A Consistent Set of Empirical Scaling Relations for Spiral Galaxies: The (v max , M oM )–(σ 0 , M BH , ϕ) 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) black hole mass, (ii) central velocity dispersion, σ 0 , and (iii) spiral-arm pitch angle, ϕ . We find that M BH ∝ v max 10.62 ± 1.37 ∝ M DM 4.35 ± 0.66 , significantly steeper than previously reported, and with a total 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 ∝ M * , tot 3.05 ± 0.53 relation observed in spiral galaxies. We also find that σ 0 ∝ v max 1.55 ± 0.25 ∝ M DM 0.63 ± 0.11 , consistent with past studies connecting stellar bulges (with σ 0  ≳ 100 km s −1 ), dark matter halos, and a nonconstant v max / σ 0 ratio. Finally, we report that tan ∣ ϕ ∣ ∝ ( − 1.18 ± 0.19 ) log v max ∝ ( − 0.48 ± 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 – 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.