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.Comment: Accepted for publication in Cel. Mech. and Dyn. Astron. A basic Fortran 90 program is available as a supplementary resource linked to the online version of the paper (or upon request, by email

Secular resonance sweeping and orbital excitation in decaying disks

“…Knowing ρ(r, z) we can compute the mass contained in each cell and sum up the individual contributions from all cells, to compute the acceleration in a given cell. The singularity related to a cell's own gravity can be removed with the use of elliptic integrals, as described in Hur (2012) and Hur et al (2014). Note that the interaction between massive planets and the gas disk leads to transport of angular momentum from the inner to the outer part of the disk (Lin & Papaloizou, 1979;Goldreich & Tremaine, 1980), which tends to open a gap at the position of the planet.…”

A Fast Poisson Solver of Second-order Accuracy for Isolated Systems in Three-dimensional Cartesian and Cylindrical Coordinates

“…As the reference potential, we take Equation (20) of Katz et al (2016) for the gravitational potential of a uniform cube. There is no algebraic expression for the potential of a rectangular torus, but Huré et al (2014) provided a closed-form expression, in terms of line integrals with smooth integrands, in their Equation (29). We use the Romberg's method with a relative tolerance 10 −10 to ensure that the numerical integrations are accurate enough to serve as a reference solution.…”

A truly Newtonian softening length for disc simulations

“…We perceive dipole and quadrupole features which are very difficult to catch by analytical means 2 . Inside the cell, λ is real and close to 1 In practical, the potential ψ cell is determined from the contour integral reported by Huré et al (2014). In the paper throughout, we use this accurate method to generate reference values.…”

“…1 In practical, the potential ψ cell is determined from the contour integral reported by Huré et al (2014). In the paper throughout, we use this accurate method to generate reference values.…”