An iterative method for the construction of N-body galaxy models in collisionless equilibrium

Abstract

(Abridged) We describe a new iterative approach for the realization of equilibrium N-body systems for given density distributions. Our method uses elements of Schwarzschild's technique and of the made-to-measure method, but is based on a different principle. Starting with some initial assignment of particle velocities, we calculate their orbits using a flexible tree-based force algorithm in the stationary potential of the target density distribution. The difference of the time-averaged density response produced by these orbits with respect to the initial density configuration is characterized through a merit function, and a stationary solution of the collisionless Boltzmann equation is found by minimizing this merit function directly by iteratively adjusting the initial velocities. Because the distribution function is in general not unique for a given density structure, we augment the merit function with additional constraints that single out a desired target solution. We do this for broad classes of axisymmetric density distributions by numerically solving the Jeans equations to obtain the second velocity moments, which are then imposed as further constraints in the optimization process. The velocity adjustment is carried out with a stochastic process in which new velocities are randomly drawn from an approximate solution of the distribution function, but are kept only when they improve the fit. Our method converges rapidly and is flexible enough to allow the construction of solutions with third integrals of motion, including disk galaxies with realistic velocity distributions. We demonstrate that the new method reproduces analytic distribution functions where they are known (such as the Hernquist sphere), and that it yields very stable N-body realizations for a variety of examples of compound galaxy models, considerably improving on widely used moment-based approaches.