as possible to unveil possible intrinsic differences in the circumplanetary disc mass and temperature between the two formation mechanisms. We found that the circumplanetary discs' mass linearly scales with the circumstellar disc mass. Therefore, in an equally massive protoplanetary disc, the circumplanetary discs formed in the disc instability model can be only a factor of 8 more massive than their core-accretion counterparts. On the other hand, the bulk circumplanetary disc temperature differs by more than an order of magnitude between the two cases. The subdiscs around planets formed by gravitational instability have a characteristic temperature below 100 K, while the core-accretion circumplanetary discs are hot, with temperatures even greater than 1000 K when embedded in massive, optically thick protoplanetary discs. We explain how this difference can be understood as the natural result of the different formation mechanisms. We argue that the different temperatures should persist up to the point when a full-fledged gas giant forms via disc instability; hence, our result provides a convenient criterion for observations to distinguish between the two main formation scenarios by measuring the bulk temperature in the planet vicinity.
11. High Energy Neutrinos Produced in the Accretion Disks by Neutrons from Nuclei Disintegrated in the AGN Jets
Bednarek, W.
2016-12-01
We investigate the consequences of acceleration of nuclei in jets of active galaxies not far from the surface of an accretion disk. The nuclei can be accelerated in the re-connection regions in the jet and/or at the jet boundary, between the relativistic jet and its cocoon. It is shown that the relativistic nuclei can efficiently fragment onto specific nucleons in collisions with the disk radiation. Neutrons, directed toward the accretion disk, take a significant part of energy from the relativistic nuclei. These neutrons develop a cascade in the dense accretion disk. We calculate the neutrino spectra produced in such a hadronic cascade within the accretion disk. We propose that the neutrinos produced in such a scenario, from the whole population of super-massive black holes in active galaxies, can explain the extragalactic neutrino background recently measured by the IceCube neutrino detector, provided that a 5% fraction of galaxies have an active galactic nucleus and a few percent of neutrons reach the accretion disk. We predict that the neutrino signals in the present neutrino detectors, produced in terms of such a model, will not be detectable even from the nearby radio galaxies similar to M87.
12. PARTICLE TRAPPING AND STREAMING INSTABILITY IN VORTICES IN PROTOPLANETARY DISKS
SciTech Connect
Raettig, Natalie; Klahr, Hubert; Lyra, Wladimir E-mail: [email protected]
2015-05-01
We analyze the concentration of solid particles in vortices created and sustained by radial buoyancy in protoplanetary disks, e.g., baroclinic vortex growth. Besides the gas drag acting on particles, we also allow for back-reaction from dust onto the gas. This becomes important when the local dust-to-gas ratio approaches unity. In our two-dimensional, local, shearing sheet simulations, we see high concentrations of grains inside the vortices for a broad range of Stokes numbers, St. An initial dust-to-gas ratio of 1:100 can easily be reversed to 100:1 for St = 1.0. The increased dust-to-gas ratio triggers the streaming instability, thus counter-intuitively limiting the maximal achievable overdensities. We find that particle trapping inside vortices opens the possibility for gravity assisted planetesimal formation even for small particles (St = 0.01) and a low initial dust-to-gas ratio of 1:10{sup 4}, e.g., much smaller than in the previously studied magnetohydrodynamic zonal flow case.
13. Lyapunov instability of rough hard-disk fluids.
PubMed
van Meel, Jacobus A; Posch, Harald A
2009-07-01
The dynamical instability of rough hard-disk fluids in two dimensions is characterized through the Lyapunov spectrum and the Kolmogorov-Sinai entropy h{KS} for a wide range of densities and moments of inertia I . For small I the spectrum separates into translation-dominated and rotation-dominated parts. With increasing I the rotation-dominated part is gradually filled in at the expense of translation until such a separation becomes meaningless. At any density, the rate of phase-space mixing, given by h{KS} , becomes less and less effective the more the rotation affects the dynamics. However, the degree of dynamical chaos, measured by the maximum Lyapunov exponent, is only enhanced by the rotational degrees of freedom for high-density gases but is diminished for lower densities. Surprisingly, no traces of Lyapunov modes were found in the spectrum for larger moments of inertia. The spatial localization of the perturbation vector associated with the maximum exponent however persists for any I .
14. The Accretion Disk of the Lithium-Depleted Young Binary St 34
NASA Technical Reports Server (NTRS)
Hartmann, Lee; Calvet, Nuria; Watson, Dan M.; D'Alessio, P.; Furlan, E.; Sargent, B.; Forrest, W. J.; Uchida, K. I.; Green, J. D.; Sloan, G. C.; Chen, C. H.; Najita, J.; Kemper, F.; Herter, T. L.; Morris, P.; Barry, D. J.; Hall, P.
2005-01-01
We presented the infrared spectrum of the young binary system St 34 obtained with the Infrared Spectrograph (IRS) on the Spitzer Space Telescope. The IRS spectrum clearly shows excess dust emission, consistent with the suggestion of White & Hillenbrand that St 34 is accreting from a circumbinary disk. The disk emission of St 34 is low in comparison with the levels observed in typical T Tauri stars; silicate features at 10 and 20 microns are much weaker than typically seen in T Tauri stars; and excess emission is nearly absent at the shortest wavelengths observed. These features of the infrared spectrum suggest substantial grain growth (to eliminate silicate features) and possible settling of dust to the disk midplane (to reduce the continuum excess emission levels), along with a relatively evacuated inner disk, as expected due to gravitational perturbations by the binary system. Although the position of St 34 in the H-R diagram suggests an age of 8f Myr, assuming that it lies at the distance of the Taurus-Auriga molecular clouds, White & Hillenbrand could not detect any Li I absorption, which would indicate a Li depletion age of roughly 25 Myr or more. We suggest that St 34 is closer than the Taurus clouds by about 30-40 pc and has an age roughly consistent with Li depletion models. Such an advanced age would make St 34 the oldest known low-mass pre-main-sequence object with a dusty accretion disk. The persistence of optically thick dust emission well outside the binary orbit may indicate a failure to make giant planets that could effectively remove dust particles.
15. Observable Consequences of Merger-driven Gaps and Holes in Black Hole Accretion Disks
Gültekin, Kayhan; Miller, Jon M.
2012-12-01
We calculate the observable signature of a black hole (BH) accretion disk with a gap or a hole created by a secondary BH embedded in the disk. We find that for an interesting range of parameters of BH masses (~106-109 M ⊙), orbital separation (~1 AU to ~0.1 pc), and gap width (10-190 disk scale heights), the missing thermal emission from a gap manifests itself in an observable decrement in the spectral energy distribution (SED). We present observational diagnostics in terms of power-law forms that can be fit to line-free regions in active galactic nucleus (AGN) spectra or in fluxes from sequences of broad filters. Most interestingly, the change in slope in the broken power law is almost entirely dependent on the width of the gap in the accretion disk, which in turn is uniquely determined by the mass ratio of the BHs, such that it scales roughly as q 5/12. Thus, one can use spectral observations of the continuum of bright AGNs to infer not only the presence of a closely separated BH binary, but also the mass ratio. When the BH merger opens an entire hole (or cavity) in the inner disk, the broadband SED of the AGNs or quasar may serve as a diagnostic. Such sources should be especially luminous in optical bands but intrinsically faint in X-rays (i.e., not merely obscured). We briefly note that viable candidates may have already been identified, though extant detailed modeling of those with high-quality data have not yet revealed an inner cavity.
16. Hybrid accretion disks in active galactic nuclei. I - Structure and spectra
NASA Technical Reports Server (NTRS)
Wandel, Amri; Liang, Edison P.
1991-01-01
A unified treatment is presented of the two distinct states of vertically thin AGN accretion disks: a cool (about 10 to the 6th K) optically thick solution, and a hot (about 10 to the 9th K) optically thin solution. A generalized formalism and a new radiative cooling equation valid in both regimes are introduced. A new luminosity limit is found at which the hot and cool alpha solutions merge into a single solution of intermediate optical depth. Analytic solutions for the disk structure are given, and output spectra are computed numerically. This is used to demonstrate the prospect of fitting AGN broadband spectra containing both the UV bump as well as the hard X-ray and gamma-ray tail, using a single accretion disk model. Such models are found to make definite predictions about the observed spectrum, such as the relation between the hard X-ray spectral index, the UV-to-X-ray luminosity ratio, and a feature of about 1 MeV.
17. A spectroscopic study of the radial velocity variations and accretion disks found in four dwarf novae
Stover, R. J.
Time resolved spectroscopic observations of the four dwarf novae SS Cyg, EM Cyg, U Gem, and RU Peg are presented. Although these systems were studied previously, all of the spectroscopic studies were done photographically. A linear response, digital detector is employed. Analytic techniques to the study of the radial velocity variations and emission line profiles found in dwarf novae are applied. In the study of SS Cyg cross-correlation techniques were used for the first time to measure the radial velocity variations of the secondary star absorption lines. In the study of U Gem, analysis of the accretion disk emission lines showed that the motion of the material in the disk cannot be described accurately by orbits defined within the three-body approximation. The observations of EM Cyg reveal an unstable accretion disk, with emission lines that vary erratically on timescales of minutes to days. New measurements of the radial velocity variations of the emission and absorption lines found in the spectrum of RU Peg agree with previous measurements but have a higher accuracy.
18. Consequences of Relativistic Neutron Outflow beyond the Accretion Disks of Active Galaxies
Ekejiuba, I. E.; Okeke, P. N.
1993-05-01
Three channels of relativistic electron injection in the jets of extragalactic radio sources (EGRSs) are discussed. With the assumption that an active galactic nucleus (AGN) is powered by a spinning supermassive black hole of mass ~ 10(8) M_⊙ which sits at the center of the nucleus and ingests matter and energy through an accretion disk, a model for extracting relativistic neutrons from the AGN is forged. In this model, the inelastic proton--proton and proton--photon interactions within the accretion disk, of relativistic protons with background thermal protons and photons, respectively, produce copious amounts of relativistic neutrons. These neutrons travel ballistically for ~ 10(3gamma_n ) seconds and escape from the disk before they decay. The secondary particles produced from the neutron decays then interact with the ambient magnetic field and/or other particles to produce the radio emissions observed in the jets of EGRSs. IEE acknowledges the support of the World Bank and the Federal University of Technology, Yola, Nigeria as well as the hospitality of Georgia State University.
19. Astro-1 and ground-based observations of Markarian 335: Evidence for an accretion disk
NASA Technical Reports Server (NTRS)
Zheng, W.; Kriss, G. A.; Davidsen, A. F.; Lee, G.; Code, A. D.; Bjorkma, K. S.; Smith, P. S.; Weistrop, D.; Malkan, M. A.; Baganoff, F. K.
1995-01-01
Simultaneous UV and optical observations of the Seyfert galaxy Markarian 335 (z = 0.026) during the Astro-1 mission yield a spectrum spanning the wavelength range of 912-8410 A. In the sub-Ly alpha region a prominent blended emission feature of O VI lambda lambda 1032, 1038, and Ly beta is nearly as strong as C IV wavelength 1549. The continuum flux extends beyond the redshifted Luman limit without a noticeable discontinuity, but a siginificant change in slope exists near the redhsifted Lyman edge. We suggest that such a change may be the signature of a Lyman edge in an accretion disk seen at a small inclination angle. Using a disk model including such an edge, we fit the spectrum with a central black hole mass of 5 x 10(exp 7) solar mass, an accretion rate of 0.07 solar mass/yr, and an optical depth at the Lyman edge of 0.4. To account for the strong O VI emission as well as the soft X-ray excess, we consider the effects of Comptonization on the disk spectrum, which can produce a high-energy tail for the UV bump and also smooth the Lyman edge feature.
20. THE EFFECTS OF MAGNETIC FIELDS AND INHOMOGENEITIES ON ACCRETION DISK SPECTRA AND POLARIZATION
SciTech Connect
Davis, Shane W.; Blaes, Omer M.; Hirose, Shigenobu; Krolik, Julian H.
2009-09-20
We present the results of one- and three-dimensional radiative transfer calculations of polarized spectra emerging from snapshots of radiation magnetohydrodynamical simulations of the local vertical structure of black hole accretion disks. The simulations cover a wide range of physical regimes relevant for the high/soft state of black hole X-ray binaries. We constrain the uncertainties in theoretical spectral color correction factors due to the presence of magnetic support of the disk surface layers and strong density inhomogeneities. For the radiation-dominated simulation, magnetic support increases the color correction factor by about 10%, but this is largely compensated by a 10% softening due to inhomogeneities. We also compute the effects of inhomogeneities and Faraday rotation on the resulting polarization. Magnetic fields in the simulations are just strong enough to produce significant Faraday depolarization near the spectral peak of the radiation field. X-ray polarimetry may therefore be a valuable diagnostic of accretion disk magnetic fields, being able to directly test simulations of magnetorotational turbulence.
1. The pulse amplitude variation with QPO frequency in SAX J1808.4-3658: Resonances with the accretion disk
Caliskan, Sirin; Alpar, Mehmet Ali; Sasmaz Mus, Sinem
2016-07-01
SAX J1808.4-3658 is an accreting millisecond pulsar with a spin period of 401 Hz. The pulsed amplitudes of this source vary with its kHz QPO frequencies (Bult & van der Klis 2015). The pulsed amplitude peaks at certain upper kHz QPO frequencies which we associate with boundary layer modes of the viscous accretion disk (Erkut et al. 2008). We model this as peaks in the energy dissipation rate at the accretion caps due to resonances between the accretion column and the driving modes of the boundary layer.
2. The response of relativistic outflowing gas to the inner accretion disk of a black hole.
PubMed
Parker, Michael L; Pinto, Ciro; Fabian, Andrew C; Lohfink, Anne; Buisson, Douglas J K; Alston, William N; Kara, Erin; Cackett, Edward M; Chiang, Chia-Ying; Dauser, Thomas; De Marco, Barbara; Gallo, Luigi C; Garcia, Javier; Harrison, Fiona A; King, Ashley L; Middleton, Matthew J; Miller, Jon M; Miniutti, Giovanni; Reynolds, Christopher S; Uttley, Phil; Vasudevan, Ranjan; Walton, Dominic J; Wilkins, Daniel R; Zoghbi, Abderahmen
2017-03-01
The brightness of an active galactic nucleus is set by the gas falling onto it from the galaxy, and the gas infall rate is regulated by the brightness of the active galactic nucleus; this feedback loop is the process by which supermassive black holes in the centres of galaxies may moderate the growth of their hosts. Gas outflows (in the form of disk winds) release huge quantities of energy into the interstellar medium, potentially clearing the surrounding gas. The most extreme (in terms of speed and energy) of these-the ultrafast outflows-are the subset of X-ray-detected outflows with velocities higher than 10,000 kilometres per second, believed to originate in relativistic (that is, near the speed of light) disk winds a few hundred gravitational radii from the black hole. The absorption features produced by these outflows are variable, but no clear link has been found between the behaviour of the X-ray continuum and the velocity or optical depth of the outflows, owing to the long timescales of quasar variability. Here we report the observation of multiple absorption lines from an extreme ultrafast gas flow in the X-ray spectrum of the active galactic nucleus IRAS 13224-3809, at 0.236 ± 0.006 times the speed of light (71,000 kilometres per second), where the absorption is strongly anti-correlated with the emission of X-rays from the inner regions of the accretion disk. If the gas flow is identified as a genuine outflow then it is in the fastest five per cent of such winds, and its variability is hundreds of times faster than in other variable winds, allowing us to observe in hours what would take months in a quasar. We find X-ray spectral signatures of the wind simultaneously in both low- and high-energy detectors, suggesting a single ionized outflow, linking the low- and high-energy absorption lines. That this disk wind is responding to the emission from the inner accretion disk demonstrates a connection between accretion processes occurring on very different
3. The response of relativistic outflowing gas to the inner accretion disk of a black hole
Parker, Michael L.; Pinto, Ciro; Fabian, Andrew C.; Lohfink, Anne; Buisson, Douglas J. K.; Alston, William N.; Kara, Erin; Cackett, Edward M.; Chiang, Chia-Ying; Dauser, Thomas; De Marco, Barbara; Gallo, Luigi C.; Garcia, Javier; Harrison, Fiona A.; King, Ashley L.; Middleton, Matthew J.; Miller, Jon M.; Miniutti, Giovanni; Reynolds, Christopher S.; Uttley, Phil; Vasudevan, Ranjan; Walton, Dominic J.; Wilkins, Daniel R.; Zoghbi, Abderahmen
2017-03-01
The brightness of an active galactic nucleus is set by the gas falling onto it from the galaxy, and the gas infall rate is regulated by the brightness of the active galactic nucleus; this feedback loop is the process by which supermassive black holes in the centres of galaxies may moderate the growth of their hosts. Gas outflows (in the form of disk winds) release huge quantities of energy into the interstellar medium, potentially clearing the surrounding gas. The most extreme (in terms of speed and energy) of these—the ultrafast outflows—are the subset of X-ray-detected outflows with velocities higher than 10,000 kilometres per second, believed to originate in relativistic (that is, near the speed of light) disk winds a few hundred gravitational radii from the black hole. The absorption features produced by these outflows are variable, but no clear link has been found between the behaviour of the X-ray continuum and the velocity or optical depth of the outflows, owing to the long timescales of quasar variability. Here we report the observation of multiple absorption lines from an extreme ultrafast gas flow in the X-ray spectrum of the active galactic nucleus IRAS 13224‑3809, at 0.236 ± 0.006 times the speed of light (71,000 kilometres per second), where the absorption is strongly anti-correlated with the emission of X-rays from the inner regions of the accretion disk. If the gas flow is identified as a genuine outflow then it is in the fastest five per cent of such winds, and its variability is hundreds of times faster than in other variable winds, allowing us to observe in hours what would take months in a quasar. We find X-ray spectral signatures of the wind simultaneously in both low- and high-energy detectors, suggesting a single ionized outflow, linking the low- and high-energy absorption lines. That this disk wind is responding to the emission from the inner accretion disk demonstrates a connection between accretion processes occurring on very
4. Fine-Tuning the Accretion Disk Clock in Hercules X-1
NASA Technical Reports Server (NTRS)
Still, M.; Boyd, P.
2004-01-01
RXTE ASM count rates from the X-ray pulsar Her X-1 began falling consistently during the late months of 2003. The source is undergoing another state transition similar to the anomalous low state of 1999. This new event has triggered observations from both space and ground-based observatories. In order to aid data interpretation and telescope scheduling, and to facilitate the phase-connection of cycles before and after the state transition, we have re-calculated the precession ephemeris using cycles over the last 3.5 years. We report that the source has displayed a different precession period since the last anomalous event. Additional archival data from CGRO suggests that each low state is accompanied by a change in precession period and that the subsequent period is correlated with accretion flux. Consequently our analysis reveals long-term accretion disk behaviour which is predicted by theoretical models of radiation-driven warping.
5. Hall instability of a weakly ionized, rotating disk with equilibrium pressure stratification and thermal loss
SciTech Connect
Bora, Madhurjya P.; Buzar Baruah, Manasi
2011-01-15
A linear stability analysis of a thin rotating Keplerian disk is presented in the framework of Hall-magnetohydrodynamics with equilibrium pressure stratification and radiative cooling. Anisotropic pressure is considered in view of a stronger axial magnetic field. The analysis is relevant in studying the stability of protoplanetary disks. It has been shown that the equilibrium pressure stratification determines the growth rate of the Hall instability. With radiative loss, the thermal modes are affected by the Hall mode and the classical instability conditions.
6. Accretion disks around neutron and strange stars in R+aR2 gravity
Staykov, Kalin V.; Doneva, Daniela D.; Yazadjiev, Stoytcho S.
2016-08-01
We study the electromagnetic spectrum of accretion disks around neutron and strange stars in R+aR2 gravity. Both static and rapidly rotating models are investigated. The results are compared with the General Relativistic results. We found difference between the results in both theories of about 50% for the electromagnetic flux and about 20% in the luminosity for models with equal mass and angular velocity in both theories. The observed differences are much lower for models rotating with Keplerian velocity and with equal masses.
7. The Evolutionary Pathways of Tidal Disruption Events: From Stars to Debris Streams, Accretion Disks, and Relativistic Jets
Coughlin, E. R.
Tidal disruption events, which occur when a star is destroyed by the gravitational field of a supermassive black hole, are unique probes of the inner regions of galaxies. In this thesis we explore various stages of the tidal disruption process, in an attempt to relate the observable signatures of tidal disruption events to the properties of the disrupted star and the black hole. We use numerical techniques to study the long-term evolution of the debris streams produced from tidal disruption events, showing that they can be gravitationally unstable and, as a result of the instability, fragment into small-scale, localized clumps. The implications of this finding are discussed, and we investigate how the thermodynamic properties of the gas comprising the stream affect the nature of the instability. We derive an analytic model for the structure of tidally-disrupted, stellar debris streams, and we compare the predictions of our model to numerical results. We present a model for the accretion disk that forms from a tidal disruption event when the accretion rate surpasses the Eddington limit of the supermassive black hole, showing that these disks are puffed up into quasispherical envelopes that are threaded by bipolar, relativistic jets. We compare the predictions of this model to observations of the jetted tidal disruption event Swift J1644+57. Finally, we derive, from the relativistic Boltzmann equation, the general relativistic equations of radiation hydrodynamics in the viscous limit, which characterize the interaction between radiation and matter when changes in the fluid over the photon mean free path are small. Our results demonstrate that, in contrast to previous works, a radiation-dominated fluid does in fact possess a finite bulk viscosity and a correction to the comoving energy density. Using the general relativistic equations of radiation hydrodynamics in the viscous limit, we present two models to describe the interaction between a relativistic jet launched
8. Beltrami state in black-hole accretion disk: A magnetofluid approach.
PubMed
Bhattacharjee, Chinmoy; Das, Rupam; Stark, David J; Mahajan, S M
2015-12-01
Using the magnetofluid unification framework, we show that the accretion disk plasma (embedded in the background geometry of a black hole) can relax to a class of states known as the Beltrami-Bernoulli (BB) equilibria. Modeling the disk plasma as a Hall magnetohydrodynamics (MHD) system, we find that the space-time curvature can significantly alter the magnetic (velocity) decay rates as we move away from the compact object; the velocity profiles in BB states, for example, deviate substantially from the predicted corresponding geodesic velocity profiles. These departures imply a rich interplay of plasma dynamics and general relativity revealed by examining the corresponding Bernoulli condition representing "homogeneity" of total energy. The relaxed states have their origin in the constraints provided by the two helicity invariants of Hall MHD. These helicities conspire to introduce an oscillatory length scale into the system that is strongly influenced by relativistic and thermal effects.
9. Reprocessing of Soft X-ray Emission Lines in Black Hole Accretion Disks
SciTech Connect
Mauche, C W; Liedahl, D A; Mathiesen, B F; Jimenez-Garate, M A; Raymond, J C
2003-10-17
By means of a Monte Carlo code that accounts for Compton scattering and photoabsorption followed by recombination, we have investigated the radiation transfer of Ly{alpha}, He{alpha}, and recombination continua photons of H- and He-like C, N, O, and Ne produced in the photoionized atmosphere of a relativistic black hole accretion disk. We find that photoelectric opacity causes significant attenuation of photons with energies above the O VIII K-edge; that the conversion efficiencies of these photons into lower-energy lines and recombination continua are high; and that accounting for this reprocessing significantly (by factors of 21% to 105%) increases the flux of the Ly{alpha} and He{alpha} emission lines of H- and He-like C and O escaping the disk atmosphere.
10. Vertical Advection Effects on Hyper-accretion Disks and Potential Link between Gamma-Ray Bursts and Kilonovae
Yi, Tuan; Gu, Wei-Min; Yuan, Feng; Liu, Tong; Mu, Hui-Jun
2017-02-01
Recent simulations on super-Eddington accretion flows have shown that, apart from the diffusion process, the vertical advection based on magnetic buoyancy can be a more efficient process to release the trapped photons in the optically thick disk. As a consequence, the radiative luminosity from the accretion disk can be far beyond the Eddington value. Following this spirit, we revisit the structure and radiation of hyper-accretion disks with mass accretion rates in the range of {10}-3∼ 10 {M}ȯ {{{s}}}-1. Our results show that, due to the strong cooling through the vertical advection, the disk temperature becomes lower than that in the classic model without the vertical advection process, and therefore the neutrino luminosity from the disk is lower. On the other hand, the gamma-ray photons released through the vertical advection can be extremely super-Eddington. We argue that the large amount of escaped gamma-ray photons may have more significant contribution to the primordial fireball than the neutrino annihilation, and may hint at a link between gamma-ray bursts and kilonovae in the black hole hyper-accretion scenario.
11. Dissipation and Vertical Energy Transport in Radiation-dominated Accretion Disks
Blaes, Omer; Krolik, Julian H.; Hirose, Shigenobu; Shabaltas, Natalia
2011-06-01
12. DISSIPATION AND VERTICAL ENERGY TRANSPORT IN RADIATION-DOMINATED ACCRETION DISKS
SciTech Connect
Blaes, Omer; Shabaltas, Natalia; Krolik, Julian H.; Hirose, Shigenobu
2011-06-01
13. Zombie Vortex Instability. I. A Purely Hydrodynamic Instability to Resurrect the Dead Zones of Protoplanetary Disks
Marcus, Philip S.; Pei, Suyang; Jiang, Chung-Hsiang; Barranco, Joseph A.; Hassanzadeh, Pedram; Lecoanet, Daniel
2015-07-01
There is considerable interest in hydrodynamic instabilities in dead zones of protoplanetary disks as a mechanism for driving angular momentum transport and as a source of particle-trapping vortices to mix chondrules and incubate planetesimal formation. We present simulations with a pseudo-spectral anelastic code and with the compressible code Athena, showing that stably stratified flows in a shearing, rotating box are violently unstable and produce space-filling, sustained turbulence dominated by large vortices with Rossby numbers of order ˜0.2-0.3. This Zombie Vortex Instability (ZVI) is observed in both codes and is triggered by Kolmogorov turbulence with Mach numbers less than ˜0.01. It is a common view that if a given constant density flow is stable, then stable vertical stratification should make the flow even more stable. Yet, we show that sufficient vertical stratification can be unstable to ZVI. ZVI is robust and requires no special tuning of boundary conditions, or initial radial entropy or vortensity gradients (though we have studied ZVI only in the limit of infinite cooling time). The resolution of this paradox is that stable stratification allows for a new avenue to instability: baroclinic critical layers. ZVI has not been seen in previous studies of flows in rotating, shearing boxes because those calculations frequently lacked vertical density stratification and/or sufficient numerical resolution. Although we do not expect appreciable angular momentum transport from ZVI in the small domains in this study, we hypothesize that ZVI in larger domains with compressible equations may lead to angular transport via spiral density waves.
14. Do Circumnuclear Dense Gas Disks Drive Mass Accretion onto Supermassive Black Holes?
Izumi, Takuma; Kawakatu, Nozomu; Kohno, Kotaro
2016-08-01
We present a positive correlation between the mass of dense molecular gas ({M}{{dense}}) of ˜100 pc scale circumnuclear disks (CNDs) and the black hole mass accretion rate ({\\dot{M}}{{BH}}) in a total of 10 Seyfert galaxies, based on data compiled from the literature and an archive (median aperture θ med = 220 pc). A typical {M}{{dense}} of CNDs is 107-8 {M}⊙ , estimated from the luminosity of the dense gas tracer, the HCN(1-0) emission line. Because dense molecular gas is the site of star formation, this correlation is virtually equivalent to the one between the nuclear star-formation rate and {\\dot{M}}{{BH}} revealed previously. Moreover, the {M}{{dense}}{--}{\\dot{M}}{{BH}} correlation was tighter for CND-scale gas than for the gas on kiloparsec or larger scales. This indicates that CNDs likely play an important role in fueling black holes, whereas greater than kiloparesec scale gas does not. To demonstrate a possible approach for studying the CND-scale accretion process with the Atacama Large Millimeter/submillimeter Array, we used a mass accretion model where angular momentum loss due to supernova explosions is vital. Based on the model prediction, we suggest that only the partial fraction of the mass accreted from the CND ({\\dot{M}}{{acc}}) is consumed as {\\dot{M}}{{BH}}. However, {\\dot{M}}{{acc}} agrees well with the total nuclear mass flow rate (i.e., {\\dot{M}}{{BH}} + outflow rate). Although these results are still tentative with large uncertainties, they support the view that star formation in CNDs can drive mass accretion onto supermassive black holes in Seyfert galaxies.
15. The structure and stability of transonic accretion disks surrounding black holes
NASA Technical Reports Server (NTRS)
Chen, Xingming; Taam, Ronald E.
1993-01-01
Stationary transonic alpha-viscosity models of accretion disks surrounding nonrotating black holes have been investigated. The viscosity is modified such that it vanishes in the supersonic region to ensure its effect does not violate the causality condition. In contrast to previous studies, the viscous stress is taken to be explicitly proportional to the angular velocity gradient and is not assumed to depend solely on the local pressure in the disk. The numerical results reveal that the structure of the innermost regions of the disk are more sensitive to the modified form of the viscosity than to the form of the viscous stress. The critical sonic point is located inside the innermost stable circular orbit of a test particle at 3 Schwarzschild radii. In these solutions, the transition from subsonic to supersonic flow results from pressure effects and not viscous effects. The linear stability of these disks has been examined in the local approximation. It is found that radiative energy transport and viscous stresses in the radial direction can have important effects. As a result, it is shown that the growth rate of the inertial-acoustic mode reaches a maximum at a critical wavelength.
16. Broadband Spectral Energy Distributions of Active Galactic Nuclei from an Accretion Disk with Advective Coronal Flow
Kawaguchi, Toshihiro; Shimura, Toshiya; Mineshige, Shin
2001-01-01
Recent multiwaveband observations of Seyfert nuclei and QSOs established significant deviations in the spectral shape of the big blue bump from a blackbody spectral shape; soft X-ray excess has a spectral index α (Fν~ν-α) of 1.6 and hard X-ray tail with α of ~0.7. We construct a disk-corona model which accounts for such broadband spectral properties. We study the emission spectrum emerging from a vertical disk-corona structure composed of two-temperature plasma by solving hydrostatic equilibrium and radiative transfer self-consistently. A fraction f of viscous heating due to mass accretion is assumed to be dissipated in a corona with a Thomson optical depth of τc, where advective cooling is also included, and a remaining fraction, 1-f, dissipates within a main body of the disk. Our model can nicely reproduce the soft X-ray excess with a power-law shape and the hard tail extending to ~50 keV. The different spectral slopes (α~1.5 below 2 keV and ~0.5 above) are the results of different emission mechanisms and different sites; the former slope is due to unsaturated Comptonization from the innermost zone, and the latter is due to a combination of the Comptonization, bremsstrahlung, and a reflection of the coronal radiation at the disk-corona boundary from the inner to surrounding zone (<=300 Schwarzschild radii). The emergent optical spectrum is redder (α~0.3) than that of the standard disk (α~-0.3), being consistent with observations, due to the different efficiencies of spectral hardening of disk emission at different radii. Further, we find that the cutoff frequency of the hard X-ray (~coronal electron temperature) and broadband spectral shape are insensitive to the black hole mass, while the peak frequency of the big blue bump is sensitive to the mass as the peak frequency ~M-1/4BH.
17. A Steady-state Alignment Front in an Accretion Disk Subjected to Lense-thirring Torques
Krolik, Julian H.; Hawley, John F.
2015-06-01
Using only physical mechanisms, i.e., 3D magnetohydrodynamics (MHD) with no phenomenological viscosity, we have simulated the dynamics of a moderately thin accretion disk subject to torques whose radial scaling mimics those produced by lowest-order post-Newtonian gravitomagnetism. In this simulation, we have shown how, in the presence of MHD turbulence, a time-steady transition can be achieved between an inner disk region aligned with the equatorial plane of the central mass’s spin and an outer region orbiting in a different plane. The position of the equilibrium orientation transition is determined by a balance between gravitomagnetic torque and warp-induced inward mixing of misaligned angular momentum from the outer disk. If the mixing is interpreted in terms of diffusive transport, the implied diffusion coefficient is ≃(0.6-0.8)cs2/{Ω } for sound speed cs and orbital frequency Ω. This calibration permits estimation of the orientation transition’s equilibrium location given the central mass, its spin parameter, and the disk’s surface density and scaleheight profiles. However, the alignment front overshoots before settling into an equilibrium, signaling that a diffusive model does not fully represent the time-dependent properties of alignment fronts under these conditions. Because the precessional torque on the disk at the alignment front is always comparable to the rate at which misaligned angular momentum is brought inward to the front by warp-driven radial motions, no break forms between the inner and outer portions of the disk in our simulation. Our results also raise questions about the applicability to MHD warped disks of the traditional distinction between “bending wave” and “diffusive” regimes.
18. Photoionization Models for the Inner Gaseous Disks of Herbig Be Stars: Evidence against Magnetospheric Accretion?
Patel, P.; Sigut, T. A. A.; Landstreet, J. D.
2017-02-01
We investigate the physical properties of the inner gaseous disks of three hot Herbig B2e stars, HD 76534, HD 114981, and HD 216629, by modeling CFHT-ESPaDOns spectra using non-LTE radiative transfer codes. We assume that the emission lines are produced in a circumstellar disk heated solely by photospheric radiation from the central star in order to test whether the optical and near-infrared emission lines can be reproduced without invoking magnetospheric accretion. The inner gaseous disk density was assumed to follow a simple power-law in the equatorial plane, and we searched for models that could reproduce observed lines of H i (Hα and Hβ), He i, Ca ii, and Fe ii. For the three stars, good matches were found for all emission line profiles individually; however, no density model based on a single power-law was able to reproduce all of the observed emission lines. Among the single power-law models, the one with the gas density varying as ∼10‑10(R */R)3 g cm‑3 in the equatorial plane of a 25 R * (0.78 au) disk did the best overall job of representing the optical emission lines of the three stars. This model implies a mass for the Hα-emitting portion of the inner gaseous disk of ∼10‑9 M *. We conclude that the optical emission line spectra of these HBe stars can be qualitatively reproduced by a ≈1 au, geometrically thin, circumstellar disk of negligible mass compared to the central star in Keplerian rotation and radiative equilibrium. Based on observations obtained at the Canada–France–Hawaii Telescope (CFHT) which is operated by the National Research Council of Canada, the Institut National des Sciences de l”Univers of the Centre National de la Recherche Scientique of France, and the University of Hawaii.
19. Evolution of Warped Accretion Disks in Active Galactic Nuclei. I. Roles of Feeding at the Outer Boundaries
Li, Yan-Rong; Wang, Jian-Min; Cheng, Cheng; Qiu, Jie
2013-02-01
We investigate the alignment processes of spinning black holes and their surrounding warped accretion disks in a frame of two different types of feeding at the outer boundaries. We consider (1) fixed flows in which gas is continually fed with a preferred angular momentum, and (2) free flows in which there is no gas supply and the disks diffuse freely at their outer edges. As expected, we find that for the cases of fixed flows the black hole disk systems always align on timescales of several 106 yr, irrespective of the initial inclinations. If the initial inclination angles are larger than π/2, the black hole accretion transits from retrograde to prograde fashion, and the accreted mass onto the black holes during these two phases is comparable. On the other hand, for the cases of free flows, both alignments and anti-alignments can occur, depending on the initial inclinations and the ratios of the angular momentum of the disks to that of the black holes. In such cases, the disks will be consumed within timescales of 106 yr by black holes accreting at the Eddington limit. We propose that there is a close connection between the black hole spin and the lifetime for which the feeding persists, which determines the observable episodic lifetimes of active galactic nuclei. We conclude that careful inclusion of the disk feeding at the outer boundaries is crucial for modeling the evolution of the black hole spin.
20. Galactic Disk Warps due to Intergalactic Accretion Flows onto the Disk
López-Corredoira, M.; Betancort-Rijo, J.; Beckman, J. E.
2008-06-01
The accretion of the intergalactic medium onto the gaseous disc is used to explain the generation of galactic warps. A cup-shaped distortion is expected, due to the transmission of the linear momentum; but, this effect is small for most incident inflow angles and the predominant effect turns out to be the transmission of angular momentum, i.e. a torque giving an integral-sign shaped warp. The torque produced by a flow of velocity ˜ 100 km/s and baryon density ˜ 10-25 kg/m3, which is within the possible values for the intergalactic medium, is enough to generate the observed warps and this mechanism offers quite a plausible explanation. The inferred rate of infall of matter, ˜ 1 M⊙/yr, to the Galactic disc that this theory predicts agrees with the quantitative predictions of chemical evolution resolving key issues, notably the G-dwarf problem. Sánchez-Salcedo (2006) suggests that this mechanism is not plausible because it would produce a dependence of the scaleheight of the disc with the Galactocentric azimuth in the outer disc, but rather than being an objection this is another argument in favour of the mechanism because this dependence is actually observed in our Galaxy.
1. V3885 SAGITTARIUS: A COMPARISON WITH A RANGE OF STANDARD MODEL ACCRETION DISKS
SciTech Connect
Linnell, Albert P.; Szkody, Paula; Godon, Patrick; Sion, Edward M.; Hubeny, Ivan; Barrett, Paul E. E-mail: [email protected] E-mail: [email protected] E-mail: [email protected]
2009-10-01
A chi-tilde{sup 2} analysis of standard model accretion disk synthetic spectrum fits to combined Far Ultraviolet Spectroscopic Explorer and Space Telescope Imaging Spectrograph spectra of V3885 Sagittarius, on an absolute flux basis, selects a model that accurately represents the observed spectral energy distribution. Calculation of the synthetic spectrum requires the following system parameters. The cataclysmic variable secondary star period-mass relation calibrated by Knigge in 2006 and 2007 sets the secondary component mass. A mean white dwarf (WD) mass from the same study, which is consistent with an observationally determined mass ratio, sets the adopted WD mass of 0.7 M {sub sun}, and the WD radius follows from standard theoretical models. The adopted inclination, i = 65 deg., is a literature consensus, and is subsequently supported by chi-tilde{sup 2} analysis. The mass transfer rate is the remaining parameter to set the accretion disk T {sub eff} profile, and the Hipparcos parallax constrains that parameter to M-dot=(5.0+-2.0) x 10{sup -9} M odot yr{sup -1} by a comparison with observed spectra. The fit to the observed spectra adopts the contribution of a 57, 000 +- 5000 K WD. The model thus provides realistic constraints on M-dot and T {sub eff} for a large M-dot system above the period gap.
2. V3885 Sagittarius: A Comparison With a Range of Standard Model Accretion Disks
NASA Technical Reports Server (NTRS)
Linnell, Albert P.; Godon, Patrick; Hubeny, Ivan; Sion, Edward M; Szkody, Paula; Barrett, Paul E.
2009-01-01
A chi-squared analysis of standard model accretion disk synthetic spectrum fits to combined Far Ultraviolet Spectroscopic Explorer and Space Telescope Imaging Spectrograph spectra of V3885 Sagittarius, on an absolute flux basis, selects a model that accurately represents the observed spectral energy distribution. Calculation of the synthetic spectrum requires the following system parameters. The cataclysmic variable secondary star period-mass relation calibrated by Knigge in 2006 and 2007 sets the secondary component mass. A mean white dwarf (WD) mass from the same study, which is consistent with an observationally determined mass ratio, sets the adopted WD mass of 0.7M(solar mass), and the WD radius follows from standard theoretical models. The adopted inclination, i = 65 deg, is a literature consensus, and is subsequently supported by chi-squared analysis. The mass transfer rate is the remaining parameter to set the accretion disk T(sub eff) profile, and the Hipparcos parallax constrains that parameter to mas transfer = (5.0 +/- 2.0) x 10(exp -9) M(solar mass)/yr by a comparison with observed spectra. The fit to the observed spectra adopts the contribution of a 57,000 +/- 5000 K WD. The model thus provides realistic constraints on mass transfer and T(sub eff) for a large mass transfer system above the period gap.
3. Reverse Radiative Shock Experiments Relevant to Accreting Stream-Disk Impact in Interacting Binaries
Krauland, Christine; Drake, R. P.; Kuranz, C. K.; Huntington, C. M.; Grosskopf, M. J.; Marion, D. C.; Young, R.; Plewa, T.
2011-05-01
In many Cataclysmic Binary systems, mass onto an accretion disk produces a `hot spot’ where the infalling supersonic flow obliquely strikes the rotating accretion disk. This collision region has many ambiguities as a radiation hydrodynamic system, but shock development in the infalling flow can be modeled. Depending upon conditions, it has been argued (Armitage & Livio, ApJ 493, 898) that the shocked region may be optically thin, thick, or intermediate, which has the potential to significantly alter the hot spot's structure and emissions. We report the first experimental attempt to produce colliding flows that create a radiative reverse shock at the Omega-60 laser facility. Obtaining a radiative reverse shock in the laboratory requires producing a sufficiently fast flow (> 100 km/s) within a material whose opacity is large enough to produce energetically significant emission from experimentally achievable layers. We will discuss the experimental design, the available data, and our astrophysical context. Funded by the NNSA-DS and SC-OFES Joint Prog. in High-Energy-Density Lab. Plasmas, by the Nat. Laser User Facility Prog. in NNSA-DS and by the Predictive Sci. Acad. Alliances Prog. in NNSA-ASC, under grant numbers are DE-FG52-09NA29548, DE-FG52-09NA29034, and DE-FC52-08NA28616.
4. Variability of the Accretion Disk of V926 Sco Inferred from Tomographic Analysis
Connolly, S. D.; Peris, C. S.; Vrtilek, S. D.
2013-11-01
We present phase-resolved spectroscopic observations of the low-mass X-ray binary V926 Sco (4U 1735-44), covering the orbital period of 0.23 days, obtained with the Walter Baade 6.5 m Magellan Telescope at the Las Campanas Observatory in 2010 June and 2011 June. We use Hα radial velocities to derive a systemic velocity of -109 ± 4 km s-1. The FWHM of the lines observed in common with previous authors are significantly lower during our observations suggesting much reduced velocities in the system. The equivalent width of the Bowen fluorescence lines with respect to He II λ4686 are factors of two or more lower during our observations in comparison to those previously reported for the system, suggesting reduced irradiation of the secondary. Doppler and modulation tomography of Hα and He II λ4686 show asymmetric emission that can be attributed to a bulge in the accretion disk, as inferred from He II observations by previous authors. The X-ray fluxes from the source at times concurrent with the optical observations are significantly lower during our observations than during optical observations taken in 2003. We suggest that the system is in a lower accretion state compared to earlier observations; this explains both the lower velocities observed from the disk and the reduction of emission due to Bowen fluorescence detected from the secondary.
5. ACCRETION KINEMATICS THROUGH THE WARPED TRANSITION DISK IN HD 142527 FROM RESOLVED CO(6–5) OBSERVATIONS
SciTech Connect
Casassus, S.; Marino, S.; Pérez, S.; Plas, G. van der; Christiaens, V.; Montesinos, Matías; Roman, P.; Dunhill, A.; Cuadra, J.; Cieza, L.; Moral, Victor; Armitage, P. J.; Wootten, A.
2015-10-01
The finding of residual gas in the large central cavity of the HD 142527 disk motivates questions regarding the origin of its non-Keplerian kinematics and possible connections with planet formation. We aim to understand the physical structure that underlies the intra-cavity gaseous flows, guided by new molecular-line data in CO(6–5) with unprecedented angular resolutions. Given the warped structure inferred from the identification of scattered-light shadows cast on the outer disk, the kinematics are consistent, to first order, with axisymmetric accretion onto the inner disk occurring at all azimuths. A steady-state accretion profile, fixed at the stellar accretion rate, explains the depth of the cavity as traced in CO isotopologues. The abrupt warp and evidence for near free-fall radial flows in HD 142527 resemble theoretical models for disk tearing, which could be driven by the reported low-mass companion, whose orbit may be contained in the plane of the inner disk. The companion’s high inclination with respect to the massive outer disk could drive Kozai oscillations over long timescales; high-eccentricity periods may perhaps account for the large cavity. While shadowing by the tilted disk could imprint an azimuthal modulation in the molecular-line maps, further observations are required to ascertain the significance of azimuthal structure in the density field inside the cavity of HD 142527.
