Toy Stars in two dimensions

Toy Stars are gas masses where the compressibility is treated without approximations but gravity is replaced by a force which, for any pair of masses, is along their line of centres and proportional to their separation. They provide an invaluable resource for testing the suitability of numerical codes for astrophysical gas dynamics. In this paper, we derive the equations for both small-amplitude oscillations and non-linear solutions for rotating and pulsating Toy Stars in two dimensions, and show that the solutions can be reduced to a small number of ordinary differential equations. We compare the accurate solutions of these equations with Smoothed Particle Hydrodynamics (SPH) simulations. The two-dimensional Toy Star solutions are found to provide an excellent benchmark for SPH algorithms, highlighting many of the strengths and also some weaknesses of the method.