The Astrophysical Journal, 877:64 (15pp), 2019 May 20 https://doi.org/10.3847/1538-4357/ab1aa4 © 2019. The American Astronomical Society. All rights reserved. A Consistent Set of Empirical Scaling Relations for Spiral Galaxies: The (vmax, MoM)–(σ0, MBH, f) Relations Benjamin L. Davis1 , Alister W. Graham1 , and Françoise Combes2,3 1 Centre for Astrophysics and Supercomputing, Swinburne University of Technology, Hawthorn, VIC 3122, Australia; [email protected] 2 Observatoire de Paris, LERMA, CNRS, PSL Université, Sorbonne Université, F-75014 Paris, France 3 Collège de France, 11 Place Marcelin Berthelot, F-75005 Paris, France Received 2019 January 19; revised 2019 April 4; accepted 2019 April 16; published 2019 May 24 Abstract Using the latest sample of 48 spiral galaxies having a directly measured supermassive black hole mass, MBH,we determine how the maximum disk rotational velocity, vmax (and the implied dark matter halo mass, MDM), correlates with the (i) black hole mass, (ii) central velocity dispersion, σ0,and(iii) spiral-arm pitch angle, f.Wefind that 10.62 1.37 4.35 0.66 fi MvBH µµmax MDM ,signi cantly steeper than previously reported, and with a total root mean square scatter (0.58 dex) similar to that about the MBH–σ0 relation for spiral galaxies—in stark disagreement with claims that MBH does not correlate with disks. Moreover, this MBH–vmax relation is consistent with the unification of the Tully–Fisher relation (involving the total stellar mass, M ) and the steep MMµ 3.05 0.53 relation observed in spiral galaxies. We also *,tot BH *,tot fi 1.55 0.25 0.63 0.11 ( σ −1) nd that s0 µµvMmax DM , consistent with past studies connecting stellar bulges with 0 100 km s , dark matter halos, and a nonconstant vmax/σ0 ratio. Finally, we report that tan∣∣f µ- ( 1.18 0.19 ) logvMmaxµ-() 0.48 0.09 log DM, providing a novel formulation between the geometry (i.e., the logarithmic spiral-arm pitch angle) and kinematics of spiral galaxy disks. While the vmax–f relation may facilitate distance estimations to face-on spiral galaxies through the Tully–Fisher relation and using f as a proxy for vmax,theMDM–f 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. Key words: black hole physics – dark matter – galaxies: evolution – galaxies: fundamental parameters – galaxies: spiral – galaxies: structure 1. Introduction for reviews on Sagittarius A*), with modern measurements (Gravity Collaboration et al. 2018a, 2018b;Amorimetal.2019) Building on the possibility of unseen mass in the solar even detecting the effects of general relativity (Einstein 1916; neighborhood (Jeans 1922;Kapteyn1922; Lindblad 1926;Oort Schwarzschild 1916;Kerr1963; Bardeen et al. 1972) in its 1932), dark matter has been considered by many astronomers to be vicinity, and providing direct imaging via very long baseline prevalent in galaxy clusters since the 1930s (Zwicky 1933; interferometry (Issaoun et al. 2019).5 In contrast to dark matter, Smith 1936;Zwicky1937;Schwarzschild1954;Rood1965).In SMBHs residing at the centers of galaxies are thought to constitute addition, the study of galaxy rotation curves provided strong a tiny fraction of the universe’stotalmass(Graham et al. evidence for the existence of dark matter (Babcock 1939;Oort 2007;Vikaetal.2009; Davis et al. 2014;Mutlu-Pakdiletal. 1940;Freeman1970;Rubin&Ford1970; Rogstad & Shostak 2016). 1972; Roberts & Rots 1973;Rubinetal.1977, 1978;Krumm& Astronomers have long been comfortable with the idea that Salpeter 1979; Rubin et al. 1980;Bosma1981;Persic& SMBH masses (M ) should correlate with properties of their Salucci 1988; Broeils 1992). The notion of nonbaryonic dark BH host galactic bulges, as evidenced by a vast literature dedicated matter subsequently grew as a mechanism to explain this to the study of correlations with a bulge’s stellar velocity anomalous gravitational phenomenon (Gershtein & Zel’dovich dispersion (σ ), (baryonic and total) mass, stellar luminosity, 1966; Marx & Szalay 1972; Cowsik & McClelland 1972;Szalay 0 Sérsic index, etc. (see Graham 2016, for a review), although &Marx1976). Intriguingly, dark matter is believed to account for many theoretical models advocate that the total gravitational 84%±1% of the total mass in the universe (Planck Collaboration mass of a galaxy or its most dominant component, the mass of et al. 2018), but it remains elusive despite considerable efforts to its dark matter halo (M ), should dictate the formation of achieve direct instrumental detection of its theorized particles (e.g., DM SMBHs (e.g., Loeb & Rasio 1994; Haehnelt et al. 1998; Silk & Tan et al. 2016;Akeribetal.2017; Aprile et al. 2018).4 The Rees 1998; Cattaneo et al. 1999; Haehnelt & Kauffmann 2000; concept of supermassive black holes (SMBHs) lurkinginthe Monaco et al. 2000; Adams et al. 2001). The rotation curve center