6. Accretion Kinematics through the Warped Transition Disk in HD142527 from Resolved CO(6-5) Observations
Casassus, S.; Marino, S.; Pérez, S.; Roman, P.; Dunhill, A.; Armitage, P. J.; Cuadra, J.; Wootten, A.; van der Plas, G.; Cieza, L.; Moral, Victor; Christiaens, V.; Montesinos, Matías
2015-10-01
The finding of residual gas in the large central cavity of the HD 142527 disk motivates questions regarding the origin of its non-Keplerian kinematics and possible connections with planet formation. We aim to understand the physical structure that underlies the intra-cavity gaseous flows, guided by new molecular-line data in CO(6-5) with unprecedented angular resolutions. Given the warped structure inferred from the identification of scattered-light shadows cast on the outer disk, the kinematics are consistent, to first order, with axisymmetric accretion onto the inner disk occurring at all azimuths. A steady-state accretion profile, fixed at the stellar accretion rate, explains the depth of the cavity as traced in CO isotopologues. The abrupt warp and evidence for near free-fall radial flows in HD 142527 resemble theoretical models for disk tearing, which could be driven by the reported low-mass companion, whose orbit may be contained in the plane of the inner disk. The companion’s high inclination with respect to the massive outer disk could drive Kozai oscillations over long timescales; high-eccentricity periods may perhaps account for the large cavity. While shadowing by the tilted disk could imprint an azimuthal modulation in the molecular-line maps, further observations are required to ascertain the significance of azimuthal structure in the density field inside the cavity of HD 142527.
7. Stellar parameters and accretion rate of the transition disk star HD 142527 from X-shooter
SciTech Connect
Mendigutía, I.; Fairlamb, J.; Oudmaijer, R. D.; Montesinos, B.; Najita, J. R.; Brittain, S. D.; Van den Ancker, M. E.
2014-07-20
HD 142527 is a young pre-main-sequence star with properties indicative of the presence of a giant planet and/or a low-mass stellar companion. We have analyzed an X-Shooter/Very Large Telescope spectrum to provide accurate stellar parameters and accretion rate. The analysis of the spectrum, together with constraints provided by the spectral energy distribution fitting, the distance to the star (140 ± 20 pc), and the use of evolutionary tracks and isochrones, led to the following set of parameters: T{sub eff} = 6550 ± 100 K, log g = 3.75 ± 0.10, L{sub *}/L{sub ☉} = 16.3 ± 4.5, M{sub *}/M{sub ☉} = 2.0 ± 0.3, and an age of 5.0 ± 1.5 Myr. This stellar age provides further constraints to the mass of the possible companion estimated by Biller et al., being between 0.20 and 0.35 M{sub ☉}. Stellar accretion rates obtained from UV Balmer excess modeling and optical photospheric line veiling, and from the correlations with several emission lines spanning from the UV to the near-IR, are consistent with each other. The mean value from all previous tracers is 2 (±1) × 10{sup –7} M{sub ☉} yr{sup –1}, which is within the upper limit gas flow rate from the outer to the inner disk recently provided by Cassasus et al.. This suggests that almost all gas transferred between both components of the disk is not trapped by the possible planet(s) in between but fall onto the central star, although it is discussed how the gap flow rate could be larger than previously suggested. In addition, we provide evidence showing that the stellar accretion rate of HD 142527 has increased by a factor ∼7 on a timescale of 2 to 5 yr.
8. DETERMINATION OF CENTRAL ENGINE POSITION AND ACCRETION DISK STRUCTURE IN NGC 4261 BY CORE SHIFT MEASUREMENTS
SciTech Connect
Haga, Takafumi; Doi, Akihiro; Murata, Yasuhiro; Sudou, Hiroshi; Kameno, Seiji; Hada, Kazuhiro
2015-07-01
We report multifrequency phase-referenced observations of the nearby radio galaxy NGC 4261, which has prominent two-sided jets, using the Very Long Baseline Array at 1.4–43 GHz. We measured radio core positions showing observing frequency dependences (known as “core shift”) in both approaching jets and counterjets. The limit of the core position as the frequency approaches infinity, which suggests a jet base, is separated by 82 ± 16 μas upstream in projection, corresponding to (310 ± 60)R{sub s} (R{sub s}: Schwarzschild radius) as a deprojected distance, from the 43 GHz core in the approaching jet. In addition, the innermost component at the counterjet side appeared to approach the same position at infinity of the frequency, indicating that cores on both sides are approaching the same position, suggesting a spatial coincidence with the central engine. Applying a phase-referencing technique, we also obtained spectral index maps, which indicate that emission from the counterjet is affected by free–free absorption (FFA). The result of the core shift profile on the counterjet also requires FFA because the core positions at 5–15 GHz cannot be explained by a simple core shift model based on synchrotron self-absorption (SSA). Our result is apparently consistent with the SSA core shift with an additional disk-like absorber over the counterjet side. Core shift and opacity profiles at the counterjet side suggest a two-component accretion: a radiatively inefficient accretion flow at the inner region and a truncated thin disk in the outer region. We proposed a possible solution about density and temperature profiles in the outer disk on the basis of the radio observation.
9. Vortices in stratified protoplanetary disks. From baroclinic instability to vortex layers
Barge, P.; Richard, S.; Le Dizès, S.
2016-08-01
Context. Large-scale vortices could play a key role in the evolution of protoplanetary disks, particularly in the dead-zone where no turbulence associated with magnetic field is expected. Their possible formation by the subcritical baroclinic instability is a complex issue because of the vertical structure of the disk and the elliptical instability. Aims: In 2D disks the baroclinic instability is studied as a function of the thermal transfer efficiency. In 3D disks we explore the importance of radial and vertical stratification on the processes of vortex formation and amplification. Methods: Numerical simulations are performed using a fully compressible hydrodynamical code based on a second-order finite volume method. We assume a perfect gas law in inviscid disk models in which heat transfer is due to either relaxation or diffusion. Results: In 2D, the baroclinic instability with thermal relaxation leads to the formation of large-scale vortices, which are unstable with respect to the elliptic instability. In the presence of heat diffusion, hollow vortices are formed which evolve into vortical structures with a turbulent core. In 3D, the disk stratification is found to be unstable in a finite layer which can include the mid-plane or not. When the unstable layer contains the mid-plane, the 3D baroclinic instability with thermal relaxation is found to develop first in the unstable layer as in 2D, producing large-scale vortices. These vortices are then stretched out in the stable layer, creating long-lived columnar vortical structures extending through the width of the disk. They are also found to be the source of internal vortex layers that develop across the whole disk along baroclinic critical layer surfaces, and form new vortices in the upper region of the disk. Conclusions: In 3D disks, vortices can survive for a very long time if the production of vorticity by the baroclinic amplification balances the destruction of vorticity by the elliptical instability
10. The production of turbulence by gravitational instabilities in self-gravitating differentially rotating disks
Ebert, R.
1994-06-01
Interstellar molecular clouds can collapse under their selfgravity. As numerical calculations have shown, the process of collapse mostly leads to differentially rotating self-gravitating disks without any central condensation. Our calculation shows that these disks are mostly gravitational unstable and can build up turbulent motion by these instabilities. From the conditions of gravitational instabilities we calculate the dispersion relation and determine the most unstable wavelength. The mean turbulent velocity and the mixing length are calculated from these wavelengths. Turbulent viscosity which results from the turbulent motion determine the angular momentum transportation out of the disk and therefore the time scale of mass concentration of the whole disk. The source of turbulent energy in this case is the gravitational energy in the rotating disk.
11. ROSSBY WAVE INSTABILITY AT DEAD ZONE BOUNDARIES IN THREE-DIMENSIONAL RESISTIVE MAGNETOHYDRODYNAMICAL GLOBAL MODELS OF PROTOPLANETARY DISKS
SciTech Connect
Lyra, Wladimir; Mac Low, Mordecai-Mark E-mail: [email protected]
2012-09-01
It has been suggested that the transition between magnetorotationally active and dead zones in protoplanetary disks should be prone to the excitation of vortices via Rossby wave instability (RWI). However, the only numerical evidence for this has come from alpha disk models, where the magnetic field evolution is not followed, and the effect of turbulence is parameterized by Laplacian viscosity. We aim to establish the phenomenology of the flow in the transition in three-dimensional resistive-magnetohydrodynamical models. We model the transition by a sharp jump in resistivity, as expected in the inner dead zone boundary, using the PENCIL CODE to simulate the flow. We find that vortices are readily excited in the dead side of the transition. We measure the mass accretion rate finding similar levels of Reynolds stress at the dead and active zones, at the {alpha} Almost-Equal-To 10{sup -2} level. The vortex sits in a pressure maximum and does not migrate, surviving until the end of the simulation. A pressure maximum in the active zone also triggers the RWI. The magnetized vortex that results should be disrupted by parasitical magneto-elliptic instabilities, yet it subsists in high resolution. This suggests that either the parasitic modes are still numerically damped or that the RWI supplies vorticity faster than they can destroy it. We conclude that the resistive transition between the active and dead zones in the inner regions of protoplanetary disks, if sharp enough, can indeed excite vortices via RWI. Our results lend credence to previous works that relied on the alpha-disk approximation, and caution against the use of overly reduced azimuthal coverage on modeling this transition.
12. The effect of anisotropic emission from thick accretion disks on the luminosity functions of active galactic nuclei
NASA Technical Reports Server (NTRS)
Urry, C. M.; Marziani, P.; Calvani, M.
1991-01-01
High-luminosity active galactic nuclei (AGNs) powered by accretion onto a massive black hole (or other compact object) may have bolometric luminosities dominated by thermal emission from a geometrically thick accretion disk. Radiation from these disks is strongly anisotropic, which has important consequences for the observed luminosity distribution, and therefore for systematic biases in flux-limited samples. The effect of anisotropic emission from an ensemble of AGNs with random oriented thick disks radiating at or near the Eddington limit is calculated. Because of their higher luminosities, it is predicted face-on disks should constitute an increasing fraction of observed high-redshift, high-luminosity AGNs. Comparison of the results with observed quasar luminosity functions suggests a narrow mass distribution with an upper limit of about a billion solar masses for high-redshift quasars.
13. Accretion disk coronae of intermediate polar cataclysmic variables. 3D magnetohydrodynamic modelling and thermal X-ray emission
Barbera, E.; Orlando, S.; Peres, G.
2017-04-01
Context. Intermediate polar cataclysmic variables (IPCV) contain a magnetic, rotating white dwarf surrounded by a magnetically truncated accretion disk. To explain their strong flickering X-ray emission, accretion has been successfully taken into account. Nevertheless, observations suggest that accretion phenomena might not be the only process behind it. An intense flaring activity occurring on the surface of the disk may generate a corona, contribute to the thermal X-ray emission, and influence the system stability. Aims: Our purposes are: investigating the formation of an extended corona above the accretion disk, due to an intense flaring activity occurring on the disk surface; studying the effects of flares on the disk and stellar magnetosphere; assessing its contribution to the observed thermal X-ray flux. Methods: We have developed a 3D magnetohydrodynamic (MHD) model of a IPCV system. The model takes into account gravity, disk viscosity, thermal conduction, radiative losses, and coronal flare heating through heat injection at randomly chosen locations on the disk surface. To perform a parameter space exploration, several system conditions have been considered, with different magnetic field intensity and disk density values. From the results of the evolution of the model, we have synthesized the thermal X-ray emission. Results: The simulations show the formation of an extended corona, linking disk and star. The flaring activity is capable of strongly influencing the disk configuration and possibly its stability, effectively deforming the magnetic field lines. Hot plasma evaporation phenomena occur in the layer immediately above the disk. The flaring activity gives rise to a thermal X-ray emission in both the [ 0.1-2.0 ] keV and the [ 2.0-10 ] keV X-ray bands. Conclusions: An intense coronal activity occurring on the disk surface of an IPCV can affect the structure of the disk depending noticeably on the density of the disk and the magnetic field of the central
14. Brightening of an accretion disk due to viscous dissipation of gravitational waves during the coalescence of supermassive black holes.
PubMed
Kocsis, Bence; Loeb, Abraham
2008-07-25
Mergers of supermassive black hole binaries release peak power of up to approximately 10(57) erg s(-1) in gravitational waves (GWs). As the GWs propagate through ambient gas, they induce shear and a small fraction of their power is dissipated through viscosity. The dissipated heat appears as electromagnetic (EM) radiation, providing a prompt EM counterpart to the GW signal. For thin accretion disks, the GW heating rate exceeds the accretion power at distances farther than approximately 10(3) Schwarzschild radii, independently of the accretion rate and viscosity coefficient.
15. DISK MASSES AT THE END OF THE MAIN ACCRETION PHASE: CARMA OBSERVATIONS AND MULTI-WAVELENGTH MODELING OF CLASS I PROTOSTARS
SciTech Connect
Eisner, J. A.
2012-08-10
We present imaging observations at the 1.3 mm wavelength of Class I protostars in the Taurus star-forming region, obtained with the CARMA interferometer. Of an initial sample of 10 objects, we detected and imaged millimeter wavelength emission from 9. One of the nine is resolved into two sources and detailed analysis of this binary protostellar system is deferred to a future paper. For the remaining eight objects, we use the CARMA data to determine the basic morphology of the millimeter emission. Combining the millimeter data with 0.9 {mu}m images of scattered light, Spitzer Infrared Spectrograph spectra, and broadband spectral energy distributions (all from the literature), we attempt to determine the structure of the circumstellar material. We consider models including both circumstellar disks and envelopes, and constrain the masses (and other structural parameters) of each of these components. We show that the disk masses in our sample span a range from {approx}< 0.01 to {approx}> 0.1 M{sub Sun }. The disk masses for our sample are significantly higher than for samples of more evolved Class II objects. Thus, Class I disk masses probably provide a more accurate estimate of the initial mass budget for star and planet formation. However, the disk masses determined here are lower than required by theories of giant planet formation. The masses also appear too low for gravitational instability, which could lead to high mass accretion rates. Even in these Class I disks, substantial particle growth may have hidden much of the disk mass in hard-to-see larger bodies.
16. ANTI-CORRELATED TIME LAGS IN THE Z SOURCE GX 5-1: POSSIBLE EVIDENCE FOR A TRUNCATED ACCRETION DISK
SciTech Connect
Sriram, K.; Choi, C. S.; Rao, A. R.
2012-06-01
We investigate the nature of the inner accretion disk in the neutron star source GX 5-1 by making a detailed study of time lags between X-rays of different energies. Using the cross-correlation analysis, we found anti-correlated hard and soft time lags of the order of a few tens to a few hundred seconds and the corresponding intensity states were mostly the horizontal branch (HB) and upper normal branch. The model independent and dependent spectral analysis showed that during these time lags the structure of the accretion disk significantly varied. Both eastern and western approaches were used to unfold the X-ray continuum and systematic changes were observed in soft and hard spectral components. These changes along with a systematic shift in the frequency of quasi-periodic oscillations (QPOs) made it substantially evident that the geometry of the accretion disk is truncated. Simultaneous energy spectral and power density spectral study shows that the production of the horizontal branch oscillations (HBOs) is closely related to the Comptonizing region rather than the disk component in the accretion disk. We found that as the HBO frequency decreases from the hard apex to upper HB, the disk temperature increases along with an increase in the coronal temperature, which is in sharp contrast with the changes found in black hole binaries where the decrease in the QPO frequency is accompanied by a decrease in the disk temperature and a simultaneous increase in the coronal temperature. We discuss the results in the context of re-condensation of coronal material in the inner region of the disk.
17. The Star-formation History and Accretion Disk Fraction of the Scorpius-Centaurus OB Association
Pecaut, Mark; Mamajek, E. E.
2013-01-01
We present a study of the star-formation history and accretion disk fraction of ~0.6-1.8 Msun stars in the nearest OB Association, Scorpius-Centaurus (Sco-Cen; ~10-20 Myr; 100-200 pc). We have performed a low-resolution spectroscopic survey for new, low-mass K- and M-type members of all three subgroups -- Upper Scorpius (US), Upper Centaurus-Lupus (UCL) and Lower Centaurus-Crux (LCC). We find that young, pre-main sequence stars are generally redder and hotter for a given spectral type than their main-sequence counterparts and therefore main-sequence intrinsic colors and temperatures are unsuitable for de-reddening the low-mass members of Sco-Cen and placing them on an H-R diagram. Using nearby, young moving groups within 75 pc, we derive a spectral type--intrinsic color sequence appropriate for pre-main sequence stars, and use synthetic spectral energy distribution fits to infer the proper temperature scale for these young stars. We use this new pre-main sequence intrinsic color and temperature calibration to place our ~200 newly identified members of Sco-Cen on an H-R diagram. We derive isochronal ages for the F-type members of Upper Centaurus-Lupus (UCL; 16 Myr; =142 pc) and Lower Centaurus-Crux (LCC; 17 Myr; =118 pc) which are consistent with the most recent results from the high-mass stars and the G- and K-type stars. However, our results for Upper Scorpius (US; 11 Myr; =145 pc) indicate it is a factor of two older than previously thought. Finally, we find an accretion disk fraction for UCL and LCC of ~3% for K-type stars decreasing to 2% for F-type stars at ~16-17 Myr, while US has an accretion disk fraction of 5% for K-type stars decreasing to <19% (95% C.L.) for F-type stars at ~11 Myr.
18. Accretion Disk Spectra of the Ultra-luminous X-ray Sources in Nearby Spiral Galaxies and Galactic Superluminal Jet Sources
NASA Technical Reports Server (NTRS)
White, Nicholas E. (Technical Monitor); Ebisawa, Ken; Zycki, Piotr; Kubota, Aya; Mizuno, Tsunefumi; Watarai, Ken-ya
2003-01-01
Ultra-luminous Compact X-ray Sources (ULXs) in nearby spiral galaxies and Galactic superluminal jet sources share the common spectral characteristic that they have unusually high disk temperatures which cannot be explained in the framework of the standard optically thick accretion disk in the Schwarzschild metric. On the other hand, the standard accretion disk around the Kerr black hole might explain the observed high disk temperature, as the inner radius of the Kerr disk gets smaller and the disk temperature can be consequently higher. However, we point out that the observable Kerr disk spectra becomes significantly harder than Schwarzschild disk spectra only when the disk is highly inclined. This is because the emission from the innermost part of the accretion disk is Doppler-boosted for an edge-on Kerr disk, while hardly seen for a face-on disk. The Galactic superluminal jet sources are known to be highly inclined systems, thus their energy spectra may be explained with the standard Kerr disk with known black hole masses. For ULXs, on the other hand, the standard Kerr disk model seems implausible, since it is highly unlikely that their accretion disks are preferentially inclined, and, if edge-on Kerr disk model is applied, the black hole mass becomes unreasonably large (greater than or approximately equal to 300 Solar Mass). Instead, the slim disk (advection dominated optically thick disk) model is likely to explain the observed super- Eddington luminosities, hard energy spectra, and spectral variations of ULXs. We suggest that ULXs are accreting black holes with a few tens of solar mass, which is not unexpected from the standard stellar evolution scenario, and their X-ray emission is from the slim disk shining at super-Eddington luminosities.
19. IRAS 16293-2422: Evidence for Infall onto a Counter-Rotating Protostellar Accretion Disk
NASA Technical Reports Server (NTRS)
Remijan, Anthony J.; Hollis, J. M.
2005-01-01
We report high spatial resolution VLA observations of the low-mass star-forming region IRAS 16293-2422 using four molecular probes: ethyl cyanide (CH3CH2CN)) methyl formate (CH3OCHO), formic acid (HCOOH), and the ground vibrational state of silicon monoxide (SiO). Ethyl cyanide emission has a spatial scale of approx. 20" and encompasses binary cores A and B as determined by continuum emission peaks. Surrounded by formic acid emission, methyl formate emission has a spatial scale of approx. 6" and is confined to core B. SiO emission shows two velocity components with spatial scales less than 2" that map approx. 2" northeast of the A and B symmetry axis. The redshifted SiO is approx. 2" northwest of blueshifted SiO along a position angle of approx. 135deg which is approximately parallel to the A and B symmetry axis. We interpret the spatial position offset in red and blueshifted SiO emission as due to rotation of a protostellar accretion disk and we derive approx. 1.4 Solar Mass, interior to the SiO emission. In the same vicinity, Mundy et al. (1986) also concluded rotation of a nearly edge-on disk from OVRO observations of much stronger and ubiquitous CO-13 emission but the direction of rotation is opposite to the SiO emission findings. Taken together, SiO and CO-13 data suggest evidence for a counter-rotating disk. Moreover, archival BIMA array CO-12C data show an inverse P Cygni profile with the strongest absorption in close proximity to the SiO emission, indicating unambiguous material infall toward the counter-rotating protostellar disk at a new source location within the IRAS 16293-2422 complex. The details of these observations and our interpretations are discussed.
20. A 3D Numerical Study of Gravitational Instabilities in Young Circumbinary Disks
Cai, Kai; Michael, Scott; Durisen, Richard
2013-07-01
Gravitational instabilities (GIs) in protoplanetary disks have been suggested as one of the major formation mechanisms of giant planets. Theoretical and computational studies have indicated that certain family of GIs can be excited in a circumbinary disk, which could lead to enhanced protoplanet formation (e.g., Sellwood & Lin 1989, Boss 2006). We have carried out a 3D simulation of a gravitationally unstable circumbinary disk around a young Sun-like star and a 0.02-Msun companion, both inside the central hole of the disk. Here we present a preliminary comparison between this simulation and a similarly simulated circumstellar disk around a solar-mass star but without the low-mass companion. The GIs stimulated by the binary and those that arise spontaneously are quite different in structure and strength. However, no fragmentation is observed, even after many orbital periods as measured in the outer disk.
1. Disk heating and bending instability in galaxies with counterrotation
Khoperskov, Sergey; Bertin, Giuseppe
2017-01-01
With the help of high-resolution long-slit and integral-field spectroscopy observations, the number of confirmed cases of galaxies with counterrotation is increasing rapidly. The evolution of such counterrotating galaxies remains far from being well understood. In this paper we study the dynamics of counterrotating collisionless stellar disks by means of N-body simulations. We show that, in the presence of counterrotation, an otherwise gravitationally stable disk can naturally generate bending waves accompanied by strong disk heating across the disk plane, that is in the vertical direction. Such a conclusion is found to hold even for dynamically warm systems with typical values of the initial vertical-to-radial velocity dispersion ratio σz/σR ≈ 0.5, for which the role of pressure anisotropy should be unimportant. We note that, during evolution, the σz/σR ratio tends to rise up to values close to unity in the case of locally Jeans-stable disks, whereas in disks that are initially Jeans-unstable it may reach even higher values, especially in the innermost regions. This unusual behavior of the σz/σR ratio in galaxies with counterrotation appears not to have been noticed earlier. Our investigations of systems made of two counterrotating components with different mass-ratios suggest that even apparently normal disk galaxies (i.e., with a minor counterrotating component so as to escape detection in current observations) might be subject to significant disk heating especially in the vertical direction.
2. Iron K Lines from Accretion Disks: Models for Line Production and Spectroscopic Constraints
NASA Technical Reports Server (NTRS)
Kallman, Timothy; Palmeri, Patrick
2004-01-01
Measured profiles of the iron K lines provide important dynamical information about emitting matrial in compact objects. However, much of the modeling work which has been used to infer the location and origin of line observed from AGN and galactic black hole sources is based on highly simplified assumptions about the microphysics of K line emission. In particular, many of the intrinsic line energies, widths and emissivities are based on central-field atomic calculations. We present the results of new calculations of the quantities for the entire iron isonuclear sequence, and demonstrate that the intrinsic K line spectra contain considerably more complexity than has been previously considered. We also present calculations of iron K emission and absorption spectra which include the new data, including the local spectrum radiated from an X-ray illuminated accretion disk. The implications for the interpretation of observed iron K lines from black hole sources will be discussed.
3. Low-frequency modes and nonbarotropic effects in pseudo-Newtonian accretion disks
NASA Technical Reports Server (NTRS)
Ipser, James R.
1994-01-01
A recently developed formalism is used to reexamine the question of the existence of hydrodynamical modes that pulsate with very low frequencies in the inner regions of accretion disks. The formalism is valid in an exact sense for the adiabatic pulsations of rotating Newtonian fluids that are generally nonbarotropic (such as those with 'nonadiabatic temperature gradients,' for example), and hence its application in the present context represents an improvement over previous analyses that are more approximate. The formalism is applied to thin non-self-gravitating disks, with the gravitational potential of the central source modified in the usual way in order to simulate relativistic effects. In the barotropic limit, the analyses indicate that in many cases nearly Keplerian disks exhibit nonaxisymmetric modes of pulsation that are trapped in the inner disk regions, with pulsation periods much longer than the dynamical timescale. These results are similar to those of earlier calculations that assume disks pulsate without changing the temperature distribution. A method is developed for including lowest order nonbarotropic effects. Previous analyses have been incapable of accurately treating the nonbarotropic regime. The application of the present method to the low-frequency modes reveals that, due to unexpected cancellations among terms, the nonbarotropic correction to the pusation frequency omega is only of order tilde-omega(sub BV exp 2) omega, where tilde-omega(sub BV) is the appropriate dimensionless Brunt-Vaisala frequency. This correction is much smaller than the expected correction of order tilde-omega(sub BV) Omega, where Omega is the rotation angular velocity. The important conclusion drawn from this is that nonbarotropic corrections are generally small and hence that low-frequency modes persist into the nonbarotropic regime. For disk temperatures appropriate to X-ray emission, the adiabatic frequencies of trapped modes are of the same order as the frequencies
4. MAGNETICALLY DRIVEN ACCRETION DISK WINDS AND ULTRA-FAST OUTFLOWS IN PG 1211+143
SciTech Connect
Fukumura, Keigo; Tombesi, Francesco; Kazanas, Demosthenes; Shrader, Chris; Contopoulos, Ioannis
2015-05-20
We present a study of X-ray ionization of MHD accretion-disk winds in an effort to constrain the physics underlying the highly ionized ultra-fast outflows (UFOs) inferred by X-ray absorbers often detected in various sub classes of Seyfert active galactic nuclei (AGNs). Our primary focus is to show that magnetically driven outflows are indeed physically plausible candidates for the observed outflows accounting for the AGN absorption properties of the present X-ray spectroscopic observations. Employing a stratified MHD wind launched across the entire AGN accretion disk, we calculate its X-ray ionization and the ensuing X-ray absorption-line spectra. Assuming an appropriate ionizing AGN spectrum, we apply our MHD winds to model the absorption features in an XMM-Newton/EPIC spectrum of the narrow-line Seyfert, PG 1211+143. We find, through identifying the detected features with Fe Kα transitions, that the absorber has a characteristic ionization parameter of log (ξ{sub c}[erg cm s{sup −1}]) ≃ 5–6 and a column density on the order of N{sub H} ≃ 10{sup 23} cm{sup −2} outflowing at a characteristic velocity of v{sub c}/c ≃ 0.1–0.2 (where c is the speed of light). The best-fit model favors its radial location at r{sub c} ≃ 200 R{sub o} (R{sub o} is the black hole’s innermost stable circular orbit), with an inner wind truncation radius at R{sub t} ≃ 30 R{sub o}. The overall K-shell feature in the data is suggested to be dominated by Fe xxv with very little contribution from Fe xxvi and weakly ionized iron, which is in good agreement with a series of earlier analyses of the UFOs in various AGNs, including PG 1211+143.
5. Magnetically Driven Accretion Disk Winds and Ultra-fast Outflows in PG 1211+143
Fukumura, Keigo; Tombesi, Francesco; Kazanas, Demosthenes; Shrader, Chris; Behar, Ehud; Contopoulos, Ioannis
2015-05-01
We present a study of X-ray ionization of MHD accretion-disk winds in an effort to constrain the physics underlying the highly ionized ultra-fast outflows (UFOs) inferred by X-ray absorbers often detected in various sub classes of Seyfert active galactic nuclei (AGNs). Our primary focus is to show that magnetically driven outflows are indeed physically plausible candidates for the observed outflows accounting for the AGN absorption properties of the present X-ray spectroscopic observations. Employing a stratified MHD wind launched across the entire AGN accretion disk, we calculate its X-ray ionization and the ensuing X-ray absorption-line spectra. Assuming an appropriate ionizing AGN spectrum, we apply our MHD winds to model the absorption features in an XMM-Newton/EPIC spectrum of the narrow-line Seyfert, PG 1211+143. We find, through identifying the detected features with Fe Kα transitions, that the absorber has a characteristic ionization parameter of log (ξc[erg cm s-1]) ≃ 5-6 and a column density on the order of NH ≃ 1023 cm-2 outflowing at a characteristic velocity of vc/c ≃ 0.1-0.2 (where c is the speed of light). The best-fit model favors its radial location at rc ≃ 200 Ro (Ro is the black hole’s innermost stable circular orbit), with an inner wind truncation radius at Rt ≃ 30 Ro. The overall K-shell feature in the data is suggested to be dominated by Fe xxv with very little contribution from Fe xxvi and weakly ionized iron, which is in good agreement with a series of earlier analyses of the UFOs in various AGNs, including PG 1211+143.
6. An ALMA Constraint on the GSC 6214-210 B Circum-Substellar Accretion Disk Mass
Bowler, Brendan P.; Andrews, Sean M.; Kraus, Adam L.; Ireland, Michael J.; Herczeg, Gregory; Ricci, Luca; Carpenter, John; Brown, Michael E.
2015-06-01
We present Atacama Large Millimeter/submillimeter Array (ALMA) observations of GSC 6214-210 A and B, a solar-mass member of the 5-10 Myr Upper Scorpius association with a 15 ± 2 MJup companion orbiting at ≈ 330 AU (2.″2). Previous photometry and spectroscopy spanning 0.3-5 μm revealed optical and thermal excess as well as strong Hα and Pa β emission originating from a circum-substellar accretion disk around GSC 6214-210 B, making it the lowest-mass companion with unambiguous evidence of a subdisk. Despite ALMA’s unprecedented sensitivity and angular resolution, neither component was detected in our 880 μm (341 GHz) continuum observations down to a 3σ limit of 0.22 mJy/beam. The corresponding constraints on the dust mass and total mass are <0.15 M⨁ and <0.05 MJup, respectively, or <0.003% and <0.3% of the mass of GSC 6214-210 B itself assuming a 100:1 gas-to-dust ratio and characteristic dust temperature of 10-20 K. If the host star possesses a putative circum-stellar disk then at most it is a meager 0.0015% of the primary mass, implying that giant planet formation has certainly ceased in this system. Considering these limits and its current accretion rate, GSC 6214-210 B appears to be at the end stages of assembly and is not expected to gain any appreciable mass over the next few megayears.
7. Modeling MHD accretion-ejection: episodic ejections of jets triggered by a mean-field disk dynamo
SciTech Connect
Stepanovs, Deniss; Fendt, Christian; Sheikhnezami, Somayeh E-mail: [email protected]
2014-11-20
We present MHD simulations exploring the launching, acceleration, and collimation of jets and disk winds. The evolution of the disk structure is consistently taken into account. Extending our earlier studies, we now consider the self-generation of the magnetic field by an α{sup 2}Ω mean-field dynamo. The disk magnetization remains on a rather low level, which helps to evolve the simulations for T > 10, 000 dynamical time steps on a domain extending 1500 inner disk radii. We find the magnetic field of the inner disk to be similar to the commonly found open field structure, favoring magneto-centrifugal launching. The outer disk field is highly inclined and predominantly radial. Here, differential rotation induces a strong toroidal component, which plays a key role in outflow launching. These outflows from the outer disk are slower, denser, and less collimated. If the dynamo action is not quenched, magnetic flux is continuously generated, diffuses outward through the disk, and fills the entire disk. We have invented a toy model triggering a time-dependent mean-field dynamo. The duty cycles of this dynamo lead to episodic ejections on similar timescales. When the dynamo is suppressed as the magnetization falls below a critical value, the generation of the outflows and also accretion is inhibited. The general result is that we can steer episodic ejection and large-scale jet knots by a disk-intrinsic dynamo that is time-dependent and regenerates the jet-launching magnetic field.
8. MEASURING THE DIRECTION AND ANGULAR VELOCITY OF A BLACK HOLE ACCRETION DISK VIA LAGGED INTERFEROMETRIC COVARIANCE
SciTech Connect
Johnson, Michael D.; Loeb, Abraham; Shiokawa, Hotaka; Chael, Andrew A.; Doeleman, Sheperd S.
2015-11-10
We show that interferometry can be applied to study irregular, rapidly rotating structures, as are expected in the turbulent accretion flow near a black hole. Specifically, we analyze the lagged covariance between interferometric baselines of similar lengths but slightly different orientations. For a flow viewed close to face-on, we demonstrate that the peak in the lagged covariance indicates the direction and angular velocity of the emission pattern from the flow. Even for moderately inclined flows, the covariance robustly estimates the flow direction, although the estimated angular velocity can be significantly biased. Importantly, measuring the direction of the flow as clockwise or counterclockwise on the sky breaks a degeneracy in accretion disk inclinations when analyzing time-averaged images alone. We explore the potential efficacy of our technique using three-dimensional, general relativistic magnetohydrodynamic simulations, and we highlight several baseline pairs for the Event Horizon Telescope (EHT) that are well-suited to this application. These results indicate that the EHT may be capable of estimating the direction and angular velocity of the emitting material near Sgr A*, and they suggest that a rotating flow may even be utilized to improve imaging capabilities.
9. EXCITATION OF TRAPPED WAVES IN SIMULATIONS OF TILTED BLACK HOLE ACCRETION DISKS WITH MAGNETOROTATIONAL TURBULENCE
SciTech Connect
Henisey, Ken B.; Blaes, Omer M.; Fragile, P. Chris; Ferreira, Barbara T.
2009-11-20
We analyze the time dependence of fluid variables in general relativistic, magnetohydrodynamic simulations of accretion flows onto a black hole with dimensionless spin parameter a/M = 0.9. We consider both the cases where the angular momentum of the accretion material is aligned with the black hole spin axis (an untilted flow) and where it is misaligned by 15 deg. (a tilted flow). In comparison to the untilted simulation, the tilted simulation exhibits a clear excess of inertial variability, that is, variability at frequencies below the local radial epicyclic frequency. We further study the radial structure of this inertial-like power by focusing on a radially extended band at 118(M/10 M{sub sun}){sup -1} Hz found in each of the three analyzed fluid variables. The three-dimensional density structure at this frequency suggests that the power is a composite oscillation whose dominant components are an over dense clump corotating with the background flow, a low-order inertial wave, and a low-order inertial-acoustic wave. Our results provide preliminary confirmation of earlier suggestions that disk tilt can be an important excitation mechanism for inertial waves.
10. Modeling High-resolution Spectra from X-ray Illuminated Accretion Disks
Garcia, Javier; Kallman, T.
2010-01-01
This work is focused on the study of X-ray illuminated accretion disks around compact objects by modeling their structure and reprocessed spectra. Use of low-accuracy and incomplete atomic data is a key limitation in models which have been calculated so far. We remedy this situation by incorporating data for line energies, transition probabilities and photoionization cross sections taken from various sources, most notably calculations using the R-matrix suite of codes. We also implement a self-consistent approach for the radiative transfer of X-rays and the heating and ionization of the gas. These promise to lead to significant improvements in the understanding of the X-ray observations of active galactic nuclei (AGN), X-ray binaries and galactic black holes. By performing detailed radiative transfer calculations we have computed the reflected spectra from constant density slabs for different input parameters (e.g., density, strength of incident X-rays, iron abundance), including the redistribution of photons due to Compton scattering. Although broad and skewed iron emission lines observed in many accreting systems are often attributed to the Doppler effect and gravitational redshift, our results show that Comptonization can be responsible for a significant fraction of the line broadening. By analyzing simulated Suzaku observations from our models, we provide equivalent and physical widths and line centroid energies for atomic lines, absorption edges and recombination continua (among other features). These are provided in tabular and graphical form that can be used directly in the interpretation of observational data.
11. Measuring the Direction and Angular Velocity of a Black Hole Accretion Disk via Lagged Interferometric Covariance
Johnson, Michael D.; Loeb, Abraham; Shiokawa, Hotaka; Chael, Andrew A.; Doeleman, Sheperd S.
2015-11-01
We show that interferometry can be applied to study irregular, rapidly rotating structures, as are expected in the turbulent accretion flow near a black hole. Specifically, we analyze the lagged covariance between interferometric baselines of similar lengths but slightly different orientations. For a flow viewed close to face-on, we demonstrate that the peak in the lagged covariance indicates the direction and angular velocity of the emission pattern from the flow. Even for moderately inclined flows, the covariance robustly estimates the flow direction, although the estimated angular velocity can be significantly biased. Importantly, measuring the direction of the flow as clockwise or counterclockwise on the sky breaks a degeneracy in accretion disk inclinations when analyzing time-averaged images alone. We explore the potential efficacy of our technique using three-dimensional, general relativistic magnetohydrodynamic simulations, and we highlight several baseline pairs for the Event Horizon Telescope (EHT) that are well-suited to this application. These results indicate that the EHT may be capable of estimating the direction and angular velocity of the emitting material near Sgr A*, and they suggest that a rotating flow may even be utilized to improve imaging capabilities.
12. Modelling the Accretion History of the Galactic Disk (and the Gravitational Lensing of a High-z Galaxy)
2015-01-01
Over its long history, the Milky Way is expected to have accreted many dwarf galaxies. The debris from the destruction of most of these dwarf galaxies will by now be fully phase-mixed throughout the Galaxy and hence undetectable as local over-densities in position-space. However, the debris from these systems could have distinct kinematic signatures that may help distinguish these stars from, for example, the Galactic disk. We aim to construct a reliable method of determining the contributions to the Milky Way disk from accreted structures that could be applied to current kinematic data sets, such as SDSS's APOGEE survey. In an effort to mimic the kinematic traits of an accreted satellite, we construct single-orbit models to compare to a cosmologically motivated simulation of satellite accretion. We find that these orbit models adhere to the kinematic signatures of certain types of accreted galaxies better than others, giving us insight on which parameters to trust when searching for accreted populations. As a bonus, we describe a separate project in which we attempt to deduce the intrinsic properties of the 8 o'clock arc, a gravitationally lensed Lyman break galaxy at redshift 2.73. Using the lensmodel code and its pixel-based source reconstruction extension pixsrc, we derive a de-lensed image of the galaxy in the source plane.
13. BINSYN: A Publicly Available Program for Simulating Spectra and Light Curves of Binary Systems with or without Accretion Disks
Linnell, Albert P.; DeStefano, Paul; Hubeny, Ivan
2012-08-01
The BINSYN program suite, a collection of programs for analysis of binary star systems with or without an optically thick accretion disk, is available for download from a wiki. This article describes the package, including download instructions. BINSYN produces synthetic spectra of individual binary star components plus a synthetic spectrum of the system. If the system includes an accretion disk, BINSYN also produces a separate synthetic spectrum of the disk face and rim. A system routine convolves the synthetic spectra with filter profiles of several photometric standards to produce absolute synthetic photometry output. The package generates synthetic light curves and determines an optimized solution for system parameters. This article includes illustrative literature references that have used the suite, including mass transfer rates in several cataclysmic binary systems.
14. DO MAGNETIC FIELDS DESTROY BLACK HOLE ACCRETION DISK g-MODES?
SciTech Connect
Ortega-Rodríguez, Manuel; Solís-Sánchez, Hugo; Arguedas-Leiva, J. Agustín; Wagoner, Robert V.; Levine, Adam
2015-08-10
Diskoseismology, the theoretical study of normal-mode oscillations in geometrically thin, optically thick accretion disks, is a strong candidate for explaining some quasi-periodic oscillations in the power spectra of many black hole X-ray binary systems. The existence of g-modes, presumably the most robust and visible of the modes, depends on general relativistic gravitational trapping in the hottest part of the disk. As the existence of the required cavity in the presence of magnetic fields has been put into doubt by theoretical calculations, we will explore in greater generality what effect the inclusion of magnetic fields has on the existence of g-modes. We use an analytical perturbative approach on the equations of MHD to assess the impact of such effects. Our main conclusion is that there appears to be no compelling reason to discard g-modes. In particular, the inclusion of a non-zero radial component of the magnetic field enables a broader scenario for cavity non-destruction, especially taking into account recent simulations’ saturation values for the magnetic field.
15. A Newly Forming Cold Flow Protogalactic Disk, a Signature of Cold Accretion from the Cosmic Web
Martin, D. Christopher; Matuszewski, Mateusz; Morrissey, Patrick; Neill, James D.; Moore, Anna; Steidel, Charles C.; Trainor, Ryan
2016-06-01
How galaxies form from, and are fueled by, gas from the intergalactic medium (IGM) remains one of the major unsolved problems in galaxy formation. While the classical Cold Dark Matter paradigm posits galaxies forming from cooling virialized gas, recent theory and numerical simulations have highlighted the importance of cold accretion flows—relatively cool (T ˜ few × 104 K) unshocked gas streaming along filaments into dark matter halos, including hot, massive, high-redshift halos. These flows are thought to deposit gas and angular momentum into the circumgalactic medium resulting in disk- or ring-like structures, eventually coalescing into galaxies forming at filamentary intersections. We earlier reported a bright, Lyα emitting filament near the QSO HS1549+19 at redshift z = 2.843 discovered with the Palomar Cosmic Web Imager. We now report that the bright part of this filament is an enormous (R > 100 kpc) rotating structure of hydrogen gas with a disk-like velocity profile consistent with a 4 × 1012 M ⊙ halo. The orbital time of the outer part of the what we term a “protodisk” is comparable to the virialization time and the age of the universe at this redshift. We propose that this protodisk can only have recently formed from cold gas flowing directly from the cosmic web.
16. Stronger Reflection from Black Hole Accretion Disks in Soft X-Ray States
Steiner, James F.; Remillard, Ronald A.; García, Javier A.; McClintock, Jeffrey E.
2016-10-01
We analyze 15,000 spectra of 29 stellar-mass black hole (BH) candidates collected over the 16 year mission lifetime of Rossi X-ray Timing Explorer using a simple phenomenological model. As these BHs vary widely in luminosity and progress through a sequence of spectral states, which we broadly refer to as hard and soft, we focus on two spectral components: the Compton power law and the reflection spectrum it generates by illuminating the accretion disk. Our proxy for the strength of reflection is the equivalent width of the Fe-K line as measured with respect to the power law. A key distinction of our work is that for all states we estimate the continuum under the line by excluding the thermal disk component and using only the component that is responsible for fluorescing the Fe-K line, namely, the Compton power law. We find that reflection is several times more pronounced (˜3) in soft compared to hard spectral states. This is most readily caused by the dilution of the Fe line amplitude from Compton scattering in the corona, which has a higher optical depth in hard states. Alternatively, this could be explained by a more compact corona in soft (compared to hard) states, which would result in a higher reflection fraction.
17. X-ray Fe-lines from Relativistic Accretion Disks Around Neutron Stars and Black Holes
Stella, Luigi
2013-01-01
The Gas Scintillation Proportional Counter (GSPC) on board the European X-ray Satellite EXOSAT (1983-1986) provided detections of Fe K-alpha emission features around 6-7 keV in the X-ray spectra of accreting neutron star and black hole candidates in X-ray binaries. Surprisingly the width of these lines was found to be broader than the GSPC resolution 10% at 6 keV): it could not be explained by thermal broadening, nor blending of (unresolved) lines from different ionization stages of Fe; very large Doppler shifts and, perhaps, thermal Comptonisation provided more promising interpretations. In 1989 Nick White and I developed the first general relativistic model for the Fe-line profile that is produced by matter orbiting in an accretion disk. By fitting the GSPC Fe-line of the black hole candidate Cyg X-1 with our model we inferred an emitting line region extending to a few tens Schwarzschild radii from the black hole, where matter orbits at ~0.1-0.2 the speed of light and effects such as relativistic Doppler shifts and boosting, as well as gravitational and transverse redshifts are conspicuous. We joined forces with Andy Fabian and Martin Rees, who were working on the same interpretation, and published the results in a MNRAS paper. The relativistic disk interpretation of the broad Fe-lines gave rise to much interest on the possibility of measuring black hole mass and spin and probing the innermost regions of accretion flows and the very strong gravitational fields close to compact objects. Very broad and sometimes highly redshifted Fe-lines have been studied by now in tens of X-ray binaries and bright Active Galactic Nuclei with the CCD detectors of the Chandra and XMM/Newton X-ray telescopes; in some cases the line profile implies the presence of a fast spinning black hole. The potential of the Fe-line diagnostics remains to be largely exploited. Moreover some alternative interpretations are not yet ruled out. An X-ray instrument with a broad energy response
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https://www.tutorialspoint.com/discrete_mathematics/discrete_mathematical_induction.htm | # Mathematical Induction
Mathematical induction, is a technique for proving results or establishing statements for natural numbers. This part illustrates the method through a variety of examples.
