Self-gravity in curved mesh elements

Abstract

The local character of self-gravity along with the number of spatial dimensions are critical issues when computing the potential and forces inside massive systems like stars and disks. This appears from the discretisation scale where each cell of the numerical grid is a self-interacting body in itself. There is apparently no closed-form expression yet giving the potential of a three-dimensional homogeneous cylindrical or spherical cell, in contrast with the Cartesian case. By using Green's theorem, we show that the potential integral for such polar-type 3D sectors -- initially, a volume integral with singular kernel -- can be converted into a regular line-integral running over the lateral contour, thereby generalising a formula already known under axial symmetry. It therefore is a step towards the obtention of another potential/density pair. The new kernel is a finite function of the cell's shape (with the simplest form in cylindrical geometry), and mixes incomplete elliptic integrals, inverse trigonometric and hyperbolic functions. The contour integral is easy to compute; it is valid in the whole physical space, exterior and interior to the sector itself and works in fact for a wide variety of shapes of astrophysical interest (e.g. sectors of tori or flared discs). This result is suited to easily providing reference solutions, and to reconstructing potential and forces in inhomogeneous systems by superposition. The contour integrals for the 3 components of the acceleration vector are explicitely given.