Applying the Tremaine-Weinberg Method to Nearby Galaxies: Stellar Mass-Based Pattern Speeds, and Comparisons with ISM Kinematics
Thomas G. Williams, Eva Schinnerer, Eric Emsellem, Sharon Meidt, Miguel Querejeta, Francesco Belfiore, Ivana Bešli?, Frank Bigiel, Mélanie Chevance, Daniel A. Dale, Simon C. O. Glover, Kathryn Grasha, Ralf S. Klessen, J. M. Diederik Kruijssen, Adam K. Leroy, Hsi-An Pan, Jérôme Pety, Ismael Pessa, Erik Rosolowsky, Toshiki Saito, Francesco Santoro, Andreas Schruba, Mattia C. Sormani, Jiayi Sun, Elizabeth J. Watkins
DD RAFT VERSION F EBRUARY
3, 2021Typeset using L A TEX twocolumn style in AASTeX63
Applying the Tremaine-Weinberg Method to Nearby Galaxies:Stellar Mass-Based Pattern Speeds, and Comparisons with ISM Kinematics T HOMAS
G. W
ILLIAMS , E VA S CHINNERER , E RIC E MSELLEM ,
2, 3 S HARON M EIDT , M IGUEL Q UEREJETA , F RANCESCO B ELFIORE , I VANA B E ˇ SLI ´ C , F RANK B IGIEL , M ´
ELANIE C HEVANCE , D ANIEL
A. D
ALE , S IMON
C. O. G
LOVER , K ATHRYN G RASHA , R ALF
S. K
LESSEN ,
10, 12
J. M. D
IEDERIK K RUIJSSEN , A DAM
K. L
EROY , H SI -A N P AN , J ´ ER ˆ OME P ETY ,
14, 15 I SMAEL P ESSA , E RIK R OSOLOWSKY , T OSHIKI S AITO , F RANCESCO S ANTORO , A NDREAS S CHRUBA , M ATTIA
C. S
ORMANI , J IAYI S UN , AND E LIZABETH
J. W
ATKINS Max Planck Institut f¨ur Astronomie, K¨onigstuhl 17, 69117 Heidelberg, Germany European Southern Observatory, Karl-Schwarzschild-Str. 2, 85748 Garching, Germany Universit´e Lyon 1, ENS de Lyon, CNRS, Centre de Recherche Astrophysique de Lyon UMR5574, 69230 Saint-Genis-Laval, France Sterrenkundig Observatorium, Universiteit Gent, Krijgslaan 281 S9, B-9000 Gent, Belgium Observatorio Astron´omico Nacional (IGN), C/Alfonso XII 3, Madrid E-28014, Spain INAF – Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50157, Firenze, Italy Argelander-Institut f¨ur Astronomie, Universit¨at Bonn, Auf dem H¨ugel 71, 53121 Bonn, Germany Astronomisches Rechen-Institut, Zentrum f¨ur Astronomie der Universit¨at Heidelberg, M¨onchhofstraße 12-14, 69120 Heidelberg, Germany Department of Physics & Astronomy, University of Wyoming, 1000 East University Avenue, Laramie, WY 82070, USA Universit¨at Heidelberg, Zentrum f¨ur Astronomie, Institut f¨ur Theoretische Astrophysik, Albert-Ueberle-Str. 2, 69120 Heidelberg, Germany Research School of Astronomy and Astrophysics, Australian National University, Canberra, ACT 2611, Australia Universit¨at Heidelberg, Interdisziplin¨ares Zentrum f¨ur Wissenschaftliches Rechnen, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany Department of Astronomy, The Ohio State University, 140 West 18 th Avenue, Columbus, OH 43210, USA IRAM, 300 rue de la Piscine, F-38406 Saint Martin d’H´eres, France
15 LERMA, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universit´es, 75014 Paris, France Max-Planck-Institut f¨ur Extraterrestrische Physik, Giessenbachstraße 1, D-85748 Garching bei M¨unchen, Germany (Received February 3, 2021; Revised February 3, 2021; Accepted February 3, 2021)
Submitted to AJABSTRACTWe apply the Tremaine-Weinberg method to 19 nearby galaxies using stellar mass surface densities and veloc-ities derived from the PHANGS-MUSE survey, to calculate (primarily bar) pattern speeds ( Ω P ). After qualitychecks, we find that around half (10) of these stellar mass-based measurements are reliable. For those galaxies,we find good agreement between our results and previously published pattern speeds, and use rotation curves tocalculate major resonance locations (co-rotation radii and Lindblad resonances). We also compare these stellar-mass derived pattern speeds with H α (from MUSE) and CO( J = 2 − ) emission from the PHANGS-ALMAsurvey. We find that in the case of these clumpy ISM tracers, this method erroneously gives a signal that issimply the angular frequency at a representative radius set by the distribution of these clumps ( Ω clump ), andthat this Ω clump is significantly different to Ω P ( ∼
20% in the case of H α , and ∼
50% in the case of CO). Thus,we conclude that it is inadvisable to use “pattern speeds” derived from ISM kinematics. Finally, we compareour derived pattern speeds and co-rotation radii, along with bar properties, to the global parameters of thesegalaxies. Consistent with previous studies, we find that galaxies with a later Hubble type have a larger ratio ofco-rotation radius to bar length, more molecular-gas rich galaxies have higher Ω P , and more bulge-dominatedgalaxies have lower Ω P . Unlike earlier works, however, there are no clear trends between the bar strength and Ω P , nor between the total stellar mass surface density and the pattern speed. Corresponding author: Thomas G. [email protected] a r X i v : . [ a s t r o - ph . GA ] F e b W ILLIAMS ET AL . Keywords:
Galaxies (573) – Galaxy dynamics (591) – Galaxy structure (622) INTRODUCTIONA fundamental, and currently open question in galaxy for-mation and evolution is how spiral arms and bars are formedand how they evolve. A natural question to ask is how long-lived these structures are, but the answer to this question re-mains elusive; even “transient” structures in terms of galaxiesmay persist for millions of years (e.g. Bournaud et al. 2005;Grand et al. 2012), and so are far beyond the timescales thatwe are able to observe. The most common theory for the for-mation of these structures (Lindblad 1963; Lin & Shu 1966,although see Sellwood 2013 and sect. 6.4.2 of Binney &Tremaine 2008 for alternative theories) is that density wavespropagating through galaxies act on the gas, forming starseither along spiral shock lines (forming grand-design spiralarms). Bars can form from disk instabilities, even in the ab-sence of gas (see sect. 6.3 of Binney & Tremaine 2008). Akey prediction of this density wave theory is that the wavespropagating through these morphological features will havea roughly invariant angular velocity across a large range ofgalactic radii. These angular velocities are referred to as pat-tern speeds, Ω P (whether this is true in the case of spiral armsis disputed, see review by Dobbs & Baba 2014).The pattern speed of a spiral arm or a bar is a key param-eter of the structure, and is associated with the evolution ofthe galaxy it is present within. For instance, a bar can onlygrow self-consistently if it lies within the co-rotation radius(i.e. where the stars move at the same speed as the den-sity wave; Contopoulos 1980). Density waves driving spiralarms have been shown to trigger star formation (e.g. Rand1993; Knapen et al. 1996), and the interface of a bar anda spiral arm can also trigger massive starburst events (e.g.Beuther et al. 2017), so galaxy evolution determines Ω P .Furthermore, when combined with the rotation curve of agalaxy, these speeds will set the location of resonances withina galaxy (the co-rotation radii, and Lindblad resonances),which in turn maintain and regulate the density wave (Lin1970), and can have significant effects on the distributions ofstars (e.g. Fragkoudi et al. 2019).However, the density wave speed is not directly observ-able, so we must turn to indirect methods to infer this param-eter. There are a number of ways to do this. For example, byidentifying resonance locations one can predict the radius ofco-rotation, and thus the pattern speed (e.g. Elmegreen et al.1989, 1996). This method is limited by an uncertain conver-sion from resonance radius to co-rotation radius (e.g. Kranzet al. 2003), and by the fact that given the particular patternspeed and galaxy rotation curve, certain resonances may notexist. Another option is to match simulations (where the pat-tern speed is directly known) to galaxies (e.g. Garcia-Burillo et al. 1993; Rautiainen & Salo 1999; Weiner et al. 2001a,b;Sormani et al. 2015). The selection of the “best” model hereis somewhat qualitative, as direct comparisons between ob-servations and simulations are difficult. This method has thebenefit of being more direct, but requires a suite of tailoredsimulations for each individual object (selecting a value ofthe stellar mass-to-light ratio, the shape of the dark matterhalo and its mass, etc.), and so is only feasible for small sam-ples of galaxies.Due to these difficulties, studies are typically limited to asingle galaxy (or a very small number of them). This meansthat the literature contains heterogeneous measures of pat-tern speeds using different methods. Given that these differ-ent methods have different systematics (or, perhaps are moresensitive to different pattern speeds within the galaxy), mak-ing direct comparisons between these various literature val-ues is difficult. Furthermore, applying the same method todifferent kinematic tracers may yield differing results (e.g.stars and H I , see Westpfahl 1998). To draw statistically ro-bust conclusions about the distributions of pattern speeds inthe local galaxy population, it is important to have homoge-neous measures of these quantities – not only in technique,but in tracer, too.Ideally, to extend studies of pattern speeds (and resonancelocations) to large samples of galaxies, we desire a methodof pattern speed determination that is widely applicable,which is data (rather than model or simulation) driven, andas quantitative as possible (i.e. with minimal reliance on by-eye feature classification). For this, the Tremaine-Weinberg(Tremaine & Weinberg 1984) method stands out as one ofthe most popular approaches, due to its minimal model as-sumptions, and the fact that it is purely based on the observedkinematics. We will describe this method in more detail inSect. 3.1. As a brief historical introduction, this method wasoriginally designed for long-slit spectroscopy and applied tomeasure bar pattern speeds of dust-free, early-type galaxies.It is sensitive to non-axisymmetry along a number of (ideallyinfinitely long) slits positioned along the major axis of thegalaxy. By taking the slope of the intensity-weighted velocitywith respect to the intensity-weighted position for a numberof slits, the velocity component mis-alignment with the lineof nodes (i.e. the major axis of the galaxy) can be quantified.The commonly used form of the method was first formulatedby Merrifield & Kuijken (1995), and has since been appliedto a number of other galaxies, including late-type galaxies(e.g. Gerssen et al. 1999; Debattista et al. 2002; Aguerri et al.2003; Corsini et al. 2003, 2007; Guo et al. 2019, amongstothers). With the advent of integral field unit (IFU) spec-troscopy, it is now possible to apply the Tremaine-WeinbergHANGS P ATTERN S PEEDS α (e.g. Emsellem et al. 2006; Fathi et al. 2009) or occasionallyCO (e.g. Rand & Wallin 2004; Zimmer et al. 2004). The un-derlying assumption, in such cases, is that this gas roughlyobeys continuity when there is little chemical transformationbetween the gas and other phases of the ISM (i.e. molecularto atomic hydrogen, or dust), and that the chemical abun-dance of the tracers remains constant. In the case of CO,Rand & Wallin (2004) argued that it should remain a validtracer when the ISM is molecule-dominated (so there is littleconversion from molecular gas to and from atomic gas) andwhen the star formation rate is low to moderate (so that thereis little molecular gas lost to star formation, or expelled byfeedback; Chevance et al. 2020). However, as discussed inRand & Wallin (2004), the clumpiness of ISM tracers likeCO and H α can pose a further issue: clumpy, highly asym-metric disks introduce a fake signal in Tremaine-Weinbergintegrals. This effect becomes especially pronounced at highresolution, when CO and H α morphology becomes charac-terised by clumpiness organised around, for example, H II