## Definition
Mathematical Induction is a mathematical technique which is used to prove a statement, a formula or a theorem is true for every natural number.
The technique involves two steps to prove a statement, as stated below −
Step 1(Base step) − It proves that a statement is true for the initial value.
Step 2(Inductive step) − It proves that if the statement is true for the nth iteration (or number n), then it is also true for (n+1)th iteration ( or number n+1).
## How to Do It
Step 1 − Consider an initial value for which the statement is true. It is to be shown that the statement is true for n = initial value.
Step 2 − Assume the statement is true for any value of n = k. Then prove the statement is true for n = k+1. We actually break n = k+1 into two parts, one part is n = k (which is already proved) and try to prove the other part.
### Problem 1
$3^n-1$ is a multiple of 2 for n = 1, 2, ...
Solution
Step 1 − For $n = 1, 3^1-1 = 3-1 = 2$ which is a multiple of 2
Step 2 − Let us assume $3^n-1$ is true for $n=k$, Hence, $3^k -1$ is true (It is an assumption)
We have to prove that $3^{k+1}-1$ is also a multiple of 2
$3^{k+1} - 1 = 3 \times 3^k - 1 = (2 \times 3^k) + (3^k - 1)$
The first part $(2 \times 3k)$ is certain to be a multiple of 2 and the second part $(3k -1)$ is also true as our previous assumption.
Hence, $3^{k+1} – 1$ is a multiple of 2.
So, it is proved that $3^n – 1$ is a multiple of 2.
### Problem 2
$1 + 3 + 5 + ... + (2n-1) = n^2$ for $n = 1, 2, \dots$
Solution
Step 1 − For $n=1, 1 = 1^2$, Hence, step 1 is satisfied.
Step 2 − Let us assume the statement is true for $n=k$.
Hence, $1 + 3 + 5 + \dots + (2k-1) = k^2$ is true (It is an assumption)
We have to prove that $1 + 3 + 5 + ... + (2(k+1)-1) = (k+1)^2$ also holds
$1 + 3 + 5 + \dots + (2(k+1) - 1)$
$= 1 + 3 + 5 + \dots + (2k+2 - 1)$
$= 1 + 3 + 5 + \dots + (2k + 1)$
$= 1 + 3 + 5 + \dots + (2k - 1) + (2k + 1)$
$= k^2 + (2k + 1)$
$= (k + 1)^2$
So, $1 + 3 + 5 + \dots + (2(k+1) - 1) = (k+1)^2$ hold which satisfies the step 2.
Hence, $1 + 3 + 5 + \dots + (2n - 1) = n^2$ is proved.
### Problem 3
Prove that $(ab)^n = a^nb^n$ is true for every natural number $n$
Solution
Step 1 − For $n=1, (ab)^1 = a^1b^1 = ab$, Hence, step 1 is satisfied.
Step 2 − Let us assume the statement is true for $n=k$, Hence, $(ab)^k = a^kb^k$ is true (It is an assumption).
We have to prove that $(ab)^{k+1} = a^{k+1}b^{k+1}$ also hold
Given, $(ab)^k = a^k b^k$
Or, $(ab)^k (ab) = (a^k b^k ) (ab)$ [Multiplying both side by 'ab']
Or, $(ab)^{k+1} = (aa^k) ( bb^k)$
Or, $(ab)^{k+1} = (a^{k+1}b^{k+1})$
Hence, step 2 is proved.
So, $(ab)^n = a^nb^n$ is true for every natural number n.
## Strong Induction
Strong Induction is another form of mathematical induction. Through this induction technique, we can prove that a propositional function, $P(n)$ is true for all positive integers, $n$, using the following steps −
• Step 1(Base step) − It proves that the initial proposition $P(1)$ true.
• Step 2(Inductive step) − It proves that the conditional statement $[P(1) \land P(2) \land P(3) \land \dots \land P(k)] → P(k + 1)$ is true for positive integers $k$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.963671088218689, "perplexity": 330.7457150869648}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347417746.33/warc/CC-MAIN-20200601113849-20200601143849-00014.warc.gz"} |
http://nag.com/numeric/fl/nagdoc_fl24/html/G12/g12aaf.html | G12 Chapter Contents
G12 Chapter Introduction
NAG Library Manual
# NAG Library Routine DocumentG12AAF
Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.
## 1 Purpose
G12AAF computes the Kaplan–Meier, (or product-limit), estimates of survival probabilities for a sample of failure times.
## 2 Specification
SUBROUTINE G12AAF ( N, T, IC, FREQ, IFREQ, ND, TP, P, PSIG, IWK, IFAIL)
INTEGER N, IC(N), IFREQ(*), ND, IWK(N), IFAIL REAL (KIND=nag_wp) T(N), TP(N), P(N), PSIG(N) CHARACTER(1) FREQ
## 3 Description
A survivor function, $S\left(t\right)$, is the probability of surviving to at least time $t$ with $S\left(t\right)=1-F\left(t\right)$, where $F\left(t\right)$ is the cumulative distribution function of the failure times. The Kaplan–Meier or product limit estimator provides an estimate of $S\left(t\right)$, $\stackrel{^}{S}\left(t\right)$, from sample of failure times which may be progressively right-censored.
Let ${t}_{i}$, $i=1,2,\dots ,{n}_{d}$, be the ordered distinct failure times for the sample of observed failure/censored times, and let the number of observations in the sample that have not failed by time ${t}_{i}$ be ${n}_{i}$. If a failure and a loss (censored observation) occur at the same time ${t}_{i}$, then the failure is treated as if it had occurred slightly before time ${t}_{i}$ and the loss as if it had occurred slightly after ${t}_{i}$.
The Kaplan–Meier estimate of the survival probabilities is a step function which in the interval ${t}_{i}$ to ${t}_{i+1}$ is given by
$S^t=∏j=1i nj-djnj ,$
where ${d}_{j}$ is the number of failures occurring at time ${t}_{j}$.
G12AAF computes the Kaplan–Meier estimates and the corresponding estimates of the variances, $\stackrel{^}{\text{var}}\left(\stackrel{^}{S}\left(t\right)\right)$, using Greenwood's formula,
$var^S^t=S^ t 2∑j=1idjnjnj-dj .$
## 4 References
Gross A J and Clark V A (1975) Survival Distributions: Reliability Applications in the Biomedical Sciences Wiley
Kalbfleisch J D and Prentice R L (1980) The Statistical Analysis of Failure Time Data Wiley
## 5 Parameters
1: N – INTEGERInput
On entry: the number of failure and censored times given in T.
Constraint: ${\mathbf{N}}\ge 2$.
2: T(N) – REAL (KIND=nag_wp) arrayInput
On entry: the failure and censored times; these need not be ordered.
3: IC(N) – INTEGER arrayInput
On entry: ${\mathbf{IC}}\left(\mathit{i}\right)$ contains the censoring code of the $\mathit{i}$th observation, for $\mathit{i}=1,2,\dots ,{\mathbf{N}}$.
${\mathbf{IC}}\left(i\right)=0$
The $i$th observation is a failure time.
${\mathbf{IC}}\left(i\right)=1$
The $i$th observation is right-censored.
Constraint: ${\mathbf{IC}}\left(\mathit{i}\right)=0$ or $1$, for $\mathit{i}=1,2,\dots ,{\mathbf{N}}$.
4: FREQ – CHARACTER(1)Input
On entry: indicates whether frequencies are provided for each time point.
${\mathbf{FREQ}}=\text{'F'}$
Frequencies are provided for each failure and censored time.
${\mathbf{FREQ}}=\text{'S'}$
The failure and censored times are considered as single observations, i.e., a frequency of $1$ is assumed.
Constraint: ${\mathbf{FREQ}}=\text{'F'}$ or $\text{'S'}$.
5: IFREQ($*$) – INTEGER arrayInput
Note: the dimension of the array IFREQ must be at least ${\mathbf{N}}$ if ${\mathbf{FREQ}}=\text{'F'}$ and at least $1$ if ${\mathbf{FREQ}}=\text{'S'}$.
On entry: if ${\mathbf{FREQ}}=\text{'F'}$, ${\mathbf{IFREQ}}\left(i\right)$ must contain the frequency of the $i$th observation.
If ${\mathbf{IFREQ}}=\text{'S'}$, a frequency of $1$ is assumed and IFREQ is not referenced.
Constraint: if ${\mathbf{FREQ}}=\text{'F'}$, ${\mathbf{IFREQ}}\left(\mathit{i}\right)\ge 0$, for $\mathit{i}=1,2,\dots ,{\mathbf{N}}$.
6: ND – INTEGEROutput
On exit: the number of distinct failure times, ${n}_{d}$.
7: TP(N) – REAL (KIND=nag_wp) arrayOutput
On exit: ${\mathbf{TP}}\left(\mathit{i}\right)$ contains the $\mathit{i}$th ordered distinct failure time, ${t}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{n}_{\mathrm{d}}$.
8: P(N) – REAL (KIND=nag_wp) arrayOutput
On exit: ${\mathbf{P}}\left(\mathit{i}\right)$ contains the Kaplan–Meier estimate of the survival probability, $\stackrel{^}{S}\left(t\right)$, for time ${\mathbf{TP}}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,{n}_{d}$.
9: PSIG(N) – REAL (KIND=nag_wp) arrayOutput
On exit: ${\mathbf{PSIG}}\left(\mathit{i}\right)$ contains an estimate of the standard deviation of ${\mathbf{P}}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,{n}_{d}$.
10: IWK(N) – INTEGER arrayWorkspace
11: IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{ or }1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{ or }\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit: ${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).
## 6 Error Indicators and Warnings
If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
${\mathbf{IFAIL}}=1$
On entry, ${\mathbf{N}}<2$.
${\mathbf{IFAIL}}=2$
On entry, ${\mathbf{FREQ}}\ne \text{'F'}$ or $\text{'S'}$.
${\mathbf{IFAIL}}=3$
On entry, ${\mathbf{IC}}\left(i\right)\ne 0$ or $1$, for some $i=1,2,\dots ,{\mathbf{N}}$.
${\mathbf{IFAIL}}=4$
On entry, ${\mathbf{FREQ}}=\text{'F'}$ and ${\mathbf{IFREQ}}\left(i\right)<0$, for some $i=1,2,\dots ,{\mathbf{N}}$.
## 7 Accuracy
The computations are believed to be stable.
## 8 Further Comments
If there are no censored observations, $\stackrel{^}{S}\left(t\right)$ reduces to the ordinary binomial estimate of the probability of survival at time $t$.
## 9 Example
The remission times for a set of $21$ leukaemia patients at $18$ distinct time points are read in and the Kaplan–Meier estimate computed and printed. For further details see page 242 of Gross and Clark (1975).
### 9.1 Program Text
Program Text (g12aafe.f90)
### 9.2 Program Data
Program Data (g12aafe.d)
### 9.3 Program Results
Program Results (g12aafe.r) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 87, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9991593360900879, "perplexity": 3332.3030195370384}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560281084.84/warc/CC-MAIN-20170116095121-00464-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/intermediate-algebra-for-college-students-7th-edition/chapter-7-section-7-3-multiplying-and-simplifying-radical-expressions-exercise-set-page-531/72 | ## Intermediate Algebra for College Students (7th Edition)
$2\sqrt[4]{2}$
RECALL: For any non-negative real numbers a and b, $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}$ Use the rule above to obtain: $=\sqrt[4]{(4)(8)} \\=\sqrt[4]{32}$ Factor the radicand (expression inside the radical sign) so that at least one factor is a perfect fourth root, and then simplify to obtain: $\\=\sqrt[4]{16(2)} \\=\sqrt[4]{2^4(2)} \\=2\sqrt[4]{2}$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8044009208679199, "perplexity": 403.7855047831753}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195531106.93/warc/CC-MAIN-20190724061728-20190724083728-00363.warc.gz"} |
http://en.wikipedia.org/wiki/Talk:Scalar_curvature | # Talk:Scalar curvature
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## Notation -- R vs S
I originally switched from S to R because every other article on WP that talks about scalar curvature uses R not S (since S is already taken, used to denote the "action").
Also: I have yet to see any book where the scalar curvature is denoted by S. Neither of the two mathbooks I have on Riemannian geometry call it that. So in what sense is "S" the "usual" notation? linas 22:18, 7 October 2005 (UTC)
It is usual in Riemannian geometry, and this is an article in Riemanninan geometry. Almost any modern book use S or Sc. Ricci flow guys use and inex oriented intos use R. See Curvature of Riemannian manifolds and Weyl curvature. Also if you do not use index notation it has no sense to use R. Tosha 22:50, 9 October 2005 (UTC)
## Pseudo-Riemannian case
It ought to be mentioned that scalar curvature is defined for pseudo-Riemannian metrics as well. There is at least one sentence in the article which implies scalar curvature is specific to Riemannian metrics only. Jjauregui (talk) 20:01, 7 April 2008 (UTC)
## Section on 2-dimensional case needs fine tuning
The section titled "2 dimensions" starts as follows:
"In 2 dimensions, scalar curvature is exactly twice the Gauss curvature:
S = 2/(ρ1 ρ2)
where ρ1, ρ2 are principal radii of the surface."
This isn't quite describing the case of "2 dimensions" but rather the case where a 2-dimensional manifold is assumed to have an embedding in 3-space. So -- either the equation should not assume the surface is embedded in 3-space, or the title of this section should be changed to reflect this assumption. (Or else both the embedded and intrinsic cases could be addressed.)Daqu (talk) 19:50, 12 March 2009 (UTC)
## Definition via triangles?
I don't know differential geometry. I've heard of a scalar-valued quantity related to curvature defined something like this: Take the sum of the angles in an equilateral triangle. Divide by the length of the side. (Or do we need the square of the length?) Take the limit as the triangle shrinks.
How is that related to the scalar curvature defined in this article? Michael Hardy (talk) 03:01, 30 August 2009 (UTC)
Up to a constant, that would be the sectional curvature. It's not really a scalar because it depends on the directions of the legs of the triangle, even if they are tending to zero in magnitude. There are, of course, relationships between the two things, but it is not quite so simple. Sławomir Biały (talk) 03:05, 30 August 2009 (UTC)
Actually, I take the last part back. Up to a factor of the dimension, the scalar curvature is the average value of the sectional curvature. So, perhaps, if one selected a triangle "at random", then the scalar curvature represents the expectation value of the construction you mention. Sławomir Biały (talk) 03:22, 30 August 2009 (UTC)
In a space with constant curvature Κ, the sum of the interior angles of a triangle is equal to π+ΚA where A is the area of the triangle. See spherical excess. From this one can show that the sum of the interior angles of an n-gon is (n-2)π+ΚA. This can be generalized to arbitrary closed curves in a space with variable curvature. See Gauss–Bonnet theorem. JRSpriggs (talk) 14:17, 1 September 2009 (UTC)
Just to clarify the above post, as I think JRSpriggs may have misunderstood Michael Hardy's original misapprehension: What he said holds for constant sectional curvature, which is not the same thing as the scalar curvature (the subject of this article). The purpose of my post was to indicate how the sectional curvature (that the angular excess measures) is related to the scalar curvature. Sławomir Biały (talk) 15:25, 1 September 2009 (UTC)
I think I should have said the amount by which the sum exceeds π, rather than just the sum. Michael Hardy (talk) 14:24, 1 September 2009 (UTC)
To Sławomir Biały: As the article says, "in two dimensions, scalar curvature is exactly twice the Gaussian curvature". And as the article on sectional curvature says, "it is the Gaussian curvature of that section". JRSpriggs (talk) 16:48, 1 September 2009 (UTC)
Of course, in two dimensions no one ever talks about "scalar curvature", as that is completely redundant. To focus on that case in the article would indeed be perverse. It is only in higher dimensions in which scalar curvature becomes an interesting object of study independently of other notions of curvature. The question is: how is the scalar curvature of a Riemannian manifold related to its sectional curvature. See my original post for an explanation. Here is slightly more detail:
$S_p = (const.)\int_{\operatorname{Gr}_2(T_pM)} K_p(\sigma)\,d\mu(\sigma)$
where the integral is taken with respect to the unit normalized rotationally-invariant measure on the Grassmannian of two-planes in the tangent space of M at p. Or, more explicitly, in an orthonormal frame
$S_p = \sum_{i,j} K_p(e_i\wedge e_j).$
--Sławomir Biały (talk) 17:17, 1 September 2009 (UTC)
Yes, that is true. But the question was about triangles, and they live in two dimensions.
Perhaps it would have been better if I had given the formula in terms of exterior angles. In that case, the sum of the exterior angles of a closed polygon (with winding number 1) is 2π-ΚA. JRSpriggs (talk) 17:50, 1 September 2009 (UTC)
Of course, but one can also make sense of triangles in higher dimensions: they simply live on a two dimensional subvariety. When understood in this way, the angle excess does not measure the scalar curvature, rather it measures the sectional curvature. Sławomir Biały (talk) 20:11, 1 September 2009 (UTC)
OK. JRSpriggs (talk) 12:36, 3 September 2009 (UTC)
## Direct geometric interpretation
The formula:
$\frac{\operatorname{Vol} (B_\varepsilon(p) \subset M)}{\operatorname{Vol} (B_\varepsilon(0)\subset {\mathbb R}^n)}= 1- \frac{S}{6(n+2)}\varepsilon^2 + O(\varepsilon^4)$
is somehow not derived comprehensible. Can someone show how this comes out using
$d\mu_g = \Big[ 1 - \frac{1}{6}R_{jk}x^jx^k+ O(|x|^3) \Big] d\mu_{{\rm Euclidean}}$
from the corresponding article of the Ricci-Tensor? —Preceding unsigned comment added by Dbfrosch (talkcontribs)
Hint. If A is a symmetric n x n matrix, then the average value of xTAx for x in the unit ball of Euclidean space is tr A/(n+2). 71.182.247.220 (talk) 21:40, 27 October 2009 (UTC)
Details. Integrate xTAx in spherical coordinates over the unit ball. Denote by T(A) the integral
$T(A) = \int_{SO(n)} R^TAR\, dR$
taken with respect to the invariant probability measure on the special orthogonal group SO(n). Then T(A) commutes with all rotations, and therefore by Schur's lemma,
$T(A) = \lambda I.$
Taking a trace on both sides gives
$\operatorname{tr} A = n\lambda.$
so
$T(A) = \frac{\operatorname{tr} A}{n} I.$
Now, let ωn−1 be the (n−1)-volume of the (n−1)-sphere. The average value of xTAx over the unit ball is given by
$\frac{\omega_{n-1}\int_0^1 t^{n+1} e_1^TT(A)e_1\, dt}{\omega_{n-1}\int_0^1 t^{n-1}\, dt} = \frac{\operatorname{tr} A}{n+2},$
as required. 71.182.247.220 (talk) 16:27, 28 October 2009 (UTC)
## Presentation is a bit confusing
The first equation defines S = tr[g] Ric, but doesn't define Ric. The second equation gives another expression for S, which doesn't use Ric, but where Ric is defined. I'm sure the experts here understand what is going on, but to me, trying to figure out, it is confusing. I also don't understand what it means to take the trace of the tensor with respect to the metric. The reference to the trace article only refers to the trace with respect to a basis. Is the metric a basis? I thought it was a bilinear form. Is their an obvious interpretation of a bilinear form as a set of basis vectors? The columns of the matrix? --Thinkor (talk) 22:36, 9 January 2010 (UTC)
Ric is not defined anywhere. It is the Ricci tensor. There is a link to follow. Sławomir Biały (talk) 15:41, 11 January 2010 (UTC) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 9, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9480075240135193, "perplexity": 657.5820213193842}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207929230.43/warc/CC-MAIN-20150521113209-00151-ip-10-180-206-219.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/174644/is-it-true-that-if-alpha-in-operatornamefraca-and-s-alpha-in-a-then?answertab=active | # Is it true that if $\alpha \in \operatorname{Frac}(A)$ and $s\alpha \in A$, then $\alpha \in S^{-1}A$?
In the proof of Proposition 1.9 in Chapter VII of Algebra by Serge Lang, it seems to me that the following property is used.
Let $A$ be a commutative entire ring, $S$ a multiplicative subset of $A$, $0 \not \in S$. Let $\alpha$ be an element of the quotient field of $A$. If $s \alpha \in A$ for some $s \in S$, then $\alpha \in S^{-1}A$.
(in a proof that $A$ integrally closed implies $S^{-1}A$ integrally closed).
Is this bold statement true ?
Since $\alpha$ is expressed as $\alpha=a/s'$, where $a, s' \in A, s' \neq 0$, $s \alpha \in A$ implies that $s a = s' b$ for some $b \in A$.
From this, how can I prove that $s' \in S$ ?
Any help would be appreciated.
-
Which edition of Algebra? – Jonas Meyer Jul 24 '12 at 13:28
It is the revised third edition. – Aki Jul 24 '12 at 13:32
You can't prove $s'\in S$ because it is false in general: see my answer. – Georges Elencwajg Jul 24 '12 at 13:40
What is $\,S^{-1}A\,$ ? By definition, $$S^{-1}A:=\{a/s\;:\;a\in A\,,\,s\in S\}\,$$with the "usual" operations (the definition is there).
Thus, we can in fact say that for $\,\alpha:=\frac{a}{b}\in F\,\,,\,a,b\in A\,,\,b\neq 0$ , as $$\alpha\in S^{-1}A\Longleftrightarrow \frac{as}{b}=\alpha s\in A\,\,,\,\text{for some}\,\,s\in S$$
In fact, there's hardly anything to prove here...
Added: Let's see if the following clarifies a little:
$$\exists\,\,s\in S\,\,s.t.\,\,s\alpha=a\in A\Longrightarrow \alpha=\frac{a}{s}\in S^{-1}A\,\,,\,\text{per definition of fractions ring}$$ and we don't care what the "original" form of $\,\alpha\,$, as element of the fractions field of $\,A\,$ , is/was.
-
I think I get it thanks to your additional comment "we don't care what the original form of $\alpha$ is/was". – Aki Jul 24 '12 at 14:15
Lang is right (see the other answers) but not what you suggest, namely that if $\alpha=a/b, \;b\neq 0$ and $s\alpha \in A$ for some $s\in S$, then you can deduce $b\in S$.
Indeed you can always write $1=1\cdot b/b\in A$ for any $b\in A\setminus \lbrace 0\rbrace$ and if what you suggest were true, then (since $1\in S$) you would always have $b\in S$, i.e. $S=A\setminus \lbrace 0\rbrace$: an absurd statement.
NB I have changed your $s'$ into $b$. I think your notation is confusing because it tends to suggest $s'\in S$ and that this partly created your difficulty.
-
Thank you, @GeorgesElencwajg. Now, I see that my assumption was wrong. – Aki Jul 24 '12 at 14:03
Great, Aki, bravo! – Georges Elencwajg Jul 24 '12 at 14:59
You don't need to show that $s'\in S$, because you have already shown $\alpha \in S^{-1}A$. As you said, $sa=s'b$, therefore $\alpha=a/s'=b/s\in S^{-1}A$.
-
I think you misread the question. $\alpha$ a priori is an element of the field of fractions of the integral domain $\alpha$ and not necessarily an element of the localisation. – user38268 Jul 24 '12 at 13:06
Yes, I did not see "entire" at all the first time around. I think the number of authors using "entire" instead of "domain" must be countable on one of Homer Simpson's hands. – rschwieb Jul 24 '12 at 13:19
I think Lang is the only one. I always found it funny that in a book so terse he spends a paragraph or so defending the terminology, although his argument was very convincing to me until I realized that no one would ever know what I was talking about unless I said “domain”. – Dylan Moreland Jul 24 '12 at 14:18
Lang's mother tongue was French. The French terminology for domain is anneau intègre. Since the adjective integral was already used for another concept in ring theory, I suppose Lang settled for entire as the closest substitute. – Georges Elencwajg Jul 24 '12 at 15:05
Suppose that for some $s \in S$ we have $\alpha s \in A$ . Then $\alpha s = a$ for some $a \in A$ and so viewing everything as happening inside the fraction field of $A$ we can do division so that
$$\alpha =\frac{\alpha}{1} = \frac{a}{s}.$$
But then by definition the guy on the right is in the localisation from which it follows that $\alpha \in S^{-1}A$.
- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9764286875724792, "perplexity": 362.36304671100334}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440645281325.84/warc/CC-MAIN-20150827031441-00046-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://cbyrohl.de/tags/workshop/ | # workshop
## Clustering Distortions from Lyman-alpha Radiative Transfer
Introduction Lyman-$\alpha$ emitters and their intensity map are powerful probes of the large-scale structure. Given typically very high optical depths, scatterings with neutral hydrogen have a non-negligible impact on the observed distribution of Lyman-$\alpha$ photons. To study this in detail, we run a suit of radiative transfer (RT) simulations on the Illustris simulation and investigate two possible distortion effects arising from RT in real space as well as in redshift space for the two-point correlation function. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9473928809165955, "perplexity": 2224.1548853621885}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195523840.34/warc/CC-MAIN-20190715175205-20190715200247-00013.warc.gz"} |
https://math.stackexchange.com/questions/3160007/question-on-a-double-integral-with-change-of-variables | # Question on a double integral with change of variables
Solve the system $$u = 3x + 2y, v = x + 4y$$ to find expressions for $$x$$ and $$y$$ in terms of $$u$$ and $$v$$.
Use these expressions to find the Jacobian $$∂(x, y)/∂(u, v)$$.
Hence evaluate the integral $$\iint(3x + 2y)(x + 4y) dx dy$$ for the region $$R$$ bounded by the lines $$y = −(3/2)x + 1,\ y = −(3/2)x + 3$$ and $$y = −(1/4)x,\ y = −(1/4)x + 1$$
So I computed the Jacobian = 1/10, I solved the equations to find $$x = h(u,v)$$ and $$y = g(u,v)$$ and then I substituted $$h$$ and $$g$$ in the equations for the boundaries ,which gave me $$u = 2$$ and $$u = 6$$, and $$v = 0$$ and $$v = 4$$. So the integral I have to compute is equal to $$\iint uvJ(u,v)dudv$$ with the above boundaries?
• My comment was the one that was mistaken, sorry. Your notes are probably right. Mar 24, 2019 at 1:55
• So is the integral I have to compute equal to $∬uvJ(u,v)dudv$, with the boundaries I found?
– user600210
Mar 24, 2019 at 1:58
Yes, it is correct. The answer is $$12.8$$.
Direct calculation:
$$\hspace{1cm}$$
$$\int_{0}^{0.8}\int_{-\frac32x+1}^{-\frac14x+1} (3x+2y)(x+4y) dydx+\\ \int_{0.8}^{1.6}\int_{-\frac14x}^{-\frac14x+1} (3x+2y)(x+4y) dydx+\\ \int_{1.6}^{2.4}\int_{-\frac14x}^{-\frac32x+3} (3x+2y)(x+4y) dydx=\\ \frac{44}{15}+\frac{104}{15}+\frac{44}{15}=\frac{192}{15}=12.8.$$
Your method: $$\int_{0}^{4}\int_{2}^{6} \frac1{10}uv dudv=\int_0^4 \frac85vdv=12.8.$$
• I got the same result, thanks for the work!
– user600210
Mar 25, 2019 at 22:19
• You are welcome. Good luck. Mar 26, 2019 at 2:11 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 23, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9759656190872192, "perplexity": 170.9593634157138}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104364750.74/warc/CC-MAIN-20220704080332-20220704110332-00200.warc.gz"} |
https://www.gradesaver.com/textbooks/science/physics/physics-principles-with-applications-7th-edition/chapter-3-kinematics-in-two-dimensions-vectors-problems-page-71/43 | ## Physics: Principles with Applications (7th Edition)
a. 10.4m/s, 17$^o$ above the horizontal. b. 10.4m/s, 17$^o$ below the horizontal.
a. Call the upward direction positive. The velocity of the ball relative to the ground is the vector sum of the horizontal and vertical motions. From the given information, we see that the velocity relative to the ground is (10 m/s, 3.0 m/s). Find the magnitude using the Pythagorean Theorem and the direction using the definition of the tangent. $$v_{BG} = 10.4 m/s$$ The angle is at 17$^o$ above the horizontal. b. The only change is that the balloon is descending, so the velocity relative to the ground is (10 m/s, -3.0 m/s). Find the magnitude using the Pythagorean Theorem and the direction using the definition of the tangent. $$v_{BG} = 10.4 m/s.$$ The angle is now at 17$^o$ below the horizontal. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.977590799331665, "perplexity": 337.75755079424056}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794864405.39/warc/CC-MAIN-20180521142238-20180521162238-00448.warc.gz"} |
http://advances.sciencemag.org/content/4/11/eaat8632 | Research ArticleMATERIALS SCIENCE
# Breakdown of the Stokes-Einstein relation above the melting temperature in a liquid phase-change material
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Vol. 4, no. 11, eaat8632
## Abstract
The dynamic properties of liquid phase-change materials (PCMs), such as viscosity η and the atomic self-diffusion coefficient D, play an essential role in the ultrafast phase switching behavior of novel nonvolatile phase-change memory applications. To connect η to D, the Stokes-Einstein relation (SER) is commonly assumed to be valid at high temperatures near or above the melting temperature Tm and is often used for assessing liquid fragility (or crystal growth velocity) of technologically important PCMs. However, using quasi-elastic neutron scattering, we provide experimental evidence for a breakdown of the SER even at temperatures above Tm in the high–atomic mobility state of a PCM, Ge1Sb2Te4. This implies that although viscosity may have strongly increased during cooling, diffusivity can remain high owing to early decoupling, being a favorable feature for the fast phase switching behavior of the high-fluidity PCM. We discuss the origin of the observation and propose the possible connection to a metal-semiconductor and fragile-strong transition hidden below Tm.
## INTRODUCTION
Phase-change materials (PCMs) can be reversibly switched between their glassy and crystalline states by heating with a voltage or laser pulse (1). The strong optical/electrical contrast between these two states makes PCMs highly interesting for data storage applications (e.g., encoding “0” and “1”). An extremely fast phase switching on a time scale of nanoseconds is a requirement for high read/write speed. However, the fast atomic dynamics inherent to PCMs seems to be at odds with the concomitant requirement of good amorphous phase stability for data retention (1). Typical PCMs include the Ge-Sb-Te alloys, especially those along the GeTe-Sb2Te3 tie line, and doped Sb2Te alloys such as Ag-In-Sb-Te (1). Physical understanding of PCMs has been mainly centered around features of their crystalline states [e.g., bonding (2)], while the liquid-state behavior was considered “ordinary” or less explored, probably because a large portion of the (supercooled) liquid state is obscured by fast crystallization.
Requiring a critical cooling rate of ~109 K s−1 for vitrification, PCMs are generally recognized as poor glass formers. The amorphous phase is so prone to crystallization that no glass transition (Tg) can be observed in a differential scanning calorimeter (DSC) before crystallization sets in upon heating (3). Thus, the broad supercooled liquid regime, ΔT = TmTx, between the melting temperature (e.g., Tm ~ 903 K for Ge1Sb2Te4) and the crystallization temperature upon heating (typically Tx ~ 400 K for GeTe-Sb2Te3 alloys) is experimentally inaccessible using standard techniques. For this reason, it has been a long-standing challenge to characterize the liquid-state behavior of PCMs—specifically, the liquid fragility that has been recently given much importance by Orava et al. (4). Fragility, defined as m = dlogη/d(Tg/T)|T=Tg, where Tg is the “standard” value (where the viscosity η reaches the value 1012 Pa·s) (5), describes the deviation of the temperature dependence of viscosity from the Arrhenius law. Fragility has been recognized as a useful parameter for understanding crystallization kinetics and the stability of amorphous states (6).
The Stokes-Einstein relation (SER) is frequently used to calculate η from D (or vice versa) in technologically important PCMs at “sufficiently” high temperature(1)where kB is the Boltzmann constant, T is the absolute temperature, and rH is the effective hydrodynamic radius. For instance, Orava et al. (4) assumed a valid SER at Tm for deriving the absolute values of crystal growth velocity of Ge2Sb2Te5 from the Kissinger-type analysis using ultrafast DSC data. Salinga et al. (7) used crystal growth velocity data from laser reflectivity measurements to determine the fragility of Ag-In-Sb-Te (m ~ 190), assuming a valid SER over a wide temperature range well below Tm. Schumacher et al. (8) compared the experimental viscosity data for Ge2Sb2Te5 at T > Tm to the values derived from the SER based on simulated self-diffusion coefficients and observed a nonnegligible discrepancy.
In general, liquids at high temperature are expected to obey the SER, as they do not feel the energy landscape, and single-particle dynamics follow the same temperature dependence as the collective macroscopic stress relaxation processes. When the temperature approaches Tg on cooling, D progressively decouples from η in fragile liquids such as o-terphenyl (OTP), phenolphthaleine dimethyl ether (PDE), and salol, beginning at ~1.2 Tg (9), supposedly due to the dynamic heterogeneity (9). A SER breakdown in PCM GeTe was also asserted by Sosso et al. (10) based on ab initio simulations, which occurs in the supercooled liquid attributed to dynamic heterogeneities. Such a breakdown in supercooled Ge2Sb2Te5 and GeTe nanoparticles was also taken into account by Chen et al. (11) and Orava et al. (4), where the necessity of using a fractional SER to describe the supercooled liquid was emphasized. Also, the crystal growth kinetic coefficient Ukin decouples from η following the Ediger et al. relation (6) Ukin ∝ η−ξ* (ξ* < 1 depending on fragility). Detailed experimental studies of multicomponent bulk metallic glass-forming liquids with high atomic packing fractions (12) (φ ~ 0.51 to 0.55) revealed a clear breakdown of the SER close to, or even well above, the critical temperature Tc of mode-coupling theory (13, 14). Liquid PCMs, on the other hand, represent p-electron bonded (15) fragile glass formers (4) with low atomic packing fractions (φ ~ 0.3 to 0.4). There has been unexpectedly little interest in, or experimental data related to, the SER above Tm for PCMs. However, the recent discussion concerning the likely existence of liquid-liquid transitions (LLTs) in PCMs suggests the probability of complex dynamical behavior, including a breakdown of the SER in these liquids (16).
In this work, we probe the microscopic dynamics in the liquid state of a typical PCM Ge1Sb2Te4 using quasi-elastic neutron scattering (QENS), which permits direct determination of both the structural α-relaxation time (proportional to shear viscosity η) and the self-diffusion coefficient D on the same sample under identical conditions. Our results question the validity of the commonly used SER in those technologically important PCMs, even well above Tm. We discuss the origin of the breakdown of SER and its relation to a possible metal-semiconductor (also fragile-strong liquid) transition hidden below Tm, which may play a critical role in speeding up crystallization kinetics, before restraining the atomic rearrangements through the fragile-strong transition. The fundamental importance of these phenomena to the technical performance of PCMs has been stressed elsewhere (16, 17).
## RESULTS
### α-Relaxation time
We obtain relaxation times from the decay of the intermediate scattering function (ISF) S(q,t), which describes the decay of microscopic density fluctuations in the liquid and was obtained according to the procedure outlined in Materials and Methods. Figure 1A shows S(q,t) taken at the first structure factor maximum q0 = 2.0 Å−1 of the liquid at different temperatures above Tm = 903 K. The data are best fitted with a simple exponential function, S(q,t)/S(q,0) = fq exp(−tq), where fq is a constant accounting for atomic vibrations and τq is the structural relaxation time. In the case of q0 = 2.0 A−1, that is, the position of the structure factor maximum, the fitting yields a collective structural relaxation time, or α-relaxation time τα, as shown in Fig. 1B, as the quasi-elastic signal at q0 arises predominantly from the coherent scattering contribution. τα is associated with the shear viscosity η in the viscoelastic model of Maxwell, which establishes a proportional relation via η = G·τα, where G is the infinite frequency shear modulus measured on time scales very short with respect to τα. This proportionality has been directly tested by combining QENS and viscosity measurements on various glass-forming melts (18).
### Self-diffusivity
Self-diffusion coefficients were determined from the QENS signal in the low-q range, where the signal is dominated by the incoherent scattering of both Ge and Te atoms and reflects their single-particle dynamics on long length and time scales. Given the incoherent cross sections of each species and their relative concentration in the alloy melt, the measured self-diffusion coefficient of Ge1Sb2Te4 represents a mean value weighted by roughly 1/3 Ge and 2/3 Te (the incoherent scattering cross section of Sb is negligibly small). As shown in the inset of Fig. 2, the incoherent relaxation times τinc follow a 1/q2 dependence at low q2 ≤ 0.6 Å−2, which is characteristic of long-range atomic diffusion in liquids in the hydrodynamic limit as q → 0 Å−1 (19). This thus allows us to derive a mean Ge/Te self-diffusion coefficient via DGe/Te = 1/(τincq2). In Fig. 2, the resulting DGe/Te values are fitted with the Arrhenius law, yielding an activation energy Ea,D = 26.41 ± 0.89 kJ mol−1 and a pre-exponent D0 = 1.4 × 10−7 m2 s−1. To our knowledge, there are no experimental diffusivity data available for liquid PCMs. Some partial atomic diffusion coefficients are available from ab initio computer simulations (20) DGe = 4.04 × 10−9 m2 s−1 and DTe = 4.06 × 10−9 m2 s−1 at 1000 K for the same composition, which are close to our value (D ≈ 5.7 × 10−9 m2 s−1) at 1003 K.
## DISCUSSION
### The breakdown of the SER
According to the SER, the product (D·η)/T and, hence, (D·τα)/T should remain constant as a function of T (Eq. 1). For the liquid PCM Ge1Sb2Te4, this is evidently not the case, as highlighted in Fig. 3A. A marked deviation is observed at 1050 K on approaching Tm (903 K) during cooling, indicating a breakdown of the SER well above the melting point up to at least 1050 K, ~1.16 Tm. For T > 1050 K, the SER seems to hold, although the limited temperature range does not allow for a conclusive assessment for even higher temperatures. Note that we take the SER in its form of D ∝ (τα/T)−1. If G in the Maxwell relation is temperature dependent (as it certainly is for fragile liquids), then the temperature of SER breakdown, in its original form with viscosity, might differ somewhat from the breakdown temperature observed here. In either case, it should occur at relaxation times on the order of 1 ps (from Figs. 1 and 3B), far shorter than in any normal liquid where the breakdown only occurs at nanosecond relaxation times.
In Fig. 3B, by fitting the data with a fractional SER of the form (21)(2)where 0 < ξ ≤ 1, we see that the high-temperature liquid for T ≥ 1050 K closely follows the SER with an exponent ξ ≈ 0.97 ± 0.11, while ξ ≈ 0.60 ± 0.03 is obtained for T ≤ 1050 K, indicating a strong deviation from the SER. The latter can be related to the decoupling of crystal growth coefficient Ukin and viscosity η for fragile liquids, which is described by the form Ukin ∝ η−ξ* given by Ediger et al. (6). ξ* = 0.67 in a similar PCM Ge2Sb2Te5, estimated by Orava et al. (4) from an empirical correlation with fragility, is close to our ξ ≈ 0.6.
The breakdown of the SER at such high temperatures (>Tm), and short relaxation times, is an important observation because (i) the SER has provided the basis for calculating viscosity and fragility from simulated self-diffusion coefficients and/or crystal growth velocities (or vice versa) of PCMs near Tm (4, 7, 8); (ii) it occurs at temperatures where diffusivities are nearly four orders of magnitude higher (~5 × 10−9 m2 s−1) than where the SER breakdown is observed in conventional glass formers [for instance, for the typical fragile molecular liquid OTP, the SER remains valid down to the much lower diffusivity D ≈ 1.3 × 10−13 m2 s−1 (22) where η ≈ 7.7 Pa·s]; and (iii) it is a feature favorable for fast phase switching behavior required for PCM functions in memory devices.
For phase-change memory devices, the PCMs must have ultrafast crystallization kinetics to crystallize in a few nanoseconds. The early (high temperature) fractional SER behavior inevitably leads to a higher diffusivity than that expected from the SER. With decreasing temperature, this difference may develop up to a few orders of magnitude in the supercooled liquid. In other words, the viscous flow may have slowed down, while the atomic diffusion could remain much faster, which would facilitate the diffusion-controlled nucleation and growth process. Hence, this can be one of the favorable factors governing the fast phase switching at a moderate temperature when heated by a “set” electric pulse in the memory devices.
### The origin of the SER violation and its related phenomenology
The origin of the breakdown is usually explained by dynamical heterogeneities for viscous liquids. In our work, we have been struck by the fact that the onset temperature for deviation from the SER occurs in a temperature domain where S(q,t) is still exponential, with no sign, even at temperatures very near the melting point, of the sort of shoulder usually associated with stretching of the exponential and the development of the dynamic heterogeneity [seen, for instance, in (23)]. Then, how can the observed SER breakdown be attributed to heterogeneity? Poole and coworkers (24) demonstrated the presence of dynamic and structural heterogeneity in a model molecular liquid in a single instantaneous configuration, where a two-step relaxation was not observable due to the loss of information during averaging over initial configurations. Moreover, our experiment, probing the short time scale (a few picoseconds), has a limited energy resolution, which does not allow us to resolve even shorter time scales. Note that the height of the plateau in the decay is less than 1, which might already signify the existence of the first-step relaxation (shoulder in the decay of ISF) even when not being undercooled. The molecular dynamics (MD) simulation study of the ISF shows the presence of dynamic heterogeneity in the supercooled GeTe (25), which, originating from structural heterogeneities due to chains of Ge–Ge homopolar bonds, may explain the breakdown of the SER in GeTe in the viscous regime below Tm (26). From our measurement, despite the observed exponential decay in ISF, the dynamic heterogeneity may still be present in the PCM above Tm. If it is the case, then the underlying structural signature and physical origin remain to be determined.
Our observation indicates that the SER is violated in the PCM well above Tm up to at least 1050 K. The limited temperature range of the data does not allow us to conclusively assert the validity of the SER at even higher temperatures. Whether the SER holds in any temperature in the PCM at all needs further experimental and, perhaps more helpful, simulation studies. Using MD simulations, Horbach and Kob (27) showed that the SER does not hold, even at very high temperatures above 3 Tm in the network glass-forming liquid silica. This raises the question whether the breakdown in the PCM is similar to the ones of directionally bonded, network glass-forming liquids. However, such a scenario is difficult to reconcile with the distinct characteristics of PCMs. PCMs have fast dynamics with a high fragility of m = 90 or higher (4, 7), whereas liquid silica has slow dynamics with a fragility of m = 20 near the strong liquid limit. The viscosity of PCMs is about ~2 mPa s at Tm, which is nine orders of magnitude lower than that of liquid silica (~106 Pa·s) at its Tm. PCMs are covalently bonded in the semiconducting solid amorphous phase but defy the description of a typical network structure by deviating from Zachariasen’s glass picture (1, 2). Above the melting point, their liquid state becomes metallic (16). Thus, neither slow dynamics nor a typical network structure is a prerequisite for the breakdown of the SER in the PCM. We note that a simulation study of a similar PCM GeTe, computing viscosity with the Green-Kubo formula and self-diffusivity, shows that the SER holds well at high temperature (10).
The observation in Ge1Sb2Te4 more resembles the case of the “most anomalous liquid”—supercooled water. For bulk water, the recent data of Dehaoui et al. (28) showed a crossover in the fractional SER behavior from ξ ≈ 1 at high temperature to ξ ≈ 0.8 at low temperature. The breakdown temperature TSE of ~340 K, ~ 1.25 Tm, and the diffusivity D ≈ 5 ×10−9 m2 s−1 at TSE (where viscosity η ≈ 0.4 mPa s) are comparable to those of Ge1Sb2Te4 (see Fig. 3A, inset). Note that the deviation from the SER in water appears well above other known anomalies such as density maximum (277 K), rapid Cp increase, and sharp viscosity rising (below Tm). As is much discussed, the anomalies of water are thought to be related to an LLT and possibly a nearby, but hidden, second critical point suggested by some computational models (29). Errington and Debenedetti (30) showed that a “Russian doll” of nested kinetic and thermodynamic anomalies exists in water and the same has since been found for other water-like systems [e.g., Si (31)]. The breakdown of the SER appears to be the anomaly persisting to the highest temperature and a much more sensitive signaler of impending anomalous character than any of the other signals yet studied.
