Generation of potential/surface density pairs in flat disks

Power law distributions

¹ Université Bordeaux 1/CNRS/OASU/UMR 5804/LAB, 2 rue de l'Observatoire, BP 89, 33270 Floirac, France e-mail: jean-marc.hure@obs.u-bordeaux1.fr
² LUTh/Observatoire de Paris-Meudon-Nancay, Place Jules Janssen, 92195 Meudon Cedex, France e-mail: didier.pelat@obspm.fr
³ Astronomy Unit, Queen Mary, University of London, Mile end Road, London E1 4NS, UK e-mail: a.pierens@qmul.ac.uk

Received: 24 November 2006
Accepted: 5 June 2007

Abstract

Aims.We report a simple method to generate potential/surface density pairs in flat axially symmetric finite size disks.

Methods.Potential/surface density pairs consist of a “homogeneous” pair (a closed form expression) corresponding to a uniform disk, and a “residual” pair. This residual component is converted into an infinite series of integrals over the radial extent of the disk. For a certain class of surface density distributions (like power laws of the radius), this series is fully analytical.

Results.The extraction of the homogeneous pair is equivalent to a convergence acceleration technique, in a matematical sense. In the case of power law distributions (i.e. surface densities of the form ), the convergence rate of the residual series is shown to be cubic inside the source. As a consequence, very accurate potential values are obtained by low order truncation of the series. At zero order, relative errors on potential values do not exceed a few percent typically, and scale with the order N of truncation as . This method is superior to the classical multipole expansion whose very slow convergence is often critical for most practical applications.