re-gions, GMCs, sharp spiral arms (e.g. Kreckel et al. 2018;Schinnerer et al. 2019; S. Meidt et al. in prep.) We revisit thisin more detail in the context of our sample in Sect. 4. Becauseof these potential shortcomings, fully sampled stellar kine-matics are preferable for applying the Tremaine-Weinbergmethod.Recent works have combined a number of literature val-ues of pattern speeds to attempt to perform statistical analy-ses (Cuomo et al. 2020). Our study complements and buildsupon this earlier work by measuring pattern speeds homo-geneously for a number of galaxies. In this work, we ap-ply the Tremaine-Weinberg method to two of the Physics atHigh Angular resolution in Nearby GalaxieS (PHANGS )surveys, namely to observations taken using the Multi UnitSpectroscopic Explorer (MUSE) instrument on the VLT (re-ferred to as PHANGS-MUSE; P.I. E. Schinnerer; E. Em-sellem et al. in prep.), and the Atacama Large Millime-ter/Submillimeter Array (ALMA) instrument (referred to asPHANGS-ALMA; P.I. E. Schinnerer, and from pilot pro-posals with P.I.s G. Blanc and A. K. Leroy; Leroy et al.2020a). Our work focuses on pattern speeds from the stel-lar mass surface density ( Σ ∗ ) from MUSE, but we also studythe application of this method to ISM tracers – ionised gas phangs.org (H α ) from MUSE, and cold, molecular gas [CO( J = 2 − )]emission from ALMA. With 19 galaxies mapped as part ofPHANGS-MUSE, and 84 with PHANGS-ALMA, this givesus an unprecedented opportunity not only to measure patternspeeds homogeneously for a large sample of galaxies at high( (cid:46) pc) resolution, but also to perform vital cross-checksbetween different kinematic tracers for many galaxies, whichare currently poorly explored in the literature. This workderives pattern speeds, along with resonance locations forthese galaxies, which are tabulated in Table 2, and are alsomade available online in a machine-readable format . Forfuture works that may have improved distance or orientationmeasurements, this table also includes these parameters asused in our work, to allow for simple rescaling in subsequentworks.The structure of this paper is as follows: we providean overview of the PHANGS programmes and data prod-ucts (Sect. 2), before summarising the Tremaine-Weinbergmethod, our tests of its efficacy on the data, and our appli-cation to the entire PHANGS-MUSE and PHANGS-ALMAdata sets (Sect. 3). In Sect. 4, we present our tests showingthat applying the Tremaine-Weinberg method to ISM tracerscan yield erroneous signals, and showing that this is gener-ally an issue in our sample. We present an overview of ourderived pattern speeds in the context of previous work, andmake comparison between our pattern speeds and previouslyderived pattern speeds for the same galaxies in the literature(Sect. 5). We then calculate the radii of major resonances forthe entire PHANGS sample (Sect. 6). We study the potentialcorrelations between these derived dynamical properties ofthe galaxy and some of its global parameters (such as Hub-ble type, molecular gas fraction; Sect. 7). We discuss theimplications of these results (Sect. 8), before presenting asummary of our work, along with future prospects for largestudies of pattern speeds in the future (Sect. 9). DATAThis work uses data from two of the PHANGS large pro-grammes: PHANGS-MUSE and PHANGS-ALMA. For de-tails on the data reduction and product creation, we referthe readers to E. Emsellem et al. (in prep.) for PHANGS-MUSE, and Leroy et al. (2020b) and Leroy et al. (2020a)for the PHANGS-ALMA data processing pipeline and sur-vey description, respectively. We provide here only a briefdescription of the data products we use in the proceeding sec-tions, which are the surface brightness and velocity maps foreach tracer (MUSE stellar mass and velocity, MUSE H α , andALMA CO, collectively referred to as “kinematic maps”),along with associated error maps. URL available upon publishing W ILLIAMS ET AL .Along with these kinematic maps, we use orientation pa-rameters (the galaxy position angle, inclination, and systemicvelocities) from Lang et al. (2020). We also make use ofdistances from Anand et al. (2020, and references therein)to calculate physical pattern speeds in km s − kpc − , ratherthan in km s − arcsec − . Finally, we make use of cata-logues from the Spitzer Survey of Stellar Structure in Galax-ies (S G; Sheth et al. 2010) – in particular, bar strengths(D´ıaz-Garc´ıa et al. 2016), bar orientations (Herrera-Endoquiet al. 2015), and bulge to total flux ratios and disk scalelengths (Salo et al. 2015).2.1.
PHANGS-MUSE
Our primary sample in this study consists of MUSE opticalspectroscopy for 19 galaxies as part of the PHANGS-MUSEdata release (DR) 2.0 (E. Emsellem et al., in prep.). Thereduction is performed using standard MUSE recipes (e.g.wavelength and flux calibration, cosmic ray rejection, mo-saicking), run through
PYMUSEPIPE . These reduced prod-ucts are then run through the MUSE data analysis pipeline(DAP), which is run in three stages: firstly, stellar kinemat-ics are measured (the stellar velocity and higher order mo-ments); next, the properties of stellar populations are esti-mated (e.g. age, stellar metallicity, stellar mass). Both ofthese stages are performed on Voronoi binned data to a stel-lar continuum signal-to-noise ratio (S/N) of 35, to maximisereliability. The fit is performed via pPXF (Cappellari & Em-sellem 2004; Cappellari 2017), and makes use of E-MILES(Vazdekis et al. 2016) simple stellar population models ofeight ages (0.15 - 14 Gyr) and four metallicities ([Z/H] = [-1.5,-0.35,0.06,0.4]). Only the wavelength range 4850-7000 ˚Aare used in the fit, in order to avoid strong sky residuals in thereddest part of the MUSE spectral range. Finally, for indi-vidual spaxels the properties of emission lines are measured(fluxes and kinematics), via a simultaneous fit of continuumand emission lines also performed via pPXF. The DAP fitsonly a single stellar population, so the stellar masses and ve-locities are an average of young and older stars. We primarilywant to apply the method to the old stellar population, as wedo not expect young stars to obey continuity. As the stellarmass is dominated by old stars (typically, the mass-weightedage per spaxel is of the order of Gyr), and the velocities ofyoung and old stars are typically similar (Rosado-Belza et al.2020; Shetty et al. 2020), therefore, averaging these valuesover stellar populations will not bias our results. Finally, forindividual spaxels, the properties of emission lines are mea-sured (fluxes and kinematics). We make use of the stellarmass surface density and velocity maps (from the first twoprocessing stages), and the H α flux and velocity maps (fromthe final processing stage). https://github.com/emsellem/pymusepipe PHANGS-ALMA
We make use of CO( J = 2 − ) maps (hereafter “CO” forbrevity) from the PHANGS-ALMA survey. Whilst the ob-servations and data reduction are detailed separately (Leroyet al. 2020a,b), we provide a brief summary of the productsused in this study.These CO maps are provided as part of the internalPHANGS-ALMA DR v3.4. This includes a total of 84 galax-ies (of which two are omitted due to no detected emission inthe cubes). We use cubes combined with all available datafrom the 12m and 7m arrays, and total power (TP) obser-vations. This means that we maximise the resolution of theobservations, whilst still being sensitive to extended emissionon all scales. In total, 69 of our galaxies have 12m+7m+TPdata, 7 have 12m+7m, 7 have 7m+TP, and 1 has 7m only.Typically, the data without 12m antenna configurations arefor the nearest galaxies, so the spatial resolution remains rea-sonably consistent. As shown in the right-hand panel of Fig.4, we do not expect the particular antenna configuration tohave a strong impact on our results.The 12m+7m data are imaged simultaneously, using amulti-scale clean followed by a single-scale clean. This com-bined cube is then corrected for primary beam pickup, andconvolved to have a round synthesised beam. The TP dataare imaged separately using the method described in Her-rera et al. (2020), and these two cubes are then combined inFourier space (“feathered”). The cubes are then collapsedinto standard moment 0 (integrated intensity) and moment 1(intensity-weighted velocity), along with error maps. Thereare two types of masks used to create these maps: firstly,strict masks that grow out any voxels with S / N > in 2or more successive channels down to voxels with S / N > in 2 or more successive channels. Secondly, there are broadmasks that take the union of the highest resolution strict maskfrom a strict mask generated at a spatial resolution of 500pc,which captures lower level, extended emission. For more de-tails of this masking, see Leroy et al. (2020b). We opt to usebroad, rather than strict maps, to maximise our completeness(at the cost of increased noise). We find a negligible impacton our results using these different masking schemes in mostcases. This choice only matters for the low surface brightnessgalaxies in the sample, for which we typically find a poor fitin any case.We use maps at their highest resolution ( . +0 . − . arcsec,corresponding to +31 − pc, whereby the native resolutionmaps are smoothed to a common circular beam for eachgalaxy. Combined with the MUSE stellar and H α maps, wehave three different kinematic tracers, which affords us use-ful cross-checks between derived pattern speeds for the sam-ple of 19 galaxies where the observations overlap. CALCULATING PATTERN SPEEDSHANGS P
ATTERN S PEEDS h m s s m s s D ec ( J ) NGC3351, M h m s s m s s D ec ( J ) Velocity x > ( )051015 < v > ( k m s ) P = 42.90 ± 0.89 km s kpc Figure 1.
Top left : Stellar mass surface density map of NGC 3351 shown in greyscale, with Tremaine-Weinberg integral slits of 1 (cid:48)(cid:48) width,oriented parallel to the major axis overlaid. Only one in every four slits is shown, due to the slit density, and are coloured according to theirposition along the kinematic minor axis.For this galaxy, the quality flag, Q = 1 (see Sect. 3.5). Top right : Stellar velocity map for the samegalaxy, with a dashed black line showing the kinematic major axis, passing through the galaxy centre.
Bottom : intensity-weighted velocity ( (cid:104) v (cid:105) )versus intensity-weighted position ( (cid:104) x (cid:105) ) for each of the slits (the colour corresponds to the slit colour in the above top-left panel). The blackline shows the best fit, and the grey shaded region the errors on the fit (in this case, this region is extremely small). One point has an extremelylarge uncertainty in this panel, and the error bar extends across the entire range of (cid:104) v (cid:105) shown. In this section, we present a brief overview of theTremaine-Weinberg method (Sect. 3.1). We detail our treat-ments of the various uncertainties associated with the method(Sect. 3.2), and then investigate the effect of slit lengths andwidths on the recovered pattern speeds (Sect. 3.3). We thenapply the method to our data set (Sect. 3.4), and performa posteriori checks on the pattern speeds (Sect. 3.5). Westress that we only provide one pattern speed; for barredgalaxies, we expect this to be the pattern speed of the bar,and for non-barred galaxies this will be a pattern speed for thespiral arms. As an umbrella term, we will refer to these mea-sured pattern speeds as “primary” pattern speeds through-out this work, as these structures will dominate the non-axisymmetry the Tremaine-Weinberg method is sensitive to when we are in a regime with multiple pattern speeds present(we discuss this further in Sect. 3.1).3.1.
