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Asvuunetric depolarization is not observed. which could be explained if he main source axis were orieuted approximately parallel o the plane of the sky.
Asymmetric depolarization is not observed, which could be explained if the main source axis were oriented approximately parallel to the plane of the sky.
This oricutation would also be consistent with the large size of the source aud with the ow flux density of the core.
This orientation would also be consistent with the large size of the source and with the low flux density of the core.
The conditions wader which a galaxy can change its degree of nuclear activity remain unknown.
The conditions under which a galaxy can change its degree of nuclear activity remain unknown.
Interaction )etxceen galaxies is often invoked as a reason for trigecrine melear activity.
Interaction between galaxies is often invoked as a reason for triggering nuclear activity.
We find hints that interaction of the ealaxy hosting the radio core with nearby galaxies might )o takingplace. but this poiut has to beconfirmed through
We find hints that interaction of the galaxy hosting the radio core with nearby galaxies might be takingplace, but this point has to beconfirmed through
a factor of 2 due to attenuation by the waveplate. and additionally some blazars are themselves highly polarized at submillimetre wavelengths (227). to the extent that it becomes difficult to accurately correct the data for this effect.
a factor of 2 due to attenuation by the waveplate, and additionally some blazars are themselves highly polarized at submillimetre wavelengths \citep{1996ApJ...462L..23S,1998MNRAS.297..667N,2007AJ....134..799J} to the extent that it becomes difficult to accurately correct the data for this effect.
All observations with he polarimeter in the beam have thus been discarded.
All observations with the polarimeter in the beam have thus been discarded.
In data aken since 2002 August 8 the data headers indicate whether he polarimeter is connected and these data are easily discarded.
In data taken since 2002 August 8 the data headers indicate whether the polarimeter is connected and these data are easily discarded.
Between 1999 July 6 and 2002 August 8. the presence of the xolarimeter must be inferred by using other metadata.
Between 1999 July 6 and 2002 August 8, the presence of the polarimeter must be inferred by using other metadata.
In cases where it is inferred that the polarimeter has been used all pointing data from the night are discarded.
In cases where it is inferred that the polarimeter has been used all pointing data from the night are discarded.
This does leave open the rare yossibility that the polarimeter is in the beam for a setup pointing and then removed prior to doing any polarimeter observations and his ease can only be tested by examining the resulting flux density data.
This does leave open the rare possibility that the polarimeter is in the beam for a setup pointing and then removed prior to doing any polarimeter observations and this case can only be tested by examining the resulting flux density data.
For observations taken during the polarimeter commissioning yeriod (1998 May to 1999 July 6) the of presence polarimeter is harder to infer and13 can depend on observation log theentries made by the observer.
For observations taken during the polarimeter commissioning period (1998 May 13 to 1999 July 6) the presence of the polarimeter is harder to infer and can depend on observation log entries made by the observer.
This approach is fairly inaccurate and results in manual removal of observations for this period.
This approach is fairly inaccurate and results in manual removal of observations for this period.
For nights on which the Moon was observed (seee.g.ο). yoINtiIng observations are ignored because the SCUBA sensitivity must be adjusted in order to observe this bright source.
For nights on which the Moon was observed \citep[see e.g.,][]{2000JCMTNews15NJ}, pointing observations are ignored because the SCUBA sensitivity must be adjusted in order to observe this bright source.
That adjustment is not present in the data headers and it is easiest for the archive search to remove them.
That adjustment is not present in the data headers and it is easiest for the archive search to remove them.
This only accounts for 15 nights of data during the period covered by this paper.
This only accounts for 18 nights of data during the period covered by this paper.
Although the data reduction attempts to be insensitive to small focus changes. large focus shifts can still be problematic especially given that a pointing observation is always done prior to a focus (and also prior to the first focus of the evening).
Although the data reduction attempts to be insensitive to small focus changes, large focus shifts can still be problematic especially given that a pointing observation is always done prior to a focus (and also prior to the first focus of the evening).
The archive extraction routine disgards all pointings that are taken before the first focus of the night. all pointings that are taken more than 1.5 hours after the last focus of the night. and all pointings that are followed by a focus that changes by more than 0.2 mm unless they are closer to the previous focus than they are to the next.
The archive extraction routine disgards all pointings that are taken before the first focus of the night, all pointings that are taken more than 1.5 hours after the last focus of the night, and all pointings that are followed by a focus that changes by more than 0.2 mm unless they are closer to the previous focus than they are to the next.
