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

We study the evolution of binary systems of Ultra-luminous 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 $20M_{\odot}$ and $100M_{\odot}$, and donor masses between $8M_{\odot}$ and $25M_{\odot}$. Super-critical 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 $12-20 M_{\odot}$ donors accreting supercritically are characterized by blue colors (F450W -- F555W $\simeq - 0.2 : +0.1$) and high luminosity ($M_{V} \simeq - 4 : - 6.5$). Systems with more massive black holes accreting supercritically from evolved donors of similar mass have comparable colours but can reach $M_V \simeq - 8$. 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 a $\sim 20M_{\odot}$ black hole, while NGC4559 X-7 requires a significantly more massive black hole.