Secular diffusion in discrete self-gravitating tepid discs II. Accounting for swing amplification via the matrix method^⋆,⋆⋆

¹ Institut d’Astrophysique de Paris and UPMC, CNRS (UMR 7095), 98bis Boulevard Arago, 75014 Paris, France
e-mail: fouvry@iap.fr
² Institute of Astronomy & KICC, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK
³ Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Keble Road, Oxford OX1 3RH, UK
⁴ Laboratoire de Physique Théorique (IRSAMC), CNRS and UPS, Univ. de Toulouse, 31062 Toulouse, France

Received: 24 July 2015
Accepted: 25 August 2015

Abstract

The secular evolution of an infinitely thin tepid isolated galactic disc made of a finite number of particles is investigated using the inhomogeneous Balescu-Lenard equation expressed in terms of angle-action variables. The matrix method is implemented numerically in order to model the induced gravitational polarisation. Special care is taken to account for the amplification of potential fluctuations of mutually resonant orbits and the unwinding of the induced swing amplified transients. Quantitative comparisons with N-body simulations yield consistent scalings with the number of particles and with the self-gravity of the disc: the fewer the particles and the colder the disc, the faster the secular evolution. Secular evolution is driven by resonances, but does not depend on the initial phases of the disc. For a Mestel disc with Q ~ 1.5, the polarisation cloud around each star boosts its secular effect by a factor of a thousand or more, accordingly promoting the dynamical relevance of self-induced collisional secular evolution. The position and shape of the induced resonant ridge are found to be in very good agreement with the prediction of the Balescu-Lenard equation, which scales with the square of the susceptibility of the disc. In astrophysics, the inhomogeneous Balescu-Lenard equation may describe the secular diffusion of giant molecular clouds in galactic discs, the secular migration and segregation of planetesimals in proto-planetary discs, or even the long-term evolution of population of stars within the Galactic centre. It could be used as a valuable check of the accuracy of N-body integrators on secular timescales.