Establishing a Relation between the Mass and the Spin of Stellar-Mass Black Holes

Stellar mass black holes (SMBHs), forming by the core collapse of very massive, rapidly rotating stars, are expected to exhibit a high density accretion disk around them developed from the spinning mantle of the collapsing star. A wide class of such disks, due to their high density and temperature, are effective emitters of neutrinos and hence called neutrino cooled disks. Tracking the physics relating the observed (neutrino) luminosity to the mass, spin of black holes (BHs) and the accretion rate ($\dot{M}$) of such disks, here we establish a correlation between the spin and mass of SMBHs at their formation stage. Our work shows that spinning BHs are more massive than nonspinning BHs for a given $\dot{M}$. However, slowly spinning BHs can turn out to be more massive than spinning BHs if $\dot{M}$ at their formation stage was higher compared to faster spinning BHs.

Figure 1

Mass as a function of the BH’s spin for a collapsar I disk using $\dot{M}$ in units of $M_{\odot }{\mathrm{s}}^{-1}$ as the parameter, labeled in each contour.

Figure 2

Mass as a function of $\dot{M}$ in units of $M_{\odot }{\mathrm{s}}^{-1}$ for a collapsar I disk using the BH’s spin as the parameter. From the bottom to top, the contours correspond to $a_{*}=0$, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.998.

Figure 3

Mass as a function of the BH’s spin for a collapsar II disk using $\dot{M}$ in units of $M_{\odot }{\mathrm{s}}^{-1}$ as the parameter, labeled in each contour.