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The inner galaxy wake is weaker past pericenter (phase 7/2).
The inner galaxy wake is weaker past pericenter (phase $\pi/2$ ).
In the absence of the halo. the first multipole contributing o the dillerential ordal acceleration. of the disk is at cquacdrupole (f= 2) order.
In the absence of the halo, the first multipole contributing to the differential or acceleration of the disk is at quadrupole $l=2$ ) order.
Lt is straightforward to convince oneself of this fact: the f=0 term is constant vielding no orce. the /2T term is proportional to i vielding a spatially constant force. and therefore the /=2 term. provides the owest order cilferential force.
It is straightforward to convince oneself of this fact: the $l=0$ term is constant yielding no force, the $l=1$ term is proportional to $r$ yielding a spatially constant force, and therefore the $l=2$ term provides the lowest order differential force.
Because the warp has m=1 symmetry. the dominant warp inducing term will be /=2. m{=1.
Because the warp has $m=1$ symmetry, the dominant warp inducing term will be $l=2$, $|m|=1$.
Similar svmmetries apply for the action of the »vrturbed. halo on the disk.
Similar symmetries apply for the action of the perturbed halo on the disk.
Including the halo warp. the dipole still can not produce the classic odd. integral-sign warp but causes an even distortion.
Including the halo warp, the dipole still can not produce the classic odd integral-sign warp but causes an even distortion.
Phe lowest order halo wake that can excite an integral-sign warp is also the /—2.|m|=1 component.
The lowest order halo wake that can excite an integral-sign warp is also the $l=2, |m|=1$ component.
To illustrate the three-dimensional structure. Figure 4 renders the isosurface corresponding to of peak amplitude wake contoured in Figure 3..
To illustrate the three-dimensional structure, Figure \ref{fig:render} renders the isosurface corresponding to of peak amplitude wake contoured in Figure \ref{fig:halowake2}.
The wake is symmetric about the satellites orbital plane aud this plane is easily. visualized.
The wake is symmetric about the satellite's orbital plane and this plane is easily visualized.
A wake must asymmetric about the z axis to cause a cdillerential vertical acceleration of the disk: in other words. a satellite in the disk plane produces no warp.
A wake must asymmetric about the $z$ axis to cause a differential vertical acceleration of the disk; in other words, a satellite in the disk plane produces no warp.
Phe maximum vertical force occurs when the pattern shown in Figures 3. and 4. is oriented perpendicular to the
The maximum vertical force occurs when the pattern shown in Figures \ref{fig:halowake2} and \ref{fig:render} is oriented perpendicular to the
of re-analyses of the data with different. assumptions. and simulations.
of re-analyses of the data with different assumptions and simulations.
To asses the validity of determining the physical parameters of 4436 and its planet along with the flux normalizations we used a simulation technique.
To asses the validity of determining the physical parameters of 436 and its planet along with the flux normalizations we used a simulation technique.
We generated ten simulated data sets with model parameters from the averaged results of G07. DO7. and S08. random noise properties similar to our observed data. and random offsets in the visit subsets.
We generated ten simulated data sets with model parameters from the averaged results of G07, D07, and S08, random noise properties similar to our observed data, and random offsets in the visit subsets.
Wiσα fitted these data using the MCMC procedure described abov and examined whether there was a systematic deviation in the determined parameters compared to the known underlying parameters.
We fitted these data using the MCMC procedure described above and examined whether there was a systematic deviation in the determined parameters compared to the known underlying parameters.
We found no systematic trend in the deviations. and their distribution followed that expected from the adopted parameter uncertainties.
We found no systematic trend in the deviations, and their distribution followed that expected from the adopted parameter uncertainties.
We also carried out a number of analyses on different realizations of the primary data set to ascertain whether some particular treatment of the data influenced. our results.
We also carried out a number of analyses on different realizations of the primary data set to ascertain whether some particular treatment of the data influenced our results.
We repeated our analysis on subsets created by removing one of the visit sets to test the robustness of the results.
We repeated our analysis on subsets created by removing one of the visit sets to test the robustness of the results.
We repeated this analysis six times. each time removing a different visit set.
We repeated this analysis six times, each time removing a different visit set.
To ascertain whether the reduced time resolution from the ss binning had an affect. we analyzed data sets where the HHz samples were binned to smaller ranges (10. 15. and 30ss).
To ascertain whether the reduced time resolution from the s binning had an affect, we analyzed data sets where the Hz samples were binned to smaller ranges (10, 15, and s).