Some nights have known problems with the secondary mirror (such as a failure of one axis) or problems with the dish shape resulting in very large variations in beam quality and calibration factors.
Some nights have known problems with the secondary mirror (such as a failure of one axis) or problems with the dish shape resulting in very large variations in beam quality and calibration factors.
These nights can not be detected automatically but are removed using a look-up table.
These nights can not be detected automatically but are removed using a look-up table.
Less than 10 nights were affected by these problems.
Less than 10 nights were affected by these problems.
Finally. all observations with a zenith sky opacity at 225 GHz greater than 0.30 are discarded (seee.g..2).
Finally, all observations with a zenith sky opacity at 225 GHz greater than 0.30 are discarded \citep[see e.g.,][]{2002MNRAS.336....1A}.
Calibration is based on long-term observations of Uranus and. to a lesser extent Mars. thereby producing a time-dependent flux calibration factor (FCF) for SCUBA.
Calibration is based on long-term observations of Uranus and, to a lesser extent Mars, thereby producing a time-dependent flux calibration factor (FCF) for SCUBA.
These have been very stable over the period. changes due to upgrades of the instrument are clearly seen and reflected in the changing FCF.
These have been very stable over the period, changes due to upgrades of the instrument are clearly seen and reflected in the changing FCF.
The values determined for Paper I were checked before re-processing the data and it was determined that the accuracy for the old narrow-band filter could be improved by calculating the FCF over shorter periods of a few months at a time rather than taking yearly averages.
The values determined for Paper I were checked before re-processing the data and it was determined that the accuracy for the old narrow-band filter could be improved by calculating the FCF over shorter periods of a few months at a time rather than taking yearly averages.
These calibration changes mean that for some periods the newly calculated flux densities can differ by up to 10 per cent from the results previously published in Paper I. In most cases the calibration difference is no more than 5 per cent.
These calibration changes mean that for some periods the newly calculated flux densities can differ by up to 10 per cent from the results previously published in Paper I. In most cases the calibration difference is no more than 5 per cent.
The calibration accuracy is shown in reftig:fef.. where data for the best secondary calibrator. 6618. have been processed using the same recipes used to process the pointing data.
The calibration accuracy is shown in \\ref{fig:fcf}, where data for the best secondary calibrator, 618, have been processed using the same recipes used to process the pointing data.
The light-curve is flat with a Gaussian of 4730d330 mmJy fitting the distribution. corresponding to a calibration accuracy of ET per cent over the 8 year period. agreeing with the accuracy demonstrated in 2..
The light-curve is flat with a Gaussian of $4730\pm 330$ mJy fitting the distribution, corresponding to a calibration accuracy of $\pm7$ per cent over the 8 year period, agreeing with the accuracy demonstrated in \cite{2002MNRAS.336...14J}.
Data uncertainties were calculated as deseribed in Paper I. The output data from the pipeline for any source are first averaged over an individual night: there has been no attempt to determine variability within a single night.
Data uncertainties were calculated as described in Paper I. The output data from the pipeline for any source are first averaged over an individual night; there has been no attempt to determine variability within a single night.
The nightly averaged data are first viewed to determine whether there are any obviously erroneous points.
The nightly averaged data are first viewed to determine whether there are any obviously erroneous points.
While this is easy to accomplish for a calibration source. or a source that is not variable. for these variable extragalactic radio-sources. this can introduce a level of subjectivity.
While this is easy to accomplish for a calibration source, or a source that is not variable, for these variable extragalactic radio-sources, this can introduce a level of subjectivity.
As described in section 2. early polarimeter observations are problematic and must be removed by inspection of the light curves.
As described in section \ref{sect:obssel} early polarimeter observations are problematic and must be removed by inspection of the light curves.
In general the light curves were inspected and in cases where a point was obviously discrepant or isolated. the processed map and observing log were inspected to decide whether the point was valid.
In general the light curves were inspected and in cases where a point was obviously discrepant or isolated, the processed map and observing log were inspected to decide whether the point was valid.
Manual removal of points mainly occurred for the following reasons:
Manual removal of points mainly occurred for the following reasons:
with ὃς denoting the sound speed which is specified as a fixed function of the cvlndzrical radius i»; Expressious (1)) (6)) eive the basic equations used for the simulations.
with $c_s$ denoting the sound speed which is specified as a fixed function of the cylindrical radius $r.$ Expressions \ref{cont}) ) – \ref{eq-state}) ) give the basic equations used for the simulations.
