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9 depends on f+ (ef
9 depends on $f_{e^{\pm}}$ (cf.
Eq.
Eq.
2).
2).
In section 32 we discuss the possible values of {1-.
In section 3 we discuss the possible values of $f_{e^{\pm}}$.
IL outergap exists in MSPs. fp~30. which gives Z,;e10C'eV. which is less than the fitting values by a factor of 2 for Optical photons.
If outergap exists in MSPs, $f_{e^{\pm}}\sim 30$, which gives $E_{inj}\sim 10 GeV$, which is less than the fitting values by a factor of 2 for Optical photons.
On the other hand. if pairs are only produced in polar gap. we have estimated in section 3 that [ως~I. which gives E;~2x0GeV.
On the other hand, if pairs are only produced in polar gap, we have estimated in section 3 that $f_{e^{\pm}}\sim 1$, which gives $E_{inj}\sim 2\times 10^2 GeV$.
This estimate is consistent with the fitting values for IR as soft photons.
This estimate is consistent with the fitting values for IR as soft photons.
However. if (he optical photons are the soft photons then Chis estimate is higher than the fitting values bv a [actor of 10 in general.
However, if the optical photons are the soft photons then this estimate is higher than the fitting values by a factor of 10 in general.
This may imply outergap exist but
This may imply outergap exist but
of the single particle IBilbert spaces: so that the dimension of His V.=Nat. and the logiitblinie upper bound to coniplexitv cau be increased linearly by a realistic. linear increase of the umber I of particles: to paraphrase Saunders Mac Lane.
of the single particle Hilbert spaces: so that the dimension of ${\cal H}$ is ${\cal N} = N n^I$, and the logarithmic upper bound to complexity can be increased linearly by a realistic, linear increase of the number $I$ of particles: to paraphrase Saunders Mac Lane,.
space, Certainly. this is ouly necessary and not sufficient to insure that the quautuii Arnold cat will physically behave in the limit as its classical relative.
Certainly, this is only necessary and not sufficient to insure that the multi-particle quantum Arnol'd cat will physically behave in the limit as its classical relative.
The nest (but not final) requirement is that the wnoreanized information preseut iu the reservoir of sinall particles be organized by the dynamics iu order to produce information at the classical rate given by the cutropy.
The next (but not final) requirement is that the unorganized information present in the reservoir of small particles be organized by the dynamics in order to produce information at the classical rate given by the Kolmogorov-Sinai entropy.
Let us therefore present the result of ummerical experiments.
Let us therefore present the result of numerical experiments.
We first show the effect of coupling to the small particles.
We first show the effect of coupling to the small particles.
Figure 2. displays the cutropics SCJ) versus J and V. the scattering coupling coustaut. for the case of a large particle of mass Af=24h interacting with 3 small particles of mass 2/7.
Figure \ref{fig-sjv} displays the entropies $S(J)$ versus $J$ and $V$, the scattering coupling constant, for the case of a large particle of mass $M=2^4h$ interacting with 3 small particles of mass $2h$ .
Recall that
Recall that
we can account for these errors by assuming the following form for the probability (ppδρ. that a galaxy of actual apparent axial ratio p’ is instead measured to have an axial ratio p: In our case. we estimate the measurement errors by fitting a straight line to the distribution of errors in the axial ratio pas reported by the isophote fitting package.
we can account for these errors by assuming the following form for the probability $E(p|p')\delta p$, that a galaxy of actual apparent axial ratio $p'$ is instead measured to have an axial ratio $p$: In our case, we estimate the measurement errors by fitting a straight line to the distribution of errors in the axial ratio $p$ as reported by the isophote fitting package.
This gives: and projection kernel hence becomes. The algorithm now involves using A(p|q) instead of A(pq) in equations 3 and 4.. for which the range of integration now becomes 0 to I. since now A'(p|q)40 for p« q.
This gives: and projection kernel hence becomes, The algorithm now involves using $K(p|q)$ instead of $k(p|q)$ in equations \ref{eqn:a5} and \ref{eqn:a6}, for which the range of integration now becomes 0 to 1, since now $K(p|q)\not=0$ for $p<q$ .
One would expect that in successive iterations. the approximation Φ(ρ) to the observed distribution improve. and this can be used to decide when to terminate the algorithm.
One would expect that in successive iterations, the approximation $\Phi(p)$ to the observed distribution improve, and this can be used to decide when to terminate the algorithm.
