A new equation for the mid-plane potential of power law discs. II. Exact solutions and approximate formulae

TL;DR

This paper derives exact and approximate solutions for the mid-plane gravitational potential in power-law discs, providing simple formulas and algorithms for astrophysical applications where disc gravity is significant.

Contribution

It presents exact series solutions and simplified approximate formulas for the gravitational potential in power-law discs, improving computational efficiency and accuracy.

Findings

Exact series solutions for the potential are derived.

Simple approximate formulas achieve a few percent accuracy.

The methods are applicable to astrophysical disc models and simulations.

Abstract

The first-order ordinary differential equation (ODE) that describes the mid-plane gravitational potential in flat finite size discs in which the surface density is a power-law function of the radius R with exponent s (Hur\'e & Hersant 2007) is solved exactly in terms of infinite series. The formal solution of the ODE is derived and then converted into a series representation by expanding the elliptic integral of the first kind over its modulus before analytical integration. Inside the disc, the gravitational potential consists of three terms: a power law of radius R with index 1+s, and two infinite series of the variables R and 1/R. The convergence of the series can be accelerated, enabling the construction of reliable approximations. At the lowest-order, the potential inside large astrophysical discs (s ~ -1.5 +/- 1) is described by a very simple formula whose accuracy (a few percent typically) is easily increased by considering successive orders through a recurrence. A basic algorithm is given. Applications concern all theoretical models and numerical simulations where the influence of disc gravity must be checked and/or reliably taken into account.