Correlations between X-ray properties and Black Hole Mass in AGN: towards a new method to estimate black hole mass from short exposure X-ray observations

Several investigations of the X-ray variability of active galactic nuclei (AGN) using the normalised excess variance (\({\sigma^2_{\rm NXS}}\)) parameter have shown that variability has a strong anti-correlation with black hole mass (\(M_{\rm BH}\)) and X-ray luminosity (\(L_{\rm X}\)). In this study we confirm these previous correlations and find no evidence of a redshift evolution. Using observations from XMM-Newton, we determine the \({\sigma^2_{\rm NXS}}\) and \(L_{\rm X}\) for a sample of 1091 AGN drawn from the XMM-Newton Cluster Survey (XCS) - making this the largest study of X-ray spectral properties of AGNs. We created light-curves in three time-scales; 10 ks, 20 ks and 40 ks and used these to derive scaling relations between \({\sigma^2_{\rm NXS}}\), \(L_{\rm X}\) (2.0-10 keV range) and literature estimates of \(M_{\rm BH}\) from reverberation mapping. We confirm the anti-correlation between \(M_{\rm BH}\) and \({\sigma^2_{\rm NXS}}\) and find a positive correlation between \(M_{\rm BH}\) and \(L_{\rm X}\). The use of \({\sigma^2_{\rm NXS}}\) is practical only for pointed observations where the observation time is tens of kiloseconds. For much shorter observations one cannot accurately quantify variability to estimate \(M_{\rm BH}\). Here we describe a method to derive \(L_{\rm X}\) from short duration observations and used these results as an estimate for \(M_{\rm BH}\). We find that it is possible to estimate \(L_{\rm X}\) from observations of just a few hundred seconds and that when correlated with \(M_{\rm BH}\), the relation is statistically similar to the relation of \(M_{\rm BH}\)-\(L_{\rm X}\) derived from a spectroscopic analysis of full XMM observations. This method may be particularly useful to the eROSITA mission, an all-sky survey, which will detect \(>\)10\(^{6}\) AGN.