Two different initial guesses for G7 (9)) were tried. the first was peaked around g~(0.6. the second was constant for all q.
Two different initial guesses for $\psi^0(q)$ ) were tried, the first was peaked around $q \sim 0.6$, the second was constant for all $q$.
These two initial guesses thus cover two extremes of possiblec(q).
These two initial guesses thus cover two extremes of possible.
. For both initial guesses. quickly converges to the form shown in Fig. 2..
For both initial guesses, quickly converges to the form shown in Fig. \ref{fig:psi}.
To determine the indicative errors in the determination of. the confidence limit histograms shown in Fig.
To determine the indicative errors in the determination of, the confidence limit histograms shown in Fig.
|. were fit with polynomials and Lucy deconvolution was applied to these polynomials.
\ref{fig:bind} were fit with polynomials and Lucy deconvolution was applied to these polynomials.
The resulting intrinsic distribution functions are also shown in Fig. 2..
The resulting intrinsic distribution functions are also shown in Fig. \ref{fig:psi},
and the shaded area hence represents the confidence interval forο).
and the shaded area hence represents the $\sim$ confidence interval for.
.. The final (p) (e. the final approximation to the observed distribution function) is shown in Fig. |.
The final $\Phi(p)$ (i.e. the final approximation to the observed distribution function) is shown in Fig. \ref{fig:bind},
and as it can be seen. it fits with the observed distribution within the error bars.
and as it can be seen, it fits with the observed distribution within the error bars.
The shown in Fig.
The shown in Fig.
2 has <gq=0.57. standard deviation 06=0.164. and skewness. 54,=q'r=0.62.
\ref{fig:psi} has $<q>~=~0.57$, standard deviation $\sigma~=~0.164$, and skewness, $\gamma_1~=~q^3/\sigma^3~=~-0.62$.
Eqn.
Eqn.
|. tand hence results obtained from inverting it also) applies only for a sample of randomly oriented oblate spheroids.
\ref{eqn:i1} (and hence results obtained from inverting it also) applies only for a sample of randomly oriented oblate spheroids.
The basic assumption hence is that the galaxy sample we are working with has an unbiased distribution of inclination angles.
The basic assumption hence is that the galaxy sample we are working with has an unbiased distribution of inclination angles.
The selection criteria for the FIGGS sample (from which the current sample is drawn) includes a requirement that the optical major axis of the galaxy be >>1.
The selection criteria for the FIGGS sample (from which the current sample is drawn) includes a requirement that the optical major axis of the galaxy be $> 1^{'}$.
Dwarf galaxies are generally dust poor (foreg.seeWalteretal.2007:Galametz 2009).. and hence to a good approximation have optically thin dises.
Dwarf galaxies are generally dust poor \citep[for eg. see ][]{wal07,gal09}, and hence to a good approximation have optically thin discs.
Highly inclined. galaxies will hence be over represented in a diameter limited sample. i.e. our sample is biased towards edge on dises.
Highly inclined galaxies will hence be over represented in a diameter limited sample, i.e. our sample is biased towards edge on discs.
This means that the true mean intrinsic axial ratio would be even larger than what we have estimated above.
This means that the true mean intrinsic axial ratio would be even larger than what we have estimated above.
It is worth noting that as the intrinsic axial ratio gets closer to [.0. the magnitude of this bias decreases. and hence the bias in our estimate should not be large.
It is worth noting that as the intrinsic axial ratio gets closer to 1.0, the magnitude of this bias decreases, and hence the bias in our estimate should not be large.
The mean intrinsic axial ratio that we obtain. viz.
The mean intrinsic axial ratio that we obtain, viz.
<qον is substantially larger than the value of 0.2 usually adopted for the discs of large spiral galaxies.
$<q> \sim 0.57$ is substantially larger than the value of $0.2$ usually adopted for the discs of large spiral galaxies.
The value from our sample is more than twice as large as older measurements of <q>> in stellar dises of Magellanie irregular galaxies by Heidmann.Hei-dmann.&deVaucouleurs(1972). (ranging from 0.20 to 0.24).
The value from our sample is more than twice as large as older measurements of $<q>$ in stellar discs of Magellanic irregular galaxies by \citet{heidmann72} (ranging from 0.20 to 0.24).
Consistent with this. our sample contains no very flat galaxies.
