The Sérsic R ^ { 1 / n } model is the best approximation known to date for describing the light distribution of stellar spheroidal and disk components , with the Sérsic index n providing a direct measure of the central radial concentration of stars .
The Sérsic 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 Sérsic 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 ~ { } \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 ~ { } 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 .