Volume 541, May 2012
|Number of page(s)||6|
|Section||Numerical methods and codes|
|Published online||14 May 2012|
A substitute for the singular Green kernel in the Newtonian potential of celestial bodies
1 Univ. Bordeaux, LAB, UMR 5804, 33270 Floirac, France
2 CNRS, LAB, UMR 5804, 33270 Floirac, France
3 ELSA, Physikalisches Institut der Universität Bonn, Nussallee 12, 53115 Bonn, Germany
Received: 13 November 2011
Accepted: 1 March 2012
The “point mass singularity” inherent in Newton’s law for gravitation represents a major difficulty in accurately determining the potential and forces inside continuous bodies. Here we report a simple and efficient analytical method to bypass the singular Green kernel 1/|r − r′| inside the source without altering the nature of the interaction. We build an equivalent kernel made up of a “cool kernel”, which is fully regular (and contains the long-range − GM/r asymptotic behavior), and the gradient of a “hyperkernel”, which is also regular. Compared to the initial kernel, these two components are easily integrated over the source volume using standard numerical techniques. The demonstration is presented for three-dimensional distributions in cylindrical coordinates, which are well-suited to describing rotating bodies (stars, discs, asteroids, etc.) as commonly found in the Universe. An example of implementation is given. The case of axial symmetry is treated in detail, and the accuracy is checked by considering an exact potential/surface density pair corresponding to a flat circular disc. This framework provides new tools to keep or even improve the physical realism of models and simulations of self-gravitating systems, and represents, for some of them, a conclusive alternative to softened gravity.
Key words: gravitation / methods: analytical / methods: numerical
© ESO, 2012
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.