Given that “water-like” anomalies such as density maxima, and diverging (or peaked) heat capacities, occur in supercooled Te, Ge, and Si (3234); in Ge15Te85 just above the eutectic temperature (35); and in As2Te3 somewhat above its Tm (36), we should expect the unusual behavior in the PCMs at lower temperatures (16). The thermodynamic response function maxima in the abovementioned chalcogenides are also associated with liquid metal-semiconductor transitions (16). Pressure-induced polyamorphic transitions between high- and low-density amorphous states (which are also metallic and semiconducting states) have been found in both Ge1Sb2Te4 and Ge2Sb2Te5 (37, 38), and these closely parallel the polyamorphism in amorphous silicon (39) and vitreous ice (40)—except that the latter obviously does not have a semiconductor-metal transition. On the basis of all of the above, the proposed LLT scenario is illustrated in the pressure-temperature (P-T) diagram for liquid and metastable liquid states of Ge1Sb2Te4 in fig. S3, which shows a conjectured LLT regime, together with available literature data. Its analogy to that of water is provided by the inset. Tanaka’s two-order-parameter model already predicts that a “V-shape” P-T phase diagram (as is the case for our PCM) is directly related to thermodynamic and dynamic anomalies similar to those of water (41). The fact that we observe the same sort of SER breakdown for the PCM as for water (see fig. S3) lends credence to the suggested phase diagram and its implication of a submerged LLT. An LLT scenario is consistent with the evidence of a fragile-strong crossover/transition found recently in a similar composition Ge2Sb2Te5 below Tm, manifesting as a continuous crossover argued by Chen et al. (11) and as a singular temperature (792 K) argued by Flores-Ruiz et al. (42).
It is important that the phenomena described in this work are not confused with recent reports of SER breakdown above “Tm” (a ternary eutectic temperature) in certain glass-forming metallic mixtures, such as ZrCuAl, where the species Cu decouples from the Zr-Al matrix (43, 44). The latter phenomenon is more closely related to the case of Cu in amorphous silicon, where DCu can be four orders of magnitude greater than that of the host (Si) atoms (45). A related phenomenon, also quite different from our conjecture, is the superionicity of Cu (or Ag) cations in many superionic glass formers, where the mobile ion decoupling is observed well above any liquidus temperature (and also above 2 Tg), and the Stokes-Einstein discrepancy at Tg can reach 11 orders of magnitude (46).
## SUMMARY
We have performed neutron scattering studies of diffusion and relaxation times in the PCM Ge1Sb2Te4 and identified a breakdown in the SER well above the Tm, which lies in a relaxation time domain 104 times shorter than that in normal liquids. The high-temperature deviation from the SER, characterized by a fractional SER with an exponent ξ ≈ 0.60 ± 0.03, implies a high diffusivity, which is favorable for fast phase switching for PCM functions. We link our finding to the behavior observed in liquid silicon, germanium, and water, where it is seen as a consequence of a submerged LLT, which provokes facile crystallization and fragile-strong transitions when ultrafast cooling preserves the liquid state. The exploration of PCMs’ anomalous liquid-state behavior will be an essential step toward understanding the fast phase switching behavior in this class of material.
## MATERIALS AND METHODS
### Sample preparation
Ge1Sb2Te4 was prepared using the Ge, Sb, and Te elements with purities ranging from 99.999 to 99.9999 atomic %. The elements were sealed under vacuum (10−6 mbar) in a fused quartz tube with an internal diameter of 5 mm and synthesized in a rocking furnace for homogenization at 900°C for 15 hours.
### QENS measurements and data analysis
The sample, sealed in the fused quartz tube, was loaded into a thin-walled Al2O3 container for the QENS measurements, which were carried out at the time-of-flight spectrometer TOFTOF at the Heinz Maier-Leibnitz neutron source (FRM II) in Munich (47, 48). Two incident neutron wavelengths λi = 4.4 and 7 Å were used to obtain a broad q and energy transfer range along with a high resolution of about 90 μeV (full width at half maximum).
Spectra were collected as a function of temperature in a high-vacuum, high-temperature Nb furnace. Raw time-of-flight data were normalized to a vanadium standard and interpolated to constant q to obtain the dynamic structure factor S(q,ω) using the FRIDA-1 software (see http://sourceforge.net/projects/frida/ for source code). All spectra were found to be well described by a model composed of the quasi-elastic scattering from the alloy melt and a flat background to approximate the processes too fast to be accurately measured by the spectrometer. The S(q,ω) obtained in the measurements, where λi = 7 Å, was additionally modeled to include the elastic scattering from the container. In general, the model S(q,ω) readswhere R(q,ω) is the instrumental resolution function, N is a normalization factor, A0 is the magnitude of the elastic scattering, and b(q,ω) is a constant but q-dependent background. The symbol ⨂ denotes a numerical convolution. The quasi-elastic scattering was found to be best described with a single Lorentzian of the form (see fig. S1)where Γ is the half width at half maximum. Below q2 ~ 0.6 Å−2, the incoherent scattering from Ge and Te dominates and the coherent contributions from thermal diffusion (Rayleigh line) and acoustic modes are effectively contained in the flat background of the observed quasi-elastic spectra (49). A mean Ge/Te self-diffusion coefficient was determined via
An analysis was also carried out in the time domain first by obtaining the ISF S(q,t) (or density correlation function) via cosine Fourier transform of the measured S(q,ω) and normalizing to the instrumental resolution function R(q,t). In general, the data were then fitted with a simple exponential decay aswhere f(q) is the amplitude, τ is the structural relaxation time, and the constant c is an offset that takes care of any remaining elastic scattering. It should be noted that this is in line with the model used for S(q,ω), as the Fourier transform of a Lorentzian is a simple exponential. To ensure consistency of the analyses in both energy transfer and time domain, we restricted the fitting range in the energy transfer domain to [−1,1] meV and in the time domain to the data points after 0.65 ps. At higher energy transfers and shorter times, the spectra are dominated by phononic vibrations and fast relaxation processes. The self-diffusion coefficient was obtained from the time domain analysis via
The values of DGe/Te reported in the manuscript represent an average of the values obtained in both analyses.
## SUPPLEMENTARY MATERIALS
Fig. S1. The dynamic structure factor S(q,ω) in the energy transfer domain (ℏω) obtained from QENS.
Fig. S2. The “effective” diffusion coefficient Dq as a function of q derived at a given temperature of 1103 K using the relation Dq = 1/(τqq2).
Fig. S3. P-T metastable liquid phase diagram conjectured for Ge1Sb2Te4, in analogy to that of water (inset).
References (5063)
This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is not for commercial advantage and provided the original work is properly cited.
## REFERENCES AND NOTES
Acknowledgments: We acknowledge the beamtime at the TOFTOF instrument operated by FRM II at the Heinz Maier-Leibnitz Zentrum (MLZ), Garching, Germany. Funding: We acknowledge financial support provided by FRM II to perform the QENS measurements. S.W. acknowledges support from Feodor Lynen Postdoctoral Research Fellowship of the Alexander von Humboldt Foundation, the Place-to-be RWTH Start-Up fund, and the DFG within SFB917. P.L. acknowledges financial support from NSF-EFRI award no. 1640860. C.A.A. acknowledges support from National Science Foundation Research grant no. CHE-1213265. Author contributions: C.A.A., P.L., Z.E., and S.W. initiated the project. M.S. and Z.E. performed experiments with samples contributed by S.W. and P.L. Z.E. and S.W. analyzed the data. S.W., C.A.A., and Z.E. wrote the manuscript with specific contributions from M.S. and P.L. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.
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http://swmath.org/software/14812 | # HiggsSignals
HiggsSignals: confronting arbitrary higgs sectors with measurements at the tevatron and the LHC. HiggsSignals is a Fortran90 computer code that allows to test the compatibility of Higgs sector predictions against Higgs rates and masses measured at the LHC or the Tevatron. Arbitrary models with any number of Higgs bosons can be investigated using a model-independent input scheme based on HiggsBounds. The test is based on the calculation of a chi-squared measure from the predictions and the measured Higgs rates and masses, with the ability of fully taking into account systematics and correlations for the signal rate predictions, luminosity and Higgs mass predictions. It features two complementary methods for the test. First, the peak-centered method, in which each observable is defined by a Higgs signal rate measured at a specific hypothetical Higgs mass, corresponding to a tentative Higgs signal. Second, the mass-centered method, where the test is evaluated by comparing the signal rate measurement to the theory prediction at the Higgs mass predicted by the model. The program allows for the simultaneous use of both methods, which is useful in testing models with multiple Higgs bosons. The code automatically combines the signal rates of multiple Higgs bosons if their signals cannot be resolved by the experimental analysis. We compare results obtained with HiggsSignals to official ATLAS and CMS results for various examples of Higgs property determinations and find very good agreement. A few examples of HiggsSignals applications are provided, going beyond the scenarios investigated by the LHC collaborations. For models with more than one Higgs boson we recommend to use HiggsSignals and HiggsBounds in parallel to exploit the full constraining power of Higgs search exclusion limits and the measurements of the signal seen at around 125.5 GeV.
## Keywords for this software
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http://social.technet.microsoft.com/wiki/contents/articles/2106.differencing-in-the-arima-formula-for-the-microsoft-time-series-algorithm.aspx | Resources For IT Professionals
# Differencing in the ARIMA formula for the Microsoft Time Series algorithm
Recently a customer discovered an issue with the ARIMA formula used in the Microsoft Time Series algorithm for data mining. She was trying to back-test the results by computing the formula and couldn't get the results to match. This can be maddening, especially when the formulas are so complex, but it also exposed a problem with the documentation. We don't really describe differencing and don't provide enough of the actual formulas used. We hope to gradually rectify that for all model types, but in the meantime here is a proposed addition to the time series technical reference that explains differencing. Suggestions and corrections are welcome! (The introduction has also been updated slightly to better explain how the two algorithms fit together.)
Many thanks to Yimin Wu for patiently explaining the implementation; any errors in describing it are mine.
{updated intro}
Implementation of the Microsoft Time Series Algorithm
Microsoft Research developed the original ARTXP algorithm that was used in SQL Server 2005, basing the implementation on the Microsoft Decision Trees algorithm. Therefore, the ARTXP algorithm can be described as an autoregressive tree model for representing periodic time series data. This algorithm relates a variable number of past items to each current item that is being predicted. The name ARTXP derives from the fact that the autoregressive tree method (an ART algorithm) is applied to multiple unknown prior states. For a detailed explanation of the ARTXP algorithm, see Autoregressive Tree Models for Time-Series Analysis, (http://go.microsoft.com/fwlink/?LinkId=45966).
The ARIMA algorithm was added to the Microsoft Time Series algorithm in SQL Server 2008 to improve long-term prediction. It is an implementation of the process for computing autoregressive integrated moving averages that was described by Box and Jenkins. The ARIMA methodology makes it possible to determine dependencies in observations taken sequentially in time, and can incorporate random shocks as part of the model. The ARIMA method also supports multiplicative seasonality. Readers who want to learn more about the ARIMA algorithm are encouraged to read the seminal work by Box and Jenkins; this section is intended to provide specific details about how the ARIMA methodology has been implemented in the Microsoft Time Series algorithm.
By default, the Microsoft Time Series algorithm uses both methods, ARTXP and ARIMA, and blends the results to improve prediction accuracy. If you want to use only a specific method, you can set the algorithm parameters to use only ARTXP or only ARIMA, or to control how the results of the algorithms are combined. Note that the ARTXP algorithm supports cross-prediction, but the ARIMA algorithm does not. Therefore, cross-prediction is available only when you use a blend of algorithms, or when you configure the model to use only ARTXP.
{new section}
Understanding ARIMA Difference Order
This section introduces some terminology needed to understand the ARIMA model, and discusses the specific implementation of differencing in the Microsoft Time Series algorithm. For a full explanation of these terms and concepts, we recommend a review of Box and Jenkins.
• A term is a component of a mathematical equation. For example, a term in a polynomial equation might include a combination of variables and constants.
• The ARIMA formula that is included in the Microsoft Time Series algorithm uses both autoregressive and moving average terms.
• Time series models can be stationary or nonstationary. Stationary models are those that revert to a mean, though they might have cycles, whereas nonstationary models do not have a focus of equilibrium and are subject to greater variance or change introduced by shocks, or external variables.
• The goal of differencing is to make a time series stabilize and become stationary.
• The order of difference represents the number of times that the difference between values is taken for a time series.
The Microsoft Time Series algorithm works by taking values in a data series and attempting to fit the data to a pattern. If the data series is are not already stationary, the algorithm applies an order of difference. Each increase in the order of difference tends to make the time series more stationary.
For example, if you have the time series (z1, z2, …, zn) and perform calculations using one order of difference, you obtain a new series (y1, y2,…., yn-1), where yi = zi+1-zi. When the difference order is 2, the algorithm generates another series (x1, x2, …, xn-2), based on the y series that was derived from the first order equation. The correct amount of differencing depends on the data. A single order of differencing is most common in models that show a constant trend; a second order of differencing can indicate a trend that varies with time.
By default, the order of difference used in the Microsoft Time Series algorithm is -1, meaning that the algorithm will automatically detect the best value for the difference order. Typically, that best value is 1 (when differencing is required), but under certain circumstances the algorithm will increase that value to a maximum of 2.
The Microsoft Time Series algorithm determines the optimal ARIMA difference order by using the autoregression values. The algorithm examines the AR values and sets a hidden parameter, ARIMA_AR_ORDER, representing the order of the AR terms. This hidden parameter, ARIMA_AR_ORDER, has a range of values from -1 to 8. At the default value of -1, the algorithm will automatically select the appropriate difference order.
Whenever the value of ARIMA_AR_ORDER is greater than 1, the algorithm multiplies the time series by a polynomial term. If one term of the polynomial formula resolves to a root of 1 or close to 1, the algorithm attempts to preserve the stability of the model by removing the term and increasing the difference order by 1. If the difference order is already at the maximum, the term is removed and the difference order does not change.
For example, if the value of AR = 2, the resulting AR polynomial term might look like this:
1 –1.4B + .45B^2 = (1- .9B) (1- 0.5B)
Note the term (1- .9B) which has a root of about 0.9. The algorithm eliminates this term from the polynomial formula but cannot increase the difference order by one because it is already at the maximum value of 2.
It is important to note that the only way that you can force a change in difference order is to use the unsupported parameter, ARIMA_DIFFERENCE_ORDER. This hidden parameter controls how many times the algorithm performs differencing on the time series, and can be set by typing a custom algorithm parameter. However, we do not recommend that you change this value unless you are prepared to experiment and are familiar with the calculations involved. Also note that there is currently no mechanism, including hidden parameters, to let you control the threshold at which the increase in difference order is triggered.
Finally, note that the formula described above is the simplified case, with no seasonality hints. If seasonality hints are provided, then a separate AR polynomial term is added to the left of the equation for each seasonality hint, and the same strategy is applied to eliminate terms that might destabilize the differenced series.
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Comments
• I have some real technical issues with these statements.....I'm not sure if you can change your approach, but here are some thoughts....
Identification of differencing orders is being done on the premise that 1) there are no pulses, level shifts, seasonal pulses and/or local time trends AND that one model is adequate for the entire time range (constancy of parameters assumption) AND that the variance of the error process is homogeneous (constancy of variance assumption) And that the underlying ARIMA process is white noise. For example a series that has a level shift will exhibit an ACF that suggests non-stationarity BUT the remedy is not to difference.
You say "The Microsoft Time Series algorithm works by taking values in a data series and attempting to fit the data to a pattern. If the data series is are not already stationary, the algorithm applies an order of difference. Each increase in the order of difference tends to make the time series more stationary."
I think this is not robust enough. Over-differencing can induce non-stationarity se the Slutzky effect en.wikipedia.org/.../Slutsky_equation
In general, Over-differencing yields non-invertible MA structure often suggesting equation simplification is in order.
Tom Reilly
www.autobox.com
• I have some real technical issues with these statements.....I'm not sure if you can change your approach, but here are some thoughts....
Identification of differencing orders is being done on the premise that 1) there are no pulses, level shifts, seasonal pulses and/or local time trends AND that one model is adequate for the entire time range (constancy of parameters assumption) AND that the variance of the error process is homogeneous (constancy of variance assumption) And that the underlying ARIMA process is white noise. For example a series that has a level shift will exhibit an ACF that suggests non-stationarity BUT the remedy is not to difference.
You say "The Microsoft Time Series algorithm works by taking values in a data series and attempting to fit the data to a pattern. If the data series is are not already stationary, the algorithm applies an order of difference. Each increase in the order of difference tends to make the time series more stationary."
I think this is not robust enough. Over-differencing can induce non-stationarity se the Slutzky effect en.wikipedia.org/.../Slutsky_equation
In general, Over-differencing yields non-invertible MA structure often suggesting equation simplification is in order.
Tom Reilly
www.autobox.com
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https://www.thestudentroom.co.uk/showthread.php?t=76287&page=6 | # Lets revise physicsA2 togather ultimate threadWatch
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This discussion is closed.
14 years ago
#101
(Original post by Mr Tom)
same... wish i hadn't buggered it in january can't do the ole wave-particle duality nonsense
still its all better than PHY6
So u took phy4 in january , lucky our school dont do january exams .
what did u get in the exam if u dont mind me askin and was it a hard paper?
I dont kno what to expect from the paper
any topics that came up alot in the jan paper? (Does anyone have that paper ?)
0
14 years ago
#102
(Original post by bassman)
wat does it mean by de broglie wavelength?
is it that a particle travels the same shape as a wave (sinusodially) or the particle has a wavelength? bit confused as i aint revised unit 4 for a very long time
the de Broglie equation is used for the wave behaviour of sub-atomic particles like electrons.
the particle doesn't travel in a sinosoidal shape (cos it IS the shape), rather the particle, for eg. an electron in an atom, is trapped inside the atom like a stationary wave is trapped in a rope.
as particles can behave like waves...the greater the velocity of the particle, the shorter will be its wavelength.
0
Thread starter 14 years ago
#103
(Original post by lilbug)
the de Broglie equation is used for the wave behaviour of sub-atomic particles like electrons.
the particle doesn't travel in a sinosoidal shape (cos it IS the shape), rather the particle, for eg. an electron in an atom, is trapped inside the atom like a stationary wave is trapped in a rope.
as particles can behave like waves...the greater the velocity of the particle, the shorter will be its wavelength.
also the mass,so thats way we dont see the effect of a travelling ball
0
14 years ago
#104
(Original post by sumitk87)
So u took phy4 in january , lucky our school dont do january exams .
what did u get in the exam if u dont mind me askin and was it a hard paper?
I dont kno what to expect from the paper
any topics that came up alot in the jan paper? (Does anyone have that paper ?)
I did PHY4 back in January, and I messed up.
Looking back(I got the exam paper back), it wasnt a difficult paper at all. It was slightly confusing at the time, but if I was presented with the same paper now, I could handle it much better. Its amazing how much difference 4 months can make.
Here is one of the questions that I really messed up on. Its a 7 mark explaination question!!
You are provided with a double slit whose slip separation is known to be 0.20mm. Describe how you would use it to measure the wavelength of the light from a monochromatic source. Mention any step you would take to make the measurement as accurate as possible.
You should include a diagram of the experimental arrangement, and state a suitable value for any other important dimensions.
(I drew a screen, and the monochromatic source, with the double slits in front of it. And I labled the distance as 6m. I only got 1/2 marks on this part)
On the explaination, I wrote(copying from my answer book):
Move the source next to the slits, so light travels through both. At the slits the light will defract and produce two coherent sources. The light from each slit will superpose, and form an interference pattern. This pattern can be observed on the screen at a pattern of high intensity and low intensity strips. The high intensities correspond to the maximas, which from when the waves at that point construct, due to being in phase. The low intensities correspond to the minimas where the wave destruct, due to being in anti-phase. There will be a centre maxima on the screen where the light from each slit would have travelled the same distance. Adjacent to this would be subsequant maximas on either side.
Using a ruler, measuring the distance between 3 or more maximas from the centre, you can find the distance between two maximas, S.(total distance/no. of maximas). Repeat and Average. Since Landa=xs/D, the wavelength can be measuring the distance between the slits and screen for D, and measuring the distance between the centre of each slit for x.
I got 2/7 for this question!
Anyone else think I got screwed over?
0
14 years ago
#105
Screwed by the exam board definately.
Personally I would have used a spectrometer.
0
Thread starter 14 years ago
#106
(Original post by SinghFello)
I did PHY4 back in January, and I messed up.
Looking back(I got the exam paper back), it wasnt a difficult paper at all. It was slightly confusing at the time, but if I was presented with the same paper now, I could handle it much better. Its amazing how much difference 4 months can make.
Here is one of the questions that I really messed up on. Its a 7 mark explaination question!!
You are provided with a double slit whose slip separation is known to be 0.20mm. Describe how you would use it to measure the wavelength of the light from a monochromatic source. Mention any step you would take to make the measurement as accurate as possible.
You should include a diagram of the experimental arrangement, and state a suitable value for any other important dimensions.
(I drew a screen, and the monochromatic source, with the double slits in front of it. And I labled the distance as 6m. I only got 1/2 marks on this part)
On the explaination, I wrote(copying from my answer book):
Move the source next to the slits, so light travels through both. At the slits the light will defract and produce two coherent sources. The light from each slit will superpose, and form an interference pattern. This pattern can be observed on the screen at a pattern of high intensity and low intensity strips. The high intensities correspond to the maximas, which from when the waves at that point construct, due to being in phase. The low intensities correspond to the minimas where the wave destruct, due to being in anti-phase. There will be a centre maxima on the screen where the light from each slit would have travelled the same distance. Adjacent to this would be subsequant maximas on either side.
Using a ruler, measuring the distance between 3 or more maximas from the centre, you can find the distance between two maximas, S.(total distance/no. of maximas). Repeat and Average. Since Landa=xs/D, the wavelength can be measuring the distance between the slits and screen for D, and measuring the distance between the centre of each slit for x.
I got 2/7 for this question!
Anyone else think I got screwed over?
I don't know why did they gave you 2/7 on this question I mean apart from not drowing the patter produced and indicate what the source is ie laser,you almost wrote down everything
0
14 years ago
#107
(Original post by habosh)
I don't know why did they gave you 2/7 on this question I mean apart from not drowing the patter produced and indicate what the source is ie laser,you almost wrote down everything
i think its to do with who marks your paper. some are v strict and sum are leanient.
i thought i messed up completely i didnt answer loads of questions in unit4 2005 i was expecting around 50ish but i ended up with 70, whereas my friend who described all the questions got 50 something.
0
Thread starter 14 years ago
#108
(Original post by bassman)
i think its to do with who marks your paper. some are v strict and sum are leanient.
i thought i messed up completely i didnt answer loads of questions in unit4 2005 i was expecting around 50ish but i ended up with 70, whereas my friend who described all the questions got 50 something.
that is honestly cruel I think the mark scheme says exactly what the dude wrote down I guess maybe he means it's what he would write now and not 4 months ago maybe
0
14 years ago
#109
(Original post by SinghFello)
You are provided with a double slit whose slip separation is known to be 0.20mm. Describe how you would use it to measure the wavelength of the light from a monochromatic source. Mention any step you would take to make the measurement as accurate as possible.
You should include a diagram of the experimental arrangement, and state a suitable value for any other important dimensions.
(I drew a screen, and the monochromatic source, with the double slits in front of it. And I labled the distance as 6m. I only got 1/2 marks on this part)
On the explaination, I wrote(copying from my answer book):
Move the source next to the slits, so light travels through both. At the slits the light will defract and produce two coherent sources. The light from each slit will superpose, and form an interference pattern. This pattern can be observed on the screen at a pattern of high intensity and low intensity strips. The high intensities correspond to the maximas, which from when the waves at that point construct, due to being in phase. The low intensities correspond to the minimas where the wave destruct, due to being in anti-phase. There will be a centre maxima on the screen where the light from each slit would have travelled the same distance. Adjacent to this would be subsequant maximas on either side.
Using a ruler, measuring the distance between 3 or more maximas from the centre, you can find the distance between two maximas, S.(total distance/no. of maximas). Repeat and Average. Since Landa=xs/D, the wavelength can be measuring the distance between the slits and screen for D, and measuring the distance between the centre of each slit for x.
I got 2/7 for this question!
Anyone else think I got screwed over?
well there are a few things you've missed - in the diagram you need a single slit between the source and the double slits to produce a coherent source so that's why you dropped a mark there. You should probably talk about path difference and phase difference, and at the maximas of intensity, the waves from the slits have a path difference of so they're in phase and constructively interfere, and at minima, the path difference is , they're anti-phase so destructively interfere (they might not have liked you just saying construct and destruct as the wave doesn't really make or destroy itself!). You also just said you'd get an interference pattern but didn't really describe that it was succesive lines of high and low intensity - you could maybe have drawn a picture. It also asked for any other dimensions - you need to say the slit width is on the scale of the wavelength of light. When it comes to measuring the fringe spacing, they might have wanted you to say you'd measure between the middles of the maxima or something.
Although 2/7 seems fairly harsh, you missed a few important points and they'll have had a markscheme which they have to follow
0
14 years ago
#110
If thats exactly what singhfello wrote in the exam and only got 2/7 , He got screwed and im gettn scared, cos that was quite a gd answer and yet only 2/7, scary
0
Thread starter 14 years ago
#111
Ok,here is the markscheme the final thingie
suitable source,laser,or filament lamp,light source,monochromatic source plus a single slit 2 marks
double slit plus screen,or a travelling microscope (unless laser used)
measure distance from slits to screen(or focus plane of microscope)
measure spacing between centres of bright and dark fringes
subtitute in lambda=xs/D
3marks
they ask for precautions I guess
so measure distance across several fringes and find avarage x or maximise D to give maximium x
1 mark
values of D
laser=1-10 m
filament lamp 1-2m
travelling microscope 0.1-2m
so I guess one mark for not writing precautions,and another half for not indicating the source.mmm ok lets say a mark so why did he lose the other 5 marks??? ok and another for not drowing the patter for example ...that gives him minimum 4/7
0
14 years ago
#112
hey habosh do u have the jan05 phy paper?
have u got in on yr computer or somin, if so is it possible to send it to me or post it on TSR. Or have u got the atcual paper
Thanks
0
Thread starter 14 years ago
#113
I got the paper as a mock exam]
Edit:I cant find the paper anywhere at the momenti'm so messy and my room is upside down
0
14 years ago
#114
(Original post by habosh)
I got the paper as a mock exam]
Edit:I cant find the paper anywhere at the momenti'm so messy and my room is upside down
not to worry, can u remember what topics came up roughly and what were the hard topics in it?
0
14 years ago
#115
Its kind of annoying when they dont give out marks.(BTW, I did say the precautions).
4 extra marks take you up one grade!
I also didnt get marks for the following parts:
(Haboosh, can you give the answers for these, because I wasnt given the answer sheet)
The final part of Q4 - Determine the amplitude of each of the travelling wave. I put 5mm.
Q5 - The reason why the graph confirms that the frequancy is inversly propotional to wavelength. Still not sure about this one?
0
Thread starter 14 years ago
#116
no it's 2.5mm ,the amplitude of the antinode is the vector sum of the amplitudes of bothwaves so the amplitude of each spearate wave is 2.5 also as it's a stationary wave then the amplitude is equal..part 5 it's because the graph is a stright line of negative gradient you can find the gradient which is the constant of the relationship lambdaf
0
Thread starter 14 years ago
#117
(Original post by sumitk87)
not to worry, can u remember what topics came up roughly and what were the hard topics in it?
magnetism..electrical field hard one i guess unless you practiced for these types of questions you'd find it easy..two chagrges
+(3uC) X - (1uC)
and then the girve a relation and prove the electrical field at that point is equal to that relation ...arrghh..I'm sos orry I cant concentrate any more I feel my brain is going to exploade
0
14 years ago
#118
(Original post by SinghFello)
Its kind of annoying when they dont give out marks.(BTW, I did say the precautions).
4 extra marks take you up one grade!
I also didnt get marks for the following parts:
(Haboosh, can you give the answers for these, because I wasnt given the answer sheet)
The final part of Q4 - Determine the amplitude of each of the travelling wave. I put 5mm.
Q5 - The reason why the graph confirms that the frequancy is inversly propotional to wavelength. Still not sure about this one?
the stationary wave is the result of the two waves superposing hence the amplitude of the single wave is half.
the graph was a log graph and straight line with negative gradient hence it was inversely proportional.
if u tell me the equation i can explain it to u
0
14 years ago
#119
Hey, doin Edexcel A2 physics and just discovered this wonderful thread. Did a few of the questions but I'm struggling on a lot of Unit 4 stuff. Just been brwosing through a few unit4 papers and got stuck on a few. Was wonderin if some ppl could help me out:
1) Basically there was a question on equipment to produce a stationary wave using a signal generator. Then it says:
How is this phenomenon used to describe the behaviour of the electron in a hydrogen atom?
2) A student observes LED from a distance of 0.20m. The pupil of her eye has a diameter of 6.0mm. Calculate the number of photons which enter her eye per second.
0
14 years ago
#120
(Original post by haz136)
Hey, doin Edexcel A2 physics and just discovered this wonderful thread. Did a few of the questions but I'm struggling on a lot of Unit 4 stuff. Just been brwosing through a few unit4 papers and got stuck on a few. Was wonderin if some ppl could help me out:
1) Basically there was a question on equipment to produce a stationary wave using a signal generator. Then it says:
How is this phenomenon used to describe the behaviour of the electron in a hydrogen atom?
2) A student observes LED from a distance of 0.20m. The pupil of her eye has a diameter of 6.0mm. Calculate the number of photons which enter her eye per second.
1) An electron is described as oving on or in a stationary wave and so it does not loose any energy and spiral into the nucleus. It makes sence to an extent
2) is that all the info ure given??
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https://stats.stackexchange.com/questions/309586/pooled-variance-of-correlated-samples | # Pooled variance of correlated samples
I got two samples originating from the following multivariate
$$(X_1, X_2) \sim \mathcal{N}\left(0, \left[ \begin{matrix} \sigma^2 & \rho\sigma^2 \\ \rho\sigma^2 & \sigma^2 \end{matrix} \right] \right)$$
(I am using this multivariate normal to simulate an autoregressive process)
What I am trying to check is what happens to the total variance of the pooled sample $X$ when considering $X_1$ and $X_2$ independent instead of correlated by $\rho$.
I can compute the pooled variance of two independent samples pretty easily using the weighted mean of variance of each sample
$$\sigma_X = \frac{n\sigma + n\sigma}{2n} = \sigma$$
But I can't find any lead on how to compute the pooled variance when the samples are correlated. I tried finding a solution using the general expression of the variance of a sample but I just end up with the weighted mean of variances. I am missing the moment where the independence of samples is assumed, could someone help me with this ?
Here are my computation
Let's have $(X_1, X_2)$ two samples from a multivariate with means $(0,0)$, variances $(\sigma_1^2, \sigma_2^2)$, covariance $\sigma_{1,2}$ and of sample size $(n, m)$. Let's now have $X$ the pooled sample of size $p = n + m$ ordered so that elements $1:n$ are elements of $X_1$ and elements $n+1:n+m$ are elements of $X_2$. I am trying to estimate $\sigma_X$ the variance of $X$
\begin{align} \sigma_X &= \frac{1}{p}\sum_{i = 1}^{p} (x_i - \mu)^2 \\ &= \frac{1}{p}\sum_{i = 1}^{p} x_i^2 \\ &= \frac{1}{p}\left( \sum_{i = 1}^{n} x_i^2 + \sum_{i = n+1}^{p=n+m} x_i^2 \right) \\ &= \frac{1}{p}\left( n\sigma_1 + m\sigma_2 \right)\\ &= \frac{n\sigma_1 + m\sigma_2}{p} \end{align}
Using the assumption that the mean of $X$ is zero because
\begin{align} \mu &= \frac{1}{p}\sum_{i=1}^{p}x_i \\ &= \frac{1}{p} \left( \sum_{i=1}^{n}x_i + \sum_{i=n+1}^{p=n+m}x_i\right) \\ &= \frac{1}{p} \left( n*0 + m*0\right) \\ &= 0 \end{align}
• Are you looking for a theoretically calculated deviation $\sigma$ of the distribution or an expectation value for the estimated variance $s$ of the sample? The correlation will interfere with the latter but not the former. – Sextus Empiricus Oct 26 '17 at 15:21
• I would like the expectation value for the estimated variance of the grand sample but if you can provide it I would also be very interested in knowing why the correlation would not interfere with the theoretical $\sigma$ of the grand sample – Riff Oct 27 '17 at 13:05
Your problem is that your $\mu$ is really an estimator $\mu=\bar x_i$, not the expectation. As an estimator it is not equal to zero. Yes, in average $E[\mu]=0$ but not the realization in a given sample.
So, when you calculate $(x_i - \mu)^2$, you can't set $\mu=0$, then you don't have $(x_i - \mu)^2\ne x_i^2$. Instead you plug your $\mu=\frac{1}{p} \left( \sum_{i=1}^{n}x_i + \sum_{i=n+1}^{p=n+m}x_i\right)$, and a get a bunch of cross terms in the sums when squaring the deviations from the average $\mu$. These cross terms will bring up the correlation $\rho$
• To me the OP is actually confusing. I wonder what the problem is in the calculations. It seems unclear whether he is (wishes) to calculata a population based or sample based statistic. He kicks of with a known distribution,$$\mathcal{N}\left(0, \left[ \begin{matrix} \sigma^2 & \rho\sigma^2 \\ \rho\sigma^2 & \sigma^2 \end{matrix} \right] \right)$$for which you *can* say that the mean is zero. However indeed in the second half he makes calculations based on the sample $x_i$. The population based statistic would be easy to calculate. However the sample is different, due to correlation. – Sextus Empiricus Oct 26 '17 at 15:53
• @Riff, how do you have different $n\ne m$ when you draw from multivariate normal? Shouldn't you have always pairs of $X_1,X_2$? – Aksakal Oct 27 '17 at 14:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 2, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9997497200965881, "perplexity": 376.2910403150223}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987829507.97/warc/CC-MAIN-20191023071040-20191023094540-00551.warc.gz"} |
https://physics.stackexchange.com/questions/173020/what-does-the-t-stand-for-in-t-duality | # What does the “T” stand for in T-duality?
First of all, I am not a physicist. I'm a graduate math student and recently I came across the concept of T-duality. Actually I'm studying generalized complex geometry, which according to this paper (Generalized complex geometry and T-duality) can be used to get some nice interpretations of T-duality.
After reading some of it, I thought the "T" stand for torus, because the the fiber are torus. But then I found another paper (Spherical T-duality) defining T-duality for $SU(2)$-bundles. At first I asked myself "Why keep calling it T-duality, if the fiber are not torus anymore?" Probably it means something else...
So, what does the "T" stand for in T-duality?
The name T-duality stands for Target-space duality, see e.g. this preprint.
T duality and S duality come from the $T$ transformation and $S$ transformation which generate the modular group $PSL(2,\mathbb{Z})$.
It comes from S matrix theory, long before quarks were imagined, S,T and U characterize the type of exchange in the Feynman diagrams entering the S matrix calculation, and they are called Mandelstam variables.
s channel-------------------------- t channel------------------------u channel
duality meant that the sums could be done either in S channel or T channel and the result should be the same.
There is a resurrection of the scattering matrix for strings and I guess the definition has morphed to what you find in the link you gave.
It is also used in nuclear physics, where dualities are useful.
The above is history of theory in particle physics, and it was T because it came after the S of Scattering ( like y after x). | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8251197934150696, "perplexity": 476.6424555937824}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195525973.56/warc/CC-MAIN-20190719012046-20190719034046-00288.warc.gz"} |
http://claesjohnson.blogspot.com/2013/02/simple-model-of-global-energy-balance.html | ## onsdag 20 februari 2013
### Illuminating Model of Global Energy Balance
The radiative heat transfer from the Earth surface to outer space via the atmosphere can for resonant frequencies of the atmosphere be illustrated in a simple water flow model for two connected containers, with container 2 representing the Earth surface with the water level H_2 representing temperature, and container 1 representing the atmosphere of height/temperature H_1 < H_2 with an outlet representing outer space.
If the channel connecting 2 with 1 has the same dimension as the outlet of 1 and the channel flow Q is proportional to level difference, we have by conservation of water, normalizing the constant of proportionality to one:
• Q = H_2 - H_ 1
• Q = H_1
and thus Q = 0.5 x H_2. We compare with the situation with container 2 directly pouring into outer space (no atmosphere) which would give the double outlet flow 2 Q = H_2, as illustrated on top right. This would correspond to non-resonant frequencies for which the atmosphere is transparent.
Introducing 1 (atmosphere) between 2 (Earth surface) and outer space thus reduces the flow with a factor 2, the reduction coming from requiring the water to pass two channels instead of one.
The model exhibits the following fundamental aspects of heat transfer in the Earth-atmosphere system:
• One water flow from high (2) to low level (1): One-way heat transfer from warm Earth surface to colder atmosphere: No "back radiation".
• 1 as passive mediator between 2 and outer space reduces the flow: Decrease of outgoing long wave radiation OLR for resonant frequencies of the atmosphere. No decrease for non-resonant frequencies.
• 1 is a passive mediator in the sense that whatever it absorbs from 2 is emitted into outer space.
The total reduction of OLR or "radiative forcing" caused by radiation through the atmosphere, is then determined by the denseness of the resonant frequencies of the atmosphere. The trace gas CO2 has a very sparse spectrum which gives small "radiative forcing" (small emissivity), as shown in previous posts.
Summary:
1. Warming effect of the atmosphere, acting as a passive intermediate "blanket" between the Earth surface and pure space, for resonant frequencies.
2. Non-warming effect for non-resonant frequencies.
3. Total warming effect dependent on denseness of resonant frequencies.
4. One-flow of heat energy from warm to cold.
5. Not two-way flow of heat energy carried by "photons" traveling back and forth.
For a new approach to radiative heat transfer connecting to this post, see Computational Blackbody Radiation.
CO2 alarmism is based on the hypothesis that the spectrum of the atmosphere with CO2 as a trace gas is dense in the entire wave number band 600 - 800, with a suggested "radiative forcing" of 3.7 W/m2 upon doubling of the concentration from preindustrial level to 600 ppm.
But the spectrum of CO2 as atmospheric trace gas is not dense but very sparse except in the narrow band 667 - 669, and thus the basis of "radiative forcing" of 3.7 W/m2 appears to be grossly incorrect, probably a factor 10 too big. Without this "radiative forcing" of 3.7 W/m2 CO2 alarmism collapses. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8271876573562622, "perplexity": 2010.8045802201095}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267160620.68/warc/CC-MAIN-20180924165426-20180924185826-00137.warc.gz"} |
https://www.lessonplanet.com/teachers/lesson-plan-plants-and-oxygen-breathing | # Plants and Oxygen: Breathing
Second graders gain an understanding of how plants produce oxygen and that the oxygen we breathe comes from trees and plants. After a lecture/demo, 2nd graders discuss the ways plants produce oxygen and the detrimental effects of cutting down too many trees. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9718044400215149, "perplexity": 4145.357626398794}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145897.19/warc/CC-MAIN-20200224040929-20200224070929-00552.warc.gz"} |
https://www.degruyter.com/document/doi/10.1515/jci-2018-0017/html | # Approximate Kernel-Based Conditional Independence Tests for Fast Non-Parametric Causal Discovery
Eric V. Strobl, Kun Zhang and Shyam Visweswaran
# Abstract
Constraint-based causal discovery (CCD) algorithms require fast and accurate conditional independence (CI) testing. The Kernel Conditional Independence Test (KCIT) is currently one of the most popular CI tests in the non-parametric setting, but many investigators cannot use KCIT with large datasets because the test scales at least quadratically with sample size. We therefore devise two relaxations called the Randomized Conditional Independence Test (RCIT) and the Randomized conditional Correlation Test (RCoT) which both approximate KCIT by utilizing random Fourier features. In practice, both of the proposed tests scale linearly with sample size and return accurate p-values much faster than KCIT in the large sample size context. CCD algorithms run with RCIT or RCoT also return graphs at least as accurate as the same algorithms run with KCIT but with large reductions in run time.
## 1 The problem
Constraint-based causal discovery (CCD) algorithms such as Peter-Clark (PC) and Fast Causal Inference (FCI) infer causal relations from observational data by combining the results of many conditional independence (CI) tests [1]. In practice, a CCD algorithm can easily request p-values from thousands of CI tests even with a sparse underlying graph. Developing fast and accurate CI tests is therefore critical for maximizing the usability of CCD algorithms across a wide variety of datasets.
Investigators have developed many fast parametric methods for testing CI. For example, we can use partial correlation to test for CI under the assumption of Gaussian variables [2], [3]. We can also consider testing for unconditional independence XY|Z=z for each constant value z when Z is discrete and P(Z=z)>0. The chi-squared test for instance utilizes this strategy when both X and Y are also discrete [4]. Another permutation-based test generalizes the same strategy even when X and Y are not necessarily discrete [5].
Testing for CI in the non-parametric setting generally demands a more sophisticated approach. One strategy involves discretizing continuous conditioning variables Z as Z˘ in some optimal fashion and assessing unconditional independence Z˘=z˘ [6], [7]. Discretization however suffers severely from the curse of dimensionality because consistency arguments demand smaller bins with increasing sample size, but the number of cells in the associated contingency table increases exponentially with the conditioning set size. A second method involves measuring the distance between estimates of the conditional densities f(X|Y,Z) and f(X|Z), or their associated characteristic functions, by observing that f(X|Y,Z)=f(X|Z) when XY|Z [8], [9]. However, the power of these tests also deteriorates quickly with increases in the dimensionality of Z.
Several investigators have since proposed reproducing kernel-based CI tests in order to tame the curse of dimensionality. Indeed, kernel-based methods in general are known for their strong empirical performance in the high dimensional setting. The Kernel Conditional Independence Test (KCIT) for example assesses CI by capitalizing on a characterization of CI in reproducing kernel Hilbert spaces (RKHSs; [10]). Intuitively, KCIT works by testing for vanishing regression residuals among functions in RKHSs. Another kernel-based CI test called the Permutation Conditional Independence Test (PCIT) reduces CI testing to two-sample kernel-based testing via a carefully chosen permutation found at the solution of a convex optimization problem [11].
The aforementioned kernel-based CI tests unfortunately suffer from an important drawback: both tests scale at least quadratically with sample size and therefore take too long to return a p-value in the large sample size setting. In particular, KCIT’s bottleneck lies in the eigendecomposition as well as the inversion of large kernel matrices [10], and PCIT must solve a linear program that scales cubicly with sample size in order to obtain its required permutation [11]. As a general rule, it is difficult to develop exact kernel-based methods which scale sub-quadratically with sample size, since the computation of kernel matrices themselves scales at least quadratically.
Many investigators have nonetheless utilized random Fourier features in order to quickly approximate kernel methods in linear time with respect to the number of Fourier features. For example, Lopez-Paz and colleagues developed an unconditional independence test using statistics obtained from canonical correlation analysis with random Fourier features [12]. Zhang and colleagues have also utilized random Fourier features for unconditional independence testing but took a different approach by approximating the kernel cross-covariance operator [13]. The authors further analyzed block and Nyström-based kernel approximations to the unconditional independence testing problem. The authors ultimately concluded that the random Fourier feature and the Nyström-based approaches both outperformed the block-based approach on average. Others have analyzed the use of random Fourier features for predictive modeling (e. g., [14], [15]) or dimensionality reduction [16]. In practice, investigators have observed that methods which utilize random Fourier features often scale linearly with sample size and achieve comparable accuracy to exact kernel methods [12], [13], [14], [15], [16].
