Black Hole Spin Measurements Based on a Thin Disc Model with Finite Thickness I. An example study of MCG-06-30-15

We present a re-analysis of the XMM-Newton and NuSTAR observing campaign for the well-studied, X-ray-bright AGN MCG-06-30-15. In particular, we consider a disc model with finite thickness. By fitting the disc reflection spectra in the data, we obtain a black hole spin of 0.87--0.99 (90\% confidence range) after taking the thickness of the disc into consideration. Spectral models with a grid of mass accretion rate from 0 to $30\%\dot{M}_{\rm Edd}$ are calculated for MCG-06-30-15. This result is obtained by considering a free disc reflection fraction parameter $f_{\rm refl}$ and is consistent with previous measurements based on razor-thin disc models. Besides, an isotropic, point-like geometry, i.e. the `lamppost' geometry, is assumed for the corona in our model. We find that such a geometry overestimates $f_{\rm refl}$ in the data. Therefore, thin disc models with consistent `lamppost' values of $f_{\rm refl}$ provide a worse fit than ones with a free $f_{\rm refl}$ parameter. We discuss possible reasons for the discrepancy between the observed and theoretical values of $f_{\rm refl}$ at the end of the paper. Modifications for the over-simplified lamppost model might be needed when the thickness of the thin disc is considered in future work.