Journal Article

Generalized Schwarzschild’s method

Mir Abbas Jalali and Scott Tremaine

in Monthly Notices of the Royal Astronomical Society

Published on behalf of The Royal Astronomical Society

Volume 410, issue 3, pages 2003-2015
Published in print January 2011 | ISSN: 0035-8711
Published online January 2011 | e-ISSN: 1365-2966 | DOI:
Generalized Schwarzschild’s method

More Like This

Show all results sharing this subject:

  • Astronomy and Astrophysics


Show Summary Details


We describe a new finite element method (FEM) to construct continuous equilibrium distribution functions (DFs) of stellar systems. The method is a generalization of Schwarzschild’s orbit superposition method from the space of discrete functions to continuous ones. In contrast to Schwarzschild’s method, FEM produces a continuous DF and satisfies the intra-element continuity and Jeans equations. The method employs two finite element meshes, one in configuration space and one in action space. The DF is represented by its values at the nodes of the action-space mesh and by interpolating functions inside the elements. The Galerkin projection of all equations that involve the DF leads to a linear system of equations, which can be solved for the nodal values of the DF using linear or quadratic programming, or other optimization methods. We illustrate the superior performance of FEM by constructing ergodic and anisotropic equilibrium DFs for spherical stellar systems (Hernquist models). We also show that explicitly constraining the DF by the Jeans equations leads to smoother and/or more accurate solutions with both Schwarzschild’s method and FEM.

Keywords: methods: numerical; galaxies: elliptical and lenticular, cD; galaxies: kinematics and dynamics; galaxies: structure

Journal Article.  8949 words.  Illustrated.

Subjects: Astronomy and Astrophysics

Full text: subscription required

How to subscribe Recommend to my Librarian

Users without a subscription are not able to see the full content. Please, subscribe or login to access all content.