of galaxies has also had an interesting history of study (see informs us about the total mass of a disk galaxy and its dark Kormendy & Richstone 1995; Longair 1996, 2006; Ferrarese & matter component. Therefore, the rotational velocity profile of a Ford 2005,for reviews on SMBHs).Itwasinaspiralgalaxy,our 6 * galaxy disk, v (R), can be considered a surrogate for its M . Milky Way, where an SMBH (Sagittarius A ) was first calculated rot DM to exist (Lynden-Bell 1969; Sanders & Lowinger 1972). For half a century, confidence in its existence gradually increased from 5 See also the revolutionary imaging of the central SMBH in M87 (Event suggestion to certainty (see Alexander 2005 and Genzel et al. 2010 Horizon Telescope Collaboration et al. 2019). 6 Globular clusters have also been shown to be apropos tracers of the dark matter halo mass; both the globular cluster system mass (Spitler & Forbes 2009) 4 See Seigar (2015), Arcadi et al. (2018), Hooper (2018), and Salucci (2019) and the number of globular clusters (Burkert & Forbes 2019) of a galaxy are for reviews on searches for dark matter. directly proportional to its dark matter halo mass. 1 The Astrophysical Journal, 877:64 (15pp), 2019 May 20 Davis, Graham, & Combes Thus, a relation between MBH and some measure of vrot(R) or regression analyses and discussion for six relations: vrot–σ0, MDM should be expected in observational data. MDM–σ0, MBH–vrot, MBH–MDM, vrot–f, and MDM–f. Finally, Recently, we showed in Davis et al. (2018) that a significant Section 4 presents our interpretation of the results and correlation exists between the total stellar mass (M*,tot) of a elucidates their significance. All printed uncertainties are 1σ spiral galaxy and the mass of its central black hole. When (≈68.3%) confidence intervals. All magnitudes are quoted in combined with the well-known Tully & Fisher (1977) relation the Vega system. between the total stellar luminosity/mass of a galaxy and its rotational velocity, one obtains a relation between vrot and MBH. 2. Data A contemporary study by Tiley et al. (2019) has defined the In Davis et al. (2017), we compiled a comprehensive sample z≈0 Tully–Fisher relation between M* and v for late- ,tot rot of 44 galaxies classified as spirals and having directly measured type galaxies7 to be such that (dynamical)9 black hole masses (see Table 1). In Davis et al. 4.0 0.1 (2018, 2019), we used this sample to determine how the black Mv*,tot µ rot .1() hole mass of a spiral galaxy scales with its total stellar and ( ) In Davis et al. 2018 , we found that bulge stellar masses, respectively. Here, we use the same 3.05 0.53 sample, plus four new galaxies (NGC 613, NGC 1365, MMBH µ ()2 *,tot NGC 1566, and NGC 1672) from Combes et al. (2019), for spiral galaxies. Therefore, we should expect to find making our total sample of spiral galaxies with directly 12.2 2.1 measured black hole masses twice the size of recent studies MvBH µ rot ,3() (e.g., Sabra et al. 2015). For this expanded sample, we tabulate f σ ( which can subsequently be converted into an M –M the and 0 measurements predominantly from Davis et al. BH DM ) relation using an expression from Katz et al. (2018b) between 2017 , plus the vmax measurements that we have assembled from the literature. We use this rotational velocity to draw a vrot and MDM. In addition to our focus on black hole mass scaling relations connection to the dark matter halo mass, which dictates the maximum rotational velocity at the outer radii of a galaxy. As for spiral galaxies in this work, we also investigate scaling ( ) relations between v (and M ) with both σ and the revealed in Davis et al. 2017 , particular care was taken to rot DM 0 obtain the fundamental spiral-arm pitch angle rather than the logarithmic spiral-arm pitch angle (f).Avrot–σ0 ratio or fi harmonics, which may be a factor of two or three smaller or relation for disk galaxies was rst suggested by Whitmore et al. fi (1979) and Whitmore & Kirshner (1981). They found larger and are accidentally obtained when an insuf ciently long section of spiral arms is used. vrot/σ0∼1.7 by measuring H I line widths for local S0 and ( ) Forty-two galaxies out of our total sample of 48 galaxies have spiral galaxies. More recently, Cresci et al. 2009 found that ( gas-rich, turbulent, star-forming z∼2 disk galaxies exhibit vmax measurements in the literature. These are mostly 29 out ) 10 ( ) v /σ ∼4.4.8 The v –σ relation represents an intriguing of 42 from the HyperLeda database Paturel et al. 2003 , rot 0 rot 0 which provides homogenized maximum rotational velocities connection between the stellar bulge of a galaxy and its dark ( ) σ calculated from the 21 cm line maximum widths Wmax or matter halo, although for spiral galaxies with turbulent disks, 0 ( α ) may trace the stellar disk as much as the bulge.