The Newtonian potential of thin disks

Abstract

The one-dimensional, ordinary differential equation (ODE) by Huré & Hersant (2007) that satisfies the midplane gravitational potential of truncated, flat power-law disks is extended to the whole physical space. It is shown that thickness effects (i.e. non-flatness) can be easily accounted for by implementing an appropriate "softening length" $\lambda$. The solution of this "softened ODE" has the following properties: i) it is regular at the edges (finite radial accelerations), ii) it possesses the correct long-range properties, iii) it matches the Newtonian potential of a geometrically thin disk very well, and iv) it tends continuously to the flat disk solution in the limit $\lambda \to 0$. As illustrated by many examples, the ODE, subject to exact Dirichlet conditions, can be solved numerically with efficiency for any given colatitude at second-order from center to infinity using radial mapping. This approach is therefore particularly well-suited to generating grids of gravitational forces in order to study particles moving under the field of a gravitating disk as found in various contexts (active nuclei, stellar systems, young stellar objects). Extension to non-power-law surface density profiles is straightforward through superposition. Grids can be produced upon request.

Accepted for publication in A&A

References in corpus (4)

• Giant Planet Migration in Viscous Power-Law Discs
• Star-disc interactions in a galactic centre and oblateness of the inner stellar cluster
• A new equation for the mid-plane potential of power law discs. II. Exact solutions and approximate formulae
• A new equation for the mid-plane potential of power law disks

Cited by in corpus (9)

• Second generation planet formation in NN Serpentis?
• Dynamical mass measurements of two protoplanetary discs
• Electrostatic potential of a uniformly charged disk through Green's theorem
• Primordial dust rings, hidden dust mass, and the first generation of planetesimals in gravitationally unstable protoplanetary disks
• Two mass conjectures on axially symmetric black hole--disk systems
• Resonance capture and long-term evolution of planets in binary star systems
• The potential of discs from a "mean Green function"
• Planetary migration in precessing disks for S-type wide binaries
• Self-gravity in thin protoplanetary discs: 1. The smoothing-length approximation versus the exact self-gravity kernel