Tn this work we allow the planet orbits to evolve due to the eravitational forces they experience frou the disk. as well as the ceutral star.
In this work we allow the planet orbits to evolve due to the gravitational forces they experience from the disk as well as the central star.
For a single simulation we consider the evolution of either six or three plaucts concurrently. located at different positions within the disk.
For a single simulation we consider the evolution of either six or three planets concurrently, located at different positions within the disk.
These planets do uot interact with cach other eravitationallv. but affect each others evolution iudirectlV through perturbatious they make to the disk structure.
These planets do not interact with each other gravitationally, but affect each others evolution indirectly through perturbations they make to the disk structure.
Iu the case of low mass planets. the effects of the turbulence will be of cousiderably ereater significance.
In the case of low mass planets, the effects of the turbulence will be of considerably greater significance.
The equation of motion for planet i is:
The equation of motion for planet $i$ is: = + - 2 _f + _f _f ).
The acceleration due to the disk is eiven by Jy dye where tje integral is oorforined over the disk volume.
The acceleration due to the disk is given by = - G _V dV where the integral is performed over the disk volume.
Note that ti6 planet potentials are also cylindrical iu that there Is no vertica conrponeut of eravitv.
Note that the planet potentials are also cylindrical in that there is no vertical component of gravity.
During the simulalons we monitor the torque per unt mass experienced by the planets. which is defined by Note tha the torque per uuit nass is indepencdeit of the planet nass. andl so may be calculated for zeroLass "plauetesiniarhesnasS “ancl for Ἡuteite iamass planets.
During the simulations we monitor the torque per unit mass experienced by the planets, which is defined by _i = Note that the torque per unit mass is independent of the planet mass, and so may be calculated for zero–mass `planetesimals' and for finite mass planets.
lanet The calalatious are perforued iu a unifonuly rotating frame with angularS velocity {Q;.
The calculations are performed in a uniformly rotating frame with angular velocity $\Omega_f$.
The planet equatio|is of motion are evolved using a simple leap-frog iutegrator. with correct centering oenploved for inclusion of the Coriolis force.
The planet equations of motion are evolved using a simple leap-frog integrator, with correct centering employed for inclusion of the Corioli's force.
The nmuuerical scheme that we employ is based on a spatially secoudorder accurate method that computes the advection using the monotonic transport algorithin (Van Leer 1977).
The numerical scheme that we employ is based on a spatially second–order accurate method that computes the advection using the monotonic transport algorithm (Van Leer 1977).
The MIID section of the code uses the ucthod of characteristics constrained transport (MOCCT) as outlined in Παν]ον Stone (1995) and duplemented iu the ZEUS code.
The MHD section of the code uses the method of characteristics constrained transport (MOCCT) as outlined in Hawley Stone (1995) and implemented in the ZEUS code.
The code has been developed frou a version of NIRVANA written originally by U. Ziegler (Ziegler Yorke 1997).
The code has been developed from a version of NIRVANA written originally by U. Ziegler (Ziegler Yorke 1997).
We use units in which the central mass AL.l. the eravitational coustant Gl. and the radius y=1 corresponds to the radial location of the immer boundary vot the computational domain.
We use units in which the central mass $M_*=1$, the gravitational constant $G=1$, and the radius $R=1$ corresponds to the radial location of the inner boundary of the computational domain.
The uuit of time is ο1=VGM, BR. although we report our results iu units of the orbital period at the disk inner edge. which we denote as 1)?xO01,
The unit of time is $\Omega^{-1}=\sqrt{G M_*/R^3}$ , although we report our results in units of the orbital period at the disk inner edge, which we denote as $P(1)=2 \pi \Omega^{-1}$.
Planets are positioned in the disk between radi ry2.32 5,2.
Planets are positioned in the disk between radii $r_{pi}=2.2$ – 3.2.
We note tha the orbital period at c;=2.2 is P(2.:3,260P(1) and at Üpi=3.2 we have P(3.2)= 5.:2P(1).
We note that the orbital period at $r_{pi}=2.2$ is $P(2.2)=3.26P(1)$ and at $r_{pi}=3.2$ we have $P(3.2)=5.72P(1)$ .
