A Consistent Set of Empirical Scaling Relations for Spiral Galaxies: The $(v_{\max },M_{\mathrm{DM}})$-$({\sigma }_0,M_{\mathrm{BH}},\varphi )$ Relations

Abstract

Using the latest sample of 48 spiral galaxies having a directly measured supermassive black hole mass, $M_{BH}$, we determine how the maximum disk rotational velocity, $v_{max}$ (and the implied dark matter halo mass, $M_{DM}$), correlates with the (i) $M_{BH}$, (ii) central velocity dispersion, ${\sigma }_0$, and (iii) spiral-arm pitch angle, $\varphi$. We find that $M_{BH}\propto v_{max}^{10.62\pm 1.37}\propto M_{DM}^{4.35\pm 0.66}$, significantly steeper than previously reported, and with a root mean square scatter ($0.58$ dex) similar to that about the $M_{BH}$-${\sigma }_0$ relation for spiral galaxies - in stark disagreement with claims that $M_{BH}$ does not correlate with disks. Moreover, this $M_{BH}$-$v_{max}$ relation is consistent with the unification of the Tully-Fisher relation (involving the total stellar mass, $M_{\ast ,tot}$) and the steep $M_{BH}\propto M_{\ast ,tot}^{3.05\pm 0.53}$ relation observed in spiral galaxies. We also find that ${\sigma }_0\propto v_{max}^{1.55\pm 0.25}\propto M_{DM}^{0.63\pm 0.11}$, consistent with past studies connecting stellar bulges (with ${\sigma }_0\gtrsim 100{\mathrm{km}\mathrm{s}}^{-1}$), dark matter halos, and a nonconstant $v_{max}/{\sigma }_0$ ratio. Finally, we report that $\tan |\varphi |\propto (-1.18\pm 0.19)\log v_{max}\propto (-0.48\pm 0.09)\log M_{DM}$, providing a novel formulation between the geometry (i.e., the logarithmic spiral-arm pitch angle) and kinematics of spiral galaxy disks. While the $v_{max}$-$\varphi$ relation may facilitate distance estimations to face-on spiral galaxies through the Tully-Fisher relation and using $\varphi$ as a proxy for $v_{max}$, the $M_{DM}$-$\varphi$ relation provides a path for determining dark matter halo masses from imaging data alone. Furthermore, based on a spiral galaxy sample size that is double the size used previously, the self-consistent relations presented here provide dramatically revised constraints for theory and simulations.

25 pages, 3 figures, and 1 table