Best approximate quadratic integrals in stellar dynamics

  • de Zeeuw P
  • Lynden-Bell D
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Abstract

The nature of the integrals of motion in Stäckel potentials, for which the equations of motion separate in ellipsoidal coordinates, is elucidated. The problem of fitting a general potential with one of Stäckel form is considered, both locally and globally. In local fitting the potential is expanded around an equilibrium point and this expansion is compared term by term with the similar expansion of a Stäckel potential. This is done explicitly for potentials with three reflection symmetries. Expansions are given for the integrals of motion in the best-fitting Stäckel potential. They may be expressed directly in terms of the expansion coefficients of the potential that is fitted. The results of local fitting are applied to the gravitational potentials of ellipsoidal density distributions, and also to ellipsoidal potentials. It is shown that there is a unique inhomogeneous density distribution stratified on similar ellipsoids of arbitrary axis ratios with a potential that is exactly of Stäckel form. A method is presented for the global fitting of a triaxial potential with one of Stäckel form. The foci of the ellipsoidal coordinates play an important role in this procedure.

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de Zeeuw, P. T., & Lynden-Bell, D. (1985). Best approximate quadratic integrals in stellar dynamics. Monthly Notices of the Royal Astronomical Society, 215(4), 713–730. https://doi.org/10.1093/mnras/215.4.713

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