EDP Sciences
Free access
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. Huré1, 2, A. Pierens3, and F. Hersant1, 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

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

© ESO 2009