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Updating the (Supermassive Black Hole Mass)-(Spiral Arm Pitch Angle) Relation: a Strong Correlation for Galaxies with Pseudobulges

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Updating the (Supermassive Black Hole Mass)-(Spiral Arm Pitch Angle) Relation: a Strong Correlation for Galaxies with Pseudobulges MNRAS 000,1{18 (2017) Preprint 20 March 2018 Compiled using MNRAS LATEX style file v3.0 Updating the (supermassive black hole mass){(spiral arm pitch angle) relation: a strong correlation for galaxies with pseudobulges Benjamin L. Davis,1? Alister W. Graham1 and Marc S. Seigar2 1Centre for Astrophysics and Supercomputing, Swinburne University of Technology, Hawthorn, Victoria 3122, Australia 2Department of Physics and Astronomy, University of Minnesota Duluth, Duluth, MN 55812, USA Accepted 2017 July 13. Received 2017 June 27; in original form 2017 May 24 ABSTRACT We have conducted an image analysis of the (current) full sample of 44 spiral galaxies with directly measured supermassive black hole (SMBH) masses, MBH, to determine each galaxy's logarithmic spiral arm pitch angle, φ. For predicting black hole masses, ◦ we have derived the relation: log¹MBH/M º = ¹7:01 ± 0:07º − ¹0:171 ± 0:017º »jφj − 15 ¼. The total root mean square scatter associated with this relation is 0.43 dex in the log MBH direction, with an intrinsic scatter of 0:33 ± 0:08 dex. The MBH{φ relation is therefore at least as accurate at predicting SMBH masses in spiral galaxies as the other known relations. By definition, the existence of an MBH{φ relation demands that the SMBH mass must correlate with the galaxy discs in some manner. Moreover, with the majority of our sample (37 of 44) classified in the literature as having a pseudobulge morphology, we additionally reveal that the SMBH mass correlates with the large- scale spiral pattern and thus the discs of galaxies hosting pseudobulges. Furthermore, given that the MBH{φ relation is capable of estimating black hole masses in bulge- less spiral galaxies, it therefore has great promise for predicting which galaxies may 5 harbour intermediate-mass black holes (IMBHs, MBH < 10 M ). Extrapolating from the current relation, we predict that galaxies with jφj ≥ 26◦:7 should possess IMBHs. Key words: black hole physics { galaxies: bulges { galaxies: evolution { galaxies: fundamental parameters { galaxies: spiral { galaxies: structure 1 INTRODUCTION Not surprisingly, pitch angle is intimately related to Hubble type, although it can at times be a poor indicator due Qualitatively, pitch angle is fairly easy to determine by eye. to misclassification and asymmetric spiral arms (Ringerma- This endeavour famously began with the creation of the cher & Mead 2010). The Hubble type is also known to corre- Hubble{Jeans sequence of galaxies (Jeans 1919, 1928; Hub- late with the bulge mass (e.g. Yoshizawa & Wakamatsu 1975; ble 1926, 1936). In modern times, this legacy has been con- Graham & Worley 2008, and references therein), and more tinued on a grand scale by the Galaxy Zoo project (Lintott luminous bulges are associated with more tightly wound et al. 2008), which has been utilizing citizen scientist volun- spiral arms (Savchenko & Reshetnikov 2013). Additionally, arXiv:1707.04001v3 [astro-ph.GA] 19 Mar 2018 teers to visually classify spiral structure in galaxies on their Davis et al.(2015) present observational evidence for the spi- website.1 If one is familiar with the concept of pitch angle, ral density wave theory's (bulge mass){(disc density){(pitch and presented with high-resolution imaging of grand design angle) Fundamental Plane relation for spiral galaxies. spiral galaxies, pitch angles accurate to ±5◦ could reasonably be determined by visual inspection and comparison with ref- It has been established that bulge mass correlates well erence spirals. Fortunately, instead of relying on only the hu- with supermassive black hole (SMBH) mass (Dressler 1989; man eye, several astronomical software routines now exist to Kormendy & Richstone 1995; Magorrian et al. 1998; Marconi measure pitch angle more precisely, particularly helpful with & Hunt 2003). A connection between pitch angle and SMBH lesser quality imaging, and in galaxies with more ambiguous mass is therefore expected given the relations mentioned spiral structure (i.e. flocculent spiral galaxies). above. Moreover, pitch angle has been demonstrated to be connected to the shear rate in galactic discs, which is itself an indicator of the central mass distribution contained within ? E-mail: [email protected] a given galactocentric radius (Seigar et al. 2005, 2006). In 1 https://www.galaxyzoo.org/ fact, it is possible to derive an indirect relationship between © 2017 The Authors 2 B. L. Davis, A. W. Graham and M. S. Seigar spiral arm pitch angle and SMBH mass through a chain of relations from (spiral arm pitch angle) ! (shear) ! (bulge mass) ! (SMBH mass), jφj ! Γ ! MBulge ! MBH: From an analysis of the simulations by Grand et al.