A substitute for the singular Green kernel in the Newtonian potential of celestial bodies

TL;DR

The paper introduces an analytical method to replace the singular Green kernel in Newtonian gravity with a regularized kernel, improving the accuracy and efficiency of modeling gravitational potentials in celestial bodies.

Contribution

It presents a novel regularization technique using a 'cool kernel' and a 'hyperkernel' to handle singularities in gravitational potential calculations for continuous mass distributions.

Findings

The method effectively regularizes the Green kernel inside sources.

It maintains the correct asymptotic behavior of the potential.

Demonstrated accuracy with a flat circular disc model.

Abstract

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.