Relativistic viscous accretion flow model for black hole sources with XMM–Newton observations

We develop a model formalism to study the structure of a relativistic, viscous, optically thin, advective accretion flow around a rotating black hole in presence of radiative coolings. We use this model to examine the physical parameters of four black hole Ultra-luminous X-ray sources (BH-ULXs), namely mass (MBH), spin (ak) and accretion rate (ṁ), respectively. While doing this, we adopt a recently developed effective potential to mimic the spacetime geometry around the rotating black holes. We solve the governing equations to obtain the shock induced global accretion solutions in terms of ṁ and viscosity parameter (α). Using shock properties, we compute the Quasi-periodic Oscillation (QPO) frequency (νQPO) of the post-shock matter, when the shock front exhibits Quasiperiodic variations. We also calculate the bolometric luminosity (Lbol) of the entire disc for these shock solutions. Utilizing νQPO and Lbol, we constrain BH-ULXs mass by varying their spin (ak) and accretion rate (ṁ). We find that NGC6946 X−1 and NGC5408 X−1 seem to accrete at sub-Eddington accretion rate provided their central sources are rapidly rotating, whereas IC342 X−1 and NGC1313 X−1 can accrete in sub/super-Eddington limit irrespective to their spin values.