In this paper, we also use random Fourier features to design two fast tests called the Randomized Conditional Independence Test (RCIT) and the Randomized conditional Correlation Test (RCoT) which approximate the solution of KCIT. Simulations show that RCIT, RCoT and KCIT have comparable accuracy, but both RCIT and RCoT scale linearly with sample size in practice. As a result, RCIT and RCoT return p-values several orders of magnitude faster than KCIT in the large sample size context. Moreover, experiments demonstrate that the causal structures returned by CCD algorithms using either RCIT, RCoT or KCIT have nearly identical accuracy.
## 2 High-level summary
We now provide a high-level summary for investigators who simply wish to perform CCD with RCIT or RCoT. We want to test whether we have XY|Z in a fast and accurate manner without resorting to parametric assumptions. Previously, Zhang and colleagues introduced a non-parametric CI test called KCIT which can test whether we have XY|Z in an accurate but not fast manner by analyzing the partial cross-covariance (operator) using the kernel method [10]. Kernels however are expensive to compute because they scale at least quadratically with sample size. Another line of work has fortunately shown that we can approximate kernels by averaging over a variety of non-linear transformations called random Fourier features (e. g., [14], [15], [17]). We therefore propose to approximate KCIT by utilizing random Fourier features, specifically by analyzing the partial cross-covariance matrix of {X,Z} and Y (RCIT) or X and Y (RCoT) after subjecting the variable sets to the non-linear transformations and then non-linearly regressing out the effect of Z. Simulations show that RCIT and RCoT return p-values in a much shorter time frame while also matching or outperforming KCIT in approximating the null distribution. RCoT in particular also returns the most accurate p-values when the conditioning set size Z is large (≥4).
## 3 Preliminaries on kernels
We will deal with kernels in this paper, so we briefly review the corresponding theory; see [18] for a more extensive discussion of similar concepts. Capital letters X,Y,Z denote sets of random variables with codomains X,Y,Z, respectively. Let HX correspond to a Hilbert space of functions mapping X to R. We say that HX is more specifically a reproducing kernel Hilbert space, if the Dirac delta operator δx:HXR (which maps fHX to f(x)R) is a bounded linear functional [19]. We can associate HX with a unique positive definite kernelkX:X×XR in which the feature mapϕ(x):XHX satisfies the reproducing propertyf,ϕ(x)HX=f(x), fHX, xX by the Moore-Aronszajn Theorem. We will require that HX be separable (i. e., it must have a complete orthonormal system) throughout this paper [18]. Hein and Bousquet [20] showed that any continuous kernel on a separable space X (e. g., Rd) induces a separable RKHS. We will likewise consider other kernels such as kY on the separable space Y.
We have the following norm defined on linear operators between RKHSs:
## Definition.
[21] Denote byΣ:HXHYa linear operator. Then, provided that the sum converges, the Hilbert Schmidt (HS) norm of Σ is defined as:
(1)ΣHS2=i,jui,ΣvjHY2,
whereuiandvjare orthonormal bases ofHYandHX, respectively.
We denote the probability distribution of X as PX and that of Y as PY. We may then define the mean elementsE[ϕ(X)]=μXHX and E[ψ(Y)]=μYHY with respect to the probability distributions; here, ϕ denotes the feature map from X to HX, and ψ likewise denotes the feature map from Y to HY. We also define the quantity μXHX2 by applying the expectation twice:
(2)μXHX2=EXX[ϕ(X),ϕ(X)HX]=EXX[kX(X,X)],
where X is an independent copy of X which follows the same distribution. The mean elements μX and μY exist so long as their respective norms in HX or HY are bounded; this is true so long as the kernels kX and kY are also bounded so that EXX[kX(X,X)]< and EYY[kY(Y,Y)]< [18].
The cross-covariance operator associated with the joint probability distribution PXY over (X,Y) is a linear operator ΣXY:HYHX defined as follows:
(3)f,ΣXYgHX=EXY[f(X)g(Y)]EX[f(X)]EY[g(Y)],
for all fHX and gHY. Gretton et al. [18] showed that the HS-norm of the cross covariance operator, ΣXYHS2, can be written in terms of kernels as follows:
(4)ΣXYHS2=EXXYY[kX(X,X)kY(Y,Y)],
provided that XY, and kX as well as kY are centered (i. e., EXX[kX(X,X)]=0 and EYY[kY(Y,Y)]=0). We can therefore consider the following empirical estimate of the ΣXYHS2 using n i. i. d. samples x,y as follows, if we assume that XY [10], [18]:
(5)Txy=1n2i=1nj=1n[K˜X]ij[K˜Y]ij=1n2tr[K˜XK˜Y],
where TxypΣXYHS2 [18]; the dependent case can be found in Lemma 1 of the same paper for the interested reader. The notations K˜X and K˜Y correspond to centralized kernel matrices such that:
(6)K˜X=HKXH,
and likewise for K˜Y. Here, H=I1n11T, I denotes an n×n identity matrix, and 1 denotes a vector of ones. The notation KX denotes a kernel matrix such as the RBF kernel where [KX]ij=exp(xixj22σ) with xi,xjx. The transformation in Equation 6 ensures that 1n2i=1nj=1n[K˜X]ij=0 similar to the centered kernels kX and kY in Equation 4.
## 4 Characterizations of conditional independence
We denote the probability distribution of X as PX and the joint probability distribution of (X,Z) as PXZ. Let LX2 denote the space of square integrable functions of X, and LXZ2 that of (X,Z). Here, LX2={s(X)EX(|s|2)<} and likewise for LXZ2. Next consider a dataset of n i. i. d. samples drawn according to PXYZ.
We use the notation XY|Z when X and Y are conditionally independent given Z. Perhaps the simplest characterization of CI reads as follows: XY|Z if and only if PXY|Z=PX|ZPY|Z. Equivalently, we have PX|YZ=PX|Z and PY|XZ=PY|Z.
### 4.1 Characterization by RKHSs
A second characterization of CI is given in terms of the cross-covariance operator ΣXY on RKHSs [22]. Recall the cross-covariance operator from HY to HX in Equation 3. We may then define the partial cross-covariance operator of (X,Y) given Z by:
(7)ΣXY·Z=ΣXYΣXZΣZZ1ΣZY,
where we use the right inverse instead of the inverse, if ΣZZ is not invertible (see Corollary 3 in Fukumizu et al. [22]). Notice the similarity of the partial cross-covariance operator to the linear partial cross-covariance matrix (as well as the conditional cross-covariance matrix in the Gaussian case).[1] We can interpret the above equation as the partial covariance between {f(X),fHX} and {g(Y),gHY} given {h(Z),hHZ} (i. e., the partial covariance of X and Y given Z after passing these three variable sets through the functions in the RKHSs HX,HY and HZ).
A kernel kX is characteristic if EXPX[f(X)]=EXQX[f(X)], fHX implies PX=QX, where PX and QX are two probability distributions of X [23]; alternatively, a kernel is characteristic if equality in the mean elements under the two distributions μPX=μQX implies equality of the distributions. Two examples of characteristic kernels include the Gaussian kernel and the Laplacian kernel. Now if we use characteristic kernels in (7), then the partial cross-covariance operator is related to the CI relation via the following conclusion:
### Proposition 1.
[22] , [23] LetX¨=(X,Z)andkX¨=kXkZ. Also letHX¨represent the RKHS corresponding tokX¨. AssumeE[kX(X,X)]<andE[kY(Y,Y)]<.[2]Further assume thatkX¨kYis a characteristic kernel on(X×Y)×Z, and thatHZ+R(the direct sum of the two RKHSs) is dense inLZ2. Then
(8)ΣX¨Y·Z=0XY|Z.
Here, ΣX¨Y·Z=0 means that f,ΣX¨Y·ZgHX¨=0 for all fHX¨ and all gHY. Recall further that ΣX¨Y·Z=0ΣX¨Y·ZHS2=0 because ΣX¨Y·ZHS2=i,jui,ΣX¨Y·ZvjHX¨2, where ui and vj are orthonormal bases of HX¨ and HY, respectively, by Definition 1.
### 4.2 Characterization by L2 spaces
We also consider a different characterization of CI which is intuitively more appealing because it allows to use to directly view CI as the uncorrelatedness of functions in certain spaces rather than a norm of the partial cross-covariance operator. In particular, consider the following constrained L2 spaces:
(9)FXZ{f˜LXZ2E(f˜|Z)=0},FYZ{g˜LYZ2E(g˜|Z)=0},FY·Z{h˜h˜=h(Y)E(h|Z),hLY2}.
Here, we write E(f˜|Z=z) as shorthand for E(f˜(·,z)) when f˜LXZ2, and likewise for g˜LYZ2. Notice that we can construct the three spaces listed above using nonlinear regression. For instance, for any fLXZ2 in FXZ, we have:
(10)f˜(X¨)=f(X¨)E(f|Z)=f(X¨)h(Z),
where h(Z)LZ2 is the regression function of f(X¨) on Z.
We then have the following result:
### Proposition 2.
[24] The following conditions are equivalent:
1. 1.
XY|Z,
2. 2.
E(f˜g˜)=0,f˜FXZandg˜FYZ,
3. 3.
E(f˜g)=0,f˜FXZandgLYZ2,
4. 4.
E(f˜h˜)=0,f˜FXZandh˜FY·Z,
5. 5.
E(f˜g)=0,f˜FXZandgLY2.
The second condition means that any “residual” function of (X,Z) given Z is uncorrelated with that of (Y,Z) given Z. The equivalence also represents a generalization of the case when (X,Y,Z) is jointly Gaussian; here, XY|Z if and only if any residual function of X given Z is uncorrelated with that of Y given Z; i. e., the linear partial correlation coefficient ρXY·Z is zero.
We also encourage the reader to observe the close relationship between Proposition 1 and claim 4 of Proposition 2. Notice that claim 4 of Proposition 2 implies that we have ΣX¨Y·Z=0 in Proposition 1. Moreover, Proposition 1 only considers functions in RKHSs, while claim 4 of Proposition 2 considers functions in L2 spaces. We find Proposition 1 more useful than claim 4 of Proposition 2 because the RKHS of a characteristic kernel might be much smaller than the corresponding L2 space.
## 5 Kernel conditional independence test
We now consider the following hypotheses:
(11)H0:XY|Z,H1:X⫫̸Y|Z.
We may equivalently rewrite the above null and alternative more explicitly using Proposition 1 as follows, provided that the kernels are chosen such that the premises of that proposition are satisfied (e. g., when the kernels are Gaussian or Laplacian [22]):
(12)H0:ΣX¨Y·ZHS2=0,H1:ΣX¨Y·ZHS2>0.
The above hypothesis implies that we can test for conditional independence by testing for uncorrelatedness between functions in reproducing kernel Hilbert spaces.
Zhang et al. [ 10] exploited the equivalence between 11 and 12 in the Kernel Conditional Independence Test (KCIT), which we now describe. Consider the functional spaces fHX¨, gHY, and hHZ. We can compute the corresponding centered kernel matrices K˜X¨, K˜Y and K˜Z from n i. i. d. samples x,y and z as in 6. We can then use these matrices to perform kernel ridge regression in order to estimate the function hHZ in 10 as follows: hˆ(z)=K˜Z(K˜Z+λI)1f(x¨), where λ denotes the ridge regularization parameter and fHX¨. Consequently, we can estimate the residual function f˜ as f˜(x¨)=f(x¨)hˆ(z)=RZf(x¨) where:
(13)RZ=IK˜Z(K˜Z+λI)1=λ(K˜Z+λI)1.
Next consider the eigenvalue decomposition K˜X¨=VX¨ΛX¨VX¨. Let ϕX¨=VX¨ΛX¨1/2. Then the empirical kernel map of HX¨|Z is given by ϕ˜X¨=RZϕX¨ because K˜X¨|Z=ϕ˜X¨ϕ˜X¨T. We may similarly write K˜Y|Z=ϕ˜Yϕ˜YT. We can thus write the centralized kernel matrices corresponding to f˜ and g˜ as follows:
(14)K˜X¨|Z=RZK˜X¨RZ,K˜Y|Z=RZK˜YRZ.
We can use the above two centered kernel matrices to compute an empirical estimate of the HS norm of the partial cross covariance operator similar to how we computed the quantity TXY in Equation 5:
(15)Tx¨y·z=1n2tr(K˜X¨|ZK˜Y|Z).
Note that Tx¨y·z denotes an empirical estimate of ΣX¨Y·ZHS2. KCIT uses the statistic SK=nTx¨y·z in order to determine whether or not to reject H0; we multiply the empirical estimate of the HS norm by n in order to ensure convergence to a non-degenerate distribution.
## 6 Proposed test statistic & its asymptotic distribution
Observe that computing SK requires the inversion of large kernel matrices which scales strictly greater than quadratically with sample size; the exact complexity depends on the algorithm used to compute the matrix inverse (e. g., O(n2.376) if we use the Coppersmith-Winograd algorithm [25]). CCD algorithms run with KCIT therefore take too long to complete. In this report, we will propose an inexpensive hypothesis test that almost always rejects or fails to reject the null whenever conditional dependence or independence holds, respectively, and even outperforms 12 as illustrated in our experimental results in Section 7. We will use a strategy that has already been successfully adopted in the context of unconditional dependence testing in doing so [12], [13].
We will in particular also take advantage of the characterization of CI presented in Proposition 1. Recall that the Frobenius norm corresponds to the Hilbert-Schmidt norm in Euclidean space. We therefore consider the following hypotheses as approximations to those in 12 and 11:
(16)H0:CA¨B·ZF2=0,H1:CA¨B·ZF2>0,
where CA¨B·Z=E[(A¨iE(A¨|Z))(BiE(B|Z))T] corresponds to the ordinary partial cross covariance matrix, where E(A¨|Z) and E(B|Z) may be non-linear functions of Z. We also have A¨=f(X¨){f1(X¨),,fm(X¨)} with fj(X¨)GX¨, j. Similarly, B=h(Y){h1(Y),,hq(Y)} with hk(Y)GY, k. The terms GX¨ and GY denote two spaces of functions, which we set to be the support of the process 2cos(WT·+B), WPW, BUniform([0,2π]). In other words, we select m functions from GX¨ and q functions from GY. We select these specific spaces because we can use them to approximate continuous shift-invariant kernels. A kernel k is said to be shift-invariant if and only if, for any aRp, we have k(xa,ya)=k(x,y), (x,y)Rp×Rp; examples of shift-invariant kernels include the Gaussian kernel frequently used in KCIT or the Laplacian kernel. The following result allows us to perform the approximation using the proposed spaces:
## Proposition 3.
[14] For a continuous shift-invariant kernelk(x,y)onRp, we have:
(17)k(x,y)=RpeiwT(xy)dFw=E[ζ(x)ζ(y)],
whereFWrepresents the CDF ofPWandζ(x)=2cos(WTx+B)withWPWandBUniform([0,2π]).
The precise form of PW depends on the type of shift-invariant kernel one would like to approximate (see Figure 1 of Rahimi and Recht [14] for a list). Since investigators most frequently implement KCIT using the Gaussian kernel kσ(x,y)=exp(xy2/σ) with hyperparameter σ, we choose to approximate the Gaussian kernel by setting PW to a centered Gaussian with standard deviation σ/2.
We will use the squared Frobenius norm of the empirical partial cross-covariance matrix as the statistic for RCIT:
(18)S=nCˆA¨B·ZF2,
where CˆA¨B·Z=1n1i=1n[(A¨iEˆ(A¨|Z))(BiEˆ(B|Z))T]. Recall however that E(A¨|Z) and E(B|Z) may be non-linear functions of Z and therefore difficult to estimate. We would instead like to approximate E(A¨|Z) and E(B|Z) with linear functions. We therefore let C=g(Z){g1(Z),,gd(Z)} with gl(Z)GZ,l, where we also set GZ to be the support of the process 2cos(WT·+B), WPW, BUniform([0,2π]). We will approximate CA¨B·Z with CˆA¨B·C=CˆA¨BCˆA¨C(ΣˆCC+γI)1CˆCB similar to 7, where γ denotes a small ridge parameter; recall that this is equivalent to computing the cross-covariance matrix across the residuals of A¨ and B given C using linear ridge regression. We can justify this procedure because the particular choice of C allows us to approximate both E(A¨|Z) and E(B|Z) with linear functions of C as described below.
Let fj=fjE(fj|Z). Then E(fj|Z)=0, so fjFXZ. Moreover, hkE(hk|Z)FY·Z. Note that we can estimate E(fj|Z) with the linear ridge regression solution uˆjTg(Z) under mild conditions because we can guarantee the following:
## Proposition 4.
(Section 3.1 of Sutherland and Schneider [15]) Consider performing kernel ridge regression offjon Z. Assume that (1)i=1nfj,i=0and (2) the empirical kernel matrix of Z,kZ, only has finite entries (i. e.,kZ<). Further assume that the range of Z,ZRdZ, is compact. We then have:
(19)P[|E˘(fj|Z)uˆjTg(Z)|ε]c0ε2edε2c1,
whereE˘(fj|Z)denotes the estimate ofE(fj|Z)by kernel ridge regression, andc0andc1are both constants that do not depend on n or d.
The exponential rate with respect to d in the above proposition suggests we can approximate the output of kernel ridge regression with a small number of random Fourier features, a hypothesis which we verify empirically in Section 7. Moreover, we can estimate E(hk|Z) with uˆkTg(Z), because we can similarly guarantee that P[|E˘(hk|Z)uˆkTg(Z)|ε]0 for any fixed ε>0 at an exponential rate with respect to d.
We can therefore consider the following spaces for S which are similar to the L2 spaces used in claim 4 of Proposition 2:
(20)GˆX¨{ffj=fjE(fj|Z),fjGX¨},GˆY·Z{hhk=hkE(hk|Z),hkGY}.
We then approximate CI with S in the following sense:
1. 1.
We always have XY|ZE(fh)=0, fGˆX¨andhGˆY·Z.
2. 2.
The reverse direction may hold for a larger subset of all possible joint distributions as the values of m and q increase; this is because at least one entry of CA¨B·C will likely be greater than zero for any given distribution as the values of m,q increase.
Note the second point refers to the population CA¨B·C as opposed to its finite sample estimate. In this paper, we only deal with the classical low dimensional scenario where m,q are fixed and the sample size n. This is reasonable because nearly all CB algorithms only test for CI when X and Y each contain a single variable. We find that the second point held in all of the cases we tested in Section 7 with only m,q=5, since we were always able to reject the null H0:CA¨B·CF2=0 by generating enough samples with m,q=5 when X⫫̸Y|Z.
### 6.1 Null distribution
We now consider the asymptotic distribution of S under the null. Let Π refer to a positive definite covariance matrix of the vectorization of (A¨E(A¨|C))(BE(B|C))T. We may denote an arbitrary entry in Π as follows:
(21)ΠA¨iBj,A¨kBl=E[(A¨iE(A¨i|C))(BjE(Bj|C))(A¨kE(Ak|C))(BlE(Bl|C))].
We have the following result:
### Theorem 1.
Consider n i. i. d. samples fromPXYZ. Let{z1,,zL}denote i. i. d. standard Gaussian variables (thus{z12,,zL2}denotes i.i.dχ12variables) and λ the eigenvalues of Π. We then have the following asymptotic distribution under the null in16:
(22)nCˆA¨B·CF2di=1Lλizi2,
where L refers to the number of elements inCˆA¨B·C.
### Proof.
We may first write:
(23)nCˆA¨B·CF2=ntr(CˆA¨B·CCˆA¨B·CT)=nv(CˆA¨B·C)Tv(CˆA¨B·C),=[nv(CˆA¨B·C)]T[nv(CˆA¨B·C)],
where v(CˆA¨B·C) stands for the vectorization of CˆA¨B·C. By CLT of the sample covariance matrix (see Lemma 1 in the Appendix A) combined with the continuous mapping theorem and the null, we know that nv(CˆA¨B·C)dN(0,Π). Here, we write an arbitrary entry ΠA¨iBj,A¨kBl under the null as follows:
(24)ΠA¨iBj,A¨kBl=Cov[(A¨iE(A¨i|C))(BjE(Bj|C)),(A¨kE(A¨k|C))(BlE(Bl|C))]=E[(A¨iE(A¨i|C))(BjE(Bj|C))(A¨kE(A¨k|C))(BlE(Bl|C))].
Now consider the eigendecomposition of Π written as Π=EΛET. Then, we have ET[nv(CˆA¨B·C)]dN(0,Λ) by the continuous mapping theorem. Note that:
(25)[nv(CˆA¨B·C)]T[nv(CˆA¨B·C)]=(ET[nv(CˆA¨B·C)])T(ET[nv(CˆA¨B·C)])di=1Lλizi2.
□
We conclude that the null distribution of the test statistic is a positively weighted sum of i. i. d. χ12 random variables.
Note that multiple methods exist for estimating the conditional expectations in S and Π in the above theorem. In this report, we will obtain consistent estimates of the conditional expectations by using kernel ridge regressions with the RBF kernel; here, consistency holds so long as the conditional expectations are continuous because the RBF kernel is dense in the space of continuous functions mapping Z to R [26]. We therefore have C˘A¨B·CpCA¨B·C, where C˘A¨B·C=1n1i=1n[(A¨iE˘(A¨|C))(BiE˘(B|C))T], by the continuous mapping theorem and weak law of large numbers assuming continuity of the conditional expectations. We can similarly approximate any arbitrary entry in Π because we may write the following:
(26)1nr=1n(A¨i,rE˘(A¨i|C))(Bj,rE˘(Bj|C))(A¨k,rE˘(A¨k|C))(Bl,rE˜(Bl|C))pE[(A¨iE(A¨i|C))(BjE(Bj|C))(A¨kE(A¨k|C))(BlE(Bl|C))].
Kernel ridge regressions however scale (strictly greater than) quadratically with sample size due to the inversion of the kernel matrix, so they may not be practical in the large sample size regime. Fortunately, we will not need to perform the kernel ridge regressions directly, because we can approximate the output of kernel ridge regression to within an arbitrary degree of accuracy for any fixed sample size n using linear ridge regression with enough random Fourier features as highlighted previously in Proposition 4. In particular, Proposition 4 implies that uˆTg(Z)E˜(A¨|C)pE˘(A¨|C) at rate exponential in d for any fixed n. We can also conclude that Σ˜A¨B·CpΣ˘A¨B·C as d for any fixed n. We can finally approximate the kernel ridge regression estimate of Π because we may write the following for an arbitrary entry in Π as d:
(27)1nr=1n(A¨i,rE˜(A¨i|C))(Bj,rE˜(Bj|C))(A¨k,rE˜(A¨k|C))(Bl,rE˜(Bl|C))p1nr=1n(A¨i,rE˘(A¨i|C))(Bj,rE˘(Bj|C))(A¨k,rE˘(A¨k|C))(Bl,rE˘(Bl|C)).
We conclude that, for a dataset of fixed sample size n, we can substitute the conditional expectation estimates of kernel ridge regression with those of linear regression with random Fourier features when estimating S as well as Π for applying Theorem 1.
Unfortunately though, a closed form CDF of a positively weighted sum of chi-squared random variables does not exist in general for applying Theorem 1. We can approximate the CDF by Imhof’s method which inverts the characteristic function numerically [27]. We should consider Imhof’s method as exact, since it provides error bounds and can be used to compute the distribution at a fixed point to within a desired precision [28], [29]. However, Imhof’s method is too computationally intensive for our purposes. We can nonetheless utilize several fast methods which approximate the null by moment matching.
### 6.2 Approximating the null distribution by moment matching
We write the cumulants of a positively weighted sum of i. i. d. χ12 random variables as follows:
(28)cr=2r1(r1)!i=1Lλir,
where λ={λ1,,λL} denotes the weights. We may for example derive the first three cumulants:
(29)m1=i=1Lλi,m2=2i=1Lλi2,m3=8i=1Lλi3.
We then recover the moments from the cumulants as follows:
(30)mr=cr+i=1r1r1i1cimri,r=2,3,
Now the Satterthwaite-Welch method [30], [31], [32] represents perhaps the simplest and earliest moment matching method. The method matches the first two moments of the sum with a gamma distribution Γ(gˆ,θˆ). Zhang and colleagues adopted a similar strategy in their paper introducing KCIT [10]. Here, we have:
(31)gˆ=12c12/c2,θˆ=c2/c1.
We however find the above gamma approximation rather crude. We therefore also consider applying more modern methods to estimating the distribution of a sum of positively weighted chi-squares. Improved methods such as the Hall-Buckley-Eagleson [33], [34] and the Wood F [35] methods match the first three moments of the sum to other distributions in a similar fashion. On the other hand, the Lindsay-Pilla-Basak method [36] matches the first 2L moments to a mixture distribution.
We will focus on the Lindsay-Pilla-Basak method in this paper, since Bodenham & Adams have already determined that the Lindsay-Pilla-Basak method performs the best through extensive experimentation [37], [38]. We therefore choose to use the method as the default method for RCIT. Briefly, the method approximates the CDF under the null FH0 using a finite mixture of L Gamma CDFs FΓ(g,θi):
(32)FH0=i=1LπiFΓ(g,θi),
where π0, i=1Lπi=1, and we seek to determine the 2L+1 parameters g, θ1,,θL, and π1,,πL. The Lindsay-Pilla-Basak method computes these parameters by a specific sequence of steps that makes use of results concerning moment matrices (see Appendix II in Uspensky [39]). The sequence is complicated and beyond the scope of this paper, but we refer the reader to [36] for details.
### 6.3 Testing for conditional un-correlatedness
Strictly speaking, we must consider the extended variable set X¨ to test for conditional independence according to Proposition 1. However, we have two observations: (1) we can substitute a test for non-linear conditional uncorrelatedness with tests for conditional independence in almost all cases encountered in practice because most conditionally dependent variables are correlated after some functional transformations, and (2) using the extended variable set X¨ makes estimating the null distribution more difficult compared to using the unextended variable set X. The first observation coincides with the observations of others who have noticed that Fisher’s z-test performs well (but not perfectly) in ruling out conditional independencies with non-Gaussian data. We can also justify the first observation with the following result using the cross-covariance operator ΣXY·Z:
### Proposition 5.
[22] , [23] AssumeE[kX(X,X)]<andE[kY(Y,Y)]<. Further assume thatkXkYis a characteristic kernel onX×Y, and thatHZ+R(the direct sum of the two RKHSs) is dense inLZ2. Then
(33)ΣXY·Z=0EZ[PXY|Z]=EZ[PX|ZPY|Z].
In other words, we have:
(34)ΣXY·Z=0PXY=PX|ZPY|ZdPZ,ΣXY·Z=0EZ[PX|ZPY|Z]=EZ[PXY|Z]XY|Z.
Notice that ΣXY·Z=0 is almost equivalent to CI, in the sense that ΣXY·Z=0 just misses those rather contrived distributions where PXY=PXY|ZdPZ=PX|ZPY|ZdPZ when X⫫̸Y|Z. In other words, if PXYPX|ZPY|ZdPZ when X⫫̸Y|Z, then we have ΣXY·Z=0ΣX¨Y·Z=0 (under the corresponding additional assumptions of Propositions 1 and 5).
Let us now consider an example of a situation where PXY|ZdPZPX|ZPY|ZdPZ when X⫫̸Y|Z. Take three binary variables X,Y,Z{0,1}. Let PZ=0=0.2 and PZ=1=0.8. Also consider the four probability tables in Table 1. Here, we have chosen the probabilities in the tables carefully by satisfying the following equation:
(35)PXY|ZdPZ=PX|ZPY|ZdPZPZ=0(PX|Z=0PY|Z=0)+PZ=1(PX|Z=1PY|Z=1)=PZ=0PXY|Z=0+PZ=1PXY|Z=1.
Of course, the equality holds when we have conditional independence PXY|Z=PX|ZPY|Z. We are however interested in the case when conditional dependence holds. We therefore instantiated the values of Tables 1a and 1b as well as the second column in Table 1c (PXY|Z=0) such that PXY|Z=0PX|Z=0PY|Z=0. We then solved for PXY|Z=1 using Equation 35 in order to complete Table 1c. This ultimately yielded Table 1d.
### Table 1
Example of a situation where PXY|ZdPZ=PX|ZPY|ZdPZ when X⫫̸Y|Z using binary variables.
Notice that we obtain a unique value for PXY|Z=1 by solving Equation 35. Hence, PXY|Z=1 has Lebesgue measure zero on the interval [0,1], once we have defined all of the other variables in the equation. Thus, ΣXY·Z=0 is not always equivalent to XY|Z, but satisfying the condition PXY|ZdPZ=PX|ZPY|ZdPZ when X⫫̸Y|Z requires a very particular setup which is probably rarely encountered in practice.
The aforementioned argument motivates us to also consider a different empirical estimate of the squared Hilbert-Schmidt norm of the partial cross covariance operator:
(36)SK=nTxy·z=1ntr(K˜X|ZK˜Y|Z),
where we have replaced X¨ with X. We can approximate the null distribution of SK by utilizing the strategies presented in Sections 3.3 and 3.4 of Zhang et al. [10]. Here, we utilize the hypotheses:
(37)H0:ΣXY·ZHS2=0,H1:ΣXY·ZHS2>0.
We similarly consider a corresponding finite dimensional partial cross-covariance matrix:
(38)S=nCˆAB·CF2,
where we have replaced A¨ with A. The above statistic is a generalization of linear partial correlation, because we consider uncorrelatedness of the residuals of non-linear functional transformations after performing non-linear regression. The asymptotic distribution for S in Theorem 1 also holds for S, when we replace A¨ with A. Here, we use the hypotheses:
(39)H0:CAB·CF2=0,H1:CAB·CF2>0.
In practice, the test which uses S, which we now call the Randomized conditional Correlation Test (RCoT), usually rivals or outperforms RCIT and KCIT, because (1) nearly all conditionally dependent variables encountered in practice are also conditionally correlated after at least one functional transformation, and (2) we can easily calibrate the null distribution of the test using S even when Z has large cardinality. We will therefore find this test useful for replacing RCIT when we have large conditioning set sizes (≥4).
### 6.4 Time complexity
We now show that RCIT and RCoT have linear time complexity with respect to the sample size n. Let d denote the number of random Fourier features in C.
### Proposition 6.
If we haven>d, then RCIT and RCoT have time complexityO(d2n).
### Proof.
Wlog, we will prove the claim for RCIT (the proof for RCoT will follow analogously). Note that it suffices to show that all sub-procedures of RCIT have time complexity O(d2n) or o(d2n).
The first step of RCIT computes the random Fourier features on the support of the process 2cos(WT·+B), WPW, BUniform([0,2π]). Let aX denote the number of dimensions in X, and likewise for aY and aZ. Let m, q and d denote the number of random Fourier features in A¨,B and C, respectively. Note that computing the samples of A¨ requires O((aX+aZ)mn) time because W is a (aX+aZ)×m matrix. As a result, computing the samples of A¨ requires O(n) time with m, aZ and aX fixed. Similarly, computing the samples of B requires O(n) time with aY,q fixed and that of C requires O(dn) time with only aZ fixed.
The second step of RCIT estimates Eˆ(A¨|C) and Eˆ(B|C) using linear ridge regressions. Recall that linear ridge regression scales O(d2n) in time, where d corresponds to the number of features (assuming n>d). Now RCIT requires m+q linear ridge regressions in order to compute Eˆ(A¨|C) and Eˆ(B|C). We therefore conclude that estimating Eˆ(A¨|C) and Eˆ(B|C) can be done in O(d2n) time with m and q fixed.
Next, computing the covariance ΣˆA¨B·C requires O(mqn) time. Finally, the time complexity of all of the methods used to approximate the null distribution do not depend on d or n once given ΣˆA¨B·C; we thus conclude that all of the null distribution approximation methods have time complexity O(1) when m and q are fixed.
We have shown that all sub-procedures of RCIT scale O(d2n) or o(d2n). We therefore conclude that RCIT has time complexity O(d2n). □
The above proposition implies that RCIT and RCoT have time complexity O(n) because d is set to a fixed number regardfless of sample size. Recall that d is fixed because we have an exponential convergence rate with respect to d as highlighted previously in Proposition 4; in other words, a fixed d is reasonable so long as n does not become extremely large. In practice, we have found that setting m=5, q=5 and p=25 works well for a variety of sample sizes and dimensions of Z as highlighted in the next section. The statement holds so long as we choose a very small regularization constant λ (e. g., 1E10); note that this is different from the standard prediction regime, where we must carefully tune λ to prevent overfitting. We can utilize a small regularization constant in the proposed CI test setting because the entries of CˆA¨B·C are never seen during training time.
## 7 Experiments
We carried out experiments to compare the empirical performance of the following tests:
1. RCIT: uses S with the Lindsay-Pilla-Basak approximation,
2. RCoT: uses S with the Lindsay-Pilla-Basak approximation,
3. KCIT: uses SK with a simulated null by bootstrap.
4. KCoT: uses SK with a simulated null by bootstrap.
We estimated the conditional expectations for S and S using linear ridge regressions with random Fourier features. We also compared RCIT and RCoT against permutation tests and list the results in the Appendix A. We present the results of KCoT in Appendix A.3 as well. Note that KCIT with the gamma approximation performs slightly faster than KCIT with bootstrap,[3] but the bootstrap results in a significantly better calibrated null distribution. We focus on large sample size (≥500) scenarios because we can just apply KCIT with bootstrap otherwise. We ran all experiments using the R programming language (Microsoft R Open) on a laptop with 2.60 GHz of CPU and 16 GB of RAM.
### 7.1 Hyperparameters
We used the same hyperparameters for RCIT and RCoT. Namely, we used the median Euclidean distance heuristic across the first 500 samples of X¨, X, Y and Z for choosing the σX¨, σX, σY, and σZ hyperparameters for the Gaussian kernels kσ(x,y)=exp(xy2/σ), respectively[4] [16] , [40]. We also fixed the number of Fourier features for X¨, X and Y to 5 and the number of Fourier features for Z to 25. We standardized all original and Fourier variables to mean zero unit variance in order to help ensure numerically stable computations. Finally, we set γ to 1E10 in order to keep bias minimal. These parameters are reasonable because we designed both RCIT and RCoT for the purposes of causal discovery, where the variable set Z is small with a sparse underlying causal graph. We can therefore utilize a relatively small number of random Fourier features as compared to the sample size n. Authors who wish to apply RCIT or RCoT in the high dimensional scenario should consider utilizing more Fourier features for Z and choosing the lambda values carefully (e. g., through cross-validation or information criteria).
With KCIT, we set σ to the squared median Euclidean distance between (X,Y) using the first 500 samples times double the conditioning set size; the hyperparameters as described in the original paper, the hyperparameters in the author-provided MATLAB implementation and the hyperparameters of RCIT/RCoT all gave worse performance.
### 7.2 Type I error
We analyzed the Type I error rates of the three CI tests as a function of sample size and conditioning set size. We evaluated the algorithms using the Kolmogorov-Smirnov (KS) test statistic. Recall that the KS test uses the following statistic:
(40)K=supxR|Fˆ(x)F(x)|=FˆXFX,
where FˆX denotes the empirical CDF, and FX some comparison CDF. If the sample comes from PX, then K converges to 0 almost surely as n by the Glivenko-Cantelli theorem.
Now a good CI test controls the Type I error rate at any α value, when we have a uniform sampling distribution of the p-values over [0,1]. Therefore, a good CI test should have a small KS statistic value, when we set FX to the uniform distribution over [0,1].
To compute the KS statistic values, we generated data from 1000 post non-linear models [10], [11]. We can describe each post non-linear model as follows: X=g1(Z+ε1), Y=g2(Z+ε2), where Z,ε1,ε2 have jointly independent standard Gaussian distributions, and g1,g2 denote smooth functions. We always chose g1,g2 uniformly from the following set of functions: {(·),(·)2,(·)3,tanh(·),exp(·2)}. Thus, we have XY|Z in any case. Notice also that this situation is more general than the additive noise models proposed in Ramsey [41], where we have X=g1(Z)+ε1, Y=g2(Z)+ε2. The post non-linear models allow us to simulate heteroskedastic noise which is commonly encountered in real scenarios but not captured with additive noise models.
#### 7.2.1 Sample size
We first assess the Type I error rate as a function of sample size. We used sample sizes of 500, 1000, 2000, 5000, ten thousand, one hundred thousand and one million. A good CI test should control the Type I error rate across all α values at any sample size. Figure 1a summarizes the KS statistic values for the three different CI tests. Observe that all tests have similar KS statistic values across different sample sizes. We conclude that all three tests perform comparably in controlling the Type I error rate with a single conditioning variable at different sample sizes.
The run time results however tell a markedly different story. Both RCIT and RCoT output a p-value much more quickly than KCIT at different sample sizes (Figure 1b). Moreover, KCIT ran out of memory at 5000 samples while RCIT and RCoT handled one million samples in a little over 6 seconds. RCIT and RCoT also completed more than two orders of magnitude faster than KCIT on average at a sample size of 2000 (Figure 1c). We conclude that RCIT and RCoT are more scalable than KCIT. Moreover, the experimental results agree with standard matrix complexity theory; RCIT and RCoT scale linearly with sample size (see Proposition 6), while KCIT scales strictly greater than quadratically with sample size.
#### 7.2.2 Conditioning set size
CCD algorithms request p-values from CI tests using large conditioning set sizes. In fact, algorithms which do not assume causal sufficiency, such as FCI, often demand very large conditioning set sizes (>5). We should however also realize that CCD algorithms search for minimal conditioning sets in order to establish ancestral relations. This means that we must focus on testing for cases where X⫫̸Y|Z, but we have either XY|ZA or X⫫̸Y|ZA, where |A|=1.
We therefore evaluated the Type I error rates of the CI tests as a function of conditioning set size by fixing the sample size at 1000 and then adding 1 to 10 standard Gaussian variables into the conditioning set so that X=g1(1kj=1kZj+ϵ1), Y=g2(1kj=1kZj+ϵ2), k={1,,10} in 1000 models. Note that this situation corresponds to 1 to 10 common causes.
Figure 1d summarizes the KS statistic values in the aforementioned scenario. We see that the KS statistic values for RCoT remain the smallest for nearly all conditioning set sizes, followed by RCIT and then KCIT. This implies that RCoT best approximates the null distribution out of the three CI tests. We also provide the histograms of the p-values across the 1000 post non-linear models at a conditioning set size of 10 for KCIT, RCIT, and RCoT in Figures 1e–1g. Notice that the histograms become progressively more similar to a uniform distribution. We conclude that RCoT controls its Type I error rate the best even with large conditioning set sizes while KCIT controls its rate the worst.
Now the run times of all three tests only increased very slightly with the conditioning set size (Figure 1h). However, both RCIT and RCoT still completed 40.91 times faster than KCIT on average (95 % confidence interval: ±0.44). These results agree with standard matrix complexity theory, as we expect all tests to scale linearly with dimensionality.
### Figure 1
Experimental results of RCIT, RCoT and KCIT when conditional independence holds. (a) All tests have comparable KS statistic values as a function of sample size. (b) However, both RCIT and RCoT complete much faster than KCIT. (c) The relative difference in speed between RCIT vs. KCIT and RCoT vs. KCIT grows with increasing sample size. (d) RCoT maintains the lowest KS statistic value with increases in dimensionality. (e–g) Histograms with a conditioning set size of 10 show that KCIT, RCIT and RCoT obtain progressively more uniform null distributions. (h) Run times of all three tests scale linearly with dimensionality of the conditioning set.
### 7.3 Power
We next evaluated test power (i. e., 1-(Type II error rate)) by computing the area under the power curve (AUPC). The AUPC corresponds to the area under the empirical CDF of the p-values returned by a CI test when the null does not hold. A CI test has higher power when its AUPC is closer to one. For example, observe that if a CI test always returns a p-value of 0 in the perfect case, then its AUPC corresponds to 1.
We examined the AUPC by adding the same small error εbN(0,1/16) to both X and Y in 1000 post non-linear models as follows: X=g1(εb+ε1), Y=g2(εb+ε2), ZN(0,1). Here, we do not allow the CI tests to condition on εb, so we always have X⫫̸Y|Z; this situation therefore corresponds to a hidden common cause.
#### 7.3.1 Sample size
We first examine power as a function of sample size. We again tested sample sizes of 500, 1000, 2000, 5000, ten thousand, one hundred thousand, and one million. We have summarized the results in Figure 2a. Both RCIT and RCoT have comparable AUPC values to KCIT with sample sizes of 500, 1000 and 2000. At larger sample sizes, KCIT again did not scale due to insufficient memory, but the AUPC of both RCIT and RCoT continued to increase at similar values. We conclude that all three tests have similar power.
The run time results mimic those of Section 7.2.1; RCIT and RCoT completed orders of magnitude faster than KCIT (Figures 2b and 2c).
#### 7.3.2 Conditioning set size
We next examined power as a function of conditioning set size. To do this, we fixed the sample size at 1000 and set Z=(Z1,,Zk) with ZN(0,Ik), k={1,,10} in the 1000 post non-linear models. We therefore examined how well the CI tests reject the null under an increasing conditioning set size with uninformative variables. A good CI test should either (1) maintain its power or, more realistically, (2) suffer a graceful decline in power with an increasing conditioning set size because none of the variables in the conditioning set are informative for rendering conditional independence by design.
We have summarized the results in Figure 2d. Notice that all tests have comparable AUPC values with small conditioning set sizes (between 1 and 3), but the AUPC value of KCIT gradually increases with increasing conditioning set sizes; the AUPC value should not increase under the current setup with a well-calibrated null because the extra variables are uninformative. To determine the cause of the unexpected increase in power, we permuted the values of X in each run in order to assess the calibration of the null distribution. Figure 2f summarizes the results, where we can see that only KCIT’s KS statistic grows with an increasing conditioning set size. We can therefore claim that the increasing AUPC value of KCIT holds because of a badly calibrated null distribution with larger conditioning set sizes. We conclude that both RCIT and RCoT maintain steady power under an increasing conditioning set size while KCIT does not.
The run times in Figures 2e and 2g again mimic those in Section 7.2.2 with RCIT and RCoT completing in a much shorter time frame than KCIT.
### Figure 2
Experimental results with RCIT, RCoT and KCIT as a function of sample size and conditioning set size when conditional dependence holds. (a) All tests have comparable AUPC values as a function of sample size with a conditioning set size of one. (b–c) Both RCIT and RCoT again complete much faster than KCIT. (d) KCIT’s AUPC value unexpectedly increases with the dimensionality of the conditioning set. Associated run times for (d) in (e). (f) The cause of KCIT’s AUPC increase lies in a badly calibrated null distribution; here we see that only KCIT’s KS statistic value increases under the null. Associated run times for (f) in (g).
### 7.4 Causal structure discovery
We next examine the accuracy of graphical structures as recovered by PC [1], FCI [42] and RFCI [43] when run using RCIT, RCoT or KCIT.
We used the following procedure in Colombo et al. [43] to generate 250 different Gaussian DAGs with an expected neighborhood size E(N)=2 and v=20 vertices. First, we generated a random adjacency matrix A with independent realizations of Bernoulli(E(N)/(v1)) random variables in the lower triangle of the matrix and zeroes in the remaining entries. Next, we replaced the ones in A by independent realizations of a Uniform([1,0.1][0.1,1]) random variable. We interpret a nonzero entry Aij as an edge from Xi to Xj with coefficient Aij in the following linear model:
(41)X1=ε1,Xi=r=1v1AirXr+εi.
for i=2,,v where ε1,,εv are mutually independent standard Gaussian random variables. The variables {X1,,Xv}=X then have a multivariate Gaussian distribution with mean 0 and covariance matrix Σ=(IvA)1(IvA)T, where Iv is the v×v identity matrix. To introduce non-linearities, we passed each variable in X through a non-linear function g again chosen uniformly from the set {(·),(·)2,(·)3,tanh(·),exp(·2)}.
For FCI and RFCI, we introduced latent and selection variables using the following procedure. For each DAG, we first randomly selected a set of 0–3 latent common causes L. From the set XL, we then selected a set of 0–3 colliders as selection variables S. For each selection variable in S, we subsequently eliminated the bottom q percentile of samples, where we drew q according to independent realizations of a Uniform([0.1,0.5]) random variable. We finally eliminated all of the latent variables from the dataset.
