# A Consistent Set of Empirical Scaling Relations for Spiral Galaxies: The $(v_{\rm max},\,M_{\rm DM})$-$(\sigma_0,\,M_{\rm BH},\,\phi)$ Relations

## TL;DR

This paper establishes new, steep empirical relations linking supermassive black hole mass, dark matter halo mass, and galaxy disk properties in spiral galaxies, with implications for galaxy evolution and distance measurement.

## Contribution

It presents the first comprehensive set of consistent scaling relations connecting black hole mass, dark matter, and disk geometry in spiral galaxies, based on an expanded galaxy sample.

## Key findings

- Steeper $M_{BH}$-$v_{max}$ relation than previously reported
- Correlation between spiral-arm pitch angle and galaxy kinematics
- Dark matter halo mass can be inferred from spiral arm geometry

## 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, $\phi$. 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 $M_{BH}$-$\sigma_0$ 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, $M_{*,tot}$) and the steep $M_{BH}\propto M_{*,tot}^{3.05\pm0.53}$ relation observed in spiral galaxies. We also find that $\sigma_0\propto v_{max}^{1.55\pm0.25}\propto M_{DM}^{0.63\pm0.11}$, consistent with past studies connecting stellar bulges (with $\sigma_0\gtrsim100\,{\rm km\,s}^{-1}$), dark matter halos, and a nonconstant $v_{max}/\sigma_0$ ratio. Finally, we report that $\tan|\phi|\propto(-1.18\pm0.19)\log{v_{max}}\propto(-0.48\pm0.09)\log M_{DM}$, providing a novel formulation between the geometry (i.e., the logarithmic spiral-arm pitch angle) and kinematics of spiral galaxy disks. While the $v_{max}$-$\phi$ relation may facilitate distance estimations to face-on spiral galaxies through the Tully-Fisher relation and using $\phi$ as a proxy for $v_{max}$, the $M_{DM}$-$\phi$ 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.