Sunulation ruus tines are typically ~500 P(1). corvespoudiug to ~1532(2.2) aud yy~907202).
Simulation runs times are typically $\simeq 500$ $P(1)$, corresponding to $\sim 153 P(2.2)$ and $\sim 90 P(3.2)$.
howe take the computational radius r=2.5 to correspond to 5 AU. then a simulation run time of 500P(1) corresponds to 1335 vr. for a solar mass central star.
If we take the computational radius $r=2.5$ to correspond to 5 AU, then a simulation run time of $500 P(1)$ corresponds to $\sim 1335$ yr, for a solar mass central star.
The smaulatious presented |.here all use the sale nuderling turbulent disk model.
The simulations presented here all use the same underlying turbulent disk model.
This uodel has a constant aspect ratio IT/r—etryΩ)=0.WF. where 0is the disk angular velocity measured in the inertial frame.
This model has a constant aspect ratio $H/r \equiv c(r)/(r \Omega)=0.07$, where $\Omega$is the disk angular velocity measured in the inertial frame.
The inner radial boundary of the computaional domain is at BH,l aud the outer boundary is at RoutΞη,
The inner radial boundary of the computational domain is at $R_{in}=1$ and the outer boundary is at $R_{out}=5$.
The simmlations were performed in the rotatiue frame with OQ,=0.28668. which is the Keplerian augular velocity at a radius of r22.3.
The simulations were performed in the rotating frame with $\Omega_f=0.28668$, which is the Keplerian angular velocity at a radius of $r=2.3$.
The boundary coucitious emiploved are very simular o those described iu PN2003.
The boundary conditions employed are very similar to those described in PN2003.
Reeious of tfthe disk iu the vicinity of the inner and outer boundaries were given ronWNeplerian augular velocity profiles (uniform in the slinulations described here) that are stade to the ATRL. and which have huge values of the deusi vodu order to uaintaim racial hydrostatic equilinii.
Regions of the disk in the vicinity of the inner and outer boundaries were given non–Keplerian angular velocity profiles (uniform in the simulations described here) that are stable to the MRI, and which have large values of the density in order to maintain radial hydrostatic equilibrium.
These regions act as buffer zones that preveut the penetration of magnetic field to the radial boundaries. thus maiutaimineg the initial value of net magnetic flux iu the couputational domain.
These regions act as buffer zones that prevent the penetration of magnetic field to the radial boundaries, thus maintaining the initial value of net magnetic flux in the computational domain.
The inner buffer zoneruus from 13 tor-2 HR;
The inner buffer zoneruns from $r=1.2$ to $r=R_{in}$ .
The outer buffer zone ruus from r=L5 Qpr-— Rout.
The outer buffer zone runs from $r=4.5$ to $r=R_{out}$ .
If the simulation is carried out further. a final statistically quasi-stationary stage is attained.
If the simulation is carried out further, a final statistically quasi-stationary stage is attained.
We now analyze the behavior of the Lagrangian particles and the evolution of the distribution functions of the relativistic electrons as these particles are advected woueh the differcut stages of the instability evolution.
We now analyze the behavior of the Lagrangian particles and the evolution of the distribution functions of the relativistic electrons as these particles are advected through the different stages of the instability evolution.
We selected three typical times. #=12 in the middle of je acoustic phase. τ=16 at the end. and t=22 when ie musing is already developed: the number of shocks crossed by each particle at these eiveu times. the average compression ratios of these shocks aud the power iudex of 1e distribution functious are eiven in Table 3.
We selected three typical times, $t=12$ in the middle of the acoustic phase, $t=16$ at the end, and $t=22$ when the mixing is already developed; the number of shocks crossed by each particle at these given times, the average compression ratios of these shocks and the power index of the distribution functions are given in Table 3.
The details of the shock nuuber aud streugth for cach particle in cach phase can be interred from Fie.
The details of the shock number and strength for each particle in each phase can be inferred from Fig.
1.
1.
Tn Fie.
In Fig.
2 we show three nuages of the jet density. lua logaritlunic grev-cale. at these selected times. with superimposed the positions of the Lagrangian particles. identified bw different svinbols.
2 we show three images of the jet density, in a logarithmic grey-scale, at these selected times, with superimposed the positions of the Lagrangian particles, identified by different symbols.
The vectors eive the module aud directions of the particles’ velocities.
The vectors give the module and directions of the particles' velocities.