The Tremaine-Weinberg Method
The Tremaine-Weinberg method is a model independentmethod to calculate a pattern speed within a galaxy. It hasthree, minimal assumptions:1. The galaxy disk is flat;2. The disk contains a single, well-defined pattern speed;3. The tracer obeys the continuity equation.Assumption (1) is justified as our observations cover mainlythe inner disks of galaxies (typically around 1 R ; Lang W ILLIAMS ET AL . R / R
25, opt r ( )0100200300 P ( k m s kp c ) P ( k m s kp c ) Figure 2.
Left : Recovered pattern speed, Ω P , versus Tremaine-Weinberg integral slit length, r , for MUSE stellar mass surface density obser-vations of NGC 3351 (at this distance, (cid:48)(cid:48) corresponds to ∼ pc). The vertical dashed black line indicates the bar extent in this galaxy. Right :Recovered pattern speed versus Tremaine-Weinberg integral slit width for the same galaxy (the resolution of this map is ∼ (cid:48)(cid:48) ). As the patternspeed measured for each slit width is not independent from other slit widths (they use the same data, and larger slit widths combine informationfrom narrower slits), the scatter in each measurement is much smaller than the typical uncertainty. In both cases, the horizontal black lineindicates our fiducial pattern speed, and the dashed black lines either side of it the uncertainty on this measurement. et al. 2020), and the detected emission is less radially ex-tended than, e.g. H I . Thus, we do not expected these ob-servations will be sensitive to disk warping. We thereforeassume that this assumption holds true across all of our sam-ple.On point (2), prior studies have has shown that many galax-ies host multiple pattern speeds, corresponding to differentmorphological features (e.g. bars and spiral arms within agalaxy; Meidt et al. 2008b; Beckman et al. 2018). In thiscase, assumption (2) may be invalid. We would expect twodifferent (but linked) phenomena here. For slits that containno information from the bar, we would expect to measure adifferent pattern speed than for slits with bar information, ifthe pattern speeds are different between spiral arms and bars.This is reflected in our quality flagging (Sect. 3.5), and webelieve this to only affect a small subset of our sample.Furthermore, as we make our slits as long as possible,some slits may pass through both bars and spiral arms. Ifthese structures have different pattern speeds, then any pat-tern speed we measure will be some weighted average ofthese two distinct values. We expect this measurement tostrongly deviate from the bar pattern speed only when thespatial and kinematic non-axisymmetry of the secondary pat-tern (i.e. spiral arms) is strong relative to the primary patternspeed, and also when this secondary pattern extends over aconsiderable portion of the slits (for examples of this, seescenarios considered by Meidt et al. 2008a). For most ofthe galaxies in our sample, this does not appear to be thecase; outer spirals tend to be both substantially dimmer andexhibit weaker streaming motions compared to the bars inour sample: whereas residual velocities from the fitted ro- tation curves by Lang et al. (2020) are typically 20 km s − or greater in our sample for bars, they reach a maximum ofonly ∼ km s − in the spiral arms. Attempting to quantifythe bias this may present would require detailed simulationsof galaxies where we know a ground truth for the patternspeed. This is beyond the scope of this work, but will be re-visited in later studies. For the present work, we assume thatthe secondary pattern is weak compared to the bar pattern,and so the bias is small – much like in earlier works apply-ing the Tremaine-Weinberg method to stellar kinematics (e.g.Aguerri et al. 2015; Guo et al. 2019; Garma-Oehmichen et al.2020).Whilst assumption (3) (the tracer obeys the continuityequation) is approximately valid for the old stellar mass sur-face density distribution that we are primarily sensitive towith our MUSE observations, it may not be formally validgiven the clumpy nature of CO and H α emission, the phasetransitions from Hı to H to H II , and the way these phasesparticipate in the star formation process. We investigate thisfurther in Sect. 4, and find that this is generally an issuewithin our sample. For this reason, the bulk of our analysis isperformed on pattern speeds measured from stellar kinemat-ics.If the continuity equation is valid, then ∂ Σ ∂t + ∂∂x (Σ v x ) + ∂∂y (Σ v y ) = 0 , (1)where ( v x , v y ) is the mean velocity of the tracer at ( x, y ) , and Σ the surface brightness at ( x, y ) . In polar coordinates ( r, φ ) ,we assume that the surface brightness ( ˜Σ ) in a frame rotatingHANGS P ATTERN S PEEDS Ω P : Σ = ˜Σ( r, φ − Ω P t ) , (2)and from this we can write ∂ Σ ∂t = − Ω P ∂ ˜Σ ∂φ = Ω P (cid:18) y ∂ Σ ∂x − x ∂ Σ ∂y (cid:19) . (3)Substituting this into Eq. 1 and integrating over both x and y (see Tremaine & Weinberg 1984), we can obtain Ω P sin( i ) = (cid:82) ∞−∞ h ( y ) (cid:82) ∞−∞ Σ v LOS ( x, y ) d x d y (cid:82) ∞−∞ h ( y ) (cid:82) ∞−∞ Σ x d x d y , (4)where i is the inclination of the galaxy, v LOS the line-of-sightvelocity, and h ( y ) a weight function. In this work, we take h ( y ) to be a boxcar function from y − d y to y + d y , to rep-resent a pseudo-slit parallel to the line of nodes. Throughoutthis work, we will refer to “slits” and “integrals” interchange-ably in this context. This equation can be simplified to Ω P sin( i ) = (cid:104) v (cid:105)(cid:104) x (cid:105) , (5)by recognising that the integrals in the numerator and denom-inator are simply the intensity-weighted velocity and positionalong a slit, respectively. In this formalism, non-zero valuesof (cid:104) v (cid:105) / (cid:104) x (cid:105) are caused by non-axisymmetric structure withinthe slit. Thus, taking a number of slits and plotting (cid:104) v (cid:105) versus (cid:104) x (cid:105) yields a straight line with a slope equivalent to Ω P sin( i ) .There are some limitations to this method given the data.Firstly, the integrals formally should extend from −∞ to ∞ ,whereas in reality this is not the case due to the limited fieldof view of the observations. However, assuming the disk isaxisymmetric at large x , we can instead integrate from − x to x (where this is set by the extent of the field of view ofthe observations). Secondly, this method is less effective forgalaxies that are face-on (due to loss of kinematic informa-tion), or edge-on (due to loss of photometric information).Finally, for galaxies with bars we expect our primary patternspeed to be the bar pattern speed. If the bar is oriented alongthe galaxy minor axis, the integrals will tend to cancel outand no pattern speed will be measured.3.2. Uncertainties on the Pattern Speeds
In order to provide robust estimates on the pattern speeduncertainties, we take multiple sources of error into accountwhen calculating them. Firstly, we account for errors in both (cid:104) x (cid:105) and (cid:104) v (cid:105) separately, based on the error in each pixel alongthe slit, and summed in quadrature. These are then propa-gated into the fitting routine using Orthogonal Distance Re-gression (ODR), which allows us to effectively account forerrors in both (cid:104) x (cid:105) and (cid:104) v (cid:105) . We use the scipy implementa-tion of ODR ( SCIPY . ODR ). We find that, given the high S/N of our observations and the large numbers of pixels alongeach slit, the formal errors on these ODR fits are very small.As an example, one such fit is shown in Fig. 1, for NGC 3351.In Appendix B, we show this visualisation for all MUSEgalaxies. Even our highest quality fits have some points thatlie off the fitted line. These can be caused by multitude ofreasons, including unmasked foreground stars, small varia-tions in the galaxy position angle with galactocentric radius,or (particularly at the edges of the observations) insufficientlylong slits. However, particularly for the fits with a quality flag Q = 1 (see Sect. 3.5), the fits tend to look excellent.We also account for errors in the position angle of thegalaxy, and the galaxy centre. We do this via a Monte-Carlomethod, perturbing the line of nodes and the galaxy centre bythe measured errors (recorded in Lang et al. 2020). We use1000 bootstraps, measure the pattern speed for each of theseiterations, and quote the pattern speed as the median of thisdistribution, with the associated errors as the 16 th and 84 th percentiles, as we find these errors tend to be asymmetric.These are listed in Table 2. We find the errors on our patternspeeds to be +15 − % (the median percentage error and 16 th and 84 th percentiles), and that it is the error in the positionangle that dominates, as has been shown in previous work(Debattista 2003; Garma-Oehmichen et al. 2020). We do notinclude uncertainties from the inclination or distance in ouruncertainty for the pattern speeds. This is because many ofour comparisons are to other quantities that are inclination-and distance-dependent (see Sect. 7).3.3. Effects of slit length and width on recovered patternspeeds
Finally, we investigate the effects of both the slit lengthand slit width on recovered pattern speeds. These can af-fect the pattern speeds in two ways: firstly, the slits must besufficiently long as to reach a sufficient radius at which thedisk is roughly axisymmetric (and the effect from morpho-logical features no longer prominent). If this is not achieved,the measurement will be biased. Secondly, if the slits aretoo wide, we may have insufficient slits covering the bar toretrieve a reliable pattern speed measurement. We use a num-ber of slit lengths, from 10 (cid:48)(cid:48) to 150 (cid:48)(cid:48) ( ∼ (cid:48)(cid:48) to 10 (cid:48)(cid:48) (in general, the resolution of our data is ∼ (cid:48)(cid:48) , but the pixels oversample the beam; this corresponds to ∼
50 pc to 1 kpc). 1 (cid:48)(cid:48) corresponds to between around 25 pc forthe nearer targets in the sample, and 120 pc for the farthest.The results of this experiment for NGC 3351 are shown inFig. 2. We find that the measured pattern speeds tend to con-verge as the slits become longer (typically, slightly longerthan the bar). We also find that the slit width has a min-imal impact on the recovered pattern speed (right panel ofFig. 2). With larger slit widths, we have fewer points to fit, W
ILLIAMS ET AL .and typically the uncertainty in the pattern speed becomesslightly larger. Motivated by these results, we opt to makeeach individual slit as long as possible, and 1 (cid:48)(cid:48) wide to ap-proximately match the resolution of the data. For surveyswith larger number of galaxies, but with physical resolutionsof ∼ kpc scales, rather than ∼ pc scales (e.g. MaNGA;Bundy et al. 2015), efforts to measure pattern speeds in thesegalaxies should still produce reliable results (right panel ofFig. 2; see also work by Guo et al. 2019). We performthese slit length and slit width diagnostics for all galaxies,and these form a critical component of our diagnostic assess-ments (Sect. 3.5). 3.4. Data Preparation
We perform a number of pre-processing steps to the databefore calculating the Tremaine-Weinberg integrals. Theseare:1.
Mask foreground stars.
For the MUSE data only,foreground stars can be a contaminant. These areclearly recognisable in the data with extreme (positiveor negative) velocities (with respect to the systemic ve-locity of the galaxy). We therefore remove any pixelswith | v | >
300 km s − . This cut is arbitrary, and mayneed to be tailored for other data sets. However, wefind that the recovered pattern speeds with masked ornon-masked stars are not significantly different;2. Subtract systemic velocity.
For the ALMA data only,the systemic velocity is not subtracted in the moment 1maps. We estimate the systemic velocity using PAF IT (Krajnovi´c et al. 2006), and subtract it from the ALMAvelocity field (moment 1) map. This has the effect ofcentring the (cid:104) v (cid:105) integrals. As we are primarily inter-ested in the slope of (cid:104) v (cid:105) versus (cid:104) x (cid:105) , this value is notparticularly important, but is included so that the lineapproximately passes through the origin;3. Remove integrals that do not cross the bar.
