A new equation for the mid-plane potential of power law disks

Aims. We show that the gravitational potential ψ in the plane of an axisymmetrical flat disk where the surface density varies as a power s of the radius R obeys an inhomogeneous first-order Ordinary Differential Equation (ODE) solvable by standard techniques. Methods. The exact derivative of the midplane potential in its integral form is found to be algebrically linked to the potential itself. Results. The ODE reads dψ dR − (1 + s) ψ R =Λ (R), where Λ is fully analytical. The potential being exactly known at the origin R = 0 for any index s (and at infinity as well), the search for solutions consists of a Two-point Boundary Value Problem (TBVP) with Dirichlet conditions. The computating time is then linear with the number of grid points, instead of quadratic from direct summation methods. Complex mass distributions which can be decomposed into a mixture of power law surface density profiles are easily accessible through the superposition principle. Conclusions. This ODE definitively takes the place of the untractable bidimensional Poisson equation for planar calculations. It opens new horizons to investigate various aspects related to self-gravity in astrophysical disks (force calculations, stability analysis, etc.).