A Consistent Set of Empirical Scaling Relations for Spiral Galaxies: The $(v_{\rm max},\,M_{\rm DM})$-$(σ_0,\,M_{\rm BH},\,ϕ)$ Relations

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, $σ_0$, and (iii) spiral-arm pitch angle, $ϕ$. We find that $M_{BH}\propto v_{max}^{10.62\pm1.37}\propto M_{DM}^{4.35\pm0.66}$, significantly steeper than previously reported, and with a root mean square scatter ($0.58$ dex) similar to that about the $M_{BH}$-$σ_0$ 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, $M_{*,tot}$) and the steep $M_{BH}\propto M_{*,tot}^{3.05\pm0.53}$ relation observed in spiral galaxies. We also find that $σ_0\propto v_{max}^{1.55\pm0.25}\propto M_{DM}^{0.63\pm0.11}$, consistent with past studies connecting stellar bulges (with $σ_0\gtrsim100\,{\rm km\,s}^{-1}$), dark matter halos, and a nonconstant $v_{max}/σ_0$ ratio. Finally, we report that $\tan|ϕ|\propto(-1.18\pm0.19)\log{v_{max}}\propto(-0.48\pm0.09)\log M_{DM}$, providing a novel formulation between the geometry (i.e., the logarithmic spiral-arm pitch angle) and kinematics of spiral galaxy disks. While the $v_{max}$-$ϕ$ relation may facilitate distance estimations to face-on spiral galaxies through the Tully-Fisher relation and using $ϕ$ as a proxy for $v_{max}$, the $M_{DM}$-$ϕ$ 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.

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