We also repeated our analysis assuming a quadratic form of limb darkening (equivalent constants to the values for the non-linear model given above are y, = 0.30. y» = 0.49).
We also repeated our analysis assuming a quadratic form of limb darkening (equivalent constants to the values for the non-linear model given above are $\gamma_1$ = 0.30, $\gamma_2$ = 0.49).
Additionally. analyses were carried out by two of us separately and with slightly different methods.
Additionally, analyses were carried out by two of us separately and with slightly different methods.
In all the cases the resulting parameters did not deviate from their nominal value given in Table | by more than 1.5c.
In all the cases the resulting parameters did not deviate from their nominal value given in Table \ref{t3} by more than $\sigma$.
Assuming a constant orbital period. transit timing deviations over the course of the observations. which spanned 11 orbits of the planet. could possibly adversely affect our analysis by distorting the morphology of the folded light curve.
Assuming a constant orbital period, transit timing deviations over the course of the observations, which spanned 11 orbits of the planet, could possibly adversely affect our analysis by distorting the morphology of the folded light curve.
The transit timings we determined from our principle analysis are remarkably regular. but might not represent the true values as they depend on the physical parameters (1.e. stellar radius. and planet radius and inclination) determined from analyzing the phase folded light curve.
The transit timings we determined from our principle analysis are remarkably regular, but might not represent the true values as they depend on the physical parameters (i.e. stellar radius, and planet radius and inclination) determined from analyzing the phase folded light curve.
To study this issue we repeated the transit timing analysis with the system parameters fixed to a representative of the G07. 07. and S08 determined values and allowing the relative flux normalizations to vary.
To study this issue we repeated the transit timing analysis with the system parameters fixed to a representative of the G07, D07, and S08 determined values and allowing the relative flux normalizations to vary.
From this we found possible timing variations of up to ss and the y for the fit was 170.9.
From this we found possible timing variations of up to s and the $\chi^2$ for the fit was 170.9.
For comparison the y for our principle analysis was 168.6.
For comparison the $\chi^2$ for our principle analysis was 168.6.
for 169 degrees of freedom.
for 169 degrees of freedom.
The low value of the fit quality metric from this alternative analysis suggests that the results from our primary analysis might not represent a unique model of the data.
The low value of the fit quality metric from this alternative analysis suggests that the results from our primary analysis might not represent a unique model of the data.
However. could true transit timings of this magnitude bias the system parameters determined by analyzing a phase folded light curve to the level our results are deviant from the results?
However, could true transit timings of this magnitude bias the system parameters determined by analyzing a phase folded light curve to the level our results are deviant from the results?
To answer this question we generated simulated data similar to that described above for investigating the affect of variable flux normalizations.
To answer this question we generated simulated data similar to that described above for investigating the affect of variable flux normalizations.
In this case we used transit times for the individual samples equal to those found from holding the system parameters fixed to those determined from the data.
In this case we used transit times for the individual samples equal to those found from holding the system parameters fixed to those determined from the data.
We then analyzed the simulated data in the same manner as we analyzed the real data.
We then analyzed the simulated data in the same manner as we analyzed the real data.
From the analysis of 10 simulated data sets we found that transit timing offsets up to at least ss did not significantly bias the results.
From the analysis of 10 simulated data sets we found that transit timing offsets up to at least s did not significantly bias the results.
For all the simulations the analysis algorithm returned the parameters that were used to create the simulated data within the one sigma uncertainties and without a systematic offset.
For all the simulations the analysis algorithm returned the parameters that were used to create the simulated data within the one sigma uncertainties and without a systematic offset.
The above analysis suggests that transit timings have not caused our results to be biased.
The above analysis suggests that transit timings have not caused our results to be biased.
However. this analysis. still poses the question of whether the inverse of the previous question is true.
However, this analysis still poses the question of whether the inverse of the previous question is true.
That is. can adopting systematically different physical parameters lead to spurious transit timing offsets?
That is, can adopting systematically different physical parameters lead to spurious transit timing offsets?
To study this we again turned to a simulation analysis.
To study this we again turned to a simulation analysis.
We generated simulated data using our determined system parameters and no transit timing offsets.
We generated simulated data using our determined system parameters and no transit timing offsets.
We then repeated the analysis of fixing the parameters to those determined from the data while allowing the individual transit times and flux normalizations to vary.
