Self-gravity at the scale of the polar cell
Université de Bordeaux, Observatoire Aquitain des Sciences de l'Univers, 2 rue de l'Observatoire, BP 89, 33271 Floirac cedex, France e-mail: email@example.com
2 CNRS/INSU/UMR 5804/LAB, BP 89, 33271 Floirac Cedex, France
3 LAL-IMCCE/USTL, 1 impasse de l'Observatoire, 59000 Lille, France
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 , radial extension , and opening angle . Accurate approximations are derived in the limit of high numerical resolution at the geometrical mean 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 except if the shape factor , 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