Consistent with this, our sample contains no very flat galaxies.
In fact. as can be seen from Fig.
In fact, as can be seen from Fig.
|. there are no galaxies with apparent axial ratio <0.2 in the sample.
\ref{fig:bind} there are no galaxies with apparent axial ratio $<0.2$ in the sample.
Further. if we look at the distribution of the observed axial ratio of different classes of galaxies in the Automated Photographic Measuring (APM) survey as in Lambas.Maddox&Loveday(1992). then our histogram resembles those for ellipticals and SOs more closely than that for spirals.
Further, if we look at the distribution of the observed axial ratio of different classes of galaxies in the Automated Photographic Measuring (APM) survey as in \citet{lam92}, then our histogram resembles those for ellipticals and S0s more closely than that for spirals.
This is another qualitative indication that the underlying intrinsic distribution of axial ratios has a higher mean than is typical for spirals.
This is another qualitative indication that the underlying intrinsic distribution of axial ratios has a higher mean than is typical for spirals.
Interestingly. the value of <q> we obtain matches well with what Staveley-Smith.Davies&Kinman(1992):&Popescu(1995) derived for thestellar disces in dwarfs.
Interestingly, the value of $<q>$ we obtain matches well with what \citet{sta92,binggeli95} derived for the discs in dwarfs.
The thickness of the dises of dwarf galaxies is contrary to what one might have naively expected for a gas disc. since. in general. collisions between gas clouds should cause them to quickly settle into a thin disc.
The thickness of the discs of dwarf galaxies is contrary to what one might have naively expected for a gas disc, since, in general, collisions between gas clouds should cause them to quickly settle into a thin disc.
However. this large axial ratio is probably consistent with the large gas dispersion in comparison to the rotational velocity observed in dwarf galaxies.
However, this large axial ratio is probably consistent with the large gas dispersion in comparison to the rotational velocity observed in dwarf galaxies.
For example. Kaufmann.WheelerandBullock(2007). did particle hydrodynamic simulations. to. show that dwarf galaxies with rotational velocities 40 did not originate as thin disces but thick systems.
For example, \citet{kau07} did particle hydrodynamic simulations to show that dwarf galaxies with rotational velocities $\sim$ 40 did not originate as thin discs but thick systems.
This still leaves open the question of where the large velocity dispersion comes from.
This still leaves open the question of where the large velocity dispersion comes from.
Duttaetal.(2009). find good evidence for a scale free power spectrum of HI fluctuations in dwarf galaxies. consisent with what would be expected from a turbulent medium.
\cite{dut09} find good evidence for a scale free power spectrum of HI fluctuations in dwarf galaxies, consisent with what would be expected from a turbulent medium.
Interestingly. they also find thatthat the dwarfs must have relatively thick gas dises. similar to the conclusions reached here.
Interestingly, they also find thatthat the dwarfs must have relatively thick gas discs, similar to the conclusions reached here.
Assuming that the origin of the velocity dispersion is turbulent motions in the ISM. the timescale for dissipation is
Assuming that the origin of the velocity dispersion is turbulent motions in the ISM, the timescale for dissipation is
assumes à constant collision rate per unit. volume in the cluster cores which dominates the total collision rate.
assumes a constant collision rate per unit volume in the cluster cores which dominates the total collision rate.
It follows from. the more general. formula Poxpz bv assuming Virial equilibrium to estimate the velocity dispersion as σκply (Verbunt&πι1987)..
It follows from the more general formula $\Gamma\propto\rho_{c}^{2}r_{c}^{3}/\sigma$ by assuming Virial equilibrium to estimate the velocity dispersion as $\sigma\propto\rho_{c}^{1/2}r_{c}$ \citep{Verbunt}.
This relationship is used to calculate the collision rates presented in table 1..
This relationship is used to calculate the collision rates presented in table \ref{GC data}.
Despite studying only the confirmed: clusters in. the RBC it ds. possible that our. sample contains some contamination from stellar sources.
Despite studying only the confirmed clusters in the RBC it is possible that our sample contains some contamination from stellar sources.
We identify misclassified clusters by finding objects whose stars should have collided many times over a Llubble time.
We identify misclassified clusters by finding objects whose stars should have collided many times over a Hubble time.
We can estimate the timescale for stellar collisions [rom E by assuming the cluster is comprised of solar tvpe stars in order to find the constant in the above proportionality (seeeq.3Verbunt&Lut1987).