We then repeated the analysis of fixing the parameters to those determined from the data while allowing the individual transit times and flux normalizations to vary.
In this analysis we consistently found transit timing variations with the same magnitude (~ ss) as those from above. and a fit y close to. but still higher. than that obtained from using the correct (input) model parameters.
In this analysis we consistently found transit timing variations with the same magnitude $\sim$ s) as those from above, and a fit $\chi^2$ close to, but still higher, than that obtained from using the correct (input) model parameters.
As we know the transit timings are not correct because the simulated data were created assuming a constant period. we can be sure that transit timings determinec from an analysis with assumed improper physical parameters would give spurious results.
As we know the transit timings are not correct because the simulated data were created assuming a constant period, we can be sure that transit timings determined from an analysis with assumed improper physical parameters would give spurious results.
Another possible source of uncertainty in our analysis is that arising from. corellated. or “red.” noise (?).
Another possible source of uncertainty in our analysis is that arising from corellated, or “red,” noise \citep{pont06}.
We assessed the influence of this type of noise on our results by re-fitting different realizations of the original data adjusted with the “prayer bead" method (?)..
We assessed the influence of this type of noise on our results by re-fitting different realizations of the original data adjusted with the “prayer bead” method \citep{moutou04}.
We used a Levenberg-Marquardt algorithm to perform the parameter optimization,
We used a Levenberg-Marquardt algorithm to perform the parameter optimization.
The individual. data sets were modified by shifting.- the residuals from the principle analysis deseribed in $3.1. by à random number and adding them back to the data.
The individual data sets were modified by shifting the residuals from the principle analysis described in 3.1 by a random number and adding them back to the data.
This was repeated 100000 times.
This was repeated 000 times.
The standard deviation of the resulting parameter distributions gives the uncertainties.
The standard deviation of the resulting parameter distributions gives the uncertainties.
In this investigation we found uncertainties about half or less than those determined from the MCMC analysis.
In this investigation we found uncertainties about half or less than those determined from the MCMC analysis.
Thus. we conclude that corellated noise is not a significant source of uncertainty in the data.
Thus, we conclude that corellated noise is not a significant source of uncertainty in the data.
This is likely because the data set includes measurements from two different orbits (1.6. the two measurements are independent and uncorrelated) for almost every transit phase.
This is likely because the data set includes measurements from two different orbits (i.e. the two measurements are independent and uncorrelated) for almost every transit phase.
From all of the above investigations we conclude that our analysis is relatively unbiased. the determined parameters are a fair representation of the data. and the assigned parameter errors are realistic.
From all of the above investigations we conclude that our analysis is relatively unbiased, the determined parameters are a fair representation of the data, and the assigned parameter errors are realistic.
Our determmed mass and radius for GJ4436b is placed in context with the previous results of GO7. DO7. and SO8. the solar system ice giants. and the theoretical mass-radius
Our determined mass and radius for 436b is placed in context with the previous results of G07, D07, and S08, the solar system ice giants, and the theoretical mass-radius
which led to clarification of a number of points in our original manuscript.
which led to clarification of a number of points in our original manuscript.
This work has been parüallv supported by Che grant in aid (16540213. 17740108) of the Ministry of Education. science. Culture. and Sports in Japan.
This work has been partially supported by the grant in aid (16540213, 17740108) of the Ministry of Education, Science, Culture, and Sports in Japan.
are similar to those of MSPs with IIelium white dwarf companions (hat are found in both GCs and the Galactic disk.
are similar to those of MSPs with Helium white dwarf companions that are found in both GCs and the Galactic disk.
A main-sequence companion seems unlikely as such svstems have only been found in dense GC's 2006).
A main-sequence companion seems unlikely, as such systems have only been found in dense GCs .
. Furthermore. we see no evidence for eclipses. although we lack coverage near conjunction when eclipses would be most likely.
Furthermore, we see no evidence for eclipses, although we lack coverage near conjunction when eclipses would be most likely.
Svstems similar to NGC 5086À have been found in other GCs of comparabledensitv?.
Systems similar to NGC 5986A have been found in other GCs of comparable.
. If a precise position can be measured through future pulsar timing. a search for an optical counterpart may be feasible.
If a precise position can be measured through future pulsar timing, a search for an optical counterpart may be feasible.
