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.