We ultimately created 250 different 500 sample datasets for PC, FCI and RFCI. We then ran the sample versions of PC, FCI and RFCI using RCIT, RCoT, KCIT and Fisher’s z-test (FZT) at α=0.05. We also obtained the oracle graphs by running the oracle versions of PC, FCI and RFCI using the ground truth.
We have summarized the results as structural Hamming distances (SHDs) from the oracle graphs in Figure 3a. PC run with RCIT and PC run with RCoT both outperformed PC run with KCIT by a large margin according to paired t-tests (PC RCIT vs. KCIT, t=14.76, p<2.2E16; PC RCoT vs. KCIT, t=12.87, p<2.2E16). We found similar results with FCI and RFCI, although by only a small margin; 3 of the 4 comparisons fell below the Bonferonni corrected threshold of 0.05/6 and the other comparison fell below the uncorrected threshold of 0.05 (FCI RCIT vs. KCIT, t=2.00, p=0.047; FCI RCoT vs. KCIT t=2.96, p=0.0034; RFCI RCIT vs. KCIT, t=3.56, p=4.5E4; RFCI RCoT vs. KCIT, t=2.80, p=0.0055). All algorithms with any of the kernel-based tests outperformed the same algorithms with FZT by a large margin (p<7E14 in all cases). Finally, the run time results in Figure 3b show that the CCD algorithms run with RCIT and RCoT complete at least 13 times faster on average than those run with KCIT. We conclude that both RCIT and RCoT help CCD algorithms at least match the performance of the same algorithms run with KCIT, but RCIT and RCoT do so within a much shorter time frame than KCIT.
### Figure 3
Results of CCD algorithms as evaluated by mean (a) SHD and (b) run times. The CCD algorithms run with KCIT perform comparably (or even slightly worse) to those run with RCIT and RCoT in (a). Run times in (b) show that the CCD algorithms run with RCIT and RCoT complete at least 13 times faster on average than those with KCIT. Error bars denote 95 % confidence intervals of the mean.
### 7.5 Real data
We finally ran PC, FCI and RFCI using RCIT, RCoT, KCIT and FZT at α=0.05 on a publicly available longitudinal dataset from the Cognition and Aging USA (CogUSA) study [44], where scientists measured the cognition of men and women above 50 years of age. The dataset contains 815 samples, 18 variables and two waves (thus 18/2=9 variables in each wave) separated by two years after some data cleaning.[5] Note that we do not have access to a gold standard solution set in this case. However, we can utilize the time information in the dataset to detect false positive ancestral relations directed backwards in time.
We ran the CCD algorithms on 30 bootstrapped datasets. Results are summarized in Figure 4. Comparisons with PC did not reach the Bonferonni level among the kernel-based tests, although PC run with either RCIT or RCoT yielded fewer false positive ancestral relations on average than PC run with KCIT near an uncorrected level of 0.05 (PC RCIT vs. KCIT, t=2.76, p=9.85E3; PC RCoT vs. KCIT, t=1.99, p=0.056). However, FCI and RFCI run with either RCIT or RCoT performed better than those run with KCIT at a Bonferroni corrected level of 0.05/6 (FCI RCIT vs. KCIT, t=29.57, p<2.2E16; FCI RCoT vs. KCIT, t=17.41, p<2.2E16; RFCI RCIT vs. KCIT, t=6.50, p=4.13E7; RFCI RCoT vs. KCIT, t=7.39, p=3.85E8). The CCD algorithms run with FZT also gave inconsistent results; PC run with FZT performed the best on average, but FCI and RFCI run with FZT also performed second from the worst. Here, we should trust the outputs of FCI and RFCI more strongly than those of PC, since both FCI and RFCI allow latent common causes and selection bias which often exist in real data. Next, CCD algorithms run with RCIT performed comparably to those run with RCoT (PC RCIT vs. RCoT, t=1.05, p=0.301; FCI RCIT vs. RCoT, t=1.54, p=0.134; RFCI RCIT vs. RCoT, t=0.89, p=0.380). We finally report that the CCD algorithms run with RCIT and RCoT complete at least 40 times faster on average than those run with KCIT (Figure 4b). We conclude that CCD algorithms run with either RCIT or RCoT perform at least as well as those run with KCIT on this real dataset but with large reductions run time.
### Figure 4
Results of CCD algorithms as evaluated on real longitudinal data. Part (a) displays mean counts of the number of ancestral relations directed backwards time. We do not display 95 % confidence intervals when we computed a standard error of zero. Part (b) summarizes the mean run times.
## 8 Conclusion
We developed two statistical tests called RCIT and RCoT for fast non-parametric CI testing. Both RCIT and RCoT approximate KCIT by sampling Fourier features. Moreover, the proposed tests return p-values orders of magnitude faster than KCIT in the large sample size setting. RCoT in particular also has a better calibrated null distribution than KCIT especially with larger conditioning set sizes. In causal graph discovery, RCIT and RCoT help CCD algorithms recover graphical structures at least as accurately as KCIT but, most importantly, also allow the algorithms to complete in a much shorter time frame. We believe that the speedups provided by RCIT and RCoT will make non-parametric causal discovery more accessible to scientists who wish to apply CCD algorithms to their datasets.
Note that RCIT and RCoT may estimate the null distribution more accurately than KCIT for multiple reasons. First, RCIT and RCoT both assess for non-vanishing covariances across a smaller set of functions than KCIT. This in turn allows the two tests to more easily estimate the null distribution than KCIT because KCIT must deal with all of the functions in the associated RKHS. Second, both RCIT and RCoT utilize more advanced methods of estimating of the null distribution than KCIT. RCIT and RCoT more specifically utilize the Lindsay-Pilla-Basak method as explained in Section 6.2 as opposed to matching the first two moments of a gamma distribution. Third, RCoT in particular utilizes the variable set X rather than X¨ which allows for lower dimensional inferences as explained in Section 6.3. RCIT and RCoT thus take advantage of new technologies and additional structure inherent within real data in order to achieve better control of the Type I error rate.
Funding source: National Human Genome Research Institute
Award Identifier / Grant number: U54HG008540
Funding source: U.S. National Library of Medicine
Award Identifier / Grant number: T15LM007059
Award Identifier / Grant number: R01LM012095
Funding statement: Research reported in this publication was supported by grant U54HG008540 awarded by the National Human Genome Research Institute through funds provided by the trans-NIH Big Data to Knowledge initiative. The research was also supported by the National Library of Medicine of the National Institutes of Health under award numbers T15LM007059 and R01LM012095.
# Acknowledgment
The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.
## Appendix A
### A.1 CLT for sample covariance
We will prove the central limit theorem (CLT) for the sample covariance matrix. We first have the following sample covariance matrices with known and unknown expectation vector, respectively:
(42)C¨=1ni=1n[XiE(X)][XiE(X)]T,Cˆ=1n1i=1n[XiEˆ(X)][XiEˆ(X)]T.
Now observe that we may write:
(43)(n1)Cˆ=i=1n[XiE(X)(Eˆ(X)E(X))][XiE(X)(Eˆ(X)E(X))]T=i=1n(XiE(X))(XiE(X))T+n(Eˆ(X)E(X))(Eˆ(X)E(X))T2(Eˆ(X)E(X))i=1n(XiE(X))T=nC¨n(Eˆ(X)E(X))(Eˆ(X)E(X))T
It follows that:
(44)n(CˆC)=n(n1n1CˆC)=n(nn1C¨nn1(Eˆ(X)E(X))(Eˆ(X)E(X))TC)=nnn1C¨nnn1(Eˆ(X)E(X))(Eˆ(X)E(X))TnC=nnn1C¨nnn1(Eˆ(X)E(X))(Eˆ(X)E(X))Tn1n1nC=nnn1(C¨C)nnn1(Eˆ(X)E(X))(Eˆ(X)E(X))T+nn1C
We are now ready to state the result:
### Lemma 1.
LetX1,,Xnrefer to a sequence of i. i. d. random k-vectors. Denote the expectation vector and covariance matrix ofX1asμ1andC1, respectively. Assume thatC˘1=Cov[vu((X1μ1)(X1μ1)T)]is positive definite, wherevu(M)denotes the vectorization of the upper triangular portion of a real symmetric matrix M. Then, we have:
(45)n(vu(Cˆ)vu(C1))dN(0,C˘1).
### Proof.
Consider the quantity aT[n(vu(Cˆ)vu(C1))]=n(aTvu(Cˆ)aTvu(C1)) where aRk(k+1)/2{0}. Note that aTvu[(X1μ1)(X1μ1)T], …, aTvu[(Xnμ1)(Xnμ1)T] is a sequence of i. i. d. random variables with expectation aTvu(C1) and variance aTC˘1a. Moreover observe that C˘1< because C˘1 is positive definite. We can therefore apply the univariate central limit theorem to conclude that:
(46)n(aTvu(C¨1)aTvu(C1))dN(0,aTC˘1a),
where C¨1=1ni=1n(Xiμ1)(Xiμ1)T. We would however like to claim that:
(47)n(aTvu(Cˆ)aTvu(C1))dN(0,aTC˘1a).
In order to prove this, we use 44 and set:
(48)n(aTvu(Cˆ)aTvu(C1))=aTAn+aTBn,
where we have:
(49)An=nn1n(vu(C¨1)vu(C)),Bn=nn1vu(C1)nnn1vu[(Eˆ(X)μ1)(Eˆ(X)μ1)T].
We already know from 46 that:
(50)n(aTvu(C¨1)aTvu(C1))dN(0,aTC˘1a).
Therefore, so does aTAn by Slutsky’s lemma, when we view the sequence of constants nn1 as a sequence of random variables. For aTBn, we know that:
(51)n(aTEˆ(X)aTμ1)dN(0,aTC1a),
by viewing aTX1,,aTXn as a sequence of random variables, noting that E(X1X1T)< because C˘1 is positive definite and then applying the univariate central limit theorem. We thus have aTBnp0. We may then invoke Slutsky’s lemma again for aTAn+aTBn and claim that:
(52)n(aTvu(Cˆ)aTvu(C1))dN(0,aTC˘1a).
We conclude the lemma by invoking the Cramer-Wold device. □
### A.2 Kernel conditional correlation test (KCoT) results
We compared KCoT against RCIT, RCoT and KCIT. We report the results in Figure 5. All tests perform comparably as a function of sample size (Figures 5a and 5c). However, KCoT performs better than KCIT and underperforms RCoT and RCIT as a function of conditioning set size. In particular, RCIT and RCoT obtain smaller KS-statistic values as a function of the conditioning set size (Figure 5b). KCIT and KCoT also obtain larger AUPC values with an increasing conditioning set size because they fail to maintain a uniform distribution under the null (Figure 5d; we again permuted the values of X for Figure 5e). We therefore conclude that RCoT and RCIT control their Type I error rates better than KCIT and KCoT even with large conditioning set sizes while maintaining power.
### Figure 5
Results comparing against KCoT. Sub-figures (a) and (b) correspond to the KS-statistic values as a function of sample size and dimensions, respectively. Notice that RCIT and RCoT have progressively smaller KS-statistic values than both KCIT and KCoT with increasing dimensions. Next, sub-figures (c) and (d) correspond to the AUPC values also as a function of sample size and dimensions, respectively. KCIT and KCoT claim the largest AUPC values because both tests fail to control the Type I error rate well, as summarized by the large KS-statistic values in sub-figure (e) obtained after permuting the values of X.
### Figure 6
Results of the same KS statistic experiments as in Section 7.2 except comparing RCIT and RCoT against S-Perm and S-Perm. Subfigures (a) and (b) again vary as a function of sample size, while subfigures (c) and (d) vary as a function of conditioning set size. RCIT and RCoT are faster than the permutation tests but yield comparable average KS statistic values.
### Figure 7
Results of the same AUPC experiments as in Section 7.3 except comparing RCIT and RCoT against S-Perm and S-Perm. Subfigures (a) and (b) again vary as a function of sample size, while subfigures (c) and (d) vary as a function of conditioning set size. RCIT and RCoT are much faster than the permutation tests but yield comparable accuracy.
### A.3 Comparisons against permutation
We also compared RCIT and RCoT against permutation CI tests. Here, we estimated the null distribution of S and S with permutations and call the resultant CI tests S-Perm and S-Perm, respectively. The permutation tests specifically involve permuting the residuals of the random Fourier features of Y one thousand times in order to estimate the null distribution. We have summarized the Type I error rate results in Figure 6 and the AUPC results in Figure 7. We ran the sample size experiments up to only ten thousand samples due to the long run times of S-Perm and S-Perm. We found that RCIT and RCoT perform similarly to the permutation tests but with significantly reduced average run time.
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44. McArdle J, Rodgers W, Willis R. Cognition and aging in the usa (cogusa), 2007–2009, 2015.Search in Google Scholar | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9455341696739197, "perplexity": 2093.9699414383067}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320299894.32/warc/CC-MAIN-20220129002459-20220129032459-00180.warc.gz"} |
http://math.stackexchange.com/questions/8624/how-the-letter-pi-came-in-mathematics/8625 | # How the letter 'pi' came in mathematics?
Is $\pi$ mean $\cfrac {22}{7}$? If so, how circumference of the circle will be $2\times \cfrac{22}7$ if radius is $1$.
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-1 since it is obvious the person asking didn't search anything on the subject. – Djaian Nov 2 '10 at 15:53
@Djaian, is it necessary that people must search before posting a question? – anon Nov 2 '10 at 18:21
@anon: They can do better than this. @P. Gangamohan: The History of Mathematical Symbols – endolith May 28 '11 at 4:27
The question has 4 answers with a total 14 upvotes , maybe we should consider upvoting the question. – Tomarinator May 20 '12 at 14:08
@SauravTomar: I see no reason to upvote. In fact I did not downvote yet, so now I did. – TMM May 20 '12 at 14:15
According to wikipedia
"The constant is named "$\pi$" because "$\pi$" is the first letter of the Greek word περίμετρος (perimeter)..."
Questions that can be avoided with a quick look at a wikipedia article are typically frowned upon here.
-
Odd, I've seen "perimetron" but not "perimetros". Maybe somebody fluent in Greek should chime in. – Ｊ. Ｍ. Nov 2 '10 at 9:43
The Greek dictionaries in Perseus prefer περίμετρον. – Yuval Filmus Nov 2 '10 at 20:54
In any case, I think the first letter is the one that matters :D – crasic Nov 2 '10 at 21:38
In modern Greek, it's περίμετρος. For the word used in ancient times, I guess Perseus should be trusted more. – ypercube Mar 14 '11 at 14:58
From Pi: A Biography of the World's Most Mysterious Number by Posamentier and Lehmann:
According to the well-known mathematics historian Florian Cajori (1859-1930), the symbol $\pi$ was first used in mathematics by William Oughtred (1575-1660) in 1652 when he referred to the ratio of the circumference of a circle to its diameter as $\frac{\pi}{\delta}$, where $\pi$ represented the periphery of a circle and $\delta$ represented the diameter...
In 1706 William Jones (1675-1749) published his book Synopsis palmariorum matheseos, in which he used $\pi$ to represent the ratio of the circumference of a circle to its diameter. This is believed to have been the first time that $\pi$ was used as it is defined today... But not until he [Leonhard Euler - A.R.] used the symbol $\pi$ in his famous book Introductio in analysin infinitorum did the use of $\pi$ to represent the ratio of the circumference of a circle to its diameter become widespread.
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Oh, I've answered the question in the title rather than those in the body. – Andrey Rekalo Nov 2 '10 at 10:45
$\pi \neq \frac{22}{7}$. First of all $\frac{22}{7}$ is just an approximate value of $\pi$. Note that $\pi$ is irrational so it cannot be expressed in the form $\frac{p}{q}$ where $p$ and $q \neq 0$ are integers.
“It is clear that I employ here the letter $\pi$ to indicate the number of Ludolf of Kuelen, $3.14159265$, etc.” By this time, Euler has been using p to denote that constant for several years, but the convention will take many years more before it is universally adopted. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9389517903327942, "perplexity": 910.3895994037305}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394010564986/warc/CC-MAIN-20140305090924-00092-ip-10-183-142-35.ec2.internal.warc.gz"} |
http://clay6.com/qa/457/assume-x-y-z-w-and-p-are-matrices-of-order-2-n-3-k-2-p-n-3-and-p-k-respecti | Browse Questions
Home >> CBSE XII >> Math >> Matrices
Assume $X, Y, Z, W$ and $P$ are matrices of order $2\times n, 3\times k, 2\times p, n\times 3$ and $p\times k$, respectively. The restriction on $n, p, k$ so that $PY + WY$ will be defined are:
\begin{array}{1 1} (A) \quad k = 3, p = n & (B) \quad k \text{ is arbitrary}, p = 2 \\ (C) \quad p \text{ is arbitrary}, k = 3 & (D) \quad k = 2, p = 3 \end{array}
Toolbox:
• Multiplication of two matrices is defined only if the number of columns of the left matrix is the same as the number of rows of the right matrix.
PY can be defined if the number of columns in $P=$ the number of rows in $Y$,
We know that the order of matrix P is $p\times k$ and the order of matrix Y is $3\times k.$ Therefore, for PY to be defined, $k$ must be equal to $3$ and the order of PY is $p\times k.$
WY can be defined if the number of columns in $W=$ the number of rows in $Y$,
We know that the order of matrix W is $n\times 3$ and the order of matrix Y is $3\times k$. The number of columns in $W =$ number of rows in $Y$ = 3. and the order of WY is $n\times k.$
Matrix $PY+WY$ is defined only when the $PY$ and $WY$ are of the same order.
Since the order of PY=$p\times k$ and the order of WY=$n\times k$, for $PY+WY$ to be define, $p$ must be equal to $n$.
Hence $PY+WY$ is defined when $k=3$,$p=n$
Thus correct option is (A).
edited Mar 1, 2013 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9868162870407104, "perplexity": 121.96000354220232}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988720062.52/warc/CC-MAIN-20161020183840-00374-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/symmetry-and-hamiltonian.134470/ | # Symmetry and Hamiltonian
1. Oct 2, 2006
### Wiemster
Why do operators representing some symmetry commute with the Hamiltonian?
2. Oct 2, 2006
### StatMechGuy
This is a very deep question, and worthy of discussion. There are a few ways that you can look at this, but I'm going to look at it from the standpoint of Noether's Theorem, which states, more or less, that every continuous symmetry corresponds to a conserved quantity of the motion. This also applies to classical systems, or field theories, since the proof (as I've seen it) only relies on the presence of a lagrangian for the system and a generator of some transformation that leaves the action invariant.
As a specific example, let's look at the case of angular momentum, because it's the simplest non-trivial one. In the case of the coulomb potential, for example, the hamiltonian is given by
$$H = \frak{\mathbf{p}^2}{2m} + \frac{q_1 q_2}{r}$$
The hamiltonian above is invariant under rotation (let's pretend we're in rectangular coordinates for now), and to see this, notice simply that $$\mathbf{p}^2$$ and $$1/r$$ are both functions of the magnitude of the vector, which is invariant under rotation, as we know from linear algebra. Thus, any generator of rotation must correspond to a conserved quantity. In both the classical and quantum case, this is the angular momentum, as we interpret the angular momentum as the generator of rotations.
So in this case we find that the angular momentum is a conserved quantity. However, because of the structure of angular momentum, we have to choose $$L^2$$ and one of the angular momentum components, usually $$L_z$$, to be conserved.
Why do these commute with the hamiltonian? Well, we know that, for example, $$L_z$$ must be a conserved quantity, so the time evolution of its state kets must be zero. From the Heisenberg equations of motion, we have that
$$\dot{L_z} = \frac{i}{\hbar} [ L_z, H ]$$
Knowing that the time evolution of the operator is zero, we have that the z component of the angular momentum commutes with the hamiltonian. A similar exercise arises for $$L^2$$. This is also why the angular momentum eigenstates and energy eigenstates are product states of the two.
The fundamental physics to keep in mind, however, is Noether's Theorem, and that if you find some symmetry transformation of the hamiltonian/lagrangian that leaves the equations of motion invariant, then you've managed to find a conserved quantity.
Last edited: Oct 2, 2006
3. Oct 2, 2006
### actionintegral
My answer seems lame now! I was going to say that symmetries of the hamiltonian are eigenstates and so they are just multiplication by a constant...
4. Oct 2, 2006
### Wiemster
Okay, so basically quantummechanically symmetries give rise to conserved operators (as classically they also give rise to conserved quantities) and via the Heisenberg equation of motion this implies the operator commutes with the Hamiltonian.
Thanks a lot! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9896712303161621, "perplexity": 200.08523218419637}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676592150.47/warc/CC-MAIN-20180721012433-20180721032433-00393.warc.gz"} |
https://cs.paperswithcode.com/paper/elastic-properties-of-isotropic-discrete | # Elastic properties of isotropic discrete systems: Connections between geometric structure and Poisson's ratio
25 Dec 2019 · Eliáš Jan ·
The use of discrete material representation in numerical models is advantageous due to the straightforward way it takes into account material heterogeneity and randomness, and the discrete and orientated nature of cracks. Unfortunately, it also restricts the macroscopic Poisson's ratio and therefore narrows its applicability... The paper studies the Poisson's ratio of a discrete model analytically. It derives theoretical limits for cases where the geometry of the model is completely arbitrary, but isotropic in the statistical sense. It is shown that the widest limits are obtained for models where normal directions of contacts between discrete units are parallel with the vectors connecting these units. Any deviation from parallelism causes the limits to shrink. A comparison of the derived equations to the results of the actual numerical model is presented. It shows relatively large deviations from the theory because the fundamental assumptions behind the theoretical derivations are largely violated in systems with complex geometry. The real shrinking of the Poisson's ratio limit is less severe compared to that which is theoretically derived. read more
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https://www.physicsforums.com/threads/cmb-spherical-harmonics-and-rotational-invariance.813976/ | # CMB , Spherical Harmonics and Rotational Invariance
1. May 15, 2015
### center o bass
In Dodelson's "Modern Cosmology" on p.241 he states that the $a_{lm}$-s -- for a given $l$-- corresponding to a spherical harmonic expansion of the photon-temperature fluctuations, are drawn from the same probability distribution regardless of the value of $m$. Dodelson does not explain this any further, but other authors claim that it is due to the fact that $m$ somehow corresponds to an orientation and this should not matter as the universe is (believed to be) statistically rotational invariant.
Question:
What is the precise property of the spherical harmonic $Y_l^m$ for a given $l$ that justifies this claim?
2. May 15, 2015
### ChrisVer
That $Y_l^m \sim e^{i m \phi}$?
3. May 15, 2015
### Chalnoth
The $Y_\ell^m$ functions for a given $\ell$ can be morphed into one another through rotations in any direction. That is, if you rotate the coordinate system, the resulting $a_{\ell m}$ parameters are a linear combination of the pre-rotated $a_{\ell m}$ parameters. During this rotation, only the $a_{\ell m}$ values with the same $\ell$ are mixed.
4. May 18, 2015
### center o bass
Thanks for the reply! From what you've now said, how would one go on to argue (fairly rigorously) that the $a_{lm}$-s for a given $l$ must be drawn from the same probability distributions?
5. May 18, 2015
### Chalnoth
That's the assumption of isotropy. As the different coefficients for the same $\ell$ are just rotations of one another, assuming isotropy requires that they all have the same probability distribution (provided you make use of the appropriate normalization for the $Y_l^m$ functions).
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https://math.stackexchange.com/questions/1108127/fermat-numbers-are-pairwise-coprime-implies-infinitely-many-primes | # Fermat Numbers are pairwise coprime $\implies$ infinitely many primes
Given that the Fermat numbers $F_m$ are pairwise relatively prime.
Prove that there are infinitely many primes.
• So, each Fermat number is either prime or is divisible by a unique prime number which does not divide any other Fermat number – lab bhattacharjee Jan 17 '15 at 18:45
• Let $p_m$ be the smallest prime divisor of $F_m$. Since the $F_k$ are pairwise relatively prime, the $p_m$ are distinct. So the set of all $p_m$ is infinite. (The relative primality is not hard to prove, but it looks as if you are not expected to give a proof.) – André Nicolas Jan 17 '15 at 18:46
• @AndréNicolas If I wanted to prove its primality how would I be able to prove it? – Alexis Dailey Jan 17 '15 at 18:48
• Let $m\lt n$. Note that $2^n=(2^m)^{n-m}$ So if $x=F_m$ then $F_n=(x-1)^{2^{n-m}}+1$. Using the binomial theorem to expand, we find that $x$ divides $F_{n}-2$. So any divisor $d$ of $F_m$ divides $F_n-2$. If $d$ also divides $F_n$, then $d$ divides $2$. But $d$ is odd, so $d=1$. (You will find variant proofs on MSE, the question has been asked repeatedly. Answers are likely to be less terse than this comment.) – André Nicolas Jan 17 '15 at 19:05
• @AndréNicolas. For answers on MSE see here. So the OP already knows it. – Dietrich Burde Jan 17 '15 at 19:30 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8375709056854248, "perplexity": 192.7584000908472}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232262029.97/warc/CC-MAIN-20190527065651-20190527091651-00387.warc.gz"} |
https://www.physicsforums.com/threads/schroedinger-equation.658787/ | # Schrödinger equation
1. Dec 13, 2012
### Mr-T
Does the Schrödinger equation completely neglect the uncertainty principle? If so, wouldn't this imply that our intensity distribution has its own probability distribution?
2. Dec 14, 2012
### tom.stoer
The Schrödinger equation predicts the wave function with certainty; but from this wave function the uncertainties of observables can be derived exactly
3. Dec 14, 2012
### Staff: Mentor
The momentum-space wave function $\Phi(p,t)$ is basically the Fourier transform of the position-space wave function $\Psi(x,t)$. The uncertainty principle comes from the properties of Fourier transforms. Any pair of functions that are related by Fourier transforms has a similar uncertainty principle.
4. Dec 14, 2012
### Mr-T
I understand what both of you are saying and I appreciate the replies.
In the Schrödinger equation we input values for energy/mass assuming we know with 100% certainty what these values for energy/mass are. Due to the input of these values is where my question holds its regards.
5. Dec 14, 2012
### tom.stoer
No
The input is a wave function, the output is a wave function at a later time. This predicts with certainty that a system will be in a state A' at time t' > t provided that it was in state A at time t; A is specified by a wave function or a state vector |A>.
In case of the time-indep. SE the input is not energy, the input is nothing! The outputs are a) the allowed energy eigenvalues and b) the corresponding eigenfunctions. The SE does not tell you in which state the system is, in only tells you what the allowed state are
6. Dec 14, 2012
### Mr-T
If you are not inputting any information into the T-I SE then how do you know what particle it is talking about?!
7. Dec 14, 2012
### Jorriss
Do you mean you specify a potential, then solve the SE equation for a given potential? Or you plug in the values of the eigenvalues?
8. Dec 14, 2012
### dextercioby
The remark by Tom is an overstatement, an exaggeration. The input is the specific form of the Hamiltonian in terms of fundamental observables such as position, momentum, spin.
9. Dec 14, 2012
### Mr-T
If all direct observables have some uncertainty, won't this mess up our intensity distribution even more than the fouriers already do?
10. Dec 14, 2012
### Staff: Mentor
OK, I think I see where you were going with your original question...
In the time-dependent Schrodinger equation $H\Psi=E\Psi$ the Hamiltonian is written as if all of its inputs were exactly known. For example, if we're dealing with two charged particles, there will be a $\frac{1}{r1-r2}$ term somewhere in it, where r1 and r2 are the positions of the two particles. You should read that as saying not that the two particles are at those exact positions, but rather that if they were in those positions that would be the exact distance between them. The uncertainty principle doesn't stop us from talking about how things would be if we knew exactly where a particle was, it just forbids us from knowing exactly where it is.
Once I have the Hamiltonian written down, I solve Schrodinger's equation; and as tom.stoer said in #2, the uncertainty principle is inherent in the ψ that comes out.
11. Dec 14, 2012
### Mr-T
Ahh yes, talking in this fashion resolves my concerns.
Thank you nug
12. Dec 15, 2012
### andrien
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http://nuit-blanche.blogspot.com/2008/04/compressed-sensing-i-screwed-up.html | ## Page Views on Nuit Blanche since July 2010
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## Wednesday, April 02, 2008
### Compressed Sensing: I screwed up, Boosting in this case is not Optimal
This is the risk of an exercise like this: blogging and trying things at the same time. I think I screwed up. Thanks to a reader who mentioned he went through a similar exercise before, I went back and checked if the results of my implementation of the Subspace Pursuit algorithm and that of Wei Dai and Olgica Milenkovic were the same. Well they don't seem to give the same results. The algorithm provided by Wei Dai is in fact pretty often converging with very high accuracy whereas my implementation works very well but with a larger number of measurements. In other words, they don't seem to be the same. The idea of boosting the Subspace Pursuit algorithm was born out of the concern that one could improve the result of the Subspace Pursuit and indeed in MMA12, the boosting works well for a "deficient" subspace pursuit algorithm (mine). When switching from that deficient algorithm to that of Wei Dai and Olgica Milenkovic then boosting fails because the solution obtained by the Subspace Pursuit is already optimal.
The take away lesson from this is: Re-weighted Lp can always help suboptimal guesses :-) and I am not quite sure it cannot be salvaged. I'll put a warning on the associated entries. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8343706130981445, "perplexity": 856.7916628793477}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207929012.53/warc/CC-MAIN-20150521113209-00216-ip-10-180-206-219.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/318067/proof-strategy-for-intersecting-lines/318092 | # Proof Strategy for intersecting lines.
Given $n$ (pairwise) nonparallel lines in $\mathbb{R}^2$. $\lbrace L_1,\ldots,L_n\rbrace$. The intersection of any two lines belongs to a third line in our set of lines. I would like to show that $\cap L_i\not = \emptyset.$
My best ideas suppose there is more than one point of intersection and to define a point of intersection of minimal distance (motivated by Kelly's proof of the Sylvester-Gallai Theorem http://en.wikipedia.org/wiki/Sylvester%E2%80%93Gallai_theorem#Kelly.27s_proof ) from our original point of intersection then use the slopes of the lines to show that there is no such closest minimal intersection point by creating new points of intersection. However, there seems to be way too many cases.
Are there any nicer approaches to this problem? I was told this is an application of Sylvester-Gallai Theorem, but I don't see the connection. Any insight would be humbly appreciated.
-
I think you're on the right track. A possible proof is as follows:
By assumption every point of intersection must have at least three lines going through it. Consider all pairs of points of intersection and lines, and choose the pair such that the perpendicular distance from the point to the line is minimum. Let the point be A and the line closest to it be lclose, and three lines through A be l1,l2,l3 clockwise as shown in the diagram attached.
Now there are two possible cases for the intersection point C of lclose with l2 we consider.
1. Firstly, if it is anti-clockwise of the perpendicular bisector, then C is closer to l1 than A is to lclose, which is a contradiction. This is the case shown in the diagram.
2. Otherwise, if it is on the perpendicular or clockwise of the perpendicular bisector, then C is closer to l3 than A is to lclose, which is again a contradiction.
Edit: Note that this proof relies on the fact that the lines are pairwise nonparallel, for otherwise if lclose and l1 were parallel the reasoning in the first case would not hold. As for what people mean by the question being an application of the Sylvester-Gallai Theorem, I think that the question is simply the projective dual of the Sylvester-Gallai Theorem.
-
Is Kelly's proof restricted to the plane? – user62384 Mar 1 '13 at 18:54
@123kid If you're referring to my proof above, the proof should be valid for all spaces where nonparallel lines would intersect at one point and you can define a closest pair of points - not sure exactly what kind of space you're looking at though! – Vincent Tjeng Mar 2 '13 at 2:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9256961941719055, "perplexity": 211.556224689174}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783402699.36/warc/CC-MAIN-20160624155002-00194-ip-10-164-35-72.ec2.internal.warc.gz"} |
http://complexzeta.wordpress.com/2007/09/12/a-mathematicians-take-on-challah-braiding/ | For many years, my mother has baked a challah nearly every Friday for Shabbat. Occasionally, however, she asks me to do some portion of the challah making, possibly including the braiding. For reasons we’ll see later, I don’t like her braiding algorithm. This post includes several algorithms, written in a way that someone who knows what a braid group is can understand. (I never had much success following those series of diagrams I sometimes see; I always wished someone would write out the braiding process in terms of generators of the braid group, so that’s what I’m going to do here after I give the preliminary definitions.)
Wikipedia’s page on braid groups has lots of interesting things, so I’ll only write a few essential points here, leaving the reader to explore Wikipedia at eir leisure. I’m finding it a bit tricky to give a good informal definition of braids, so I’ll just assume that my reader knows roughly what a braid is and skip to the formal definition.
The braid group on $n$ strands is the group $B_n=\langle a_1,\ldots,a_{n-1}\mid a_ia_{i+1}a_i=a_{i+1}a_ia_{i+1} \text{ for } 1\le i\le n-2, \ a_ia_j=a_ja_i \text{ for } |i-j|>1\rangle$. In terms of actually playing with braid strands, $a_i$ means interchanging strand $i$ with strand $i+1$ by putting strand $i$ over strand $i+1$. It is pretty simple to see that these generators do indeed induce all possible braids (although I haven’t yet said what a braid is), and that the relations ought to hold. Now, of course, a braid is an element of the braid group.
The braid groups become rather complicated quite quickly. While $B_0=B_1=0$ and $B_2=\mathbb{Z}$, already $B_3$ is nonabelian, and it’s isomorphic to the fundamental group of the complement of a trefoil knot in $\mathbb{R}^3$.
Note also that there is a natural homomorphism $\pi:B_n\to S_n$ that tells us where the strand that started in the $i^\text{th}$ position ends up.
Okay, now it’s time for some challah braiding algorithms. My mother’s usual challah has four strands on the bottom and three on the top. The algorithm for the top braid is pretty natural: $(a_1a_2^{-1})^n$, where $n$ is decided by the length of the dough ropes.
I’m more concerned about the element of $B_4$ used for the bottom braid. She uses $(a_1a_2^{-1}a_3^{-1}a_2)^n$. If $n=1$, we have $\pi(a_1a_2^{-1}a_3^{-1}a_2)=(142)(3)$ (in cycle notation). This is already bad news to me: one step of the algorithm produces a single fixed point! I think one step of the algorithm ought to give an $n$-cycle (here a 4-cycle) or else a pure braid (i.e. a braid in the kernel of $\pi$). But it gets worse: the strand that starts in position 3 has no undercrossings. So when we’re done, it sits on top of every other strand.
It turns out not to be so bad because the three-strand braid sits on top of the four-strand braid, so the central portion of the four-strand braid is not visible in the finished bread. But aesthetically (and mathematically), this feels like a serious flaw to me.
Fortunately, I found an alternate algorithm for four-strand braiding that lacks these flaws: $(a_2a_1a_3^{-1})^n$. If $n=1$, $\pi(a_2a_1a_3^{-1})=(1243)$, which is nice. Furthermore, every strand has both overcrossings and undercrossings. So this is my new preferred braid.
Sometimes, however, it is preferable to braid with six strands. There was an article in the newspaper that explained how to do it, but I was unable to follow it. Fortunately, I found a YouTube video that shows someone doing it (possibly the same way; I can’t tell). I was able to transcribe this method in terms of generators of the braid group. However, I’m not quite sure where it is supposed to end, so my braid may be slightly different from the one shown in the video. The braid is the video is $(e^{-1}d^{-1}c^{-1}b^{-1}a^{-1}(bcdeabd^{-1}c^{-1}b^{-1}a^{-1}e^{-1}d^{-1})^n$, except that it might stop somewhere in the middle of the $(\cdot)^n$. I don’t really want to calculate $\pi$ of this braid (computations like this have never been that easy for me), but I would guess that it is a 6-cycle if it stops at an appropriate moment. (Also, it’s not as complicated as the formula would make it seem, since there’s a lot of stuff like moving the strand on the right all the way over to the left, and it takes a lot of generators to express that, even though it’s not complicated when you’re actually braiding dough.) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 28, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8575234413146973, "perplexity": 353.9816908008043}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1386163046801/warc/CC-MAIN-20131204131726-00060-ip-10-33-133-15.ec2.internal.warc.gz"} |
https://research.utwente.nl/en/publications/on-the-convergence-to-stationarity-of-birth-death-processes | # On the convergence to stationarity of birth-death processes
P. Coolen-Schrijner, Erik A. van Doorn
Research output: Book/ReportReportOther research output
## Abstract
Taking up a recent proposal by Stadje and Parthasarathy in the \linebreak[4] setting of the many-server Poisson queue, we consider the integral \linebreak[4] $\int_0^{\infty}[\lim_{u\to\infty} E(X(u))-E(X(t))]dt$ as a measure of the speed of convergence towards stationarity of the process $\{X(t), t \geq 0\}$, and evaluate the integral explicitly in terms of the parameters of the process in the case that $\{X(t), t \geq 0\}$ is an ergodic birth-death process on $\{0,1,\ldots\}$ starting in 0. We also discuss the discrete-time counterpart of this result, and examine some specific examples.
Original language Undefined Enschede University of Twente, Department of Applied Mathematics Published - 2000
### Publication series
Name Memorandum / Department of Applied Mathematics Department of Applied Mathematics, University of Twente 1554 0169-2690
• MSC-60J80
• IR-65741
• EWI-3374 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9805446267127991, "perplexity": 1250.8339235497976}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141186414.7/warc/CC-MAIN-20201126030729-20201126060729-00116.warc.gz"} |
https://zbmath.org/?q=ut%3Ahighest+degree+of+precision+pt%3Ar+py%3A1991 | ×
## Quadrature formulae for entire functions with 2-periodic data.(English)Zbl 0753.65019
The author studies quadrature formulae with equidistant nodes involving 2-periodic data of not necessarily consecutive derivatives. The question of existence and uniqueness of such formulae which have highest degree of precision with respect to entire functions of exponential type is considered.
Quadrature formulae of highest degree of precision are obtained without knowing the corresponding interpolation process. A representation of the remainder for functions belonging to a certain Sobolev space is given. Examples are considered.
Reviewer: D.Acu (Sibiu) | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8092982769012451, "perplexity": 527.0593513772209}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710719.4/warc/CC-MAIN-20221130024541-20221130054541-00781.warc.gz"} |
https://repository.uantwerpen.be/link/irua/99397 | Publication
Title
Dynamic response of artificial bipolar molecules
Author
Abstract
We calculate the equilibrium properties and the dynamic response of two vertically coupled circular quantum dots populated by particles of different electrical charge sign, i.e., electrons and holes. The equilibrium density profiles are obtained and used to compute the frequencies and oscillator strengths of magnetoplasma excitations. We find a strong coupling between the modes derived from the center-of-mass modes of the individual dots which leads to an anticrossing with a pronounced oscillator strength transfer from the "acoustic" to the "optical" branch. Also, due to the breaking of the generalized Kohn theorem a number of other than center-of-mass modes are excited whose oscillator strengths, however, are rather weak.
Language
English
Source (journal)
Physical review : B : condensed matter and materials physics. - Lancaster, Pa, 1998 - 2015
Publication
Lancaster, Pa : 2002
ISSN
1098-0121 [print]
1550-235X [online]
Volume/pages
66:7(2002), 9 p.
Article Reference
075311
ISI
000177969800103
Medium
E-only publicatie
Full text (Publisher's DOI)
Full text (open access)
UAntwerpen
Faculty/Department Research group Publication type Subject | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9473385214805603, "perplexity": 4057.7997282460897}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934806620.63/warc/CC-MAIN-20171122175744-20171122195744-00691.warc.gz"} |
http://www.gradesaver.com/the-maze-runner/q-and-a/who-came-to-let-thomas-out-of-the-slammer-at-the-end-of-the-day-314589 | # Who came to let Thomas out of the slammer at the end of the day
Who came to let out Thomas out of the slammer at the end of the day | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9965126514434814, "perplexity": 1041.330194944049}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501170614.88/warc/CC-MAIN-20170219104610-00048-ip-10-171-10-108.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/tb-ta.66653/ | # Tb / Ta = ?
1. Mar 9, 2005
### the_d
Tb / Ta = ??????
i have a final question, which is how do i find this. i know what the question is askin i just dont understand how to get the answer
Two planets A and B, where B has twice the
mass of A, orbit the Sun in elliptical orbits.
The semi-major axis of the elliptical orbit of
planet B is two times larger than the semi-
major axis of the elliptical orbit of planet A.
What is the ratio of the orbital period of
planet B to that of planet A?
2. Mar 9, 2005
### dextercioby
HINT:Apply the 3-rd law of Kepler...
Daniel.
3. Mar 9, 2005
### James R
What is the equation which relates the semi-major axis of an orbit to the orbital period? | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9766440987586975, "perplexity": 1185.9944356489873}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934805023.14/warc/CC-MAIN-20171118190229-20171118210229-00397.warc.gz"} |
http://www.physicsforums.com/showthread.php?t=290839 | # Finding magnitude and direction of equilibrant.
by Obama
Tags: direction, equilibrant, magnitude
P: 13 1. The problem statement, all variables and given/known data Find the magnitude and direction of the equilibrant of each of the following foces: a) forces of 32N and 48N acting at an angle of 90 degress to each other b) forces of 16N and 10N acting at an angle of 19 degrees to each other 2. Relevant equations Sine Law, Cosine law, Soh Cah Toa, Pythagorean. 3. The attempt at a solution a) Because there is a 90 degree angle, to find the missing side created to make a triangle with the two forces, I would use the pythagorean theorum. The square root of 32^2+48^2 is 57.7 Newtons. This I know is correct. However, when trying to find the direction, I've become puzzled. Since it is a right angled triangle, I used soh cah toa, in which: tanx=(32/48), x=34 degrees. So the direction would be 34 degrees to 32N, correct? The answer page is 146 to 48 N. I'm not sure if there can be two correct answers, as I have found that 180 subtract my original answer of 34 would give me 146, the "correct" answer. Could someone explain this to me? b) Since there is no right angle, I am using the Cosine law. a^2=10^2 + 16^2-2(10)(16)cos170 (By creating a parallelogram, I found the angle opposite to a is 180-10 degrees). The answer I get is 25.9N. Again, this is correct, and again, I've no clue why my direction is not the same as the answer page. Using the sine law, sinx/16N = sin170/26. The answer is 6.1 degrees to 16N. The back of the book says 174 degrees to 10N. Again, I have found that 180 subtract my answer of 6.1 degrees would give me 174, the correct answer. I have absolutely no idea why I must subtract by 180 to find the answer. Please, anyone care to explain to me? This is part of the Calculus and Vectors course, but since this is vectors, I figured it would be more appropriate to post it in the Physics section,thanks. EDIT: Actually, my teacher advised the physics method is different, so if somone can move this to the Calculus section, you are appreciated.
HW Helper
P: 5,341
There are several ways to add them.
One is to resolve them into components i,j and add the components and resolve the Resulting vector.
The other head to tail addition, which makes the parallelogram like you were doing. It all should come out to the same result.
I am using the Cosine law. a^2=10^2 + 16^2-2(10)(16)cos170
On this one I don't understand your angle.
If they are 19° from each other I would think the angle was 161° if you were adding 1 tail to head of the other.
Related Discussions Introductory Physics Homework 15 Introductory Physics Homework 5 Introductory Physics Homework 1 Introductory Physics Homework 2 Introductory Physics Homework 3 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.882005512714386, "perplexity": 425.31636442155076}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500835822.36/warc/CC-MAIN-20140820021355-00171-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://economics.stackexchange.com/questions/22124/compare-utility-functions?noredirect=1 | # Compare utility functions
I recently joined an econ class. I am so lost on how to prove their equality. As a math standpoint, these are completely different equations. Please help!
u1 (x1, x2) = x1^(2/3) x2^(1/3)
u2 (x1, x2) = 4ln (x1) + 2 ln(x2) +3
• – Herr K. May 24 '18 at 6:30
According to the logical positivist view of decision theory, utility functions are just descriptions of observable behavior and have no intrinsic meaning absent this vantage. In other words, $u(x) > u(y)$ indicates that $x$ would be chosen in favor of $y$, but the magnitude of the difference carries no addition information. If $u(x) = 1$ and $u(y) = 0$ the decision maker prefers $x$; if $u(x) = 10000$ and $u(y)= 0$ the decision maker prefers $x$. In either case, we can draw the same conclusions.
