Secular diffusion in discrete self-gravitating tepid discs

I. Analytic solution in the tightly wound limit^⋆

¹ Institut d’Astrophysique de Paris and UPMC, CNRS (UMR 7095), 98bis boulevard Arago, 75014 Paris, France
e-mail: fouvry@iap.fr; pichon@iap.fr
² Institute of Astronomy & KICC, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK
³ Laboratoire de Physique Théorique (IRSAMC), CNRS and UPS, Univ. de Toulouse, 31062 Toulouse, France
e-mail: chavanis@irsamc.ups-tlse.fr

Received: 19 February 2015
Accepted: 2 May 2015

Abstract

The secular evolution of an infinitely thin tepid isolated galactic disc made of a finite number of particles is described using the inhomogeneous Balescu-Lenard equation. Assuming that only tightly wound transient spirals are present in the disc, a WKB approximation provides a simple and tractable quadrature for the corresponding drift and diffusion coefficients. It provides insight into the physical processes at work during the secular diffusion of a self-gravitating discrete disc and makes quantitative predictions on the initial variations of the distribution function in action space. When applied to the secular evolution of an isolated stationary self-gravitating Mestel disc, this formalism predicts the initial importance of the corotation resonance in the inner regions of the disc leading to a regime involving radial migration and heating. It predicts in particular the formation of a ridge-like feature in action space, in agreement with simulations, but over-estimates the timescale involved in its appearance. Swing amplification is likely needed to resolve this discrepancy. In astrophysics, the inhomogeneous Balescu-Lenard equation and its WKB limit may also 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.