Rough flux density. estimates were obtained by assuming that the offl-pulse RAIS noise levels are described by the radiometer equation. We find 5,= 21. 27. and 45(dv at 2GIIz. 1.5Gllz. and 820MlIE. respectively. which vields a spectral index of a~ —0.9. flatter than the canonical value of —1.7.
Rough flux density estimates were obtained by assuming that the off-pulse RMS noise levels are described by the radiometer equation, We find $S_{\nu} = 21$ , $27$ , and $45\; \uJy$ at $2\; \GHz$, $1.5\; \GHz$, and $820\; \MHz$, respectively, which yields a spectral index of $\alpha \sim -0.9$ , flatter than the canonical value of $-1.7$.
We searched ten clusters for pulsars. five of which have 10<p.4x104L.pe but found only one MSP.
We searched ten clusters for pulsars, five of which have $10^3 < \rho\rmsub{c} < 4 \times 10^4\; \Lsun\, \pcubpc$, but found only one MSP.
This stands in contrast (ο previous survevs that. [ound several pulsars in low density clusters.
This stands in contrast to previous surveys that found several pulsars in low density clusters.
We therefore consider what [actors may have led to our null results.
We therefore consider what factors may have led to our null results.
Fie.
Fig.
3. shows the approximate limiting lhumünosi(v of our searches. where D7. alone with the 1.4GIIz pseudo-Iuminosities of pulsars with reliable {hix density measurements fromIIRSOT.. scaled to 2CGIlz using a spectral index of —1L.7 (NGC 59864 is also shown).
\ref{fig:sensitivity} shows the approximate limiting luminosity of our searches, where $L\rmsub{min} = S\rmsub{min} D^2$ , along with the $1.4\; \GHz$ pseudo-luminosities of pulsars with reliable flux density measurements from, scaled to $2\; \GHz$ using a spectral index of $-1.7$ (NGC 5986A is also shown).
It is immediately obvious that were more sensitive (han our searches.
It is immediately obvious that were more sensitive than our searches.
We can attribute this to wo factors.
We can attribute this to two factors.
The first is that all the GCs from had D<8 kpe. wilh the exception of M53 (D=11.8 kpc).
The first is that all the GCs from had $D < 8\; \kpc$ , with the exception of M53 $D = 17.8\; \kpc$ ).
Dy comparison. only (wo clusters in our survey have D«8kpc. while most have D>10kpc.
By comparison, only two clusters in our survey have $D < 8\; \kpc$, while most have $D > 10\; \kpc$.
The second [actor is that. were able to reach lower limiting [τικ densities using Arecibo than we could with the GBT.
The second factor is that were able to reach lower limiting flux densities using Arecibo than we could with the GBT.
Despite these factors. 3/10 of our searches should have been sensitive to the brightest e53% of a population similar to the pulsars. and 8/10 of our searches should have been sensitive to the brightest (wo pulsars.
Despite these factors, 3/10 of our searches should have been sensitive to the brightest $\sim 53\%$ of a population similar to the pulsars, and 8/10 of our searches should have been sensitive to the brightest two pulsars.
Furthermore. four out offiveof (he clusters with pe>10*L.*were among the five most sensitively searchedGCs in our sample.
Furthermore, four out offiveof the clusters with $\rho\rmsub{c} > 10^3\; \Lsun\, \pcubpc$were among the five most sensitively searchedGCs in our sample.
In other
In other
Calculations of state-to-state cross sections from initial rotational state /=0 for the excitation of CO by H on the MRCI. CCSD(T). BBH. and WKS surfaces were performed for collision energies at 400 and 800 em! as given in Figs.
Calculations of state-to-state cross sections from initial rotational state $J=0$ for the excitation of CO by H on the MRCI, CCSD(T), BBH, and WKS surfaces were performed for collision energies at 400 and 800 $^{-1}$ as given in Figs.
2 and 3. respectively.
2 and 3, respectively.
The cross sections using the BBH and WKS surfaces are In good agreement with those obtained by Green et al. (
The cross sections using the BBH and WKS surfaces are in good agreement with those obtained by Green et al. (
1995) and Balakrishnan et al. (
1995) and Balakrishnan et al. (
2002). respectively.
2002), respectively.
While the results presented here adopted the rigid-rotor approximation. we performed tests with BBH and WKS and found that the resulting cross sections using the full surfaces. Le.. with vibrational motion. are practically identical.
While the results presented here adopted the rigid-rotor approximation, we performed tests with BBH and WKS and found that the resulting cross sections using the full surfaces, i.e., with vibrational motion, are practically identical.