We can estimate the timescale for stellar collisions from $\Gamma$ by assuming the cluster is comprised of solar type stars in order to find the constant in the above proportionality \citep[see eq. 3][]{Verbunt}.
. This ogives that the timescale on which we expect a star to collide is: Ler is the total collision rate and Ny=(4Barepy, is the total number of stars in the cluster core.
This gives that the timescale on which we expect a star to collide is: $\Gamma_{T}$ is the total collision rate and $N_{\star}=(4/3)\pi r_{c}^{3}\rho_{c}$ is the total number of stars in the cluster core.
From this equation we find 5 GCs to have Zu<10vr.
From this equation we find 5 GCs to have $t_{coll}<10^{9}\rm{yr}$.
One of the clusters (D431) has a bright star contaminating its profile.
One of the clusters (B431) has a bright star contaminating its profile.
llowever the other four clusters (BOOGL). BOOGD. D292D. and 1-50) are well fit and we propose that they are likely to be misclassified stars.
However the other four clusters (B006D, B096D, B292D, and B480) are well fit and we propose that they are likely to be misclassified stars.
It has previously been suggested that D292D max not be a GC (Lluxoretal.2008). and we note that the radial velocity estimates in the RBC of three of these clusters are consistent with being Milky Way halo stars.
It has previously been suggested that B292D may not be a GC \citep{Huxor} and we note that the radial velocity estimates in the RBC of three of these clusters are consistent with being Milky Way halo stars.
We remove these clusters from the following analysis.
We remove these clusters from the following analysis.
Having identified the best fitting Wine mocel to describe each cluster we investigate the reliability of the parameters found.
Having identified the best fitting King model to describe each cluster we investigate the reliability of the parameters found.
Due to the small spatial scale of the clusters we are fitting. it is likely that the errors on these parameters will be dominated by errors in the PSE model which is convolved with the cluster.
Due to the small spatial scale of the clusters we are fitting, it is likely that the errors on these parameters will be dominated by errors in the PSF model which is convolved with the cluster.
To investigate the magnitude of the typical measurement errors. we compare the results obtained from fitting the same cluster from dillerent observations.
To investigate the magnitude of the typical measurement errors, we compare the results obtained from fitting the same cluster from different observations.
In total. 115 clusters are present in multiple images.
In total, 115 clusters are present in multiple images.
These observations have all been taken under slightly dilferent conditions ancl will have cilferent PSE models.
These observations have all been taken under slightly different conditions and will have different PSF models.
Pherefore by fitting these clusters independently and comparing the resulting parameters we can estimate the reliability of the parameters caleulated.
Therefore by fitting these clusters independently and comparing the resulting parameters we can estimate the reliability of the parameters calculated.
Figure 1. shows the cillerences between the derived. parameters for the clusters fit in more than one image.
Figure \ref{fig:compare} shows the differences between the derived parameters for the clusters fit in more than one image.
For clusters brighter than Ix-15mag we Lind good agreement between the parameters found.
For clusters brighter than K=15mag we find good agreement between the parameters found.
However it can be seen that the scatter increases significantly for clusters with Ix 15mag.
However it can be seen that the scatter increases significantly for clusters with $>$ 15mag.
This suggests that the errors on the parameters found for these faint clusters are significantly higher.
This suggests that the errors on the parameters found for these faint clusters are significantly higher.
The large deviation in the magnitudes found for these faint clusters is likely to be duc to contamination from nearby stars or surface brightness Ductuations.
The large deviation in the magnitudes found for these faint clusters is likely to be due to contamination from nearby stars or surface brightness fluctuations.
We expect that the variation observed in figure 1. gives a reasonable estimate of the errors on the parameters found for all the GC's studied.
We expect that the variation observed in figure \ref{fig:compare} gives a reasonable estimate of the errors on the parameters found for all the GCs studied.
In the following analvsis. we include all 235 GC's studied but note that the errors on individual parameters for the faintest clusters will be large.
In the following analysis, we include all 235 GCs studied but note that the errors on individual parameters for the faintest clusters will be large.
Since most of the CC's studied: ancl all X-ray GC's are brighter than this. our conclusions should not be sensitive to this increased CLLOL.
