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

J.-M Huré,A Dieckmann

doi:10.1051/0004-6361/201118443

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.

Highlights

As a direct consequence of Newton’s law for gravitation (Newton 1760; Kellogg 1929), the potential of any continuous distribution of matter inside a volume V at a point P(r) of space is given by ψ(r) = −G
The presence of the Green kernel 1/|r − r | is known to represent a difficulty in calculating ψ everywhere inside and very close to V since this function diverges as r → r
We have reformulated the Green kernel appearing in potential problems to circumvent the singularity and, at the same time, properly account for it

Summary

Introduction

The presence of the Green kernel 1/|r − r | is known to represent a difficulty in calculating ψ everywhere inside and very close to V since this function diverges as r → r This singularity is classically avoided by converting the Green function into an infinite series (e.g. Kellogg 1929; Cohl & Tohline 1999). The Poisson equation provides another other way to derive ψ numerically (Kellogg 1929; Durand 1953) This approach requires accurate boundary or interior/matching conditions only accessible through Eq (1), and complex geometries are not always easy to manage (Grandclément et al 2001). We present a new means to evaluate Eq (1) that avoids the singularity, and, at the same time, properly accounts for it This is achieved by replacing the singular kernel by an equivalent and regular, two-term form, one term being the gradient of a new scalar potential[2].

Splitting of the Green kernel

Δ2 Δ δ2

The singular term as the vertical gradient of a hyperkernel

The case of axially-symmetric bodies

Checking the accuracy

Concluding remarks