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

II. Exact solutions and approximate formulae

¹ Université de Bordeaux, LAB, 351 cours de la Libération, Talence 33405, France e-mail: [jean-marc.hure;jean-pierre.busset]@obs.u-bordeaux1.fr
² CNRS/INSU, UMR 5804/LAB, 2 rue de l'Observatoire, BP 89, 33271 Floirac Cedex, France e-mail: franck.hersant@obs.u-bordeaux1.fr

³ La Maurellerie, 37290 Bossay-sur-Claise, France e-mail: cyril.carreau@wanadoo.fr

Received: 29 February 2008
Accepted: 31 May 2008

Abstract

Aims. The first-order ordinary differential equation (ODE) that describes the mid-plane gravitational potential in flat finite size discs of surface density (Huré & Hersant 2007, A&A, 467, 907) is solved exactly in terms of infinite series.

Methods. 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.

Results. Inside the disc, the gravitational potential consists of three terms: a power law of radius R with index , and two infinite series of the variables R and . 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.

Conclusions. Applications concern all theoretical models and numerical simulations where the influence of disc gravity must be checked and/or reliably taken into account.