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

Free access article

Issue		A&A Volume 490, Number 2, November I 2008


Page(s)		477 - 486
Section		Astrophysical processes
DOI		http://dx.doi.org/10.1051/0004-6361:200809682
Published online		18 August 2008

A&A 490, 477-486 (2008)
DOI: 10.1051/0004-6361:200809682

II. Exact solutions and approximate formulae

J.-M. Huré^{1, 2}, F. Hersant², C. Carreau³, and J.-P. Busset¹

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 $\Sigma(R) \propto R^{s}$ (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 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 $\pm$ 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.

Key words: gravitation -- methods: analytical -- accretion, accretion disks

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