Forgalaxies with bars, we attempt to isolate the bar pat-tern speed from any other pattern speeds present in thegalaxy (i.e. to minimise the effects of potentially dif-ferent pattern speeds in the spiral arms). This is doneby removing any slits that do not at least partially coverthe bar. We note that these slits may cross multiple fea-tures with multiple patterns, but as the slit lengths needto be suitably long to reach the axisymmetric disk, thisis unavoidable;4.
Symmetrise the integral.
Finally, after the other pre-processing steps, we make sure each integral goes from − x to x . To do this, we find the maximum distancefrom the minor axis where both sides of each slit still Hubble N g a l All CO: N = 83 Q = 1, 2: N = 550.02.55.0 N g a l All M : N = 19 Q = 1, 2: N = 100.02.55.0 N g a l All H : N = 19 Q = 1, 2: N = 14 Figure 3.
Distribution of well-constrained Tremaine-Weinberg val-ues as a function of Hubble morphological types for top : MUSE- M ∗ measured values; middle : MUSE-H α measured values; and bottom : ALMA-CO measured values. In each case, grey indicatesall the values for each tracer (i.e. the underlying population), andthe coloured bars the well-constrained sample: red for MUSE- M ∗ ,cyan for MUSE-H α , and blue for ALMA-CO. contain data. Given the field of view of the observa-tions, and previous bar masking, this may lead to slitsthat are different lengths from each other. It is criti-cally important that each slit is symmetric, as if theyare not, this asymmetry can induce a false signal in theintegrals.Having performed this pre-processing, we place a number ofslits across the surface brightness and velocity maps, and cal-culate the intensity-weighted position and velocity of all thepixels within each slit. We do this for all 19 galaxies coveredby MUSE, and all 84 of our galaxies covered by ALMA. Wefind that all of the MUSE galaxies can be fitted, but due tono detected CO emission in the cube, the ALMA data forIC 5332 and NGC 3239 cannot be fitted. This leaves us witha total of 82 ALMA galaxies, and 18 of these overlap withMUSE observations. IC 5332 is present in both samples,but as no emission is detected in the ALMA cube, we onlypresent measurements from the MUSE observations.3.5. Quality Flagging
Having applied the Tremaine-Weinberg method for all thegalaxies (with all available tracers) in our sample, we per-form a posteriori quality flagging on these values. We in-clude all measured values in Table 2 along with these qualityflags ( Q ). We take a minimalist approach to the quality flag-ging, to provide clear delineation between the flags, whichare defined as follows:HANGS P ATTERN S PEEDS
91. Single, well defined slope: integrals have converged,pattern speed is stable with decreasing slit width;2. Clear multiple slopes visible in the (cid:104) v (cid:105) / (cid:104) x (cid:105) plot, butotherwise the fit would be a quality flag (1);3. Poor fit: integral has not converged, points do not forma clear, well-defined slope;4. Data of insufficient quality to calculate a reliable slope.Our flags span this entire range, from 1 to 4 (see Table 2).We note that these quality flags simply represent the qualityof the fits, and will not take into account the issues presentin the ISM tracers discussed in Sect. 4. We do include thesein the table with the caveats mentioned in this later section,both because we will make comparison between the tracers inSect. 5, and because future work will investigate these valuesin the context of other methods.This flagging is performed independently on each tracerby three of the authors, and our final flag is the mode of thesethree flags. In the case of disagreement between all three au-thors, we instead take the highest flag value of any author, tobe as conservative as possible. We opt for the mode to en-sure there are no fractional flags which may cause confusionas to their definition. In this work going forwards, we willdefine “well-constrained” Tremaine-Weinberg values as hav-ing Q = 1 or 2, and only use these in our analyses. The onlygalaxy we use with a quality flag of 2 is NGC 3627, for whichpoints towards the south-west appear to have a different slope(see Fig. 26).The distribution of well-constrained values is shown as afunction of Hubble morphological type (from the HyperLedadatabase; Makarov et al. 2014) in Fig. 3. Many of the galax-ies flagged as having poorly measured values are from ear-lier Hubble types. This is likely due to their lack of promi-nent morphological features, and so no strong signal in theTremaine-Weinberg plots. For the MUSE stellar mass values,we find that the fit is well-constrained for 10 of the 19 galax-ies. Of these, 9 have bars present, and 1 (NGC 0628) doesnot. The PHANGS-MUSE sample contains 15 barred galax-ies and 4 non-barred, so we preferentially measure bar pat-tern speeds. For some with a quality flag of 3 or 4, we detectno evidence for a pattern (IC 5332 and NGC 5068), but inthese cases the velocity field is very irregular (and so the po-sition angles change with radius). APPLICATION TO ISM TRACERSWhilst we base the majority of this study on pattern speedsderived from stellar kinematics, the PHANGS-ALMA dataset in particular offers a much larger sample size. The pat-tern speed for the Milky Way bar derived from gas dynamics(Sormani et al. 2015) has been found to match well with thatderived from stellar dynamics (Sanders et al. 2019), and so using the ISM may provide useful, independent pattern speedmeasurements. We therefore turn to the question of whetherISM tracers are intrinsically compromised due to their in-complete coverage, and clumpiness. If each slit goes througha single clump, even if there is no pattern present, applyingthe Tremaine-Weinberg method will give non-zero integrals,and this can lead us to measure a false pattern speed. In re-ality, in these data, each slit will pass through many of theseclumps, which are not positioned randomly (i.e. they clusteron spiral arms, or bars). We thus critically examine whetherwe can disentangle the true signal from the effect of incom-plete coverage.To test this, for each galaxy we take an axisymmetrised ve-locity and surface brightness profile, based on the CO maps.For the velocity field, we project the fits from Lang et al.(2020) into the frame of the galaxy, and for the CO data wetake the average surface brightness within a number of an-nuli. We then mimic the incomplete coverage of these mapsby blanking pixels in these axisymmetrised profiles whereemission is not detected, and apply the method describedin Sect. 3.1 to these. In this case, there is no pattern speedpresent in these maps, and any deviation is simply due tothe incomplete CO coverage and the non-axisymmetry of theCO morphology. We refer to this as Ω null , and compare thisto the “pattern speed” measured from the actual CO data, asshown in the left panel of Fig. 4. We refer to this as Ω clump .As we can see, the points tend to lie close to the rela-tion, indicating that much of the signal we measure usingthe Tremaine-Weinberg method may simply be due to theclumpy nature of the tracer itself. We also find a similar trendrepeating this test for H α . The angular velocity that we mea-sure with this experiment is some function of the underlyingrotation curve of the galaxy and the sampling of this rotationcurve by the tracer’s distribution of “clumps”. In the rightpanel of Fig. 4, we compare Ω clump from the higher reso-lution ( ∼ (cid:48)(cid:48) ) and lower resolution ( ∼ (cid:48)(cid:48) ) ALMA data. Thevalues calculated are very similar, and we find that repeatingthe Ω null test with these values calculated from lower resolu-tion data yields much the same results as the higher resolu-tion data. This indicates that this effect is endemic to the useof CO, and not simply due to the high resolution of our data.As an initial exploration into this, we take some simpledensity-wave spiral models following Binney & Tremaine(2008), with a known pattern speed, and blank regions tomimic the clumpiness of the employed tracer. We find thatmore spatially extended distributions of clumps (probing fur-ther out in galactocentric radius), produce lower Ω clump thanmore centrally concentrated arrangements. We believe thatthis leads to the results in Sect. 5.2, where Ω clump and Ω P appear systematically offset. As CO extends further out ingalaxies hosting larger bars (the outermost CO-emitting ra-dius tabulated by Lang et al. 2020 increases roughly with0 W ILLIAMS ET AL . clump (km s kpc )0255075100 nu ll ( k m s kp c ) COH clump, highres (km s kpc )0255075100 c l u m p , l o w r e s ( k m s kp c ) Figure 4.
Left:
Comparison between the Tremaine-Weinberg signal generated an axisymmetric surface brightness and velocity distributionwith incomplete map coverage ( Ω null ), and from the actual ALMA-CO (blue) and MUSE-H α (cyan) data ( Ω clump ). This test has been carriedout for all ALMA galaxies where a rotation curve has been fitted (Lang et al. 2020, 65 galaxies), and to the 13 galaxies in the MUSE DR1.0. The relation is shown as a dashed black line. Right:
Comparison between Ω clump measured for high-resolution data (generally 12m+7m+TP)and lower resolution data (7m+TP). increasing R bar ). Secondly, longer bars tend to have lowerpattern speeds than shorter bars in similar-sized galaxies (i.e. R ∼ . ). As applied to PHANGS-ALMA, we will thusfind lower Ω clump in galaxies with lower bar pattern speeds.Note that this may not always hold for other tracers, e.g. formuch more extended H I disks, when the extent of the tracer isnot tied to the morphology of the dynamical feature in ques-tion. However, it does appear that this is the case for bothour ALMA-CO and MUSE-H α data. A full exploration ofthe links between Ω clump and Ω P is beyond the scope of thiswork, and will require detailed simulations of galaxies. DERIVED PATTERN SPEEDSWith our final selection of well-constrained Tremaine-Weinberg values in hand (Table 1), we show the distributionof the 19 MUSE stellar mass pattern speeds in Fig. 5. Notethat we use KDE plots throughout this work to improve read-ability, by avoiding the ‘steps’ we would see in histogramsfor these small number of data points. However, any statis-tics are calculated from the data themselves, rather than thedistribution of the KDE. For the values we consider well-constrained, the average speed is +10 − km s − kpc − (thisis the median of the sample, along with the 16 th and 84 th percentiles of the distribution). The distribution of all thepattern speeds derived from the MUSE stellar mass maps arealso shown in this figure, and it can be seen that they occupy asimilar range of speeds ( +14 − km s − kpc − ). Thus, poorlyconstrained pattern speeds are not found at any particular lowor high values of Ω P , and so this manual quality flagging isrequired compared to simple threshold cut.5.1. Comparison to the Literature P (km s kpc )0.000.010.020.03 P r ob a b ilit y D e n s it y All Q = 1, 2 Figure 5.
The distribution of pattern speeds derived from MUSEstellar mass maps for 19 galaxies. The red line shows the Ker-nel Density Estimate (KDE) distribution for all calculated pat-tern speeds (using the optimal bandwidth formula from Silverman1986), the shaded black line the distribution of well-constrained pat-tern speeds. The KDE is normalised such that the integral of thedistribution is equal to one, so the shape of each distribution, ratherthan the absolute scaling, is what should be considered here.
Some of these galaxies have previously published patternspeeds , and we make comparison between our values andthese earlier values in Fig. 6 (scaling to our assumed galaxydistances and inclinations). These pattern speeds come froma variety of methods, but we note that the pattern speeds ofNGC 1365, NGC 3627, and NGC 4321 in the literature come NGC 0628: Mart´ınez-Garc´ıa & Puerari (2014); NGC 1300: Lindblad& Kristen (1996); NGC 1365: Speights & Rooke (2016); NGC 1433:Treuthardt et al. (2008); NGC 1512: Koribalski & L´opez-S´anchez (2009);NGC 1566: Korchagin et al. (2005); NGC 1672: D´ıaz et al. (1999);NGC 3627: Rand & Wallin (2004); NGC 4254: Egusa et al. (2004);NGC 4303: Schinnerer et al. (2002); NGC 4321: Hernandez et al. (2005)
HANGS P
ATTERN S PEEDS Table 1.