What does this mean with regard to your question? It tells us what we mean by equality of utility functions: we mean that these two different (so no definitionally equal) functions represent the same observable data. Specifically, exactly when $u_1(x) > u_1(y)$ do we also have $u_2(x) > u_2(y)$. As mentioned in the comments, this is precisely when we can compose $u_1$ with a strictly increasing function and pop out $u_2$. (This is because all we care about is the ordering between objects, and strictly increasing functions preserve order).
So, what might our strictly increasing function look like? Well, we can take the log, so that we get $\frac23 ln(x_1) + \frac13 ln(x_2)$ then multiplying by 6 and adding 3 does the trick. Each of these was strictly increasing so too is the composition: $$(x,y) \mapsto 6 (ln(x) + ln(y)) + 3$$ is our desired function. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.880990207195282, "perplexity": 373.05331749851103}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141169606.2/warc/CC-MAIN-20201124000351-20201124030351-00077.warc.gz"} |
https://brilliant.org/problems/can-solve-this-equation-with-simple-method/ | # Can solve this equation with simple method
Calculus Level 3
Find the value of $$x$$ satisfying $$x^x=100$$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9745685458183289, "perplexity": 816.7467979280548}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267861981.50/warc/CC-MAIN-20180619080121-20180619100121-00148.warc.gz"} |
https://www.francoangeli.it/riviste/SchedaRivista.aspx?IDArticolo=36277&Tipo=Articolo%20PDF&idRivista=10 | Limiti e prospettive delle fonti rinnovabili in Italia
Titolo Rivista: ECONOMIA DELLE FONTI DI ENERGIA E DELL’AMBIENTE
Autori/Curatori: Domenico Coiante
Anno di pubblicazione: 2008 Fascicolo: Lingua: Italiano
Numero pagine: 17 P. 163-179 Dimensione file: 1169 KB
DOI: 10.3280/EFE2008-002011
Il DOI è il codice a barre della proprietà intellettuale: per saperne di più: clicca qui qui
Limits and Prospects of Renewable Energy Sources in Italy - The Italian energy balance for year 2005 is discussed with particular attention on renewable energy production. The potentials of renewable sources are evaluated in terms of energy density that can be obtained from occupied plant area. About 20000 km2 of sunny barren lands are present in South of Italy, particularly suitable for photovoltaic plants and that corresponds to a potential production of 144 Mtep of primary energy. Therefore, in theory, the photovoltaic energy potential is comparable with energy balance. The grid connection limit due to intermittent power generation of photovoltaic and wind energy systems is considered in relation with the stability of grid power level. Assuming a 25% maximum grid penetration of intermittent power with respect to capacity of active thermoelectric generators, the renewable energy contribution amounts to about 2% of annual energy balance. In front of expectations for a larger contribution, the practical result is the renewable energy production of present systems is marginal, unsuitable for counteracting the global climate crisis. The conclusion is that, for exploiting the large renewable energy potential, is necessary to implement the plants with an energy storage system able to overcome the source intermittency. Without this improvement, the expectations on renewable energy sources could be disappointed.
Key words: intermittent renewable sources, energy production limit, grid connection
1. Impossibile comunicare con Crossref: The remote server returned an error: (404) Not Found.
Domenico Coiante, in "ECONOMIA DELLE FONTI DI ENERGIA E DELL’AMBIENTE" 2/2008, pp. 163-179, DOI:10.3280/EFE2008-002011 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.948769211769104, "perplexity": 4113.182217465942}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703522150.18/warc/CC-MAIN-20210121004224-20210121034224-00250.warc.gz"} |
https://pdfkul.com/euclidean-co-embedding-of-ordinal-data-for-multi-hady-w-lauw_59ba97f21723dde2a958c213.html | Euclidean Co-Embedding of Ordinal Data for Multi-Type Visualization Dung D. Le∗
Abstract Embedding deals with reducing the high-dimensional representation of data into a low-dimensional representation. Previous work mostly focuses on preserving similarities among objects. Here, not only do we explicitly recognize multiple types of objects, but we also focus on the ordinal relationships across types. Collaborative Ordinal Embedding or COE is based on generative modelling of ordinal triples. Experiments show that COE outperforms the baselines on objective metrics, revealing its capacity for information preservation for ordinal data.
1
Introduction
We are interested in embedding, a visualization that maps a high-dimensional representation of data to a lower-dimensional one. The emphasis is on its capacity to preserve as much information as possible. Each data point is represented by a coordinate in a low-dimensional Euclidean space, and the relationship among data points are visualizable through Euclidean distances in that visualization space. Most of the previous works on embedding focus on metric embedding, whose objective is to preserve the pairwise distances among data points [19, 20, 18, 4]. This is applicable when the main relationship among objects is similarity, e.g., images of handwritten digits or human faces [4]. Ordinal data refers to data where the ranking established by numerical values are more significant than the exact values. Such a representation is applicable to various domains, e.g., preferences [16], document retrieval [8]. As a focusing point, and without loss of generality, subsequently, we primarily use the example of the domain of preferences, where users express how much they like various items. For instance, after purchasing a product on Amazon, a user may leave an explicit rating. While listening to music at Spotify, a user leaves implicit traces of her liking for a track or an artist by the frequencies at which she consumes them. In both explicit and implicit cases, it is important to model the relative sense of whether an item is preferred to another. ∗ School of Information Systems, Singapore Management University. Email: [email protected] † School of Information Systems, Singapore Management University. Email: [email protected]
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Figure 1: Euclidean Embedding of Users & Items
Problem. Embedding for ordinal data seeks to preserve the ordinal relationships among data points. Our goal is ordinal co-embedding, where multiple object types are involved (e.g., users and items), and crosstype ordinal relationships are key (e.g., users express preferences over items). We discuss the scenario of a preference dataset. Suppose for each user, we are given pairwise rankings over items. A triple hu, i, ji indicates that a user u prefers an item i to a different item j. As output, every user and every item would be respectively assigned a latent coordinate (to be learned) in a Ddimensional Euclidean space. We assume D = 2 or 3 for their appropriateness for visualization. User u’s preference for item i to item j is visualizable through a shorter distance between u and i than between u and j. Figure 1 illustrates an example 2D embedding for three users (blue triangles) and three items (purple crosses), specifying their respective coordinates. Through our spatial perception of the relative distances, we can immediately tell that the user u1 prefers item i1 the most (closest), followed by item i2 , and item i3 the least (furthest). Such information leaps out at us without our having to consciously compute the distances. In addition to visualization, embedding could also enable other applications arising from its Euclidean metric properties. One potential application is retrieval for recommendation queries, such as which items are the closest (most preferred) to a user. Euclidean geometry fits the mould of spatial data management, allowing it
to benefit from such developments as spatial indexing [3] and efficient nearest-neighbor query processing [17]. For another potential application, as embedding relies on building a compact model for user preferences, it may eventually enable an interactive interface for training recommender systems. In text domain [12], we may seek an embedding that preserves the relative importance of words to a document (for summarization). Approach. While there has been prior work on ordinal embedding [11, 1, 21], our work is novel in a couple of fundamental respects. First, the “classical” ordinal embedding is formulated mainly for one object type, e.g., cities [21], images [1]. It enforces that for same-type quadruple of objects hi, j, k, li, if i is closer to j in the original data than k is to l, the same ordinal relationship should hold in the embedding space. This presumes that the primary information is similarity among objects. In contrast, our primary objective is based on ranking. More specifically, the ranking of objects of one type (e.g., items) by an object of a different type (e.g., user). For instance, it is possible for two users to be “similar”, say in terms of their demographics or their habits of watching horror movies, and yet to have different rankings over specific items. Moreover, because classical ordinal embedding deals with within-type ordinal relationships, it implicitly assumes that there is one underlying reality to approximate, e.g., distances of cities in the map [21]. However, for many ordinal datasets, there may not be a singular ground-truth reality. For preference data, each user imposes his or her own ranking on the items, and these rankings may be different and at times conflicting. This fundamental difference motivates two distinguishing aspects of our approach. Because a common embedding space needs to accommodate the diverse preferences of users, we harness the collaborative effect among users and among items. In order to capture the variance in the rankings induced by preferences of different users or items in a principled way, we also formulate our model in terms of probabilistic generative modelling. Contributions and Organization. We provide the formal problem statement in Section 2. In this paper, we make the following contributions towards the problem. First, in Section 3, we propose a new embedding model, called Collaborative Ordinal Embedding or COE. This model is notable in its generative modeling of ordinal embedding allowing various types of triples, as well as in its objective function with both a penalty component for violated observations and a reward component for preserved observations on a smooth continuous spectrum modeled by probabilistic Sigmoid or Gompertz distributions. Second, in Section 3.3, we describe COE’s learning algorithm to derive the embedding co-
ordinates that maximize the posterior probability of the generative model based on stochastic gradient ascent for both Sigmoid and Gompertz. Third, in Section 5, comprehensive experiments on publicly available datasets show that COE outperforms the baselines, both in preserving the observed pairwise comparisons and in predicting unseen pairwise comparisons expressed as relative distances in the Euclidean space. We review the related work in Section 4, and conclude in Section 6. 2
Problem Formulation
We formally define the problem addressed in this paper, which is co-embedding of objects based on cross-type ordinal relationships. Moreover, for ease of reference, we adopt the language of preference dataset, and refer to one of the types as “users”, and the other type as “items”. Note that this is merely nomenclature, and does not limit the object types in the ordinal data. Input. The set of users is U, and u or v refers to a user. The set of items is I, and i or j refers to an item. The input is a multiset of triples T = TA ∪TB , consisting of “type-A” triples TA ⊂ U × I × I and “type-B” triples TB ⊂ U × U × I. A type-A triple tuij ∈ TA relates a user u ∈ U and two different items i, j ∈ I, indicating u’s preferring i to j. A type-B tuvi ∈ TB indicates a user u has greater preference over i than user v does. Such triples form a general representation of preferences over one object type as expressed by the other object type. There are examples abound in both explicit and implicit feedback scenarios. Triples can be derived from ratings, e.g., when u assigns a higher rating to i than to j. Other than ratings, it could also model implicit feedback [16]. For cable TV, u may watch the channel i but not j, or spend a longer time watching i than j [7]. For Web search, u may click on the result i after skipping j [15]. Outside of preference domain, in text, a word i may be more frequent than another word j in document u. Alternatively, document u may be more relevant to word i than document v does. While we focus on cross-type triples, it is feasible to accommodate triples involving three objects of the same type, e.g., u is more “similar” to v than to v 0 . Here, we will not concentrate on such similarity-based triples. More generally, we can use triple form (o1τ1 , o2τ2 , o3τ3 ), where oiτi are objects of types τi , (i = 1, 2, 3) respectively, to represent ordinal relations among multiple objects. The framework can be extended naturally by adding latent variables for objects of each type. For simplicity, we only present our model with two types. Output. Given T , the goal is to assign a coordinate xu ∈ RD to each user u ∈ U, as well as a coordinate yi ∈ RD to each item i ∈ I, such that their distances in RD preserve the relative ordering indicated by the
γ (v ∈ U ) > u
u ∈U
xv
xu
cuij
cuvi
1. For each user u ∈ U: Draw u’s coordinate: xu ∼ Normal(0, γ 2 I),
yi i∈I
yj ( j ∈ I) > i
β
Figure 2: Collaborative Ordinal Embedding (COE) triples. We denote the collection of all user coordinates as X and the collection of all item coordinates as Y . The coordinates of users and items lie in the same Ddimensional Euclidean space, where D is 2 or 3. Problem 1. (Ordinal Co-Embedding) Given a set of triples T , find the set of user coordinates X and item coordinates Y , so as to meet the following respective condition for as many triples in T as possible, i.e., tuij ∈ TA ⇒||xu − yi || < ||xu − yj ||, tuvi ∈ TB ⇒||xu − yi || < ||xv − yi || 3
each triple hu, v, ii where u < v, we associate it with a variable cuvi . The state of cuij (or cuvi ) and the generation of tuij (or tuvi ) are related to user and item coordinates through the following generative process. The generative process of COE is as follows:
Methodology
We now describe our proposed model, called Collaborative Ordinal Embedding or COE. The challenge is integrating the diverse triples into the same low-dimensional Euclidean space. The input triples T may also suffer from sparsity, variance, and uncertainties, in the form of incompleteness (not all possible triples are specified), inconsistency (some triples are conflicting), and repetitions (some triples may occur more than once). Yet the final objective is a unified view for all items and users.
2. For each item i ∈ I: Draw i’s coordinate: yi ∼ Normal(0, β 2 I), 3. For each triple hu, i, ji ∈ TA : • Draw cuij ∼ Bernoulli(P(cuij = 1 | xu , yi , yj )), • If cuij = 1, generate a triple instance tuij , • Else (cuij = 0), generate a triple instance tuji . 4. For each triple hu, v, ii ∈ TB : • Draw cuvi ∼ Bernoulli(P(cuvi = 1 | xu , xv , yi )). • If cuvi = 1, generate a triple instance tuvi , • Else (cuvi = 0), generate a triple instance tvui . In Step 1 and Step 2, we generate the users’ and items’ coordinates, placing zero-mean multi-variate spherical Gaussian priors on these coordinates, with γ 2 and β 2 controlling the respective variances of the Normal distributions. I denotes the identity matrix. In Step 3, we generate type-A triples involving one user and two items, by drawing the outcome for cuij from a Bernoulli process, where the parameter is specified by the probability P(cuij = 1 | xu , yi , yj ) of generating a triple instance tuij . In Step 4, we generate type-B triples involving two users and one item. 3.2 Triple Probability Function A crucial component is how the latent coordinates of users and items would generate the pairwise comparisons in T . This bridge between the hidden variables and the observations is the triple probability function. To keep the discussion streamlined, in the following we discourse on type-A triples of the form hu, i, ji, but a similar principle applies in a symmetric manner to type-B triples. The principle in relating latent coordinates to a triple hu, i, ji is: if u prefers i to j, the distance from xu to yi is shorter than that from xu to yj . The more evidence there is that u prefers i to j, the closer xu should be to yi than to yj . To realize this intuition, we express the probability P(cuij = 1 | xu , yi , yj ) in terms of the Euclidean distances ||xu − yi || and ||xu − yj ||. Let ∆uij be a quantity expressed in terms of these distances, such that ∆uij is higher the more u prefers i to j. One realization of ∆uij is Equation 3.1.
3.1 Generative Model To achieve this, we harness the “collaborative” effect. Since item coordinates are shared across users, users with similar coordinates would have similar ordinal relationships with items. To develop this probabilistically, we design a graphical model, whose plate notation is illustrated in Figure 2. We model each user coordinate and each item coordinate as real-valued latent random variables xu and yi respectively. For each triple hu, i, ji where i < j, we associate it with a binary random variable cuij . When cuij takes on the value of 1, it corresponds to an instance of tuij ∈ T . When cuij = 0, it corresponds to an instance of tuji ∈ T . In Figure 2, cuij is shaded and lies within its own plate, i.e., it is observed and there could be multiple instances. Correspondingly, for (3.1)
∆uij = ||xu − yj || − ||xu − yi ||
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probability). Moreover, since ∆uij = 0 correlates with uncertainty of 0.5 probability, we set b = ln 2. In turn, α is a scaling parameter to be tuned. Figure 3(b) shows that the left side ∆uij < 0 has steeper drop, while the right side has gentler gain. In turn the greater α is, the steeper is the slope overall.
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3.3 Learning Algorithms Given T as input observations, our goal is to learn the latent coordinates X and Y with the highest posterior probability P(X, Y |T ). Through Bayes’ Theorem, we have Figure 3: Triple Probability Function P(X, Y |T ) = P(T , X, Y )/P(T ). Since P(T ) does not affect the model parameters, the goal is to maximize Because tuij and tuji are opposites, we have the joint probability, as shown in Equation 3.4. P(cuij = 1 | xu , yi , yj ) = 1 − P(cuij = 0 | xu , yi , yj ). ∆uij has a bearing on these probabilities. For ∆uij > 0, (3.4) arg max P(T , X, Y |γ, β) X,Y the triple tuij is more likely. For ∆uij < 0, tuji is more likely. For ∆uij = 0, the two triples are equally likely. The joint probability is decomposed into four terms To model the probabilities of triples as a function corresponding to the steps in the generative process. of ∆uij (or ∆uvi ), we identify two possible functions. Sigmoid Function. The first is Sigmoid in Equation 3.2, where λ is a scaling parameter. Figure 3(a) P(T , X, Y |γ, β) = P(X|γ) × P(Y |β) × P(T |X, Y ), Y shows that the probability that u prefers i to j tends D − 1 ||xu ||2 , P(X|γ) = (2πγ 2 )− 2 e 2γ 2 towards 1 as ∆uij → ∞, and 0 as ∆uij → −∞. *"'+,-./,0'12345,/3
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(3.2)
P(cuij = 1| xu , yi , yj ) =
1 1+
This function allows us to model both a penalty for violating observed triples (probability mass < 0.5), and a reward for preserving observed triples (probability mass > 0.5). This is different from classical ordinal embedding. For instance, the state-of-the-art SOE [21] (see Section 4) only has a penalty component, but no reward. This holds two advantages for COE. First, there is a smoother spectrum of penalty and reward over a continuous function vs. the cliff effect for SOE. Second, there is discrimination among triples with more vs. less evidence earning different probability masses. The scaling parameter λ controls the slope of the function. The greater is λ, the steeper is the penalty/reward. The λ setting may empirically tuned. Gompertz Function. Sigmoid is symmetrical, which implies that the penalty component is commensurate with the reward component. There may be instances when we seek to model penalty and reward asymmetrically. In particular, we may place greater importance on penalty, i.e., steeper slope for negative ∆uij and gentler slope for positive ∆uij . This can be modeled by the Gompertz function, as shown in Equation 3.3. (3.3)
P(Y |β) =
e−λ·∆uij
P(cuij = 1| xu , yi , yj ) = a · e−b·e
−α·∆uij
To fit the triple probability function, we set a = 1 so as to put the range of values between 0 and 1 (reflecting
Y
D
(2πβ 2 )− 2 e
− 12 ||yi ||2 2β
,
i∈I
P(TA |X, Y ) =
Y
P(cuij = 1 | xu , yi , yj ),
tuij ∈TA
P(TB |X, Y ) =
Y
P(cuvi = 1 | xu , xv , yi ).
tuvi ∈TB
Maximizing the joint probability is equivalent to maximizing its logarithm, shown below. To simplify the parameters, we set γ = β, and equate both γ12 and 1 β 2 to a common regularization parameter η. L = ln P(X|γ) + ln P(Y |β) + ln P(T |X, Y ) X X = ln P(T |X, Y ) − η ||xu ||2 − η ||yi ||2 u∈U
i∈I
To find the coordinates that maximize the joint probability, we employ stochastic gradient ascent for computationally efficiency, an important factor given the potentially huge size of pairwise comparisons. Sigmoid Function. For the Sigmoid function, the gradient of L w.r.t. each user coordinate xu is: ∂L = ∂xu +
X
{i,j: tuij ∈TA }
X {i,v: tuvi ∈TB }
+
X {i,v: tvui ∈TB }
λe−λ∆uij
xu − yj xu − yi − ||xu − yj || ||xu − yi ||
1 + e−λ∆uij λe−λ∆uvi 1 + e−λ∆uvi
yi − xu ||yi − xu ||
λe−λ∆vui 1 + e−λ∆vui
−yi + xu ||yi − xu ||
− η · xu
Algorithm 1 Stochastic Gradient Ascent for COE-S (with Sigmoid triple probability function) 1: Initialize xu for u ∈ U {u,v: tuvi ∈TB } X 2: Initialize yi for i ∈ I xu − yi λe−λ∆uij + 3: while not converged do 1 + e−λ∆uij ||xu − yi || {u,j: tuij ∈TA } 4: Draw a triple at random from T . X λe−λ∆uji −xu + yi 5: if it is a type-A triple tuij ∈ TA then + − η · y i 1 + e−λ∆uji ||xu − yi || 6: h xu ← +i · {u,j: tuji ∈TA } xu xu −yj xu −yi λe−λ∆uij − η · x − u −λ∆ Algorithm 1 describes the stochastic gradient ascent uij ||xu −yj || ||xu −yi || 1+e h −λ∆ i uij algorithm for the version COE-S with Sigmoid function. λe i 7: yi ← yi + · 1+e−λ∆uij ||xxuu −y − η · y i It first initializes the coordinates of users and items. In h −λ∆ −yi || i uij −x +y λe 8: yj ← yj + · 1+e−λ∆uij ||xuu−yjj|| − η · yj each iteration, a triple is randomly selected from T , and the model parameters are updated based on the 9: if it is a type-B triple tuvi ∈TB then i h −λ∆ gradients above, with a decaying learning rate over uvi yi −xu λe 2 2 − η · x 10: x ← x + · u u u −λ∆ time. The complexity is O(|U|×|I| +|U| ×|I|). In case h1+e−λ∆ uvi ||yi −xu || i uvi −yi +xv λe of having triples of multi-type ordinal relations among 11: xv ← xv + · 1+e −λ∆uvi ||yi −xv || − η · xv multiple objects, the complexity is still a polynomial of 12: h yi ← +i · yi variables with highest degree is 3. yi −xv yi −xu λe−λ∆uvi − − η · y i Gompertz Function. For the Gompertz function, ||yi −xv || ||yi −xu || 1+e−λ∆uvi the gradient of L w.r.t. each user coordinate xu is: 13: Return {xu }u∈U and {yi }i∈I X xu − yj xu − yi ∂L The gradient w.r.t. each item coordinate yi is: −λ∆uvi X
∂xu
yi − xv yi − xu − ||yi − xv || ||yi − xu ||
λe 1 + e−λ∆uvi
∂L = ∂yi
α ln(2)e−α∆uij
=
||xu − yj ||
{i,j: tuij ∈TA }
+
X
α ln(2)e−α∆uvi
α ln(2)e−α∆vui
{i,v: tuvi ∈TB }
+
X {i,v: tvui ∈TB }
yi − xu ||yi − xu ||
−yi + xu ||yi − xu ||
||xu − yi ||
− η · xu
The gradient w.r.t. each item coordinate yi is:
∂L = ∂yi +
X
{u,v: tuvi ∈TB }
X
α ln(2)e−α∆uij
{u,j: tuij ∈TA }
+
X
yi − xv yi − xu − ||yi − xv || ||yi − xu || xu − yi ||xu − yi || −xu + yi − η · yi ||xu − yi ||
α ln(2)e−α∆uvi
α ln(2)e−α∆uji
{u,j: tuji ∈TA }
The algorithm and the complexity for the version COE-G with Gompertz function are similar to those for COE-S, but with the corresponding gradients above. 4
Related Work
We now relate to several categories of previous work. Ordinal Embedding. Given a set of data points, ordinal embedding seeks to preserve the relative comparisons of pairwise distances among data points [11]. In Section 5, we compare to a representative: the stateof-the-art SOE [21], which was shown to be more efficient and accurate than GNMDS [1]. Our key differences from SOE include the explicit modeling of crosstype ordinal relationships, and our probabilistic modeling that has both penalty and reward components. [22] investigated embedding for similarity-based triplets.
Metric Embedding. Metric embedding seeks to preserve similarity or distance values. In working with preference data, our work is related to CFEE [10], which fits rating values. CFEE expressed a rating rˆui by user u on item i in terms of the squared Euclidean distance between xu and yi . Fitting ratings directly may not necessarily preserve the pairwise comparisons, as we will see in Section 5. In embedding two object types, our work is related to embedding co-occurrences, e.g., documents and words [6] or words and images [24]. The idea is to express co-occurrence frequencies in terms of Euclidean distances. In Section 5 we include a comparison to CODE [6] to show fitting co-occurrences may not preserve comparisons. [13] analyzes generalized convex formulation for co-embedding. Matrix Factorization. Embedding and matrix factorization are recognized as different problems. The latter’s objective is to find a latent vector U for each user and V for each item, such that the inner product U T V approximates ratings [14] or pairwise comparisons [16, 23]. A tenuous link between squared Euclidean distance and inner product, i.e., ||U − V ||2 = ||U ||2 + ||V ||2 − 2U T V , does not imply monotonicity because of the vector magnitudes. [2] proposed post facto transformation, by extending output latent vectors by one dimension and using that extra dimension to equalize the magnitude of item vectors. This could only preserve either of user-centric or item-centric triples, but not both. In Section 5, we compare to the composite of BPR [16], followed by [2]’s transformation.
Table 1: Datasets
MovieLens Netflix Last.fm 20News
5
users/ docs
items/ words
943 429,102 1,772 15,744
1,413 17,769 3,521 14,414
ratings/ observations 99,543 99,841,834 72,955 1,076,900
type-A hu, i, ji triples 7.80 × 106 2.68 × 109 1.50 × 106 5.61 × 107
type-B hu, v, ii triples 8.22 × 106 2.51 × 1011 3.87 × 106 2.19 × 108
Experiments
document length, we divide each word’s frequency by the document length, and generate triples from these normalized term frequencies. Because of the different natures of the two categories of datasets, which involve some different comparative baselines, in the following we organize the experiments into two sections, one for each dataset category. 5.1 Rating-based Datasets Because the main purpose is visualization, all comparisons are based on embedding in two-dimensional space. We experiment with two versions of our model. The first uses the Sigmoid function, referred to as COE-S. The second uses the Gompertz function, referred to as COE-G. The first baseline is a representative of the traditional ordinal embedding SOE [21]. We use the authors’ implementation5 . The second baseline is the embedding designed to fit the numerical rating values, i.e., CFEE [10]. As its authors have not made their implementation available, we implement it in Java. The third baseline is matrix factorization based on pairwise comparisons BPR [16] with one dimension, followed by [2]’s Euclidean transformation into two dimensions, denoted as BPR+. For BPR, we use the Java implementation in LibRec6 . The justifications for the baselines were discussed in Section 4. We tune the respective parameters for the best performance on each dataset. Metrics. We apply several metrics that allow an evaluation of the various methods in terms of information preservation in two-dimensional Euclidean space. As is common for dimensionality reduction [9], the primary aim is how well the reduced dimensionality preserves the observed data. The first and main metric is preservation accuracy, the extent to which the information within the observed triples is preserved by the u coordinates. For a user u, let Tobserved denote the triples involving u. For u, the preservation accuracy is defined as the fraction of her triples for which the coordinates reflect the preference direction in the triples. Overall, the preservation accuracy is the average of users’ preservation accuracies, as shown in Equation 5.5. By doing so, it is not biased towards few users with many ratings at the expense of many users with few ratings.
Our objective is to investigate the effectiveness of COE, for visualization in low-dimensional Euclidean space. Datasets. While COE assumes ordinal triples as inputs, we experiment with publicly available datasets with numerical values and derive the triples accordingly. This allows us to compare to baselines that work directly with the numerical values. We work with four datasets of two categories, and their sizes are listed in Table 1. The first category includes rating-based preference datasets: MovieLens 1 and Netflix 2 . The object types are users and movies (items). The raw observations are ratings. As in [5], we apply Z-score normalization, which compensates for different rating means and rating spreads to make ratings more comparable across users. We then generate a type-A triple tuij for each instance where a user u has higher normalized rating on an item i than on item j, and a type-B triple tuvi for each instance where a user u has higher normalized rating on i than v does. We do not generate any triple involving non-rated items. For MovieLens, Netflix, each user has been preconditioned by the original dataset to have at least 20 ratings. We further ensure that each item has at least 4 ratings. We find similar practice in other works [16]. The second category are based on cooccurrences: Last.fm 3 and 20News 4 . Last.fm contains users’ listening frequencies to music artists (items). As in above, we retain users with at least 20 items, and items with at least 4 users. To show applicability beyond preferences, we include the text-based 20News, which has documents (“users”) and words (“items”). We downloaded the dataset with stop words removed and the remaining words stemmed. Following the standard practice by the baseline [6], we filter out extremely infrequent words (less than 5 documents), and extremely frequent words (top 100 most frequent). For both datasets, the raw (5.5) u : ||xu − yi || < ||xu − yj ||}| 1 X |{tuij ∈ Tobserved observation is the term frequency of a word (or an item) u |U| |T observed | in a document (or a user). To normalize the effect of u∈U 1 http://grouplens.org/datasets/movielens/ 2 http://www.cs.uic.edu/ liub/Netflix-KDD-Cup-2007. ~ html 3 http://files.grouplens.org/datasets/hetrec2011/ hetrec2011-lastfm-2k.zip 4 http://web.ist.utl.pt/acardoso/datasets/
As mentioned in Section 2, we do not presume that the input set of triples are complete. It is therefore interesting to study how well the learnt coordinates 5 http://rpackages.ianhowson.com/cran/loe/man/SOE.html 6 http://www.librec.net/
Table 2: Rating-based Dataset (MovieLens - 100K Sample): COE vs. Ordinal Embedding COE-S COE-G SOE
Preservation Accuracy Type-A Type-B H-Mean 70.1% 57.3% 63.0% 70.0% 57.5% 63.2% 69.4% 55.9% 61.9%
Prediction Accuracy Type-A Type-B H-Mean 62.7% 57.4% 59.9% 62.8% 57.9% 60.2% 62.5% 56.0% 59.1%
could generalize to unseen triples. We introduce a secondary metric, prediction accuracy, the extent to which the coordinates can infer the preference directions of hidden triples Thidden . For an embedding solution as a whole, the prediction accuracy is derived from userlevel accuracies, as shown in Equation 5.6.
(5.6)
u : ||xu − yi || < ||xu − yj ||}| 1 X |{tuij ∈ Thidden u |U | u∈U |Thidden |
The above definitions are for type-A triples. A corresponding version is defined for type-B triples. We will present the results both types separately, as well as together by taking their harmonic mean (H-Mean). We split the ratings randomly into 80% Robserved and 20% Rhidden , in a stratified manner to maintain the same ratio for every user. The observed set of triples Tobserved are formed within Robserved . The hidden set of triples Thidden include triples formed within Rhidden , as well as triples involving one rating each from Robserved and Rhidden . Ordinal-based methods learn from Tobserved , while the rest learn from with Robserved . Both preservation and prediction accuracies range from 0% (worst) to 100% (best). For statistical significance, we average the results across 10 random (80:20) splits. These metrics are general for ordinal triples. Since the ordinal triples are derived from ratings, we include a rating-based third measure: average rating among knearest neighbors (k-NN). Intuitively, a good embedding with high preservation should place higher-rated items closer to the user. Given a user, we identify the knearest rated items based on their Euclidean distances in the embedding space, and average the user’s ratings on those items. Symmetrically, this can be measured from each item’s point of view. We average this across users and items respectively for k = 1 and k = 5. Versus Ordinal Embedding. Existing ordinal embedding packages do not scale to large datasets. The author implementation of SOE limits the number of input size to 100K. We sample 100K triples from Tobserved , and use them to compare SOE and COE. Yet, this is only applicable to MovieLens, as SOE cannot cope with the number of users and items in Netflix. Table 2 shows the performance of the methods on the 100K sample of MovieLens for both type-A and
1-NN Avg Rating Users Items H-Mean 4.38 3.66 3.99 4.41 3.67 4.01 4.29 3.44 3.82
5-NN Avg Rating Users Items H-Mean 4.24 3.48 3.82 4.24 3.48 3.82 4.22 3.38 3.75
type-B triples. Focusing on the overall figures (harmonic mean in bold), we see that the preservation accuracies of COE-S and COE-G are similar at 63.0% and 63.2%. Both are higher than SOE’s 61.9%, whose lower performance is statistically significant. For prediction accuracies, the figures are slightly lower overall, but the relative trend is the same. For visualization based on dimensionality reduction, preservation is the greater objective, as the intent is to represent the observed data. Table 2 also shows the comparison of the average rating among 1-nearest neighbors (1-NN), as well as 5-NN. Again, we take the harmonic mean (H-Mean) between users’ and items’ rating averages. Evidently, the nearest neighbors around every user or item tend to have high ratings (in the scale of 1 to 5). COE-G and COE-S are similar, while SOE is significantly lower. Versus Other Baselines. In Table 3, we employ the full data to compare to the other baselines. COE-S and COE-G have significantly higher results in Table 3, because they run with the full set of observed triples. CFEE, which fits rating values directly, generally achieves lower accuracies. Since rating and visualization spaces are distinct, forcing their unification may not obtain the best embedding to preserve the triples. BPR+, which learns matrix factorization by pairwise ranking, followed by Euclidean transformation, also achieves lower results. As mentioned in Section 4, the Euclidean transformation applied to BPR’s output could only preserve the pairwise comparisons of either type-A triples or type-B triples (not both at once). However, we present the best results for both transformations, which evidently are still lower than COE’s. This signifies that for visualization, directly modelling Euclidean distance, such as in COE, leads to better visualization. Table 4 shows the results for the much-larger Netflix dataset, which also support the major observations made above. The differences between COE’s variants and the baselines are statistically significant. Visualization. Figure 4 shows an example of three users U887 (blue), U222 (red), U903 (green) in MovieLens, and the 17 items (crosses) that all three have rated. For instance, U222 and U903 are closer to Fargo (which they rated 5) than U887 is (who rated it 2). Interestingly, U222 is closer to U903 than U222 is to U887, supported by the Pearson correlation of their
Table 3: Rating-based Dataset (MovieLens): COE vs. Other Baselines COE-S COE-G CFEE BPR+
Preservation Accuracy Type-A Type-B H-Mean 75.0% 65.0% 69.6% 75.0% 65.0% 69.6% 67.2% 62.4% 64.7% 68.4% 60.9% 64.5%
COE-S COE-G CFEE BPR+
Preservation Accuracy Type-A Type-B H-Mean 75.2% 66.3% 70.4% 74.9% 65.5% 69.9% 66.0% 62.4% 64.2% 68.2% 60.2% 64.0%
Prediction Accuracy Type-A Type-B H-Mean 64.0% 59.0% 61.4% 64.0% 59.0% 61.4% 59.7% 60.3% 60.0% 62.1% 59.1% 60.5%
1-NN Avg Rating Users Items H-Mean 4.48 3.93 4.19 4.48 3.87 4.15 4.07 3.63 3.84 4.14 3.63 3.87
5-NN Avg Rating Users Items H-Mean 4.33 3.58 3.92 4.33 3.55 3.90 4.03 3.50 3.75 4.13 3.40 3.73
Table 4: Rating-based Dataset (Netflix): COE vs. Other Baselines Prediction Accuracy Type-A Type-B H-Mean 63.3% 61.2% 62.2% 63.1% 60.7% 61.9% 58.9% 61.4% 60.2% 60.3% 58.8% 59.6%
Figure 4: Example Visualization of Users (triangles) and Items (crosses) in MovieLens
ratings on items: 0.31 between (U222, U903), and -0.21 between (U222, U887). The layout of movies are also intuitive. Horror films Scream and Island of Dr. Moreau are on the top left. Science fictions Star Wars, Return of the Jedi, and Back to the Future are at the centre. Darker dramas Fargo, Apocalypse Now are on the top right. Comedies such as Kingpin and Beavis and Butthead are on the far right. Family-oriented Searching for Bobby Fischer and Lost World are towards the bottom. Efficiency is not our major focus here. The learning algorithms can be run offline. On MovieLens and LastFM, COE takes approximately a minute on a PC with Intel Core i5 3.2GHz CPU and 12GB RAM. For 20News, the running time of COE is around 15 minutes. Our efficiency is comparable to other models running on pairwise comparisons, e.g., BPR, and is much faster than ordinal embedding, i.e., SOE.
1-NN Avg Rating Users Items H-Mean 4.63 4.06 4.32 4.66 4.05 4.34 4.15 3.93 4.04 4.07 3.16 3.56
5-NN Avg Rating Users Items H-Mean 4.51 3.74 4.09 4.52 3.72 4.08 4.10 3.74 3.91 4.00 3.15 3.52
5.2 Cooccurrence-based Datasets We now discuss the comparisons for the other two datasets based on cooccurrences: Last.fm and 20News. Here, we focus on the comparison to CODE [6], which fits co-occurrence frequencies. We use the implementation7 by its author. For the metrics, we again rely on preservation and prediction accuracies. In addition, we adapt the “average rating” concept to the cooccurrence scenario. Since the raw observation is normalized term frequency, we evaluate the average term frequencies among the knearest neighbors of a document or a word respectively. The higher it is, the more successful is the embedding in placing the closest words to a document (vice versa). Table 5 for Last.fm and Table 6 for 20News show that both COE versions have significantly higher preservation and prediction accuracies than the baseline CODE. This experiment showcases that the information within ordinal triples is not easily approximated by fitting probabilities of co-occurrences (which is semantically closer to similarity/distance-based embedding). This is also evident from the comparison of average normalized term frequencies among the k-NN. The values seem deceptively low, these frequencies are actually high, considering that each document consists of many words. For instance, in Table 6, COE achieves 0.050 for k = 1, which implies that the nearest word to a document is expected to cover 5% of the document. We have also compared to ordinal embedding SOE, and COE is also better than SOE on these datasets. 6
Conclusion
We address the problem of ordinal co-embedding based on cross-type ordinal relationships, whereby every user and every item is respectively associated with a la7 http://ai.stanford.edu/
~gal/
Table 5: Cooccurrence-based Dataset (Last.fm): COE vs. Cooccurrence Embedding COE-S COE-G CODE
Preservation Accuracy Type-A Type-B H-Mean 64.5% 85.6% 73.5% 64.0% 85.7% 73.3% 53.3% 52.8% 53.1%
COE-S COE-G CODE
Preservation Accuracy Type-A Type-B H-Mean 78.9% 90.3% 84.3% 77.0% 88.0% 82.1% 59.7% 56.2% 57.9%
Prediction Accuracy Type-A Type-B H-Mean 51.7% 63.2% 56.9% 51.4% 63.1% 56.6% 49.8% 54.7% 52.2%
1-NN Avg Frequency Users Items H-Mean 0.048 0.047 0.047 0.048 0.047 0.047 0.032 0.031 0.032
5-NN Avg Frequency Users Items H-Mean 0.041 0.032 0.036 0.040 0.032 0.036 0.032 0.032 0.032
Table 6: Cooccurrence-based Dataset (20News): COE vs. Cooccurrence Embedding Prediction Accuracy Type-A Type-B H-Mean 51.0% 69.2% 58.7% 50.8% 68.7% 58.4% 48.7% 52.8% 50.7%
tent coordinate in a low-dimensional Euclidean space. The objective is to place a user closer to a more preferred item. This accommodates datasets including ratings and co-occurrences. Experiments on public datasets show that Collaborative Ordinal Embedding or COE outperforms comparable baselines in information preservation in the low-dimensional visualization space. Acknowledgments This research is supported by the Singapore Research Foundation under its International Centre @ Singapore Funding Initiative and tered by the IDM Programme Office, Media ment Authority (MDA).
References [1] S. Agarwal, J. Wills, L. Cayton, G. Lanckriet, D. J Kriegman, and S. Belongie. Generalized non-metric multidimensional scaling. In AISTATS, 2007. [2] Y. Bachrach, Y. Finkelstein, R. Gilad-Bachrach, L. Katzir, N. Koenigstein, N. Nice, and U. Paquet. Speeding up the xbox recommender system using a euclidean transformation for inner-product spaces. In RecSys, 2014. [3] N. Beckmann, H.-P. Kriegel, R. Schneider, and B. Seeger. The R*-tree: An efficient and robust access method for points and rectangles. In SIGMOD, 1990. [4] L. Van der Maaten and G. Hinton. Visualizing data using t-SNE. JMLR, 9, 2008. [5] M. D. Ekstrand, J. T. Riedl, and J. A. Konstan. Collaborative filtering recommender systems. Foundations and Trends in Human-Computer Interaction, 4(2):81– 173, 2011. [6] A. Globerson, G. Chechik, F. Pereira, and N. Tishby. Euclidean embedding of co-occurrence data. JMLR, 8:2047–2076, 2007. [7] Y. Hu, Y. Koren, and C. Volinsky. Collaborative filtering for implicit feedback datasets. In ICDM, 2008.
1-NN Avg Frequency Docs Words H-Mean 0.050 0.049 0.050 0.049 0.047 0.048 0.035 0.022 0.027
5-NN Avg Frequency Docs Words H-Mean 0.039 0.029 0.037 0.038 0.028 0.036 0.033 0.020 0.025
[8] T. Joachims. Training linear svms in linear time. In KDD, 2006. [9] I. Jolliffe. Principal Component Analysis. Wiley Online Library, 2005. [10] M. Khoshneshin and W. N. Street. Collaborative filtering via euclidean embedding. In RecSys, 2010. [11] J. B. Kruskal. Nonmetric multidimensional scaling: a numerical method. Psychometrika, 29(2), 1964. [12] C. D. Manning, P. Raghavan, and H. Sch¨ utze. Introduction to information retrieval. 2008. [13] F. Mirzazadeh, Y. Guo, and D. Schuurmans. Convex co-embedding. In AAAI, 2014. [14] A. Mnih and R. Salakhutdinov. Probabilistic matrix factorization. In NIPS, 2007. [15] F. Radlinski and T. Joachims. Query chains: learning to rank from implicit feedback. In KDD, 2005. [16] S. Rendle, C. Freudenthaler, Z. Gantner, and L. Schmidt-Thieme. BPR: Bayesian personalized ranking from implicit feedback. In UAI, 2009. [17] N. Roussopoulos, S. Kelley, and F. Vincent. Nearest neighbor queries. In SIGMOD, 1995. [18] S. T. Roweis and L. K. Saul. Nonlinear dimensionality reduction by locally linear embedding. Science, 290, 2000. [19] R. N. Shepard. The analysis of proximities: Multidimensional scaling with an unknown distance function. i. Psychometrika, 27(2), 1962. [20] J. B. Tenenbaum, V. De Silva, and J. C. Langford. A global geometric framework for nonlinear dimensionality reduction. Science, 290, 2000. [21] Y. Terada and U. V Luxburg. Local ordinal embedding. In ICML, 2014. [22] Laurens Van der Maaten and Kilian Weinberger. Stochastic triplet embedding. In MLSP, pages 1–6, 2012. [23] M. Weimer, A. Karatzoglou, Q. V. Le, and A. Smola. Cofirank - maximum margin matrix factorization for collaborative ranking. In NIPS, 2007. [24] J. Weston, S. Bengio, and N. Usunier. Wsabie: Scaling up to large vocabulary image annotation. In IJCAI, 2011.
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... for courses available at Reynolds Community College fractal cancer 4 620 000 Google hits July ... 5 2017 Charles E Schmidt College of Science Course Descriptions Biological Sciences Chemistry .... Select a trial membership to give us a. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8334269523620605, "perplexity": 3645.61346398556}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370497171.9/warc/CC-MAIN-20200330150913-20200330180913-00164.warc.gz"} |
http://mathhelpforum.com/calculus/201889-what-average-rate-change-amplitude-sine-wave.html | # Math Help - What is the average rate of change of the amplitude of a sine wave?
1. ## What is the average rate of change of the amplitude of a sine wave?
I have previously posted this question but I accidentally put it in the wrong section.
The formula for inductive reactance is XL =2πfL
I understand why the inductive reactance increases as the inductance and frequency increase but I am not sure why "2π" is included in the formula. I suspect that π must represent the average rate of change of the amplitude of the sine wave. By calculating the average by using four values of sine, I get 4.2 units of amplitude per second but if I use eight values of sine, I get 3.6 units of amplitude per second. My guess is that were I to use ninety values of sine, the average would be very close to the numerical value of π. I assume that "2" is in the formula because the peak amplitude occurs twice in each cycle.
My question is: What is the average rate of change of the amplitude of a sine wave?
Thank you,
Percy Pimm
2. ## Re: What is the average rate of change of the amplitude of a sine wave?
