Self-gravity at the scale of the polar cell

Free access article

Issue		A&A Volume 500, Number 2, June III 2009


Page(s)		617 - 620
Section		Astrophysical processes
DOI		http://dx.doi.org/10.1051/0004-6361/200911806
Published online		08 April 2009

A&A 500, 617-620 (2009)
DOI: 10.1051/0004-6361/200911806

Self-gravity at the scale of the polar cell

J.-M. Huré^{1, 2}, A. Pierens³, and F. Hersant^{1, 2}

¹ Université de Bordeaux, Observatoire Aquitain des Sciences de l'Univers, 2 rue de l'Observatoire, BP 89, 33271 Floirac cedex, France
e-mail: jean-marc.hure@obs.u-bordeaux1.fr
² CNRS/INSU/UMR 5804/LAB, BP 89, 33271 Floirac Cedex, France 
³ LAL-IMCCE/USTL, 1 impasse de l'Observatoire, 59000 Lille, France

Received 6 February 2009 / Accepted 5 March 2009

Abstract
We present the exact calculus of the gravitational potential and acceleration along the symmetry axis of a plane, homogeneous, polar cell as a function of mean radius $\bar{a}$ , radial extension $\Delta a$ , and opening angle $\Delta \phi$ . Accurate approximations are derived in the limit of high numerical resolution at the geometrical mean $\langle a \rangle$ of the inner and outer radii (a key-position in current FFT-based Poisson solvers). Our results are the full extension of the approximate formula given in the textbook of Binney & Tremaine to all resolutions. We also clarify definitely the question about the existence (or not) of self-forces in polar cells. We find that there is always a self-force at radius $\langle a \rangle$ except if the shape factor $\rho \equiv \bar{a}\Delta \phi /\Delta a \rightarrow 3.531$ , asymptotically. Such cells are therefore well suited to build a polar mesh for high resolution simulations of self-gravitating media in two dimensions. A by-product of this study is a newly discovered indefinite integral involving complete elliptic integral of the first kind over modulus.

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

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