Roche accretion of stars close to massive black holes

In this paper, we consider Roche accretion in an extreme mass-ratio inspiral binary system formed by a star orbiting a massive black hole. The ultimate goal is to constrain the mass and spin of the black hole and provide a test of general relativity in the strong-field regime from the resultant quasi-periodic signals. Before accretion starts, the stellar orbit is presumed to be circular and equatorial, and shrinks due to gravitational radiation. New fitting formulae are presented for the inspiral time and the radiation reaction torque in the relativistic regime. If the inspiralling star fills its Roche lobe outside the innermost stable circular orbit of the hole, gas will flow through the inner Lagrange point (L1) to the hole. We give a new, accurate interpolation formula of the volume enclosed by the relativistic Roche lobe. If this mass transfer happens on a time-scale faster than the thermal time-scale but slower than the dynamical time-scale, the star will evolve adiabatically, and, in most cases, will recede from the hole while filling its Roche lobe. We calculate how the stellar orbital period and mass transfer rate change through the 'Roche evolution' for various types of stars in the relativistic regime. We envisage that the mass stream eventually hits the accretion disc, where it forms a hotspot orbiting the hole and may ultimately modulate the luminosity with the stellar orbital frequency. The observability of such a modulation is discussed along with a possible interpretation of an intermittent 1 h period in the X-ray emission of RE J1034+396.