Pattern speeds and co-rotation radii for the ten well-constrained stellar masspattern speeds.Galaxy PGC D i PA Bar? Ω P Q R CR Mpc ◦ ◦ kmkpc s kpc
NGC0628 5974 9.84 8.9 20.7 0 . +4 . − . . ± . NGC1087 10496 15.85 42.9 359.1 1 . +3 . − . . ± . NGC1433 13586 12.11 28.6 199.7 1 . +2 . − . . ± . NGC1512 14391 17.13 42.5 261.9 1 . +5 . − . . ± . NGC1672 15941 19.4 42.6 134.3 1 . +0 . − . . ± . NGC2835 26259 12.38 41.3 1.0 1 . +4 . − . . ± . NGC3351 32007 9.96 45.1 193.2 1 . +11 . − . . ± . NGC3627 34695 11.32 57.3 173.1 1 . +20 . − . . ± . NGC4303 40001 16.99 23.5 312.4 1 . +5 . − . . ± . NGC7496 70588 18.72 35.9 193.7 1 . +6 . − . . ± . P (km s kpc )NGC0628NGC1300NGC1365NGC1433NGC1512NGC1566NGC1672NGC3627NGC4254NGC4303NGC4321 LiteratureMUSE M (This work)ALMA CO (This work) Figure 6.
Comparison between derived pattern speeds for MUSEstellar mass maps (red symbols), ALMA CO-based Ω clump (bluesymbols) and previously published literature values (grey symbols).If the pattern speed is not well measured ( Q = 3 or 4), the dotis unfilled. For literature pattern speeds from applying Tremaine-Weinberg to ISM tracers, we outline these points in light green. from applying the Tremaine-Weinberg method to ISM trac- Pattern Speed Difference P r ob a b ilit y D e n s it y CO, N=9
H , N=8
Figure 7.
KDE plot showing the relative pattern speed difference(Eq. 6) for both CO (blue line) and H α (cyan line) with respectto the stellar mass pattern speed. The dashed black line indicateswhere Ω P = Ω clump . The vertical dashed blue and cyan lines showthe median of the distributions for CO and H α , respectively. ers. For completeness, we show those points with Q -valuesgreater than 2, but we stress that these do not constitute truemeasurements of pattern speeds, and are not used in any ofour analysis here or later in this work. In general, our pat-tern speeds agree well with previously published values, andtypically agree within the uncertainties (a median absolutedeviation of ∼ Comparisons Between Tracers
ILLIAMS ET AL . R / R R (kpc)050100150200 ( k m s kp c ) ± /2 R barP R CR Lang+ (2020)
Figure 8.
Angular speeds ( Ω ) versus galactic radius for NGC 3351.Angular speeds from Lang et al. (2020) are shown as black points.When one of these Ω values is consistent with the pattern speed(green line, with associated errors in green), it is highlighted inred. These points are then combined to form our estimate of the co-rotation radius (red line) and associated error (shaded red region).The bar radius is shown as a vertical, black, dot-dash line. We alsoinclude dashed, black lines to indicate Ω ± κ/ , which we use to cal-culate ILR and OLR (these values and associated errors are omittedhere to maintain readability, but are included in Table 2). h m s s m s s D ec ( J ) Figure 9.
Resonances highlighted on MUSE white light map ofNGC 3351. Co-rotation (and associated errors) is shown in red, andthe inner Lindblad radius in blue.
As the PHANGS-MUSE sample overlaps entirely with thePHANGS-ALMA sample, we can compare how closely theISM tracers (H α , CO) measured Ω clump tends to agree with Ω P . We compare values flagged as well-constrained in boththe stellar mass, and the ISM tracer in question (for both H α and CO, this leaves us with 9 galaxies). We then define a “pattern speed difference” as Pattern Speed Difference = Ω clump − Ω P Ω P , (6)and show the distributions for both ISM tracers in Fig. 7.Typically, we find that Ω clump is higher than Ω P , some-what greater than our ∼
10% errors ( ∼
40% in the case ofCO, ∼
20% in the case of H α ). Therefore, whilst applyingthe Tremaine-Weinberg method to clumpy ISM tracers mayyield reasonable results, it is clear from this exercise that Ω clump and Ω P are systematically different quantities, andthus Ω clump should not be used as a proxy for Ω P . LOCATIONS OF MAJOR RESONANCESWe next use our pattern speeds to calculate major reso-nance locations for each galaxy. In this work we focus onthe co-rotation radius, R CR , but we also provide estimatesof the outer and inner Lindblad radii (OLR and ILR). Theseresonance locations are derived with the assumption of circu-larity, and so we give a single radius for each resonance. Todo this, we make use of the rotation curves derived from thePHANGS-ALMA data in Lang et al. (2020), and rescaledto our assumed distances. These rotation curves approxi-mate the circular velocity, v circ , in a number of radial bins,where the galaxy centre, inclination, and position angle arefirst fit using a Bayesian MCMC analysis. Following this,least-squares fitting uses a harmonic decomposition to modelthe rotational velocity within a series of radial annuli. How-ever, as these curves are measured from CO, there may bedeviations from this circularity, particularly in the centres ofgalaxies, or for strong spiral or bar streaming motions. Weuse this observed velocity profile as a proxy for the true circu-lar velocity of the galaxy, and is such is a first-order approx-imation to calculate the resonance locations. Where non-axisymmetry of the potential dominates, the observed veloc-ity profile will overestimate the true circular velocity, and sothis approximation will lead to incorrect resonance locations,but assessing the impact of this requires detailed simulationsfor each galaxy, and as such is beyond the scope of this work.However, observational works have applied this method pre-viously (see, e.g. Schinnerer et al. 2000; Fathi et al. 2009),and find the location of expected structures near these res-onance locations, which would indicate this first-order ap-proximation is often close to the true value. As such, wehighlight that these resonance locations are approximations,but represent our best estimation of the true value. Through-out this work, we will refer to these resonance locations asthe co-rotation radius and ILR/OLR, but bear in mind thesecaveats.Typically, the rotation curves derived from the CO mapsare very similar to those derived from H α and stars in oursample, and so we are confident our choice of using CO-based rotation curves will not bias our results. By convertingHANGS P ATTERN S PEEDS P r ob a b ilit y D e n s it y R CR / h R P r ob a b ilit y D e n s it y Figure 10.
Left : KDE plot showing the distribution of R (the ratio of co-rotation radius to bar length) for our nine well-constrained stellarmass bar pattern speeds. Right : KDE plot showing the distribution of the ratio of the co-rotation radius to the galaxy scale length for the samesample of galaxies. In each case, we highlight the median value as a solid black vertical line, and the 16 th and 84 th percentiles as dashed blacklines (these percentiles are calculated directly from the data, rather than from the KDE). The commonly assumed literature values are shown asred, vertical lines. the rotation velocity to an angular velocity, Ω( R ) , the co-rotation radius is simply where the angular velocity is equalto the pattern speed.To calculate R CR , we use the velocities as fitted to eachradial bin. This allows us to propagate through the uncertain-ties both in the rotation curve and the pattern speed. In somecases, we have multiple regions where the angular velocitycrosses the pattern speeds, and we report all of these in Ta-ble 2. In many cases, due to the shape of the rotation curvea number of consecutive points are consistent with being co-rotation, within the errors. For these points, we associatethese to the same R CR if they are within σ , and we take themean of them as the nominal values, with the range as the un-certainty. An example of this R CR measurement is shown inFig. 8, for NGC 3351. As can be seen, our error propagationleads to reliable estimates of R CR , but quite large uncertain-ties, on the order of ∼ R CR . Thelocations of these resonances occur when Ω P = Ω( R ) ± κ/ (7)where positive is for the OLR, and negative is for the innerILR. κ is the epicyclic frequency, which is given by κ = 2Ω( R ) R dd R (cid:0) R Ω( R ) (cid:1) . (8) These curves are also highlighted in Fig. 8, and we calcu-late their average radii and uncertainties in the same way asfor R CR . The locations of the resonances are shown on theMUSE white light map of NGC 3351 in Fig. 9.With R CR directly measured, we can investigate two com-monly assumed ratios for inferring the co-rotation radiuswhen it is not directly accessible. The first is R CR / R bar (commonly referred to as R ), where we calculate the depro-jected bar length as R bar , deproj = R bar , proj × (cid:113) cos(∆PA) + (sin(∆PA) sec( i )) , (9)where R bar , proj is the projected bar length from Herrera-Endoqui et al. (2015), ∆PA is the relative alignment of thebar with the galaxy major axis, and sec( i ) is the secant func-tion of the inclination (the inverse cosine). We show the dis-tribution of R for our sample in the left panel of Fig. 10.The average value (and spread) for our sample is . +0 . − . (this spread is calculated from the data, rather than from theKDE plot in Fig. 10). Taking into account the error in co-rotation radius, along with a characteristic uncertainty in thebar length of 20% (D´ıaz-Garc´ıa et al. 2016), we find thataround half of our measured values of R are inconsistentwith the commonly assumed value of 1.2 in the literature(e.g. Elmegreen et al. 1996; Aguerri et al. 1998) to σ . Thisis well within the expectation when including Poisson noisedue to small number statistics, and so our values of R are, onaverage, consistent with the expected value of 1.2.Another commonly used ratio for inferring R CR is throughthe disk scale length, h r . We use the S G scale lengths cal-culated in Salo et al. (2015) from multiple component imagedecompositions of
Spitzer µ m images, and show the ratio4 W ILLIAMS ET AL . P ( k m s kp c ) N = 8,= 0.14 ± 0.29 25 50 75 20406080 P ( k m s kp c ) N = 9,= 0.00 ± 0.220 500510 R CR ( kp c ) N = 8,= 0.07 ± 0.14 25 50 75 0510 R CR ( kp c ) N = 9,= 0.28 ± 0.220 500.00.20.40.6 R b a r / R N = 8,= 0.00 ± 0.36 25 50 75 0.00.20.40.6 R b a r / R N = 8,= 0.00 ± 0.290 50024 N = 8,= 0.36 ± 0.14 25 50 75 024 N = 8,= 0.21 ± 0.210 50| PA| (deg)0.20.40.60.81.0 S b a r N = 6,= 0.33 ± 0.131020304050607080 i ( d e g )
25 50 75 i (deg) 0.20.40.60.81.0 S b a r N = 6,= 0.20 ± 0.27 020406080 | P A | ( d e g ) Figure 11.
Pattern speeds measured from MUSE stellar mass data and global parameters with galaxy orientation parameters.
Left : From topto bottom, the pattern speed, co-rotation radius, bar radius, R , and bar strength versus ∆PA (the difference between bar and galaxy positionangles). Points are coloured by the inclination of the galaxy. Right : Same as the left column, but now versus inclination, with points colouredby ∆PA . For galaxies without a bar, we show the pattern speed and co-rotation radius as an unfilled circle. In each case, the number of points, N , and the Kendall τ correlation is given. We also show the well-constrained PHANGS-ALMA sample as grey crosses. HANGS P
ATTERN S PEEDS h r /R CR istaken to be 3 (e.g. Kranz et al. 2003), and we find the valuefor our sample to be . +0 . − . . Again, given the small num-ber statistics we conclude our sample to be consistent with avalue for h r /R CR of 3. CORRELATIONS WITH GLOBAL GALAXYPARAMETERS6 W
ILLIAMS ET AL . . . . P (kms kpc ) N = , = . ± .
