Theory of Stellar Orbits in the Galaxy

Ollongren, Alexander

doi:10.1146/annurev.aa.03.090165.000553

Cited by 46 publications

(38 citation statements)

References 6 publications

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“…A nonresonant orbit with only these two integrals would completely fill the region of the meridional plane enclosed by the zero-velocity curve (ZVC). However, numerical studies (e.g., Contopoulos 1960;Ollongren 1962;Richstone 1982) show that most orbits also conserve a third integral, I 3 , which confines the orbit to a subset of the allowed meridional plane region. When the third integral is present, the orbit touches the ZVC at a finite number of points.…”

Section: Density and Potentialmentioning

confidence: 99%

Difficulties with Recovering the Masses of Supermassive Black Holes from Stellar Kinematical Data

Valluri

Merritt

Emsellem

2004

ApJ

167

166

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We investigate the ability of three-integral, axisymmetric, orbit-based modeling algorithms to recover the parameters defining the gravitational potential (mass-to-light ratio Ç and black hole mass M . ) in spheroidal stellar systems using stellar kinematical data. We show that the potential estimation problem is generically underdetermined when applied to long-slit kinematical data of the kind used for most black hole mass determinations to date. A range of parameters (Ç, M . ) can provide equally good fits to the data, making it impossible to assign bestfit values. The indeterminacy arises from the large variety of orbital solutions that are consistent with a given mass model. We demonstrate the indeterminacy using a variety of data sets derived from realistic models, as well as published observations of the galaxy M32. The indeterminacy becomes apparent only when a sufficiently large number of distinct orbits are supplied to the modeling algorithm; if too few orbits are used, spurious minima appear in the 2 ðÇ; M Þ contours, and these minima do not necessarily coincide with the parameters defining the gravitational potential. We show that the range of degeneracy in M . depends on the degree to which the data resolve the radius of influence r h of the black hole. For FWHM=2r h k 0:5, where FWHM refers to the instrumental resolution, we find that only very weak constraints can be placed on M . . In the case of M32, our reanalysis demonstrates that when a large orbit library is used, data published prior to 2000 (FWHM=2r h % 0:25) are equally consistent with black hole masses in the range 1:5 Â 10 6 M < M < 5 Â 10 6 M , with no preferred value in that range. Exactly the same data can reproduce previous published results with smaller orbit libraries. While the Hubble Space Telescope (HST ) Space Telescope Imaging Spectrograph (STIS) data for this galaxy (FWHM=2r h % 0:06) may overcome the degeneracy in M . , HST data for most galaxies do not resolve the black hole's sphere of influence and in these galaxies the degree of degeneracy allowed by the data may be greater than previously believed. We investigate the effect of regularization, or smoothness constraints, on the degree of degeneracy of the solutions. Enforcing smoothness reduces the range of acceptable models, but we find no indication that the true potential can be recovered simply by enforcing smoothing. For a given smoothing level, all solutions in the minimum 2 valley exhibit similar levels of noise; as the smoothing is increased, there is a systematic shift in the midpoint of the 2 valley, until at a high level of smoothing the solution is biased with respect to the true solution. These experiments suggest both that the indeterminacy is real (i.e., that it is not an artifact associated with nonsmooth solutions) and that there is no known empirical way to choose the smoothing parameter to ensure that the correct solution is selected.

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Section: Density and Potentialmentioning

confidence: 99%

Difficulties with Recovering the Masses of Supermassive Black Holes from Stellar Kinematical Data

Valluri

Merritt

Emsellem

2004

ApJ

167

166

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“…This method is independent from solving the Jeans equations. It is clear from these models (see also Ollongren (1962) for much earlier orbit calculations) that the third integral of motion has to be an essential part of the model construction.…”

Section: Introductionmentioning

confidence: 99%

Stellar kinematics using a third integral of motion: method and application on the Andromeda galaxy

Kipper

Tenjes

Tihhonova

et al. 2016

Mon. Not. R. Astron. Soc.

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We probe the feasibility of describing the structure of a multi-component axisymmetric galaxy with a dynamical model based on the Jeans equations while taking into account a third integral of motion. We demonstrate that using the third integral in the form derived by G. Kuzmin, it is possible to calculate the stellar kinematics of a galaxy from the Jeans equations by integrating the equations along certain characteristic curves. In cases where the third integral of motion does not describe the system exactly, the derived kinematics would describe the galaxy only approximately.We apply our method to the Andromeda galaxy, for which the mass distribution is relatively firmly known. We are able to reproduce the observed stellar kinematics of the galaxy rather well. The calculated model suggests that the velocity dispersion ratios σ

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“…In each case the goal was to use a Stäckel third integral as an approximate of the third quasi integral of the fitted potential, as in Wayman (1959), van de Hulst (1962), Ollongren (1962), Hori (1962), and more recently, Manabe (1979), Batsleer & Dejonghe (1994), or Famaey & Dejonghe (2003).…”

Section: Introductionmentioning

confidence: 99%

Quasi integral of motion for axisymmetric potentials

Bienaymé

Robin

Famaey

2015

A&A

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We present an estimate of the third integral of motion for axisymmetric three-dimensional potentials. This estimate is based on a Stäckel approximation and is explicitly written as a function of the potential. We tested this scheme for the Besançon Galactic model and two other disc-halo models and find that orbits of disc stars have an accurately conserved third quasi integral. The accuracy ranges from of 0.1% to 1% for heights varying from z = 0 kpc to z = 6 kpc and Galactocentric radii R from 5 to 15 kpc. We also tested the usefulness of this quasi integral in analytic distribution functions of disc stellar populations: we show that the distribution function remains approximately stationary and that it allows to recover the potential and forces by applying Jeans equations to its moments.

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Theory of Stellar Orbits in the Galaxy

Cited by 46 publications

References 6 publications

Difficulties with Recovering the Masses of Supermassive Black Holes from Stellar Kinematical Data

Difficulties with Recovering the Masses of Supermassive Black Holes from Stellar Kinematical Data

Stellar kinematics using a third integral of motion: method and application on the Andromeda galaxy

Quasi integral of motion for axisymmetric potentials

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