Modelling optical emission of Ultra-luminous X-ray Sources accreting above the Eddington limit

We study the evolution of binary systems of ultraluminous X-ray sources and compute their optical emission assuming accretion onto a black hole via a non-standard, advection-dominated slim disc with an outflow. We consider systems with black holes of 20–|100\, M|⁠, and donor masses between 8 and |25\, M|⁠. Supercritical accretion has considerable effects on the optical emission. The irradiating flux in presence of an outflow remains considerably stronger than that produced by a standard disc. However, at very high accretion rates the contribution of X-ray irradiation becomes progressively less important in comparison with the intrinsic flux emitted from the disc. After main sequence the evolutionary tracks of the optical counterpart on the colour–magnitude diagram are markely different from those computed for Eddington-limited accretion. Systems with stellar-mass black holes and |12--20 \, M| donors accreting supercritically are characterized by blue colours (F450W – F555W ≃ − 0.2: +0.1) and high luminosity (MV ≃ −4: −6.5). Systems with more massive black holes accreting supercritically from evolved donors of similar mass have comparable colours but can reach MV ≃ −8. We apply our model to NGC 1313 X-2 and NGC 4559 X-7. Both sources are well represented by a system accreting above Eddington from a massive evolved donor. For NGC 1313 X-2 the agreement is for an |~ 20\, M| black hole, while NGC 4559 X-7 requires a significantly more massive black hole.