11 0 . . . N = , = . ± .
19 2 . . . N = , = . ± .
14 7 . . . N = , = . ± .
28 200300 N = , = . ± .
29 02 N = , = . ± .
21 0 . . P (kms kpc ) N = , = . ± .
17 2 . . . R CR (kpc) N = , = . ± .
17 0 . . . N = , = . ± .
19 2 . . . N = , = . ± .
21 7 . . . N = , = . ± .
28 200300 N = , = . ± .
29 02 N = , = . ± .
29 0 . . R CR (kpc) N = , = . ± .
22 2 . . . . . . . R bar / R N = , = . ± .
29 0 . . . N = , = . ± .
27 2 . . . N = , = . ± .
29 7 . . . N = , = . ± .
29 200300 N = , = . ± .
38 02 N = , = . ± .
38 0 . . . . . . R bar / R N = , = . ± .
29 2 . . . N = , = . ± .
14 0 . . . N = , = . ± .
40 2 . . . N = , = . ± .
19 7 . . . N = , = . ± .
29 200300 N = , = . ± .
29 02 N = , = . ± .
19 0 . . N = , = . ± .
21 2 . . . T H ubb l e . . . . . S bar N = , = . ± .
27 0 . . . B / T N = , = . ± .
27 2 . . . h r ( kp c ) N = , = . ± .
27 7 . . . l og ( M R [ M kp c ]) N = , = . ± .
27 200300 v r , i n f ( k m s ) N = , = . ± .
40 02 R t ( kp c ) N = , = . ± .
13 0 . . f m o l . . . . . S bar N = , = . ± .
13 1020304050607080 i (deg) F i g u re . F r o m t op t obo tt o m :t h e p a tt e r n s p ee d , c o -r o t a ti on r a d i u s , b a rr a d i u s , R , a ndb a r s t r e ng t hv e r s u s , fr o m l e f tt o r i gh t:t h e H ubb l e m o r pho l og i ca lt yp e , bu l g e - t o - t o t a l fl ux r a ti o , g a l a xyd i s k s ca l e l e ng t h , s t e ll a r m a ss d i v i d e dby R , a s y m p t o ti c r o t a ti onv e l o c it y , g a l a xy r o t a ti on t u r nov e rr a d i u s , a nd m o l ec u l a r g a s fr ac ti on . P o i n t s a r ec o l ou r e dby t h e i n c li n a ti ono f t h e g a l a xy , g a l a x i e s w it hb a r ss ho w n a s c i r c l e s , a ndg a l a x i e s w it hou t a ss qu a r e s , a ndg a l a x i e s w it hno t e d s p i r a l a r m s a r e h i gh li gh t e d i n r e d . I n eac h s ubp l o t , t h e nu m b e r o f po i n t s , N , a nd t h e K e nd a ll τ c o rr e l a ti on i s g i v e n , a l ong w it h it s a ss o c i a t e dun ce r t a i n t y . W ea l s op l o t w e ll - c on s t r a i n e d P HANG S - A L M A v a l u e s a s g r e y c r o ss e s . HANGS P
ATTERN S PEEDS ∆PA ). In this work, we will focus on Ω P , R CR , R bar , R , and the bar strength, S bar . We take thebar lengths from Herrera-Endoqui et al. (2015), and the barstrength as the normalised m = 2 Fourier density amplitude( A max2 ) from D´ıaz-Garc´ıa et al. (2016). Following Cuomoet al. (2020), we only use points that satisfy ∆Ω P / Ω P ≤ . ,where ∆Ω P is the error in Ω P , but in this work we find thatthis removes only one galaxy (NGC 7496). This leaves uswith a maximum of 9 galaxies, but depending on the avail-able literature data, the number of galaxies compared mayvary slightly with parameter (for example, only 6 galaxieshave bar strengths in D´ıaz-Garc´ıa et al. 2016). We alsoshow the values from the ALMA points in grey, to highlightany misleading conclusions that may be drawn from using Ω clump instead of Ω P (or, equivalently, the bias of using COrather than stars).We first consider any pathological effects that may arisefrom the orientation of the galaxy. Our ability to measuremorphological properties of bars may be affected by the in-clination, or relative misalignment of the bar with the galaxyitself, and it is important to consider whether this may biasour results. Fig. 11 shows the relationships between Ω P , R CR , R bar , R , and S bar with ∆PA and the galaxy incli-nation, where we also indicate the Kendall (1938) τ correla-tion coefficient. As we are correlating a number of variablesagainst each other, by random chance we may see some “sig-nificant” correlations in random data, based simply on using p -values (i.e. for 100 random distributions, five would be ex-pected to have p < . ; see e.g. Holm 1979, as well asKruijssen et al. 2019 for a recent discussion of this). The p -value also assumes all errors are equal, which is not thecase here. Instead, we take a Monte-Carlo approach to as-sign a statistical error to the significance of the correlation.For , realisations, we perturb each value by its associ-ated errors in both x and y , and repeat the τ calculation. Wethen quote the median value of the correlation, and the σ spread in these values. If the correlation is larger than this 1 σ error, then the correlation is significant. Ideally, we should see no correlations with projection pa-rameters. Reassuringly, the only significant correlations wesee are between S bar and ∆PA , and R and ∆PA . Thesemay give some artificially more significant correlation go-ing forwards, but otherwise we are confident that the projec-tion of the galaxy will not systematically bias our results. Ifwe consider the PHANGS-ALMA points we flag as well-constrained, which cover a wider range of galaxy inclina-tions, there is clearly an anti-correlation between bar lengthand inclination and a correlation between bar length and ∆PA . As these bars are typically defined by-eye, at high in-clination recovering the true bar length becomes increasinglydifficult, and this tends to lead to the length being underesti-mated. For studies that attempt to measure such quantities inhighly inclined systems, it is therefore important to take thisbias into account.We next correlate these quantities with a number of globalgalaxy parameters. These are the Hubble morphologicaltype (from HyperLeda), the bulge-to-total flux ratio at 3.6 µ m(B/T) and galaxy disk scale length ( h r ; both from Salo et al.2015), the total stellar mass ( M ∗ ) from the z = 0 Multiwave-length Galaxy Synthesis (z0MGS; Leroy et al. 2019) nor-malised by R for a measure of the average stellar surfacedensity of the galaxy. We also take the asymptotic veloc-ity of the galaxy rotation curve ( v r, inf ) and turnover radiusbetween the rising and flat part of the rotation curve ( R t )from (Lang et al. 2020, see their Eq. 10), and the moleculargas fraction ( f mol ; Leroy et al. 2020a). These are shown inFig. 12, where we also highlight the number of points in eachcombination of these parameters, along with the Kendall τ .We colour each point by inclination to highlight any potentialdependence of these correlations with galaxy orientation (seeFig. 11).Even given our relatively small number of galaxies, we finda number of significant correlations. In terms of the patternspeeds themselves, we find higher Ω P in later-type galax-ies, and galaxies that are more molecular gas-dominated. Wealso find lower pattern speeds in galaxies that are more bulge-dominated (with a higher bulge-to-total flux ratio), and thosewith a larger scale length. We find the co-rotation radius tobe larger in more bulge-dominated galaxies, and also galax-ies with a larger scale length. We see no significant corre-lations with the R normalised bar length. We find that R tends to be higher in later-type galaxies. This appears to becontradictory to numerical simulations, showing that the barpattern speed slows down over time (Wu et al. 2018). How-ever, as we do not know the age of the bars in these galaxies,we cannot draw this conclusion from these results. Furtherwork to calculate the bar ages in these galaxies may help toanswer this question. We also find that R tends to be lowerin galaxies with a larger scale length. We also see that barstend to be stronger in galaxies with a higher R t , and a higher8 W ILLIAMS ET AL .
25 50 750510 R CR ( kp c ) N = 9,= 0.28 ± 0.2225 50 750.00.20.40.6 R b a r / R N = 8,= 0.07 ± 0.2925 50 75024 N = 8,= 0.07 ± 0.2925 50 75 P (km s kpc )0.20.40.60.81.0 S b a r N = 6,= 0.07 ± 0.27 N = 6,= 0.07 ± 0.27 N = 8,= 0.07 ± 0.36 N = 8,= 0.07 ± 0.210 5 10 R CR (kpc) N = 6,= 0.07 ± 0.27 N = 8,= 0.07 ± 0.290.00 0.25 0.50 R bar / R N = 6,= 0.07 ± 0.40 0 2 4 N = 6,= 0.20 ± 0.27 Figure 13.
