On Black Hole Mass Estimation from X-ray Spectra of Ultraluminous X-ray Sources

We propose a methodology to derive a black-hole mass for super-critical accretion flow. Here, we use the extended disk blackbody (extended DBB) model, a fitting model in which the effective temperature profile obeys the relation $T_{\rm eff} \propto r^{-p}$, with $r$ being the disk radius and $p$ being treated as a fitting parameter. We first numerically calculate the theoretical flow structure and its spectra for a given black-hole mass, $M$, and accretion rate, $\dot{M}$. Through fitting to the theoretical spectra by the extended DBB model, we can estimate the black-hole mass, $M_{\rm x}$, assuming that the innermost disk radius is $r_{\rm in}=3r_{\rm g} (\propto M_{\rm x})$, where $r_{\rm g}$ is the Schwarzschild radius. We find, however, that the estimated mass deviates from that adopted in the spectral calculations, $M$, even for low-$\dot{M}$ cases. We also find that the deviations can be eliminated by introducing a new correction for the innermost radius. Using this correction, we calculate mass correction factors, $M/M_{\rm x}$, in the super-critical regimes for some sets of $M$ and $\dot M$, finding that a mass correction factor ranges between $M/M_{\rm x} \sim$ 1.2 -- 1.6. The higher is $\dot{M}$, the larger does the mass correction factor tend to be. Since the correction is relatively small, we can safely conclude that the black holes in ULXs which Vierdayanti et al. (2006, PASJ, 58, 915) analyzed are stellar-mass black holes with the mass being $< 100M_{\odot}$.