Going back a lot of years.
d (sin x) / dx = cos(x) | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9884458780288696, "perplexity": 243.54116469850374}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-40/segments/1443737942301.73/warc/CC-MAIN-20151001221902-00040-ip-10-137-6-227.ec2.internal.warc.gz"} |
https://iwaponline.com/wst/article/60/2/483/17783/Phosphorus-removal-in-laboratory-scale-unvegetated | This research has two eventual goals: (1) To optimize performance of subsurface constructed wetlands for removal of phosphorus (P) (2) To demonstrate that dewatered alum sludge (a by-product), can be reused as a constructed wetland substrate. To achieve these, alum sludge from a water treatment plant was characterized and used as main substrate in four experimental vertical sub-surface flow constructed wetland systems treating dairy farm wastewater. Results show that the alum sludge has suitable hydraulic characteristics (uniformity coefficient = 3.6) for use as a substrate, and in the batch studies, up to 48.6 mg-P was removed by 1 g of the alum sludge at a P concentration of 360 mg-P/l and a dosage of 5 g/l. Results from the experimental systems highlight the significant P removal ability of the alum sludge. However, the inclusion of pea gravel at the infiltrative surface of some of the systems had a negative effect on the P removal performance. Sequential P-fractionation results show that there was no significant increase in the easily extractable P, but for total P, there was significant increase, although this was found to decrease with depth. This study shows that the novel use of dewatered alum sludge can bring about high P removal in vertical subsurface flow constructed wetland systems. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8427314758300781, "perplexity": 4673.45339566501}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039741628.8/warc/CC-MAIN-20181114041344-20181114062828-00027.warc.gz"} |
https://www.physicsforums.com/threads/direct-absorption-proces.224890/ | # Direct absorption proces
1. Mar 28, 2008
### ehrenfest
[SOLVED] direct absorption proces
1. The problem statement, all variables and given/known data
My solid-state physics book (Kittel) says the following in the chapter about semiconductors:
"In a direct absorption process the threshold of continuous optical absorption at frequency $\omega_g$ measure the band gap $E_g = \hbar \omega_g$",
Apparently this is a definition, so it is hard to argue with it, but can someone explain what the "threshold of continuous optical absorption" means and how that could measure the band gap?
2. Relevant equations
3. The attempt at a solution
2. Mar 28, 2008
### olgranpappy
by "threshold" of absorption he means the lowest (incoming beam) energy at which you measure absorption. below this energy there will be no absorption and the incoming beam will simply pass through the material as if it were completely transpartent.
I.e., the absorption spectrum should look something like a theta function
$$\theta(\hbar \omega - E_g)$$
times some other smooth function
in real life the absorption spectrum is not a perfect step function, but it does "turn on" fairly sharply at "threshold" (sharply enough so that we can tell what the threshold incoming energy is).
The reason he uses the word "continuous" is that there actually can be some absorption "below threshold" due to bound states, but that absorption shows up in the spectrum as discrete little peaks (like delta function, but not infinity sharp), not as a continuous spectrum.
3. Mar 28, 2008
### ehrenfest
I see, thanks. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8490235805511475, "perplexity": 1581.558165247146}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560279248.16/warc/CC-MAIN-20170116095119-00090-ip-10-171-10-70.ec2.internal.warc.gz"} |
http://sjscience.org/article?id=270 | Research article
# Critical Casimir Force between Inhomogeneous Boundaries
Version 1 Released on 26 May 2015 under Creative Commons Attribution 4.0 International License
Jerome Dubail1, Raoul Santachiara 2,*, Thorsten Emig 2,3,4,*
## Authors' affiliations
1. Institut Jean Lamour, Physique de la Matière et des Matériaux (IJL/P2M). CNRS : UMR7198 - Université de Lorraine
2. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS). CNRS : UMR8626 - Université Paris XI - Paris Sud
3. Department of Physics, School of Science - MIT
4. MultiScale Materials Science for Energy and Environment - Internation Joint Unit MIT/CNRS (UMI3466)
5. *. Unregistered author (unverified)
# Abstract
To study the critical Casimir force between chemically structured boundaries immersed in a binary mixture at its demixing transition, we consider a strip of Ising spins subject to alternating fixed spin boundary conditions. The system exhibits a boundary induced phase transition as function of the relative amount of up and down boundary spins. This transition is associated with a sign change of the asymptotic force and a diverging correlation length that sets the scale for the crossover between different universal force amplitudes. Using conformal field theory and a mapping to Majorana fermions, we obtain the universal scaling function of this crossover, and the force at short distances.
Fluctuation-induced forces are generic to all situations where fluctuations of a medium or field are confined by boundaries. Examples include QED Casimir forces [1,2], van der Walls forces [3], and thermal Casimir forces in soft matter which are most pronounced near a critical point where correlation lengths are large [4,5]. The interaction is then referred to as critical Casimir force (CCF). Analogies and differences between these variants of the common underlying effect have been reviewed in Ref. [6].
Experimentally, CCFs can be observed indirectly in wetting films of critical fluids [7], as has been demonstrated close to the superfluid transition of ${}^4$He [8] and binary liquid mixtures [9]. More recently, the CCF between colloidal particles and a planar substrate has been measured directly in a critical binary liquid mixture [10,11]. Motivated by the possibility that the lipid mixtures composing biological membranes are poised at criticality [12,13], it has been also proposed that inhomogeneities on such membranes are subject to a CCF [14] which provides an example of a 2D realisation.
The amplitude of the CCF is in general a universal scaling function that is determined by the universality classes of the fluctuating medium [15]. It depends on macroscopic properties such as the surface distance, shape and boundary conditions of the surfaces but is independent of microscopic details of the system [5]. Controlling the sign of fluctuation forces (attractive or repulsive) is important to a myriad of applications in design and manipulation of micron scale devices. While for QED Casimir forces a generalized Earnshaw's theorem rules out the possibility of stable levitation (and consequently force reversals) in most cases [16], the sign of the CCF depends on the boundary conditions at the confinement. For classical binary mixtures, surfaces have a preference for one of the two components, corresponding to fixed spin boundary conditions ($+$ or $-$) in the corresponding Ising universality class. Depending on whether the conditions are like ($++$ or $--$) or unlike ($+-$ or $-+$) on two surfaces, the CCF between them is attractive or repulsive. So-called ordinary or free spin boundary conditions are difficult to realize experimentally but can emerge due to renormalization of inhomogeneous conditions as we shall show below [17]. Motivated by their potential relevance to nano-scale devices, fluctuation forces in the presence of geometrically or chemically structured surfaces have been at the focus recently. Sign changes of CCFs due to wedge like surface structures have been reported very recently [18]. Competing boundary conditions can give rise to interesting crossover effects with respect to strength and even sign of the forces. Here we consider such a situation for the Ising universality class in 2D. At criticality, this system can be described by conformal field theory (CFT) [19,20], and CCFs are related to the central charge of the CFT [2123], and scaling dimensions of boundary operators [24].
Figure 1 Figure 1. Ising strip of width $L$ with alternating fixed spin boundary conditions on one side, with a typical spin configuation indicated by the shading.
In this Letter, we show that boundary conditions which alternate periodically between two spin states (see Fig. 1) give rise to a novel phase transition. Associated with that is a diverging correlation length that sets the scale for a sign change of the CCF on one side of the transition. We obtain the critical exponents and exact expressions for the universal scaling function of the force in the critical region. Consider the Ising model on an infinitely long strip of width $L$, and assume that the system is at its critical temperature $T_c$ so that it is conformally invariant. For homogenous, fixed spin boundary conditions $\gamma_1$, $\gamma_2 = \pm$ on the two boundaries, the critical Casimir energy per unit strip length, ${\cal F}$, is determined by CFT. Since $L$ is the only finite length scale, the energy obeys a simple power law. The amplitude is determined only by the central charge $c=1/2$ of the Ising model and the scaling dimension $h_{\gamma_1 \gamma_2}$ of the so-called boundary condition changing (BCC) operator from $\gamma_1$ to $\gamma_2$ (see below for details on the BCC operator) [24], $$\label{eq:1} {\cal F} = - \pi \left( \frac{1}{48} - h_{\gamma_1 \gamma_2} \right) \frac{1}{L}$$ where we measure here and the following energies in units of $k_B T_c$. For like boundary conditions $\gamma_1=\gamma_2 = +$ or $-$ one has $h_{++}=h_{--} = 0$ and hence an attractive force $F = - d {\cal F}/dL$. For unlike boundary conditions $\gamma_1 \neq \gamma_2$ one gets $h_{+-}=h_{-+}=1/2$ and hence a repulsive force. There is one more conformally invariant boundary condition that corresponds to free (f) spins or ordinary boundary conditions. When combined with fixed boundary conditions, the corresponding BCC operator has the scaling dimension $h_{f+}=h_{f-}=1/16$ which implies a repulsive interaction in Eq. (\ref{eq:1}). In the following we consider a strip with homogeneous $+$ spins on one boundary and alternating regions of $-$ and $+$ spins of length $a$ and $b$, respectively, on the other boundary, see Fig. 1.
If the temperature is slightly different from $T_c$, the system is in the critical region, where the free energy density can be decomposed into non-singular (${\cal F}_{ns}$) and singular (${\cal F}_{s}$) contributions, $$\label{eq:2} {\cal F}(t,L,\tau) = {\cal F}_{ns}(t,L,\tau)+ {\cal F}_s(t,L,\tau)$$ that depend on the reduced temperature $t=T/T_c-1$, the width $L$, and a scaling variable $\tau=a/b-1$ that is specific to the alternating boundary conditions in Fig. 1. While the non-singular part is an analytic function of $t$ and $\tau$, the singular part is not. For homogeneous boundary conditions, $t$ is the only relevant scaling variable, and in the critical region the singular part of the free energy density is given by a universal scaling function $\vartheta$ that depends only on $L/\xi$ [15,5] where $\xi(t\to 0^\pm) = \xi_0^\pm |t|^{-\nu}$ is the bulk correlation length with amplitude $\xi_0^\pm$ and exponent $\nu=1$ for the Ising model. As we shall see below, the same renormalization-group (RG) concepts apply to a novel, boundary induced critical region that we identify for inhomogeneous boundary conditions around $a=b$. To focus on that region, we assume in the following that the system is at its bulk critical point, $t=0$. For large $L \gg a,b$ the singular part of the free energy density can be expressed in terms of a universal scaling function of the new correlation length $\xi_c(\tau)=(a+b)|\tau|^{-\nu_c}$, $$\label{eq:3} {\cal F}_s(0,L,\tau) = \frac{1}{L} \vartheta[L/\xi_c(\tau)] \, .$$ Below we shall determine $\vartheta$ and the exponent $\nu_c$.
BCC operators have been introduced in CFT to study systems with discontinuous boundary conditions [24]. When inserted on a boundary, these local operators interpolate between the different boundary conditions on either side of the insertion point. They are highest weight states of weight $h$ and all such states may be realized by an appropriate pair of boundary conditions. For the critical Ising model, the BCC operator that takes the boundary condition from $+$ spin to $-$ spin corresponds to the chiral part of the energy operator $\epsilon(z,\bar z)$. This can be understood easily in the representation of the Ising model in terms of a free Majorana fermion field $\psi(z)$ out of which the energy operator is composed, $\epsilon(z,\bar z)=i \psi(z)\bar\psi(\bar z)$ [25]: The Jordan-Wigner transformation shows that the fermion creation and annihilation operators flip locally the spin orientation.
Now the BCC operators permit us to relate the partition function of the strip with alternating boundary conditions to a correlator for the field $\psi(z)$ at positions where the boundary conditions change. On the upper complex plane, one has $\langle \psi(z)\psi(z')\rangle = 1/(z-z')$ which yields (after a conformal map) for the partition function of the strip the Pfaffian, $$\label{eq:4} Z = Z_0 \langle \psi(w_1) \ldots \psi(w_{2N}) \rangle = Z_0 \text{Pf} (G) = Z_0 {\det}^{1/2} (G),$$ with $G = [\langle \psi(w_i)\psi(w_j) \rangle]_{i,j=1,\ldots,2N}$, where we used the Wick theorem for fermions, $w_j$ are the positions of the $2N$ BCC operators on the upper edge of the strip, and $Z_0$ is the partition function of the homogenous system with $a=0$. Due to the symmetry under translations by $a+b$, the matrix $G$ is of block Toeplitz form, $G_{ij} = g_{i-j}$, with $$\label{eq:5} g_j = \begin{pmatrix} g[j (a+b)] & g[j(a+b)-a] \\ g[j(a+b)+a] & g[j (a+b)] \end{pmatrix} \, ,$$ where $g(w)=\pi/[2L \sinh(\pi w / (2L)]$.
The free energy density can be expressed in the thermodynamic limit as $$\label{eq:6} {\cal F} = -\frac{\pi}{48} \frac{1}{L} - \lim_{N\to\infty} \frac{1}{2N(a+b)} \log \det G \, .$$ The Szegö-Widom (SW) theorem for block Toeplitz matrices states that the determinant can be expressed in terms of the matrix valued Fourier series $\varphi(\theta) = \sum_{k=-\infty}^\infty g_k e^{ik\theta}$ as [26] $$\label{eq:7} \lim_{N\to \infty} \frac{1}{2N} \log \det G = \frac{1}{4\pi} \int_0^{2\pi} d\theta \log \det \varphi(\theta)$$ where $\det$ acts now on a $2\times 2$ matrix. It turns out that this formula can be only applied for the case $a<b$. The reason for that is a subtle difference between the Toeplitz matrix $G$ and the corresponding circulant matrix $C$ that describes periodic boundary conditions along the strip. While for $a<b$ the spectra of $G$ and $C$ become equivalent for $N\to\infty$, for $a>b$ there exists a pair of eigenvalues of $GC^{-1}$ that tend to zero exponentially for $N\to\infty$, yielding an extra contribution $\delta$ that is determined by the decay of the Fourier integral $$\label{eq:8} J=\frac{1}{2\pi} \int_0^{2\pi} d\theta e^{-i j \theta} \left[\varphi^{-1} (\theta)\right]_{11} \sim e^{-j \delta} \quad \text{for} \, j\to \infty$$ and has to be subtracted from the r.h.s. of Eq. (\ref{eq:7}) for $a>b$. Here $\left[\varphi^{-1} (\theta)\right]_{11}$ denotes the $11$-element of the $2\times 2$ matrix $\varphi^{-1} (\theta)$. In the following we apply Eqs. (\ref{eq:7}) and (\ref{eq:8}) to compute the critical Casimir force in various scaling limits.
When $L\ll a,b$, the function $g(w)$ defined below Eq. (\ref{eq:5}) can be replaced by $g(w)=(\pi/L)e^{-\pi|w|/(2L)}$ which yields the exact determinant $$\label{eq:9} \det \varphi(\theta) = \frac{\cos \theta -\cosh(\pi(a-b)/(2L))}{\cos \theta -\cosh(\pi(a+b)/(2L))} \, .$$ For $a<b$, the SW theorem then yields $\frac{1}{2N}\log \det G = - (\pi a)/(2L)$. For $a>b$, this is also the correct result as it follows from subtracting the correction $\delta$ which follows from Eq. (\ref{eq:8}) and $J=e^{-\pi|a-b|j/(2L)}$ as $\delta=\pi(a-b)/(2L)$. It follows that the critical Casimir force for $L\ll a,b$ is $$\label{eq:10} F = \frac{\pi}{48} \frac{23a-b}{a+b} \frac{1}{L^2} + \ldots$$ It has an analytic amplitude that varies continously with $a/b$. This result is identical to an addition of the amplitudes from Eq. (\ref{eq:1}) for unlike and like boundary conditions, weighted by $a/(a+b)$ and $b/(a+b)$, according to their occurrence. Hence, additivity holds at short distances. This has been observed also for a 3D Ising model in the special case of boundaries with alternating stripes of equal width [17].
Next, we consider the case $L \gg a,b$. Using the Abel-Plana summation formula, it can be shown that in this limit the elements of the matrix $$\label{eq:11} \varphi(\theta) = \frac{\pi}{L} \begin{pmatrix} i \gamma_1(\theta) & \gamma_2(\theta) \\ -\gamma^*_2(\theta) & i \gamma_1(\theta) \end{pmatrix}$$ approach
$$\label{eq:12} \gamma_j(\theta) = \frac{L}{a+b} \left\{ \hat\gamma_j(\theta,\tau) - i^{j+1} \left[ \tanh(\theta L/(a+b)) +\tanh((\theta-2\pi) L/(a+b)) \right]\right\}$$ with $\hat\gamma_1(\theta,\tau) = 1-\theta/\pi$ and \begin{multline} \hat\gamma_2(\theta,\tau) = \frac{1}{\pi} \left[ -\frac{\tau+2}{\tau+1} + e^{i \theta} (\tau+2) {}_2F_1\left(1,\frac{1}{\tau+2},\frac{\tau+3}{\tau+2},e^{i \theta}\right) \right. \\ - \left. e^{-i \theta} \frac{\tau+2}{2\tau+3} {}_2F_1\left(1,\frac{2\tau+3}{\tau+2},\frac{3\tau+5}{\tau+2},e^{-i \theta}\right) \right] \, \end{multline} where ${}_2F_1$ is a hypergeometric function. For $a<b$, the SW theorem then yields the free energy density $$\label{eq:13} {\cal F} = -\frac{\pi}{48}\frac{1}{L} -\frac{1}{4\pi(a+b)} \int_0^{2\pi} \log\left\{ 1 + \Gamma(\theta,\tau) \left[ \tanh\frac{\theta L}{a+b} +\tanh\frac{(\theta-2\pi)L}{a+b} \right] \right\} d\theta \, .$$
where we have subtracted a $L$-independent contribution that does not change the force, and defined $\Gamma(\theta,\tau)=(2\hat\gamma_1+i(\hat\gamma_2-\hat\gamma^*_2)) /(|\hat\gamma_1|^2-|\hat\gamma_2|^2)$. In the evaluation of the integral, the correlation length $\xi_c(\tau)$ defined above Eq. (\ref{eq:3}) becomes important. The integrand is exponentially localized around $\theta=0, 2\pi$ over a small range $(a+b)/L$. Also, it can be shown that $\Gamma$ has the scaling property $\lim_{\tau\to 0} \Gamma( \tau^2/\zeta,\tau) = \Gamma_0(\zeta) = 1/(1+\pi^3\zeta/32)$ for any constant $\zeta$. Hence, in the critical region of small $\tau$, or $\xi_c(\tau) \gg a+b$, the proper scaling is obtained by setting $\zeta=(L\tau^2)/(a+b)=L/\xi_c$ (up to a numerical coefficient), showing that the exponent $\nu_c=2$. In the integral, $\Gamma(\theta,\tau)$ can be replaced by $\Gamma_0(\zeta)$ and one obtains after a simple integration the result for the universal scaling function of Eq. (\ref{eq:3}) when $a<b$ or $\tau<0$, $$\label{eq:14} \vartheta_-(\zeta) = \frac{1}{4\pi} \text{Li}_2\left( \frac{2}{1+\pi^3 \zeta /32}-1\right)$$ where $\text{Li}_2(x)=\sum_{k=1}^\infty x^k/k^2$ is a polylogarithm function. Outside the critical region $L\gg \xi_c$, one has $\vartheta_-(\zeta\to\infty) = -\pi/48$ so that the force is fully dominated by the boundary regions with like spins. On the contrary, for $L \ll \xi_c$, and hence $\tau\to 0^-$, the frustration between almost equal amounts of fixed $+$ and $-$ spins on the boundaries leads to a renormalization to effectively free boundary conditions with $\vartheta_-(\zeta\to 0) = \pi/24$. For $a>b$, the correction $\delta$ yields an extra contribution $\Delta\vartheta(\zeta)$ determined by $$\label{eq:15} \Delta\vartheta(\zeta) \tan[\Delta\vartheta(\zeta)] = \frac{\pi^3}{32} \zeta$$ so that the scaling function for $\tau>0$ is $\vartheta_+(\zeta) =\vartheta_-(\zeta) +\Delta\vartheta(\zeta)$. Since $\Delta\vartheta(\zeta\to 0)=0$, the scaling function is continuous around $\tau=0$. For $L \gg \xi_c$, however, $\Delta\vartheta(\zeta\to \infty)=\pi/2$ so that the system asymptotically realizes homogenous unlike boundary conditions with $\vartheta_+(\zeta\to \infty)=23\pi/48$.
Figure 2 Figure 2. Schematic overview of critical Casimir force amplitudes as function of the strip width $L$ and the ratio $a/b$. For $L\gg a,b$ the solid curves represent the diverging correlation length $\xi_c$. The horizontal dashed line indicates the cut along which the force amplitude is plotted in Fig. 3. Along the red curve the sign of the force changes whereas the blue curve indicates only a change between two universal (repulsive) limits.
Our findings can be summarized by the scheme of Fig. 2. It shows the different scaling regimes and the corresponding asymptotic amplitudes of the Casimir force. At short distance $L\ll a,b$ the amplitude varies continuously across the critical point at $a=b$, with a sign change at $b/a=23$. For $L\gg a,b$ there exist three distinct regions: around $a=b$ appears a region where $L \ll \xi_c$ where the force is repulsive and approaches for asymptotic $L$ the universal amplitude for fixed-free spin boundary conditions. For $a<b$, the force changes sign from attractive to repulsive when $L$ approaches $\xi_c$, corresponding to a stable point. For $a>b$, the force is always repulsive but the amplitude crosses over from $\pi/24$ to $23\pi/48$ under an increase of $L$ beyond $\xi_c$.
The dependence of the force $F$ on $|a-b|$ at fixed $L\gg a,b$ (see dashed horizontal line in Fig. 2) is determined by $F = -\partial {\cal F}/\partial L = \Theta(x_s) L^{-2}$ with a universal scaling function $\Theta$ of the scaling variable $x_s$ that is defined on both sides of the critical point by $x_s=\text{sign}(\tau)(L/\xi_c)^{1/2}\sim a-b$. This function is shown in Fig. 3 where we used the results for $\vartheta_\pm(L/\xi_c)$ of Eqs. (\ref{eq:14}), (\ref{eq:15}). In the critical region $|x_s| \ll 1$, one has the expansions $$\label{eq:16} \Theta(x_s) = \left\{ \begin{array}{ll} \frac{\pi}{24} - \frac{\pi^2}{64} x_s^2 + \ldots & \text{for} x_s < 0 \\ \frac{\pi}{24} + \frac{\pi^{3/2}}{8\sqrt{2}} x_s + \ldots & \text{for} x_s > 0 \end{array}\right. \, ,$$ whereas for $L$ outside the critical region, $|x_s| \gg 1$, $$\label{eq:17} \Theta(x_s) = \left\{ \! \begin{array}{ll} -\frac{\pi}{48} + \frac{32\log 2}{\pi^4} \frac{1}{x_s^{2}} + \ldots & \text{for} x_s < 0 \\ \frac{23\pi}{48} - \frac{32(\pi^2-\log 2)}{\pi^4} \frac{1}{x_s^{2}} + \ldots & \text{for} x_s > 0 \end{array}\right. .$$ We see that $\Theta(x_s)$ is not analytic around $x_s=0$ and hence constitutes the singular part of the free energy density, see Eq. (\ref{eq:2}). This resembles the singular nature of scaling functions describing the bulk transition at $T=T_c$.
Figure 3 Figure 3. Universal scaling function $\Theta(x_s)$ for the critical Casimir force as function of the scaling variable $x_s\sim a-b$.
Our results show the existence of a novel phase transition for the critical Casimir force in the 2D Ising model that is induced by inhomogeneous boundary conditions with a varying ratio of up and down spins. We obtained exact expressions for the universal scaling function of the force. Due to the observed renormalization of boundary conditions, in binary mixtures, ordinary (free spin) boundary conditions can be realized experimentally and “switched” on and off by varying the distance $L$, or an inhomogeneous surface field. The crossover between different universal amplitudes leads to a stable equilibrium point at $L\simeq \xi_c$ for $1/23<a/b<1$. The emergence of the novel phase transition at $a=b$ is related to the relevance of a surface magnetic field $\sim \tau$ at a surface with free spin boundary conditions. This can be seen from the decay of the spin correlations along a single surface, $\langle \sigma_x \sigma_{x'}\rangle \sim |x-x'|^{-\eta_\|}$ with $\eta_{\|} = 1$ for free boundary conditions [15]. Since the surface field contributes an energy $\sim \tau \int dx \sigma_x$, the scaling dimension $y_\tau = 1/\nu_c = 1-\eta_{\|}/2$ which is identical to our findings above. It is interesting to explore these concepts in general spatial dimensions for Ising and XY models, and tri-critical points which have an even richer spectrum of possible boundary conditions.
We thank M. Kardar for many fruitful discussions.
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https://kyunghyuncho.me/three-faces-of-sparsity-nonlinear-sparse-coding/ | # Three faces of sparsity: nonlinear sparse coding
it’s always puzzled me what sparsity means when computation is nonlinear, i.e., decoding the observation from a sparse code using nonlinear computation, because the sparse code can very well be turned into a dense code along the nonlinear path from the original sparse code to the observation. this made me write a short note earlier, as in a few years back, and i thought i’d share my thoughts on sparsity here with you:
in my mind, there are three ways to define sparse coding.
1. code sparsity: the code is sparse, i.e., $|z|_0 = O(1)$.
2. computational sparsity: the computation is sparse, i.e., $x = \sum_{k=1}^K z_k w_k$, where $K = O(1)$ and $w_k \in \mathbb{R}^d$.
3. noise robustness: the computation is robust to perturbation to the parameters: let $\tilde{w} = w + \epsilon$, where $\epsilon \sim \mathcal{N}(0, \sigma^2 1_{|w|})$. the MSE between $x$ and $\tilde{x} = |\sum_{k=1}^K z_k w_k – \sum_{k=1}^K z_k \tilde{w}_k|_2^2$ is $O(d \times \sigma^2)$ not $O(d’ \times d \times \sigma^2)$, because $k \ll d’$ is a constant w.r.t. $d’$.
these are equivalent if we constrain the decoder to be linear (i.e., $x = \sum_{i=1}^{d’} z_i w_i$,) but they are not with a nonlinear decoder. in particular, let us consider a neural net decoder with a single hidden layer such that $x = u \max(0, w z),$ where $u \in \mathbb{R}^{d \times d_h}$ and $w \in \mathbb{R}^{d_h \times d’}$. we can then think of how these different notions of sparsity manifest themselves and how we could encourage these different types of sparsity when training a neural net.
the amount of computation is then $O(d \times d_h + d_h \times d’)$ which reduces to $O(d \times d’)$ assuming $d_h = O(d’)$. even if we impose the code sparsity on $z$, the overall computation does not change ($O(d \times d_h + d_h \times k)$) and remain as $O(d \times d’)$. in other words, code sparsity does not imply computation sparsity, as was the case with linear sparse coding.
based on this observation, one can imagine imposing sparsity on all odd-numbered layers (counting the $z$ as the first layer) and the penultimate layer (one before $x$) in order to satisfy computational sparsity with a nonlinear decoder. in the example above, this implies that the sparsity should be imposed on both $z$ and $\max(0, wz)$.
this naive approach to computational sparsity implies noise robustness, as the number of parameters used in computation is restricted by construction. it does not mean however that there aren’t any other way to impose noise robustness. in particular, we can rewrite the whole problem of sparse coding as
$$\min_{z, w, u} \frac{1}{N} \sum_{n=1}^N |x^n – u \max(0, w z^n)|^2$$
subject to $$| \text{Jac}_{w,u} u \max(0, w z^n) |_F^2 < k d~\text{for all}~n=1,\ldots, N.$$
in other words, the influence of perturbing the parameters on the output must be bounded by a constant multiple of the output dimensionality.
of course it is not tractable to solve this problem exactly, but we can write a regularized proxy problem:
$$\min_{z, w, u} \frac{1}{N} \sum_{n=1}^N |x^n – u \max(0, w z^n)|^2 + \lambda | \text{Jac}_{w, u} u \max (0, wz^n) |_F^2,$$
where $\lambda$ is a regularization strength. in other words, we find the parameters, $w$ and $u$, that are robust to perturbation in terms of the output.
So, which sparsity are we referring to and do we desire when talking about sparsity in neural networks? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9549835920333862, "perplexity": 302.7181794627886}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948932.75/warc/CC-MAIN-20230329023546-20230329053546-00256.warc.gz"} |
http://mathhelpforum.com/statistics/208875-i-need-know-how-get.html | # Math Help - I need to know how to get this %
1. ## I need to know how to get this %
Hello,
i have a problem calculating this:
I have a bankroll of : 100 $i have 65% chance to win 10$
i have 35% chance to lose 10$and i have unlimited tries. what is the % of my bankroll to become 0 example: first hit 65% for my bankroll to become 110$ and 35% chance for my bankroll to become 90\$
thanks.
2. ## Re: I need to know how to get this %
If you win i times then you must lose i+ 10 times, out of a total of n= 2i+ 10 times or i=(n-10)/2. It looks like it is a binomial distribution with p= .65, q= .35. That is, the probability of losing all your money in n turns is $\begin{pmatrix}n \\ (n-10)/2\end{pmatrix}(.65)^{(n-10)/2}(.35)^{(n+10)/2}$. For that to be possible n must be even, say n= 2m, so the probability of eventually losing all your money is $\sum_{m=0}^\infty \begin{pmatrix} 2m \\ m- 5\end{pmatrix}(.65)^{m- 5}(.35)^{m+5}$.
3. ## Re: I need to know how to get this %
can you calculate this for me, for only 100 runs and 1000 runs
cause it's not working with me
i m trying to calculate how much bankroll i need to move to bigger stakes. with a risk of 0.1%
4. ## Re: I need to know how to get this %
it's always so close to zero, i think i m doing it right.
thanks alot hallofivy | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8405460119247437, "perplexity": 738.5297749479197}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701146241.46/warc/CC-MAIN-20160205193906-00296-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://socratic.org/questions/what-is-the-slope-of-the-line-perpendicular-to-y-5-12x-2 | Algebra
Topics
What is the slope of the line perpendicular to y=5/12x-2 ?
Jan 3, 2016
$- \frac{12}{5}$
Explanation:
Lines that are perpendicular have slopes that are opposite reciprocals.
The slope of $y = \frac{5}{12} x - 2$ is $\frac{5}{12}$.
The opposite reciprocal of $\frac{5}{12}$ is $- \frac{1}{\frac{5}{12}} = - \frac{12}{5}$.
Impact of this question
157 views around the world | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 5, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8721687197685242, "perplexity": 2206.4762874102257}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496670162.76/warc/CC-MAIN-20191119172137-20191119200137-00227.warc.gz"} |
https://physics.stackexchange.com/questions/311615/force-not-acting-on-the-center-of-mass/311628 | # Force not acting on the center of mass
There are a few questions here which are almost the same. But I still had a doubt I read that a force applied anywhere on a rigid body produces the same acceleration on the center of mass. How is that possible?
If I apply a force on the center of mass it just accelerates but if I apply a force off the center of mass it accelerates linearly and rotates. Since it rotates some of the force goes into rotating the body so the linear acceleration cannot be the same? Isn't it?
And is there a mathematical proof that a force applied any where on the rigid body produces the same acceleration? There was a proof in one the answers which used the action of force on discrete particles of the body. Somebody pls give a better proof
• There is not better proof than the one about the action of the forces on the discrete particles. The key step is newtons third law which hows that the forces between the particles cancell out when you compute the motion of the CofM. By the way: understanding the bit about off-center force causing rotation is important. You need to appreciate that change on momentum is force $\times$ time, but change in energy is force $\times$ distance. – mike stone Feb 12 '17 at 16:34
• If you wrap string round a body and pull, you will need to move you hand further to get the same change in momentum as attatching it at the cofm would give. The extra work has gone into rotation. – mike stone Feb 12 '17 at 16:42
• @mikestone Is that also true if a force is applied at the COM and the same force is applied at some other point. If the body rotates and accelerates linearly will its linear acceleration be then equal to the acceleration that it would have when the force would have been applied to the COM . Or is it true only when the body somehow just accelerates and doesn't rotate even though the force is off center. i.e when there is no rotation only then the acceleration of the body due to the force off center equal to the acceleration of the body when force in on COM it's true when there is rotation – E2n Feb 12 '17 at 18:00
• The acceleration of the cofm is the same wherever the force is applied. – mike stone Feb 12 '17 at 19:18
• -1. What was wrong with the proof mentioned in your last paragraph? Please provide a link which identifies that answer. – sammy gerbil Feb 12 '17 at 19:56
This is a result of Newton's 2nd law. Force is the time derivative of linear momentum. And momentum of a collection of particles is defined as $${\bf p} = \sum_i m_i {\bf v}_i = \left( \sum_i m_i \right) {\bf v}_C$$ where $m_i$ is the individual mass, ${\bf v}_i$ the individual velocity and ${\bf v}_C$ the velocity of the center of mass. By taking the derivative of the above you get the relationship between force ${\bf F}$ and center of mass acceleration $\dot{{\bf v}}_C$
$${\bf F} = \left( \sum_i m_i \right) \dot{{\bf v}}_C$$
The center of mass location ${\bf r}_C$ is defined by
$$\sum_i m_i {\bf r}_i = \left( \sum_i m_i \right) {\bf r}_C$$
and by direct differentiation of the above, you get
$$\sum_i m_i {\bf v}_i = \left( \sum_i m_i \right) {\bf v}_C$$
where ${\bf v}_i = \dot{{\bf r}}_i$ and ${\bf v}_C = \dot{{\bf r}}_C$
So what happens when a force is applied away from the center of mass?
The point where the force is applied will accelerate at least as much as the center of mass. In general, it will accelerate more due to the rotation. The force will feel a reduced mass given by the relationship
$$m_{eff} = \left( \frac{1}{m} + \frac{c^2}{I} \right)^{-1}$$ where $m$ is the mass, $I$ is the mass moment of inertia about the axis of rotation and $c$ is the moment arm of the force as seem by the center of mass.
Centre of mass of a body is that point where it appears the that the whole mass of the body is considered there, hence $$A_{\rm body} = F_{\rm net}/M$$ Now the force being applied on the centre of mass produces no torque because the distance of the force from centre of mass of body is 0.
The force which actually produce torque is the force of friction and only other force acting on the body, but not on the position of centre of mass. For most objects consider reference point to be centre of mass to measure the torque on the body.
Moreover, the torque which is produced would help the object to roll it not to push it. It's magnitude would depend on the frictional force and the mass of the body.
• The question doesn't talk about rolling and friction. I believe he is asking the general question about one force being off centre without any other forces acting. – Steeven Sep 18 '17 at 13:09 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8154920935630798, "perplexity": 206.01653062097236}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141182794.28/warc/CC-MAIN-20201125125427-20201125155427-00639.warc.gz"} |
http://science.sciencemag.org/content/297/5586/1497 | Special Reviews
# Dynamics of Recent Climate Change in the Arctic
See allHide authors and affiliations
Science 30 Aug 2002:
Vol. 297, Issue 5586, pp. 1497-1502
DOI: 10.1126/science.1076522
## Abstract
The pattern of recent surface warming observed in the Arctic exhibits both polar amplification and a strong relation with trends in the Arctic Oscillation mode of atmospheric circulation. Paleoclimate analyses indicate that Arctic surface temperatures were higher during the 20th century than during the preceding few centuries and that polar amplification is a common feature of the past. Paleoclimate evidence for Holocene variations in the Arctic Oscillation is mixed. Current understanding of physical mechanisms controlling atmospheric dynamics suggests that anthropogenic influences could have forced the recent trend in the Arctic Oscillation, but simulations with global climate models do not agree. In most simulations, the trend in the Arctic Oscillation is much weaker than observed. In addition, the simulated warming tends to be largest in autumn over the Arctic Ocean, whereas observed warming appears to be largest in winter and spring over the continents.
• * To whom correspondence should be addressed. E-mail: dickm{at}apl.washington.edu
View Full Text | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8921667337417603, "perplexity": 2806.7021345151843}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257645405.20/warc/CC-MAIN-20180317233618-20180318013618-00590.warc.gz"} |
http://mathonline.wikidot.com/the-matrix-form-of-the-chain-rule-for-compositions-of-differ | The Matrix Form of the Chain Rule For Compositions Of Differ
# The Matrix Form of the Chain Rule for Compositions of Differentiable Functions from Rn to Rm
Recall from The Chain Rule for Compositions of Differentiable Functions from Rn to Rm page that if $S \subseteq \mathbb{R}^n$ is open, $\mathbb{a} \in S$, $\mathbf{g} : S \to \mathbb{R}^p$, and if $\mathbf{f}$ is another function such that the composition $\mathbf{h} = \mathbf{f} \circ \mathbf{g}$ is well defined then if $\mathbf{g}$ is differentiable at $\mathbf{a}$ with total derivative $\mathbf{g}'(\mathbf{a})$ and $\mathbf{f}$ is differentiable at $\mathbf{b} = \mathbf{g}(\mathbf{a})$ with total derivative $\mathbf{f}'(\mathbf{b}) = \mathbf{f}'(\mathbf{g}(\mathbf{a}))$ then $\mathbf{h}$ is differentiable at $\mathbf{a}$ and:
(1)
\begin{align} \quad \mathbf{h}'(\mathbf{a}) = \mathbf{f}'(\mathbf{b}) \circ \mathbf{g}'(\mathbf{a}) = \mathbf{f}'(\mathbf{g}(\mathbf{a})) \circ \mathbf{g}'(\mathbf{a}) \end{align}
Also recall from earlier on The Jacobian Matrix of Differentiable Functions from Rn to Rm page that if a function is differentiable at a point then the total derivative of that function at that point is the Jacobian matrix of that function at that point. Therefore, if the composition $\mathbf{h} = \mathbf{f} \circ \mathbf{g}$ is well defined, $\mathbf{g}$ is differentiable at $\mathbf{a}$ with total derivative $\mathbf{f}'(\mathbf{a}) = \mathbf{D} \mathbf{g}(\mathbf{a})$ and $\mathbf{f}$ is differentiable at $\mathbf{b} = \mathbf{g}(\mathbf{a})$ with total derivative $\mathbf{f}'(\mathbf{b}) = \mathbf{D} \mathbf{f} (\mathbf{b})$ (i.e., $\mathbf{f}'(\mathbf{g}(\mathbf{a})) = \mathbf{D} \mathbf{f} (\mathbf{g}(\mathbf{a}))$ then from linear algebra, the matrix of a composition of two linear maps is equal to the product of the matrices of those linear maps, that is:
(2)
\begin{align} \quad \mathbf{D} \mathbf{h} (\mathbf{a}) = [\mathbf{D} \mathbf{f} (\mathbf{b})][\mathbf{D} \mathbf{g}(\mathbf{a})] = [\mathbf{D} \mathbf{f} (\mathbf{g}(\mathbf{a}))] [\mathbf{D} \mathbf{g} (\mathbf{a})] \end{align}
Furthermore, if $S \subseteq \mathbb{R}^n$ is open, $\mathbf{g} : S \to \mathbb{R}^m$ and $\mathbf{f} : R(\mathbf{g}) \to \mathbb{R}^p$, i.e.:
(3)
\begin{align} \quad (x_1, x_2, ..., x_n) \to_{\mathbf{g}} (y_1, y_2, ..., y_m) \to_{\mathbf{f}} (z_1, z_2, ..., z_p) \end{align}
Then for all $k \in \{ 1, 2, ..., p \}$ and for all $j \in \{ 1, 2, ..., n \}$ we have that:
(4)
\begin{align} \quad \frac{\partial z_k}{\partial x_j} = \sum_{i=1}^{m} \frac{\partial z_k}{\partial y_i} \frac{\partial y_i}{\partial x_j} \end{align} | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 4, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9953612089157104, "perplexity": 392.7181486718168}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585424.97/warc/CC-MAIN-20211021133500-20211021163500-00025.warc.gz"} |
https://mathematics.huji.ac.il/event/dynamics-lunch-raimundo-briceno-tau | # Dynamics Lunch: Raimundo Briceno (TAU) "A Breiman type theorem for Gibbs measures"
We will review a Breiman type theorem for Gibbs measures due to Gurevich and Tempelman. For a translation invariant Gibbs measure on a suitable translation invariant configuration set X \subset S^G, where G is an amenable group and S is a finite set, we will prove the convergence of the Shannon-McMillan-Breiman ratio on a specific subset of "generic" configurations. Provided that the above Gibbs measure exists, we also prove the convergence in the definition of pressure and the fact that this Gibbs measure is an equilibrium state.
## Date:
Tue, 09/01/2018 - 12:00 to 13:00 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9430553317070007, "perplexity": 989.2904919455844}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986672723.50/warc/CC-MAIN-20191017045957-20191017073457-00142.warc.gz"} |
http://mathhelpforum.com/math-topics/221049-normal-distribution-again.html | # Math Help - Normal distribution, again
1. ## Normal distribution, again
If X is normally distributed with a mean of 5 and a variance of 4, find the value of k so that P(-k<X<k)=0.80
P(-k<X<k)=0.80
let (k-5)/2=z
1-2P(Z>z)=0.8
P(Z>z)=0.1
(k-5)/2=1.282
k=7.564
2. ## Re: Normal distribution, again
Hey Trefoil2727.
I think the problem is that you have P(Z>z) instead of P(Z<z). Since you are looking at P(-k < X < k) then it means that you are looking at 1 - 2*P(X < k) due to symmetry of the normal distribution which means that 1 - 2*P(Z < z) = 0.8 where z = (k - 5)/2
If you use this you get using R:
> (qnorm(0.1,0,1)*2+5)
[1] 2.436897
Since I used a computer for this, if the answer is based on using tables then it will probably lose precision since the answer of 2.43 is in the tails of the distribution.
3. ## Re: Normal distribution, again
Originally Posted by chiro
Hey Trefoil2727.
I think the problem is that you have P(Z>z) instead of P(Z<z). Since you are looking at P(-k < X < k) then it means that you are looking at 1 - 2*P(X < k) due to symmetry of the normal distribution which means that 1 - 2*P(Z < z) = 0.8 where z = (k - 5)/2
If you use this you get using R:
> (qnorm(0.1,0,1)*2+5)
[1] 2.436897
Since I used a computer for this, if the answer is based on using tables then it will probably lose precision since the answer of 2.43 is in the tails of the distribution.
huh, how can this be done?
4. ## Re: Normal distribution, again
The R package allows you to do a lot of statistical computations very easily.
qnorm(0.1,0,1) returns the value of z where P(Z < z) = 0.1.
You can do the same thing by getting a statistical normal table and finding the value of z where P(Z < z).
The difference is that qnorm uses a computational algorithm where-as a table just gives values (calculating by the same kind of computer algorithm) and you look up those values on paper rather than with a computer.
5. ## Re: Normal distribution, again
You seem to have used .08 rather than .8. If you look at the Normal Distribution table here, you will see that "z= .1" corresponds to P(z)= .04, NOT .40. (Since this app is giving P(z> 0), and your problem has -.8< z< .8, you divide by 2 to get 0< z< .4.) P(z< .4) corresponds to z= 1.28, not .1.
6. ## Re: Normal distribution, again
sorry I still couldn't get it, why there's -0.8<z<0.8? i'm suppose to find the k right?
7. ## Re: Normal distribution, again
Oh, blast. I was reading the problem backwards! P(-k< z< k)= .80 is the same as P(0< z< k)= .40 and I get z= 1.28 for that.
So (z- 5)/2= 1.28, z- 5= 2(1.28)= 2.56, z= 7.56 which is the answer you got. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9035934209823608, "perplexity": 1371.1102805209957}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435376073161.33/warc/CC-MAIN-20150627033433-00104-ip-10-179-60-89.ec2.internal.warc.gz"} |
http://clay6.com/qa/21311/according-to-einstein-s-photoelectric-equation-the-plot-of-the-ke-of-the-em | Browse Questions
According to Einstein’s photoelectric equation, the plot of the KE of the emitted photoelectrons from a metal vs the frequency of the incident radiation gives a straight line whose slope
$\begin{array}{1 1} \text{(a) depends on the nature of the metal used} \\ \text{(b) depends on the intensity of the radiation} \\ \text{(c) both A & B} \\ \text{ (d) Depends on neither A nor B} \end{array}$
Can you answer this question?
Ans : (d)
Einstein’s photoelectric equation is $KE_{max} = hc \: – \phi$……..(i)
The equation of the line is $y=mx+C$………..(ii)
comparing eqns. (i) & (ii),
$m = h, c = -\phi$
So, slope is a constant and independent of intensity of radiation
answered Dec 27, 2013
edited Mar 26, 2014 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9394668936729431, "perplexity": 2115.673093433027}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698542665.72/warc/CC-MAIN-20161202170902-00169-ip-10-31-129-80.ec2.internal.warc.gz"} |
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