The largest difference was less than ~10% for the 0—| transition with WKS.
The largest difference was less than $\sim$ for the $0\rightarrow 1$ transition with WKS.
Our rigid-rotor cross sections in Figs.
Our rigid-rotor cross sections in Figs.
2 and 3 further confirm the previously noted result that the O-| transition using WKS is approximately an order of magnitude larger than that obtained with BBH.
2 and 3 further confirm the previously noted result that the $0\rightarrow 1$ transition using WKS is approximately an order of magnitude larger than that obtained with BBH.
Comparing now the rigid-rotor cross sections obtained with the new CCSD(T) and MRCI surfaces. we see that 1) the two sets of cross sections are in. very good agreement. i1) they reproduce the even AJ propensity obtained with BBH. and iii) the O— ] transition is small with a value similar to that obtained with BBH. in contrast to the WKS result.
Comparing now the rigid-rotor cross sections obtained with the new CCSD(T) and MRCI surfaces, we see that i) the two sets of cross sections are in very good agreement, ii) they reproduce the even $\Delta J$ propensity obtained with BBH, and iii) the $0\rightarrow 1$ transition is small with a value similar to that obtained with BBH, in contrast to the WKS result.
These observations can be interpreted by examining the .t=| terms displayed in Fig.
These observations can be interpreted by examining the $\lambda=1$ terms displayed in Fig.
Ib since the dominant contributions to the state-to-state cross sections originate from potential coupling matrix elements with t=[AJ].
1b since the dominant contributions to the state-to-state cross sections originate from potential coupling matrix elements with $\lambda=|\Delta J|$.
The 44=| components from the CCSD(T) and MRCI calculations are seen to be very similar. both displaying a peak near 5 aj and comparable long-range behavior.
The $\lambda=1$ components from the CCSD(T) and MRCI calculations are seen to be very similar, both displaying a peak near 5 $_0$ and comparable long-range behavior.
Conversely. for the WKS surface the t=1 component peaks at a shorter internuclear distance. closer to the barrier location. and has a different long-range behavior.
Conversely, for the WKS surface the $\lambda=1$ component peaks at a shorter internuclear distance, closer to the barrier location, and has a different long-range behavior.
Indeed. the long-range tail of the WKS potential appears to be decaying slower than the other potentials.
Indeed, the long-range tail of the WKS potential appears to be decaying slower than the other potentials.
The BBH 2t=1 component is smaller than that of the other surfaces which may explain its slightly smaller 0—| cross section.
The BBH $\lambda=1$ component is smaller than that of the other surfaces which may explain its slightly smaller $0\rightarrow 1$ cross section.
To further test the apparent discrepancy related to the t=| component of the Legendre expansion. we examined scattering results for excitation of the J=| and 2 rotational states
To further test the apparent discrepancy related to the $\lambda=1$ component of the Legendre expansion, we examined scattering results for excitation of the $J=1$ and 2 rotational states
pulsars in the EGRET data.
pulsars in the EGRET data.
Clearly. the range of absolute offset values. based on phase selected ~-ray events. is minimal in the case of the pulsars.
Clearly, the range of absolute offset values, based on phase selected $\gamma$ -ray events, is minimal in the case of the pulsars.
The 3EG source related to Vela is a very special case.
The 3EG source related to Vela is a very special case.
It is the strongest known -ray source. and one of the best localized. Ros20.0217.
It is the strongest known $\gamma$ -ray source, and one of the best localized, $R_{95} = 0.021^\circ$.
The CL contour of the EGRET detection and the offset with Vela are both one order of magnitude less than the typical values of these quantities in the 3EG.
The CL contour of the EGRET detection and the offset with Vela are both one order of magnitude less than the typical values of these quantities in the 3EG.
The misplacing for Vela occurs because the analysis technique privileges the discovery and correct detection of weaker sources. and 1t Is applied to all EGRET sources in the 3EG (2/3 of which are unidentified with no obvious candidates) identically.
The misplacing for Vela occurs because the analysis technique privileges the discovery and correct detection of weaker sources, and it is applied to all EGRET sources in the 3EG (2/3 of which are unidentified with no obvious candidates) identically.
The offset of the Vela position and. m general. of bright sources. 1s minimized by using map bins smaller than the standard 0.57 used in the 3EG.
The offset of the Vela position and, in general, of bright sources, is minimized by using map bins smaller than the standard $^\circ$ used in the 3EG.