Generation of potential/surface density pairs in flat disks. Power law distributions
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 Σ(R) ∝ R^s), 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 1/N^3. This method is superior to the classical multipole expansion whose very slow convergence is often critical for most practical applications.
 Publication:

Astronomy and Astrophysics
 Pub Date:
 November 2007
 DOI:
 10.1051/00046361:20066808
 arXiv:
 arXiv:0706.3616
 Bibcode:
 2007A&A...475..401H
 Keywords:

 gravitation;
 methods: analytical;
 accretion;
 accretion disks;
 methods: numerical;
 Astrophysics
 EPrint:
 Accepted for publication in Astronomy &