Toy stars in one dimension

This paper describes a simple model of a star where compressibility is retained but the gravitational force is replaced by a force between pairs which is directed along their line of centres and proportional to their separation. This force was analysed by Newton. It may be considered the simplest many body force because the system reduces to a set of independent particles moving in a harmonic oscillator potential. We call such models ‘toy stars’. In this paper we simplify further and focus on the one-dimensional problem. We show that a non-linear solution exists where the velocity is a linear function of the coordinate and the density a quadratic function of the coordinates. We generalize this result to include a magnetic field. Our results provide a very useful benchmark for algorithms designed to simulate gravitating gas and we show that Smoothed Particle Hydrodynamics (SPH) simulations of toy stars with or without magnetic fields give results in good agreement with theory.