Supermassive black holes and their host spheroids III. The $M_{BH} -   n_{sph}$ correlation
  The S\'ersic $R^{1/n}$ model is the best approximation known to date for describing the light distribution of stellar spheroidal and disk components, with the S\'ersic index $n$ providing a direct measure of the central radial concentration of stars. The S\'ersic index of a galaxy's spheroidal component, $n_{sph}$, has been shown to tightly correlate with the mass of the central supermassive black hole, $M_{BH}$. The $M_{BH}-n_{sph}$ correlation is also expected from other two well known scaling relations involving the spheroid luminosity, $L_{sph}$: the $L_{sph}-n_{sph}$ and the $M_{BH}-L_{sph}$. Obtaining an accurate estimate of the spheroid S\'ersic index requires a careful modelling of a galaxy's light distribution and some studies have failed to recover a statistically significant $M_{BH}-n_{sph}$ correlation. With the aim of re-investigating the $M_{BH}-n_{sph}$ and other black hole mass scaling relations, we performed a detailed (i.e.~bulge, disks, bars, spiral arms, rings, halo, nucleus, etc.) decomposition of 66 galaxies, with directly measured black hole masses, that had been imaged at $3.6\rm~\mu m$ with Spitzer. In this paper, the third of this series, we present an analysis of the $L_{sph}-n_{sph}$ and $M_{BH}-n_{sph}$ diagrams. While early-type (elliptical+lenticular) and late-type (spiral) galaxies split into two separate relations in the $L_{sph}-n_{sph}$ and $M_{BH}-L_{sph}$ diagrams, they reunite into a single $M_{BH} \propto n_{sph}^{3.39 \pm 0.15}$ sequence with relatively small intrinsic scatter ($\epsilon \simeq 0.25 \rm~dex$). The black hole mass appears to be closely related to the spheroid central concentration of stars, which mirrors the inner gradient of the spheroid gravitational potential. 
Giulia A. D. Savorgnan
astro-ph.GA