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A calibration is made for the correlation between the X-ray VariabilityAmplitude (XVA) and Black Hole (BH) mass. The correlation for 21reverberation-mapped Active Galactic Nuclei (AGN) appears very tight, with anintrinsic dispersion of 0.20 dex. The intrinsic dispersion of 0.27 dex can beobtained if BH masses are estimated from the stellar velocity dispersions. Wefurther test the uncertainties of mass estimates from XVAs for objects whichhave been observed multiple times with good enough data quality. The resultsshow that the XVAs derived from multiple observations change by a factor of 3.This means that BH mass uncertainty from a single observation is slightly worsethan either reverberation-mapping or stellar velocity dispersion measurements;however BH mass estimates with X-ray data only can be more accurate if the meanXVA value from more observations is used. Applying this relation, the BH massof RE J1034+396 is found to be $4^{+3}_{-2} \times 10^6$ $M_{\odot}$. The highend of the mass range follows the relationship between the 2$f_0$ frequenciesof high-frequency QPO and the BH masses derived from the Galactic X-raybinaries. We also calculate the high-frequency constant $C= 2.37 M_\odot$Hz$^{-1}$ from 21 reverberation-mapped AGN. As suggested by Gierli\'nski etal., $M_{\rm BH}=C/C_{\rm M}$, where $C_{\rm M}$ is the high-frequencyvariability derived from XVA. Given the similar shape of power-law dominatedX-ray spectra in ULXs and AGN, this can be applied to BH mass estimates ofULXs. We discuss the observed QPO frequencies and BH mass estimates in theUltra-Luminous X-ray source M82 X-1 and NGC 5408 X-1 and favor ULXs asintermediate mass BH systems (abridged).

astrophysical journal/the astrophysical journal

Calibrating the correlation between black hole mass and X-ray  variability amplitude: X-ray only black hole mass estimates for active  galactic nuclei and ultra-luminous X-ray sources