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

We model the multiwavelength properties of binaries accreting at super-critical rates with the aim to better understand the observational properties of Ultra-luminous X-ray Sources (ULXs). We calculate an extended grid of binary systems which evolve through Roche Lobe Overflow and undergo case A mass transfer from massive donors (up to 25 ${\rm M}_{\odot } \,$) on to massive black holes (BHs) (up to 100 ${\rm M}_{\odot } \,$). Angular momentum loss with the ejection of mass through an outflow is incorporated. We apply our super-Eddington accretion model to these systems, computing their evolutionary tracks on the colour-magnitude diagram (CMD) for the Johnson and Hubble Space Telescope photometric systems. We found that the tracks occupy specific positions on the CMD depending on the evolutionary stage of the donor and of the binary. Moreover, their shapes are similar, regardless the BH mass. More massive BHs lead to more luminous tracks. We additionally compute their optical-through-X-ray spectral energy distribution considering the effects of a Comptonizing corona that surrounds the innermost regions of the disc. We apply our model to four ULXs: NGC 4559 X-7, NGC 5204 X-1, Holmberg II X-1, and NGC 5907 ULX-2. We found that accretion on to BHs with mass in the range of 35-55 ${\rm M}_{\odot }$ is consistent with to the observational properties of these sources. We finally explore and discuss the possibility to extend our model also to ULXs powered by accreting pulsars (PULXs).