Since most of the GCs studied and all X-ray GCs are brighter than this, our conclusions should not be sensitive to this increased error.
the hours each student is awake f... Gueeligible to first order).
the hours each student is awake $f_{zzz}$ (negligible to first order).
We originally caleulated this to be unity. but later concluded that Barucs.ODrieu.Fortuey,&UWurtord(2002) did not merit recognition as a useful paper.
We originally calculated this to be unity, but later concluded that \citet{lpl02} did not merit recognition as a useful paper.
Readers ueed uot be reminded of Steward eraduate student coutributious such as the epic "Super πισο Iuterforometric Telescope: A New Paracigua In Optical luterferomnetrv (Ruduickoetal.1999).. aud the “The Effects of Moore's Law and Slacking on Large Computations” (Cottbrathetal.1999).
Readers need not be reminded of Steward graduate student contributions such as the epic “Super Huge Interferometric Telescope: A New Paradigm In Optical Interferometry” \citep{rudnick99}, and the “The Effects of Moore's Law and Slacking on Large Computations” \citep{gottbrath99}.
. Ouly 2 students joined the LPL eraduate program iu fall 2001. compared to 10 at Steward Observatory.
Only 2 students joined the LPL graduate program in fall 2001, compared to 10 at Steward Observatory.
Ouly through beseiug and pleading on behalf of LPL biewies was the Lunar Planetary Lab luilding spared reallocation of office space to new Steward eraduate studoeuts.
Only through begging and pleading on behalf of LPL bigwigs was the Lunar Planetary Lab building spared reallocation of office space to new Steward graduate students.
Barwes.οDieu.Fortuev.&Ihuford.(2002) contained πμJed words such as cCdotoral aud dlieitiniate.
\citet{lpl02} contained misspelled words such as “dotoral” and “illigitimate”.
This «emonstrates the inability to run a staπας, spoll-checker.
This demonstrates the inability to run a standard spell-checker.
We later found that their paper spell-checked. but witha Speak-And-Spell.
We later found that their paper spell-checked, but with a Speak-And-Spell.
A standard Speas-And-Spell contaiis SO vocabulary words aud Ll baruvard animal sounds. so they were unable to check the speline of their longer words.
A standard Speak-And-Spell contains 80 vocabulary words and 14 barnyard animal sounds, so they were unable to check the spelling of their longer words.
We can not rule out the possibilitv that the root othe word “dotoral” is actually "dolt. lOWOCVOT.
We can not rule out the possibility that the root of the word “dotoral” is actually “dolt”, however.
The utter relevance of LPL evacuate studeuts has been enipincally deiioustrated.
The utter irrelevance of LPL graduate students has been empirically demonstrated.
conditions.
conditions.
To begin with, we take the same approach as above.
To begin with, we take the same approach as above.
Stable multi-resonant configurations of this family are listed as the second set of entries in table (1).
Stable multi-resonant configurations of this family are listed as the second set of entries in table (1).
We simulated the evolutions of these systems with 20 integrations each.
We simulated the evolutions of these systems with 20 integrations each.
After completion, a clear boundary between initial conditions that result in smooth migration and those that result in scattering developed.
After completion, a clear boundary between initial conditions that result in smooth migration and those that result in scattering developed.
Namely, all setups where Saturn Uranus are initially in a 2:1 MMR were characterized by smooth evolutions.
Namely, all setups where Saturn Uranus are initially in a 2:1 MMR were characterized by smooth evolutions.
A similar scenario describes the fate of initial conditions where Saturn Uranus are in a 3:2 MMR while Uranus Neptune are in a 3:2 or a 4:3 MMR.
A similar scenario describes the fate of initial conditions where Saturn Uranus are in a 3:2 MMR while Uranus Neptune are in a 3:2 or a 4:3 MMR.
However in the same context, if Uranus and Neptune start out in a 5:4 or a 6:5 MMR, ice giant/gas giant scattering as well as transient phases of high eccentricities are present.
However in the same context, if Uranus and Neptune start out in a 5:4 or a 6:5 MMR, ice giant/gas giant scattering as well as transient phases of high eccentricities are present.
Particularly, for the configuration where Uranus Neptune start out in a 5:4 MMR, of the integrations were successful with of them exhibiting close encounters between an ice giant and both gas giants.
Particularly, for the configuration where Uranus Neptune start out in a 5:4 MMR, of the integrations were successful with of them exhibiting close encounters between an ice giant and both gas giants.