The X‐Ray Spectral Evolution in X‐Ray Binaries and Its Application to Constrain the Black Hole Mass of Ultraluminous X‐Ray Sources

We investigate the relationship between the hard X-ray photon index and the Eddington ratio (=Lₗ (0. 5-25 keV) /L₄₃₃) in six X-ray binaries (XRBs) with well constrained black hole masses and distances. We find that different XRBs follow different anti-correlations between and when is less than a critical value, while and generally follow the same positive correlation when is larger than the critical value. The anti-correlation and the positive correlation may suggest that they are in different accretion modes (e. g. , radiatively inefficient accretion flow (RIAF) and standard disk). We fit both correlations with the linear least-square method for individual sources, from which the crosspoint of two fitted lines is obtained. Although the anti-correlation varies from source to source, the crosspoints of all sources roughly converge to the same point with small scatter (=-2. 10. 2, =1. 5 0. 1), which may correspond to the transition point between RIAF and standard accretion disk. Motivated by the observational evidence for the similarity of the X-ray spectral evolution of ultraluminous X-ray sources (ULXs) to that of XRBs, we then constrain the black hole masses for seven ULXs assuming that their X-ray spectral evolution is similar to that of XRBs. We find that the BH masses of these seven luminous ULXs are around 10^4, which are typical intermediate-mass BHs (IMBHs). Our results are generally consistent with the BH masses constrained from the timing properties (e. g. , break frequency) or the model fitting with a multi-color disk.