On formation rate of close binaries consisting of a super-massive black hole and a white dwarf

Abstract

The formation rate of a close binary consisting of a super-massive black hole and a compact object (presumably a white dwarf) in galactic cusps is calculated with help of the so-called loss cone approximation. For a power low cusp of radius $r_{a}$, the black hole mass $M\sim 10^{6}M_{\odot}$, this rate $\dot N_{wd}\sim 4\cdot 10^{-5}K(p}\sqrt{{GM\over r_{a}^{3}}}\approx 3\cdot 10^{-9}K(p){({M\over 10^{6}M_{\odot}})}^{1/2}{({r_{a}\over 1pc})}^{-3/2}yr^{-1}$. The function $K(p)$ depends on parameter $p$ determining the cusp profile, and for the standard cusp profiles with $p=1/4$ $K(p)\sim 2$. We estimate the probability ${\it Pr}$ of finding of a compact object orbiting around a black hole with period $P$ in one particular galaxy to be ${\it Pr}\sim 10^{-7}{({P/10^{3}s\over M/10^{6}M_{\odot}})}^{8/3} {({M/10^{6}M_{\odot}\over r_{a}/ 1pc})}^{3/2}$. The object with the period $P\sim 10^{3} s$ emits gravitational waves with amplitude sufficient to be detected by LISA type gravitational wave anatenna from the distance $\sim 10^{3}Mpc$. Based on estimates of masses of super-massive black holes in nearby galaxies, we speculate that LISA would detect several such events during its mission.