Abstract
In many astrophysical problems involving discs (gaseous or particulate) orbiting a dominant central mass, gravitational potential of the disc plays an important dynamical role. Its impact on the motion of external objects, as well as on the dynamics of the disc itself, can usually be studied using secular approximation. This is often done using softened gravity to avoid singularities arising in calculation of the orbit-averaged potential — disturbing function — of a razor-thin disc using classical Laplace-Lagrange theory. We explore the performance of several softening formalisms proposed in the literature in reproducing the correct eccentricity dynamics in the disc potential. We identify softening models that, in the limit of zero softening, give results converging to the expected behavior exactly, approximately or not converging at all. We also develop a general framework for computing secular disturbing function given an arbitrary softening prescription for a rather general form of the interaction potential. Our results demonstrate that numerical treatments of the secular disc dynamics, representing the disc as a collection of N gravitationally interacting annuli, are rather demanding: for a given value of the (dimensionless) softening parameter, ς ≪ 1, accurate representation of eccentricity dynamics requires N ∼ Cς−χ ≫ 1, with C ∼ O(10), 1.5 ≲ χ ≳. In discs with sharp edges a very small value of the softening parameter ς (≲ 10−3) is required to correctly reproduce eccentricity dynamics near the disc boundaries; this finding is relevant for modelling planetary rings.
Highlights
Astrophysical discs orbiting a central mass Mc are ubiquitous in a variety of contexts – galactic, stellar, and planetary (Latter, Ogilvie & Rein 2017)
Complementary to this, we develop a general framework for computing the well-behaved secular disturbing function for a broad range of softened gravitational potentials
In Appendix A, we show that the form of the disturbing function given by equations (4)–(6) is generic for a wide class of softening models, for which the interaction potential between the two masses m1 and m2 located at r1 and r2, correspondingly, relative to the central mass, has a form3 i (r1, r2) = −Gmj (r1 − r2)2 + F (r1, r2) −1/2, (10)
Summary
Astrophysical discs orbiting a central mass Mc are ubiquitous in a variety of contexts – galactic, stellar, and planetary (Latter, Ogilvie & Rein 2017). Whenever Md Mc, particles perturbed by the disc gravity move on nearly Keplerian orbits that evolve rather slowly. The orbitaveraging procedure, known as Gauss’ method, is equivalent to calculating the time-averaged potential due to orbiting point masses by smearing them into massive elliptical ‘wires’ (having shape of their eccentric orbits) with non-uniform linear density proportional to the time spent by an object at a particular phase of its orbit. Such orbit-averaged potential, known as secular disturbing function Rd, fully determines the secular dynamics of the system
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