Corner plot of the y -axis quantities from Figs. 11 and 12. Galaxies without bars are indicated as squares (i.e., NGC 0628); galaxieswith bars are circles. In each subplot, the number of points, N , and the Kendall τ correlation is given, along with its associated uncertainty. Weobserve a weak anti-correlation between co-rotation radius and pattern speed, while all other quantities appear uncorrelated. f mol . Whilst we do see correlations between other parame-ters in this figure, the small number of galaxies means thatwe cannot robustly draw conclusions about the strength ofthe correlations between them. We see no strong dependencewith inclination in any of these correlations, and we there-fore are confident any biases seen in Fig. 11 do not stronglyaffect the results presented here. Across all of these parame-ters, our single non-barred galaxy (NGC 0628) occupies thesame parameter space as the barred galaxies, as do galaxieswith clear spiral arms versus more flocculent morphologies.The grey points indicate PHANGS-ALMA values that weconsider well-constrained (those that the straight line fits to (cid:104) v (cid:105) versus (cid:104) x (cid:105) look reasonable, although these are Ω clump values, rather than Ω P ). Notably, these values – even if theydo not measure the true pattern speed of the structure presentin the disk – appear to occupy very similar regions in param-eter space as our more robust stellar pattern speed estimates. This serves as a warning that similar measurements madewith ISM tracers may give the appearance of meaningfulcorrelations between dynamical structure and global galaxyproperties, even when none are present. Indeed, Ω clump ex-hibits a trend with stellar mass surface density, whilst wesee no significant trends with our stellar mass-based patternspeeds. Based on our studies described in Sect. 4 (whichsuggest that Ω clump is a function of both the shape of the ro-tation curve of the galaxy as well as the radial extent of theISM clumps), the trends in Fig. 12 arise as a result of a re-lation between the galaxy gravitational potential and the waythe ISM is distributed within it.Finally, we also correlate the y -axis quantities of Fig. 12with each other, and this is shown in Fig. 13. The only sig-nificant correlation we see here is unsurprising – that galaxieswith higher pattern speeds have shorter co-rotation radii. Wealso see that there is variation both in R CR and R bar /R HANGS P
ATTERN S PEEDS Ω P (although we do not find these correlations individ-ually to be significant). The decrease in R bar /R appears tobe driven mainly by the decrease in R bar , and this means wedo not see a strong variation in R across our galaxies. DISCUSSIONThis work presents homogeneous measurements of pat-tern speeds and resonance locations for ten galaxies ob-served as part of the PHANGS-MUSE survey. This sam-ple size means that we can draw exploratory conclusionsabout trends in these parameters with the global propertiesof the galaxy, and, as our methodology for the applicationof the Tremaine-Weinberg method is generally data-driven,with minimal qualitative checks (except for the quality flag-ging at the end), we believe that future studies can build onthe present work in a homogeneous way.We have first investigated the scatter in R , the ratio of co-rotation to bar length. Theoretical arguments by Contopoulos(1980) have shown this value should be a little larger than 1,and measurements of R in simulation (Athanassoula 1992;Debattista & Sellwood 2000) and in observation (e.g. Merri-field & Kuijken 1995; Gerssen et al. 1999; Font et al. 2017)have shown < R < . . This has led to, when the co-rotation radius cannot be measured from the pattern speed,a common expectation that R CR ∼ . R bar . We find forour sample that the median and 16 th /84 th percentile spreadin the sample is R = 1 . +0 . − . . The scatter on this distribu-tion is consistent with the typical uncertainty in the measure-ment of R CR . Thus, we find that our sample is compatiblewith a value of R = 1 . . We also find this to be the casefor the ratio of the co-rotation radius, R CR , to the disk scalelength, h r . This ratio is commonly assumed to be 3 (Kranzet al. 2003), with little scatter. We find a large scatter, witha median and 16 th /84 th percentile spread in the sample of R CR / h r = 2 . +0 . − . . There is some scatter here, but giventhe small number statistics of this study we find our mea-surement to be in agreement with the commonly assumed R CR / h r = 3 .8.1. Relationships between pattern speeds, co-rotationradii, and bar parameters with global galaxy properties
Next, we discuss the relationships between the patternspeeds, co-rotation radii, and bar parameters in the contextof global galaxy parameters (Fig. 12). Even given our smallnumber of galaxies, we find some significant correlationshere. For galaxies with a later Hubble type, we find a larger R (driven by shorter bar lengths). This is in agreement with,e.g. Erwin (2005), and so this finding is not surprising. Wealso find that galaxies with a longer disk scale length tend tohave a lower R . Given that we expect galaxies with higherscale lengths to have larger R CR (Kranz et al. 2003), but notnecessarily larger bars, this naturally follows. More molecu-lar gas-rich systems tend to have a higher pattern speed. This has been observed in simulations by Ghosh & Jog (2016),and also in some observational studies (Aguerri et al. 2015;Guo et al. 2019; Garma-Oehmichen et al. 2020). Galaxiesthat have larger bulges tend to have somewhat lower patternspeeds. This is seen in simulations (Kataria & Das 2019),and our results appear to confirm that. Finally, unlike Cuomoet al. (2020), we do not find a significant correlation between Ω P and the stellar mass surface density.8.2. Relationships between pattern speeds, co-rotationradii, and bar parameters
Finally, we turn to the inter-relation between our derivedparameters with various bar parameters (Fig. 13). We findonly one significant correlation: galaxies with higher patternspeeds tend to have smaller co-rotation radii. Despite this,our sample of galaxies has a roughly constant R . Although,it is also informative to look at parameters where we findno significant correlations – as in Cuomo et al. (2020), wefind no correlation between R and R bar . Their galaxy sam-ple tends to focus on earlier-type galaxies than the PHANGSsample, and so this may be a trend throughout the entiregalaxy population. Unlike previous works (e.g. Erwin 2005;D´ıaz-Garc´ıa et al. 2016; Kruk et al. 2018; Guo et al. 2019),we find no trend between bar length and bar strength. Wealso find no correlation between R and S bar , much like inCuomo et al. (2020). Finally, there is a theoretical predictionthat there should be an anti-correlation between S bar and Ω P (e.g. Debattista & Sellwood 2000; Athanassoula 2003). Asa bar loses angular momentum, it slows down and hence thepattern speed reduces. However, we do not see this in oursample. This may be due to low number statistics, or the va-riety of initial conditions in these galaxies washing out anycorrelation we may see between the bar strength and the pat-tern speed. It may be useful to study this in the context ofmore similar galaxies (e.g. age, size, morphology), but oursample has insufficient numbers to allow for this sort of bin-ning. CONCLUSIONSIn this work, we have applied the Tremaine-Weinbergmethod to stellar mass maps obtained as part of thePHANGS-MUSE survey, to measure pattern speeds for asample of galaxies. This work improves on previous stud-ies in a number of ways. These pattern speeds are calculatedhomogeneously, with a consistent methodology on a consis-tent kinematic tracer. We rigorously account for various un-certainties in our measurements of Ω P , and perform a num-ber of tests that allow us to determine whether these patternspeeds are reliable. We find that of our sample of 19 galaxies,ten have well measured pattern speeds. For the nine galax-ies that do not have well-measured pattern speeds, in two wedetect no evidence of a pattern (IC 5332 and NGC 5068),0 W ILLIAMS ET AL .and for the others we find the velocity field to be extremelymessy, meaning the galaxy position angle changes stronglywith radius.Leveraging H α maps from PHANGS-MUSE, as well asCO maps from PHANGS-ALMA, we also critically exam-ine the use of clumpy ISM tracers in determining patternspeeds. We find that the incomplete coverage can lead to afalse signal in Tremaine-Weinberg integrals (which we referto as Ω clump ), and that this Ω clump is systematically differentfrom Ω P ( ∼
40% higher for CO, ∼
20% higher for H α ). Animportant result from this paper, therefore, is that these trac-ers produce a compromised measure of the pattern speed. Wethus caution the reliability of pattern speeds in the literaturederived from ISM tracers.With these pattern speeds, combined with measured CO-based rotation curves for PHANGS galaxies by Lang et al.(2020), we calculate a number of resonances for the sample:the co-rotation radius, and the outer and inner Lindblad radii.The full list of pattern speeds, quality flags, and resonancelocations are given in Table 2, along with our adopted orien-tation parameters. We have compared the co-rotation radiusboth to the bar length and disk scale length, and find thatthese ratios are consistent with commonly assumed literaturevalues of 1.2, and 3, respectively. Given the spread in our val-ues, we would suggest taking an uncertainty in these ratios of ∼
30% and ∼ R , thatmore molecular gas-dominated galaxies have higher patternspeeds, and that more bulge-dominated galaxies have lowerpattern speeds. However, we also find an absence of corre-lations where we may expect them. In particular, we findno correlation between bar strength and pattern speed, norbetween total stellar mass surface density and pattern speed.With larger number statistics the small correlations we seehere may become more significant, and future work will beable to rule out whether these correlations truly are absent, orsimply due to small number statistics.There are a number of future studies that follow on natu-rally from this work. Firstly, it is important to note that theTremaine-Weinberg method can only recover a single patternspeed in a galaxy. We may expect different morphologicalfeatures to have different pattern speeds, which we are unableto recover in this work. We have flagged those we believe tohave strong signals of multiple pattern speeds. Applying amethod that allows for radial variation in the pattern speedhas been beyond the scope of this work, but will be the focusof future work. Secondly, our work has been applied to datawith cloud-scale resolution (1 (cid:48)(cid:48) , corresponding to ∼
100 pc).We have performed tests for a number of slit widths ranging from ∼
