Generalized Schwarzschild's method

TL;DR

This paper introduces a finite element method that generalizes Schwarzschild's orbit superposition technique to produce continuous equilibrium distribution functions for stellar systems, improving smoothness and accuracy.

Contribution

It extends Schwarzschild's method by using finite elements to generate continuous distribution functions satisfying Jeans equations, with applications to spherical stellar models.

Findings

FEM produces smoother, more accurate DFs.

Explicit Jeans equation constraints improve solutions.

Method successfully applied to Hernquist models.

Abstract

We describe a new finite element method (FEM) to construct continuous equilibrium distribution functions 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 distribution function (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.