The N-ring code, developed as a MATLAB script, can be used to simulate the long-term, secular evolution of self-gravitating particulate disks and their response to external perturbations in general (coplanar) astrophysical setups. 

This semi-analytic framework is built upon the continuum version of the classical Laplace-Lagrange theory, whereby the disk is modelled as a series of N>>1 massive rings interacting with each other and with (any) external perturbers. Unlike the classical Laplace-Lagrange theory, however, here the ring-ring interaction is softened by spatially smoothing the Newtonian point-mass potential. This is essentially done by introducing a small, but non-zero, softening length into the calculations, rendering the force between two rings finite -- rather than infinite -- at points of orbit crossings. The N-ring code adopts the softening formalism of Hahn (2003), which stems from accounting for the vertical extent of the interacting rings, and vertically averaging the resulting disturbing function over the disk. The model is fully described in Sefilian et al. (2023); see e.g. Sections 2.1 & 3 therein. An extensive report on various tests which are used for verifying the performance of the N-ring code is outlined in Appendix A of the original paper. 

Given the parameters of the star, the planet, and the disk, the MATLAB script first calculates some matrix A that encapsulates the mutual gravitational interactions among all considered rings (i.e., disk and planet), and then solves for the time evolution of the eccentricities and longitudes of pericenter of each ring. 

Enjoy simulating self-gravitating (debris) disks!

Publication to cite: Sefilian et al. (2023) [Link]