In Fig
In Fig.
3 we plot the distribution functions associated to some of the Lagrangian particles at the initial time aud after the acceleration aud loss processes. at tle same times selected above,
3 we plot the distribution functions associated to some of the Lagrangian particles at the initial time and after the acceleration and loss processes, at the same times selected above.
During the linear phase the particles advance advected by the duid aud the relativistic electrous lose cucrey irougeh adiabatic expansion aud svuchrotron radiation.
During the linear phase the particles advance advected by the fluid and the relativistic electrons lose energy gh adiabatic expansion and synchrotron radiation.
The distribution functions steepen and the maxima euergv diminishes. as can be seen. for example. frou the spectrum associated to particle 5 at time f=12 (contiuuous line in Fig.
The distribution functions steepen and the maximum energy diminishes, as can be seen, for example, from the spectrum associated to particle 5 at time $t=12$ (continuous line in Fig.
3). when this particle has not crossed anv shock vet.
3), when this particle has not crossed any shock yet.
When shocks start to form. clectrous are accelerated.
When shocks start to form, electrons are accelerated.
The firs acceleration process occurs for the distribution fiction associate to particle 1 at time f£=8.33.
The first acceleration process occurs for the distribution function associated to particle 1 at time $t=8.33$.
At time f=12 only the particles initially ocated at a clistance less than 0.7 jet radi from the jet axis have crossed a sjenificaut ( 1) umber of shocks: a this stage shocks are still weak. with compression ratios not exceediug r72.5. aud the clectrous are only weakly accelerated: from Table 3 we see that the distribution function is a power law slightly flatter than the initial one.
At time $t=12$ only the particles initially located at a distance less than 0.7 jet radii from the jet axis have crossed a significant $\sim 4$ ) number of shocks; at this stage shocks are still weak, with compression ratios not exceeding $r \sim 2.5$, and the electrons are only weakly accelerated; from Table 3 we see that the distribution function is a power law slightly flatter than the initial one.
The particles located ou the jet edees initially, iustead. cross one or two shocks at most. and their spectrum is still very steep (+~6) duc to the losses occurred in the linear phase of the evolution.
The particles located on the jet edges initially, instead, cross one or two shocks at most, and their spectrum is still very steep $\gamma \sim 6$ ) due to the losses occurred in the linear phase of the evolution.
At tine f=16 the jet is at the enc of the acoustic phase: the Lagraugiui particles lave crossed a high uiuber of shocks (ou average 6. up to 10 for particle 1).
At time $t=16$ the jet is at the end of the acoustic phase; the Lagrangian particles have crossed a high number of shocks (on average $\sim 6$, up to 10 for particle 1).
Many of these shocks are very strong. with compression ratios rcd. and the acceleration process results very efficient as can be interred from the values of the spectral index. which lavs at this stage iu the interval 1.555$2.
Many of these shocks are very strong, with compression ratios $r \sim 4$, and the acceleration process results very efficient as can be inferred from the values of the spectral index, which lays at this stage in the interval $1.5 \lapp \gamma \lapp 2$.
There are three main reasons wl particles whose initial position is located near to the jet axis undergo a lieher uuuber of shock accelerations: first. their oueitudinal velocity is higher with respect to the particles ocated on he edges of the jet: second. their trajectory is ess affected by motions iu the radial direction. since. due o the asstmed ecometry. the value of the radial velocity approaches zero near to the jet axis (see also Fig.
There are three main reasons why particles whose initial position is located near to the jet axis undergo a higher number of shock accelerations: first, their longitudinal velocity is higher with respect to the particles located on the edges of the jet; second, their trajectory is less affected by motions in the radial direction, since, due to the assumed geometry, the value of the radial velocity approaches zero near to the jet axis (see also Fig.
2): urd. the shocks that form due to the non linear erowt[um f the svuuuectrical body modes of the Ielviu-ITeliiboltz oestability are biconical shocks whose vertex is located o- je jet axis. aud so the distance between shocks is sinaller tear the axis than near the edges.
2); third, the shocks that form due to the non linear growth of the symmetrical body modes of the Kelvin-Helmholtz instability are biconical shocks whose vertex is located on the jet axis, and so the distance between shocks is smaller near the axis than near the edges.
