ASTROPHYSICAL DISK DYNAMICS

The Gravitational Potential of Astrophysical Disks: Softened

Astrophysical disks orbiting a central mass are ubiquitous in a variety of contexts – galactic, stellar, and planetary. Generally, the masses of such disks are much less than that of the central body. Nevertheless, gravity of such disks can still play an important dynamical role in the orbital evolution of their constituent particles as well as the dynamics of external objects. It is thus important to characterize the long-term  dynamical effects of disk gravity. 

One of the way in which this can be done is by using the classical Laplace-Lagrange secular theory, first developed to study the evolution of planetary orbits. Within this framework, interacting bodies are smeared into massive rings, and the resulting perturbations -- expressed mathematically in what is known as the disturbing function -- is then written as a power series in orbital eccentricities and inclinations and a Fourier series in the orbital angles. Thus, an astrophysical disk may, in principle, be modelled as a continuum of perturbers, i.e., with very large number of rings, each interacting with the others as per the classical disturbing function. This approach, however, is ill-posed from a mathematical point of view when applied to self-gravitating disks, since it would result in singularities. This singularity stems from the fact that the gravitational potential diverges at null interparticle separations. This issue is often overcome in the literature by making use of softened forms of gravity, i.e., 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. 

Graphical representation of a disk composed of N >> 1 particles which is modelled as a collection of geometrically spaced massive rings. This diagram has been adapted from Batygin (2018)

In Sefilian & Rafikov (2019), we analysed the performance of several softening prescriptions found in the literature in reproducing the expected - unsoftened - eccentricity dynamics driven by razor-thin disks. We identified softening prescriptions that, in the limit of vanishing softening, yield results converging to the expected behaviour exactly, approximately or not converging at all. 

We further developed a general analytic framework for computing the orbit-averaged disk potential given an arbitrary softening prescription. This framework accurately reproduces the expected secular dynamical behaviour for a wide class of softened gravity models. 

Theory aside, our results suggest that caution must be exercised in numerical treatments of disks which involve modelling the disk as a collection of massive, nested rings. In particular, disks should be modelled by relatively large number of rings to ensure that the correct secular behaviour is properly captured, provided a correct implementation of potential softening. We also show that this constraint is further aggravated for disks with sharp edges, such as planetary rings.

Later, in Sefilian et al. (2023), we also developed a semi-analytic code, which we refer to as the "N-ring" code, that is based on a generalized, softened version of the classical Laplace-Lagrange secular theory. This tool allows the study of the secular evolution of self-gravitating particulate disks and their response to external perturbations in general (coplanar) astrophysical setups. This code is publicly available here

Scaling of number of softened rings with softening parameter to ensure the convergence of the disk-driven free precession rate in discretized disks to the expected results in continuous softened disks. Results are shown for various softening prescriptions available in the literature, assuming different surface density profiles for the disk. It is evident that large number of rings are to be included to capture the expected long-term eccentricity dynamics in nearly-Keplerian disks. Figure taken from Sefilian & Rafikov (2019)