Gravitational softening as a smoothing operation

In self-consistent N -body simulations of collisionless systems, gravitational interactions are modified on small scales to remove singularities and simplify the task of numerically integrating the equations of motion. This ‘gravitational softening’ is sometimes presented as an ad-hoc departure from Newtonian gravity. However, softening can also be described as a smoothing operation applied to the mass distribution; the gravitational potential and the smoothed density obey Poisson’s equation precisely. While ‘softening’ and ‘smoothing’ are mathematically equivalent descriptions, the latter has some advantages. For example, the smoothing description suggests a way to set up N -body initial conditions in almost perfect dynamical equilibrium.