After time f=16 the jet material starts to mix with ie external mediuu. shocks become less frequent aud weaker (compression ratios 7— 2). until time f£=22 whe-
After time $t=16$ the jet material starts to mix with the external medium, shocks become less frequent and weaker (compression ratios $r \sim 2$ ), until time $t=22$ when
Thus. the combined effects of bound stellar orbits result in equipartition of kinetic energy. when the total mass in stars equals the mass of the dominant central object.
Thus, the combined effects of bound stellar orbits result in equipartition of kinetic energy, when the total mass in stars equals the mass of the dominant central object.
This resolves the problem. raised in Paper 1. regarding the applicability of equipartition of kinetic energy for stellar svstems bound to a massive object.
This resolves the problem, raised in Paper I, regarding the applicability of equipartition of kinetic energy for stellar systems bound to a massive object.
We evaluate Eq. (
We evaluate Eq. (
9) for the broken power law distribution of stars in the central cusp observed by Genzeletal. (2003): p(r)=L2x105(r/0.4pc).M.pe7. where a=1.4 for ro0.4 pe and à=2.0 for r>0.1 pe.
9) for the broken power law distribution of stars in the central cusp observed by \citet{Genzel03}: : $\rho(r) = 1.2 \times 10^6 (r/0.4~{\rm pc})^{-\alpha}~~\msun~{\rm pc}^{-3},$ where $\alpha=1.4$ for $r<0.4$ pc and $\alpha=2.0$ for $r\ge0.4$ pc.
This stellar mass distribution contains 4x109 within rz2 pe ofA*.
This stellar mass distribution contains $4\times10^6$ within $r\approx2$ pc of.
. We do not continue bevond this radius. as the stellar mass would start to dominate and our assumption that stars are bound to starts to break down.
We do not continue beyond this radius, as the stellar mass would start to dominate and our assumption that stars are bound to starts to break down.
Assuming a characteristic stellar mass of m=0.453.. equal to the first moment of a standard initial mass [unction (Cox2000).. and contains 4x105.. we find c?7=0.07!. which implies individual component speeds of 0.05
Assuming a characteristic stellar mass of $m=0.453$, equal to the first moment of a standard initial mass function \citep{Allen00}, and contains $4\times10^6$, we find ${<V^2>}^{1/2}=0.07$, which implies individual component speeds of 0.05.
The mass function of stars in the inner (wo parsecs is likely to be Hatter than a stancarel IME. owing to mass segregation effects.
The mass function of stars in the inner two parsecs is likely to be flatter than a standard IMF, owing to mass segregation effects.
Since the expected motion of scales as ym. we would expect a higher rms speed for than caleulated above. possibly by a factor ol two or more.
Since the expected motion of scales as $\sqrt{m}$, we would expect a higher rms speed for than calculated above, possibly by a factor of two or more.
Overall. our estimate of the motion of iserlremely conservative.
Overall, our estimate of the motion of is conservative.
The assumptions of 1) a standard IAIF. 2) a perfectly random stellar distribution (no clumping or anisolropies). 3) no contribution Irom a possible cluster of dark stellar remnants close to (see 814.4). and 4) no contribution from mass asvunnetries bevond r=2 pe (see 844.5). all contribute to give the lowest possible motion forÀ*.
The assumptions of 1) a standard IMF, 2) a perfectly random stellar distribution (no clumping or anisotropies), 3) no contribution from a possible cluster of dark stellar remnants close to (see 4.4), and 4) no contribution from mass asymmetries beyond $r=2$ pc (see 4.5), all contribute to give the lowest possible motion for.
. The analvtical approach of the previous section assumes stars of a single mass and circular orbits.
The analytical approach of the previous section assumes stars of a single mass and circular orbits.
In reality. one expects a distribution of stellar masses and. especially in the crowdecl environment of such a dense stellar svstem. a wide distribution of orbital eccenlricilies.
In reality, one expects a distribution of stellar masses and, especially in the crowded environment of such a dense stellar system, a wide distribution of orbital eccentricities.
la order to understand better the ellects of stars in the central cusp on the motion ofΑι. we carried out [ull numerical simulations of the effects of the ~10"to10* stars thought to orbit within 2 pe of the Galactic center.
In order to understand better the effects of stars in the central cusp on the motion of, we carried out full numerical simulations of the effects of the $\sim10^6~{\rm to}~10^7$ stars thought to orbit within 2 pc of the Galactic center.
We are interested in the change in position of over a time period of eight vears.
We are interested in the change in position of over a time period of eight years.