Open AccessA non‐linear theory of vertical resonances in accretion discs
Author(s) -
Ogilvie G. I.
Publication year - 2002
Publication title -
monthly notices of the royal astronomical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.058
H-Index - 383
eISSN - 1365-2966
pISSN - 0035-8711
DOI - 10.1046/j.1365-8711.2002.05270.x
Subject(s) - physics , radius , accretion (finance) , accretion disc , resonance (particle physics) , astrophysics , forcing (mathematics) , astronomy , atomic physics , atmospheric sciences , computer security , computer science
An important and widely neglected aspect of the interaction between an accretion disc and a massive companion with a coplanar orbit is the vertical component of the tidal force. As shown by Lubow, the response of the disc to vertical forcing is resonant at certain radii, at which a localized torque is exerted, and from which a compressive wave (p mode) may be emitted. Although these vertical resonances are weaker than the corresponding Lindblad resonances, the m =2 inner vertical resonance in a binary star is typically located within the tidal truncation radius of a circumstellar disc. In this paper I develop a general theory of vertical resonances, allowing for non‐linearity of the response, and dissipation by radiative damping and turbulent viscosity. The problem is reduced to a universal, non‐linear ordinary differential equation with two real parameters. Solutions of the complex non‐linear Airy equation are presented to illustrate the non‐linear saturation of the resonance and the effects of dissipation. It is argued that the m =2 inner vertical resonance is unlikely to truncate the disc in cataclysmic variable stars, but contributes to angular momentum transport and produces a potentially observable non‐axisymmetric structure.