(2013) 2 and application of the MBH{MBulge relation of Marconi & Hunt (2003), the following (black hole mass){(spiral arm pitch an- gle), MBH{φ, relation estimation is obtained: ◦ log¹MBH/M º ≈ 8:18 − 0:041 »jφj − 15 ¼ : (1) In this paper we explore and expand upon the estab- lished MBH{φ relation (Seigar et al. 2008; Berrier et al. 2013), which revealed that MBH decreases as the spiral arm pitch angle increases. One major reason to pursue such a relation is the potential of using pitch angle to predict which galax- ies might harbour intermediate-mass black holes (IMBHs, 5 MBH < 10 M ). The MBH{φ relation will additionally en- able one to probe bulge-less galaxies where the (black hole mass){(bulge mass), MBH{MBulge, and (black hole mass){ Figure 1. A logarithmic spiral (red), circle (blue), line tangent (S´ersic index), MBH{n, relations can no longer be applied. to the logarithmic spiral (magenta) at point (r; θ), line tangent Furthermore, pitch angles can be determined from images to circle (cyan) at point (r; θ) and radial line (green) passing without calibrated photometry and do not require carefully through the origin and point (r; θ). Included are the length of determined sky backgrounds. The pitch angle is also inde- r0, the angle φ and the location of the reference point (r; θ). For ◦ pendent of distance. Pitch angles have been measured for this example, r0 = 1, jφ j = 20 and point ¹r; θº = (≈ 1:33; π/4º; galaxies as distant as z > 2 (Davis et al. 2012). making the circle radius ≈ 1:33. Note that the spiral (red) radius ! 1 We present the mathematical formulae governing log- can continue outward towards infinity as θ + and continue inward towards zero as θ ! −∞. arithmic spirals in Section2. We describe our sample se- lection of all currently known spiral galaxies with directly measured black hole masses and discuss our pitch angle nuclei (AGN) central black holes (Wada et al. 2016). This measurement methodology in Section3. In Section4, we sort of expansion allows for radial growth without changing present our determination of the MBH{φ relation, includ- shape. One such special case of a logarithmic spiral is the ing additional tests upon its efficacy, and division into golden spiral (jφj ≈ 17◦: 0), which widens by a factor of the subsamples segregated by barred/unbarred and literature- golden ratio (≈ 1:618) every quarter turn, itself closely ap- assigned pseudo/classical bulge morphologies; revealing a proximated by the Fibonacci spiral (see appendix A of Davis strong MBH{φ relation amongst the pseudobulge subsam- et al. 2014). ple. Finally, we comment on the meaning and implications One can define the radius from the origin to a point of our findings in Section5, including a discussion of pitch along a logarithmic spiral at ¹r; θº as angle stability, longevity and interestingly, connections with τθ tropical cyclones and eddies in general. r = r0e ; (2) We adopt a spatially flat lambda cold dark mat- where r0 is an arbitrary real positive constant representing ter (ΛCDM) cosmology with the best-fitting Planck the radius when the azimuthal angle θ = 0, and τ is an TT+lowP+lensing cosmographic parameters estimated by arbitrary real constant (see Fig.1). If τ = 0, then one obtains the Planck mission (Planck Collaboration et al. 2016): a circle at constant radius r = r0, while if jτj = 1, one obtains ΩM = 0:308, ΩΛ = 0:692 and h67:81 = h/0:6781 = a radial ray from the origin to infinity. The parameter τ −1 −1 H0/(67:81 km s Mpc º ≡ 1. Throughout this paper, therefore quantifies the tightness of the spiral pattern. all printed errors and plotted error bars represent 1σ (≈ A logarithmic spiral is self-similar and thus always ap- 68:3 per cent) confidence levels. pears the same regardless of scale. Every successive 2π rev- olution of a logarithmic spiral grows the radius at a rate of 2 THEORY r n+1 = e2πτ; (3) r Logarithmic spirals are ubiquitous throughout nature, man- n ifesting themselves as optimum rates of radial growth for where rn is any arbitrary radius between the origin and the azimuthal winding in numerous structures such as mollusc point ¹rn; θº and rn+1 is the radius of the spiral after one shells, tropical cyclones and the arms of spiral galaxies. Ad- complete revolution, such that rn+1 is the radius between ditional astrophysical examples of the manifestation of log- the origin and the point ¹rn+1; θ + 2πº. arithmic spirals include protoplanetary discs (P´erezet al. The rate of growth (of the spiral radius as a function of 2016; Rafikov 2016), circumbinary discs surrounding merg- azimuthal angle) of such a logarithmic spiral can be defined ing black holes (Zanotti et al. 2010; Giacomazzo et al. 2012) using the derivative of equation (2), such that and the geometrically thick disc surrounding active galactic dr = r τeτθ = τr: (4) dθ 0 2 Γ ≈ − j j ¹ / º ≈ Γ dr dr 1:70 0:03 φ and log MBulge M 1:17 + 9:42. Notice that τ = dθ /r = 0 generates a circle and τ = dθ /r = MNRAS 000,1{18 (2017) The MBH{φ relation 3 1 generates a radial ray.
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