Orbital circularisation of white dwarfs and the formation of gravitational radiation sources in star clusters containing an intermediate mass black hole

¹ Department of Applied Mathematics and Theoretical Physics, CMS, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK e-mail: pbi20@damtp.cam.ac.uk
² Astro Space Center, P. N. Lebedev Physical Institute, Profsouyznaya St., 84/32 Moscow, Russia

Received: 15 January 2007
Accepted: 31 August 2007

Abstract

Aims.We consider how tight binaries consisting of a super-massive black hole of mass M = 10³-10⁴ and a white dwarf in quasi-circular orbit can be formed in a globular cluster. We point out that a major fraction of white dwarfs tidally captured by the black hole may be destroyed by tidal inflation during ongoing tidal circularisation, and therefore the formation of tight binaries is inhibited. However some fraction may survive tidal circularisation through being spun up to high rotation rates. Then the rates of energy loss through gravitational wave emission induced by tidally excited pulsation modes and dissipation through non linear effects may compete with the rate of increase of pulsation energy due to dynamic tides. The semi-major axes of these white dwarfs are decreased by tidal interaction below a “critical” value where dynamic tides decrease in effectiveness because pulsation modes retain phase coherence between successive pericentre passages.

Methods.We estimate the rate of formation of such circularising white dwarfs within a simple framework, modelling them as  polytropes and assuming that results obtained from the tidal theory for slow rotators can be extrapolated to fast rotators.

Results. We estimate the total capture rate as ~ N ~ 2.5 10^-8 yr^-1, where M₄ = M/10⁴ and r_0.1 is the radius of influence of the black hole expressed in units 0.1 pc. We find that the formation rate of tight pairs is approximately 10 times smaller than the total capture rate, for typical parameters of the problem. This result is used to estimate the probability of detection of gravitational waves coming from such tight binaries by LISA.

Conclusions. We conclude that LISA may detect such binaries provided that the fraction of globular clusters containing black holes in the mass range of interest is substantial and that the dispersion velocity of the cluster stars near the radius of influence of the black hole exceeds ~20 km s^-1.