100 pc to ∼ ∼ kpc resolution. Given that we are only confident in pat-tern speeds measured using stellar masses and velocities, thiswork is hindered by low number statistics. The possibility ofhaving reliable pattern speeds for a factor of 100 more galax-ies will present a significant increase in statistical power forstudying the links between the properties of galaxies and pat-tern speeds. ACKNOWLEDGMENTSThis work has been carried out as part of the PHANGScollaboration.The authors would like to thank the anonymous peer re-viewer for their constructive comments, that have improvedthe quality of the paper. TGW would also like to per-sonally thank P. Woolford for discussions and support dur-ing the preparation of this manuscript. TGW, ES, H-AP,TS, FS acknowledge funding from the European ResearchCouncil (ERC) under the European Union’s Horizon 2020research and innovation programme (grant agreement No.694343). IB, FB acknowledge funding from the Euro-pean Research Council (ERC) under the European Union’sHorizon 2020 research and innovation programme (grantagreement No.726384/Empire). SCOG, RSK, MCS, EJWacknowledge financial support from the German ResearchFoundation (DFG) via the Collaborative Research Center(SFB 881, Project-ID 138713538) ‘The Milky Way Sys-tem’ (subprojects A1, B1, B2, B8, and P2). SCOG andRSK also acknowledge financial support from the Heidel-berg Cluster of Excellence STRUCTURES in the frame-work of Germany’s Excellence Strategy (grant EXC-2181/1- 390900948), and from the ERC via the ERC Synergy GrantECOGAL (grant 855130). JMDK and MC gratefully ac-knowledges funding from the Deutsche Forschungsgemein-schaft (DFG) through an Emmy Noether Research Group,grant number KR4801/1-1 and the DFG Sachbeihilfe, grantnumber KR4801/2-1. JMDK gratefully acknowledges fund-ing from the European Research Council (ERC) under theEuropean Union’s Horizon 2020 research and innovationprogramme via the ERC Starting Grant MUSTANG (grantagreement number 714907). The work of AKL and JS is par-tially supported by the National Science Foundation underGrants No. 1615105, 1615109, and 1653300. ER acknowl-edges the support of the Natural Sciences and EngineeringResearch Council of Canada (NSERC), funding referencenumber RGPIN-2017-03987.This paper makes use of the following ALMA data, whichhave been processed as part of the PHANGS-ALMA survey:HANGS P ATTERN S PEEDS
NumPy (Oliphant 2006),
AstroPy (Astropy Collaboration et al. 2013; Price-Whelanet al. 2018),
SciPy (Virtanen et al. 2020),
Matplotlib (Hunter 2007), and
Seaborn (Waskom et al. 2017). Thiswork has also made use of the HyperLeda database.REFERENCES Aguerri, J. A. L., Beckman, J. E., & Prieto, M. 1998, AJ, 116, 2136Aguerri, J. A. L., Debattista, V. P., & Corsini, E. M. 2003,MNRAS, 338, 465Aguerri, J. A. L., M´endez-Abreu, J., Falc´on-Barroso, J., et al.2015, A&A, 576, A102Anand, G. S., Lee, J. C., Van Dyk, S. D., et al. 2020, arXiv e-prints,arXiv:2012.00757Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013,A&A, 558, A33Athanassoula, E. 1992, MNRAS, 259, 345—. 2003, MNRAS, 341, 1179Beckman, J. E., Font, J., Borlaff, A., & Garc´ıa-Lorenzo, B. 2018,ApJ, 854, 182Beuther, H., Meidt, S., Schinnerer, E., Paladino, R., & Leroy, A.2017, A&A, 597, A85Binney, J., & Tremaine, S. 2008, Galactic Dynamics: SecondEditionBournaud, F., Combes, F., & Semelin, B. 2005, MNRAS, 364, L18Bundy, K., Bershady, M. A., Law, D. R., et al. 2015, ApJ, 798, 7Cappellari, M. 2017, MNRAS, 466, 798Cappellari, M., & Emsellem, E. 2004, PASP, 116, 138Chevance, M., Kruijssen, J. M. D., Hygate, A. P. S., et al. 2020,MNRAS, 493, 2872Contopoulos, G. 1980, A&A, 81, 198Corsini, E. M., Aguerri, J. A. L., Debattista, V. P., et al. 2007,ApJL, 659, L121Corsini, E. M., Debattista, V. P., & Aguerri, J. A. L. 2003, ApJL,599, L29Cuomo, V., Aguerri, J. A. L., Corsini, E. M., & Debattista, V. P.2020, A&A, 641, A111 http://leda.univ-lyon1.fr Debattista, V. P. 2003, MNRAS, 342, 1194Debattista, V. P., Gerhard, O., & Sevenster, M. N. 2002, MNRAS,334, 355Debattista, V. P., & Sellwood, J. A. 2000, ApJ, 543, 704Debattista, V. P., & Williams, T. B. 2004, ApJ, 605, 714D´ıaz, R., Carranza, G., Dottori, H., & Goldes, G. 1999, ApJ, 512,623D´ıaz-Garc´ıa, S., Salo, H., & Laurikainen, E. 2016, A&A, 596, A84Dobbs, C., & Baba, J. 2014, PASA, 31, e035Egusa, F., Sofue, Y., & Nakanishi, H. 2004, PASJ, 56, L45Elmegreen, B. G., Elmegreen, D. M., Chromey, F. R.,Hasselbacher, D. A., & Bissell, B. A. 1996, AJ, 111, 2233Elmegreen, B. G., Elmegreen, D. M., & Seiden, P. E. 1989, ApJ,343, 602Emsellem, E., Fathi, K., Wozniak, H., et al. 2006, MNRAS, 365,367Erwin, P. 2005, MNRAS, 364, 283Fathi, K., Beckman, J. E., Pi˜nol-Ferrer, N., et al. 2009, ApJ, 704,1657Font, J., Beckman, J. E., Mart´ınez-Valpuesta, I., et al. 2017, ApJ,835, 279Fragkoudi, F., Katz, D., Trick, W., et al. 2019, MNRAS, 488, 3324Garcia-Burillo, S., Combes, F., & Gerin, M. 1993, A&A, 274, 148Garma-Oehmichen, L., Cano-D´ıaz, M., Hern´andez-Toledo, H.,et al. 2020, MNRAS, 491, 3655Gerssen, J., Kuijken, K., & Merrifield, M. R. 1999, MNRAS, 306,926Ghosh, S., & Jog, C. J. 2016, MNRAS, 459, 4057Grand, R. J. J., Kawata, D., & Cropper, M. 2012, MNRAS, 421,1529Guo, R., Mao, S., Athanassoula, E., et al. 2019, MNRAS, 482,1733
ILLIAMS ET AL . Hernandez, O., Wozniak, H., Carignan, C., et al. 2005, ApJ, 632,253Herrera, C. N., Pety, J., Hughes, A., et al. 2020, A&A, 634, A121Herrera-Endoqui, M., D´ıaz-Garc´ıa, S., Laurikainen, E., & Salo, H.2015, A&A, 582, A86Holm, S. 1979, Scandinavian Journal of Statistics, 6, 65Hunter, J. D. 2007, Computing in Science & Engineering, 9, 90Kataria, S. K., & Das, M. 2019, ApJ, 886, 43Kendall, M. G. 1938, Biometrika, 30, 81Knapen, J. H., Beckman, J. E., Cepa, J., & Nakai, N. 1996, A&A,308, 27Korchagin, V., Orlova, N., Kikuchi, N., Miyama, S. M., &Moiseev, A. V. 2005, arXiv e-prints, astroKoribalski, B. S., & L´opez-S´anchez, ´A. R. 2009, MNRAS, 400,1749Krajnovi´c, D., Cappellari, M., de Zeeuw, P. T., & Copin, Y. 2006,MNRAS, 366, 787Kranz, T., Slyz, A., & Rix, H.-W. 2003, ApJ, 586, 143Kreckel, K., Faesi, C., Kruijssen, J. M. D., et al. 2018, ApJL, 863,L21Kruijssen, J. M. D., Pfeffer, J. L., Crain, R. A., & Bastian, N. 2019,MNRAS, 486, 3134Kruk, S. J., Lintott, C. J., Bamford, S. P., et al. 2018, MNRAS,473, 4731Lang, P., Meidt, S. E., Rosolowsky, E., et al. 2020, ApJ, 897, 122Leroy, A., Lee, J., Van Dyk, S., et al. 2020a, (ApJ in preparation)—. 2020b, (ApJS submitted)Leroy, A. K., Sandstrom, K. M., Lang, D., et al. 2019, ApJS, 244,24Lin, C. C. 1970, in IAU Symposium, Vol. 38, The Spiral Structureof our Galaxy, ed. W. Becker & G. I. Kontopoulos, 377Lin, C. C., & Shu, F. H. 1966, Proceedings of the NationalAcademy of Science, 55, 229Lindblad, B. 1963, Stockholms Observatoriums Annaler, 5, 5Lindblad, P. A. B., & Kristen, H. 1996, A&A, 313, 733Makarov, D., Prugniel, P., Terekhova, N., Courtois, H., & Vauglin,I. 2014, A&A, 570, A13Mart´ınez-Garc´ıa, E. E., & Puerari, I. 2014, ApJ, 790, 118Meidt, S. E., Rand, R. J., Merrifield, M. R., Debattista, V. P., &Shen, J. 2008a, ApJ, 676, 899Meidt, S. E., Rand, R. J., Merrifield, M. R., Shetty, R., & Vogel,S. N. 2008b, ApJ, 688, 224Merrifield, M. R., & Kuijken, K. 1995, MNRAS, 274, 933 Oliphant, T. E. 2006, A guide to NumPy, Vol. 1 (Trelgol PublishingUSA)Price-Whelan, A. M., Sip˝ocz, B. M., G¨unther, H. M., et al. 2018,AJ, 156, 123Querejeta, M., Meidt, S. E., Schinnerer, E., et al. 2016, A&A, 588,A33Rand, R. J. 1993, ApJ, 410, 68Rand, R. J., & Wallin, J. F. 2004, ApJ, 614, 142Rautiainen, P., & Salo, H. 1999, A&A, 348, 737Rosado-Belza, D., Falc´on-Barroso, J., Knapen, J. H., et al. 2020,arXiv e-prints, arXiv:2010.11815Salo, H., Laurikainen, E., Laine, J., et al. 2015, ApJS, 219, 4Sanders, J. L., Smith, L., & Evans, N. W. 2019, MNRAS, 488,4552Schinnerer, E., Eckart, A., Tacconi, L. J., Genzel, R., & Downes,D. 2000, ApJ, 533, 850Schinnerer, E., Maciejewski, W., Scoville, N., & Moustakas, L. A.2002, ApJ, 575, 826Schinnerer, E., Hughes, A., Leroy, A., et al. 2019, ApJ, 887, 49Sellwood, J. A. 2013, Dynamics of Disks and Warps, ed. T. D.Oswalt & G. Gilmore, Vol. 5, 923Sheth, K., Regan, M., Hinz, J. L., et al. 2010, PASP, 122, 1397Shetty, S., Bershady, M. A., Westfall, K. B., et al. 2020, ApJ, 901,101Silverman, B. W. 1986, Density Estimation for Statistics and DataAnalysis (London: Chapman & Hall)Sormani, M. C., Binney, J., & Magorrian, J. 2015, MNRAS, 454,1818Speights, J. C., & Rooke, P. C. 2016, ApJ, 826, 2Tremaine, S., & Weinberg, M. D. 1984, ApJL, 282, L5Treuthardt, P., Salo, H., Rautiainen, P., & Buta, R. 2008, AJ, 136,300Vazdekis, A., Koleva, M., Ricciardelli, E., R¨ock, B., &Falc´on-Barroso, J. 2016, MNRAS, 463, 3409Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, NatureMethods, 17, 261Waskom, M., Botvinnik, O., O’Kane, D., et al. 2017,mwaskom/seaborn: v0.8.1 (September 2017)Weiner, B. J., Sellwood, J. A., & Williams, T. B. 2001a, ApJ, 546,931Weiner, B. J., Williams, T. B., van Gorkom, J. H., & Sellwood,J. A. 2001b, ApJ, 546, 916Westpfahl, D. J. 1998, ApJS, 115, 203Wu, Y.-T., Pfenniger, D., & Taam, R. E. 2018, ApJ, 860, 152Zimmer, P., Rand, R. J., & McGraw, J. T. 2004, ApJ, 607, 285
HANGS P
ATTERN S PEEDS A. DERIVED PATTERN SPEEDS AND RESONANCE LOCATIONS4 W
ILLIAMS ET AL . T a b l e . “ P a tt e r n s p ee d s ” ( Ω p f o r s t e ll a r m a ss , Ω c l u m p f o rI S M t r ace r s ) a nd i n f e rr e d r e s on a n ce l o ca ti on s f o r a l a x i e s . A d e s c r i p ti ono f t h ec o l u m nn a m e s i s g i v e n i n t h e T a b l e no t e s . G a l a xy P G C D i P A B a r ? Ω c l u m p , A Q A R C R , A Ω P , MM Q MM R C R , MM Ω c l u m p , M H α Q M H α R C R , M H α M p c ◦◦ k m k p c s k p c k m k p c s k p c k m k p c s k p c E S O - . . . − . + . − . ····················· I C . . . . + . − . . ± . ·················· I C . . . . + . − . . ± . ·················· I C . . . ········· − . + . − . ··· . + . − . ··· NG C . . . . + . − . ····················· NG C . . . . + . − . ····················· NG C . . . . + . − . . ± . . + . − . . ± . . + . − . . ± . NG C . . . . + . − . . ± . ·················· NG C . . . . + . − . . ± . . + . − . . ± . . + . − . . ± . NG C . . . . + . − . . ± . ·················· NG C . . . . + . − . . ± . . + . − . . ± . . + . − . . ± . NG C . . . . + . − . . ± . ·················· NG C . . . . + . − . . ± . . + . − . . ± . . + . − . ··· NG C . . . . + . − . . ± . . + . − . . ± . . + . − . . ± . NG C . . . . + . − . . ± . . + . − . . ± . . + . − . . ± . NG C . . . . + . − . . ± . ·················· NG C . . . . + . − . . ± . . + . − . . ± . . + . − . . ± . NG C . . . . + . − . . ± . ·················· NG C . . . . + . − . . ± . ·················· NG C . . . . + . − . . ± . . + . − . . ± . . + . − . . ± . NG C . . . . + . − . ····················· NG C . . . . + . − . . ± . . + . − . . ± . . + . − . . ± . NG C . . . . + . − . . ± . ·················· NG C . . . . + . − . . ± . ·················· NG C . . . . + . − . . ± . ·················· NG C . . - . . + . − . . ± . ·················· NG C . . . . + . − . . ± . ·················· T a b l e c on ti nu e d HANGS P
ATTERN S PEEDS T a b l e ( c on ti nu e d ) G a l a xy P G C D i P A B a r ? Ω c l u m p , A Q A R C R , A Ω P , MM Q MM R C R , MM Ω c l u m p , M H α Q M H α R C R , M H α M p c ◦◦ k m k p c s k p c k m k p c s k p c k m k p c s k p c NG C . . . . + . − . . ± . ·················· NG C . . . . + . − . . ± . . + . − . . ± . . + . − . . ± . NG C . . . . + . − . ····················· NG C . . . . + . − . . ± . ·················· NG C . . - . . + . − . . ± . ·················· NG C . . - . . + . − . . ± . ·················· NG C . . . . + . − . . ± . . + . − . . ± . . + . − . . ± . NG C . . . . + . − . ····················· NG C . . . . + . − . . ± . ·················· NG C . . . . + . − . . ± . ·················· NG C . . . . + . − . . ± . ·················· NG C . . . . + . − . . ± . ·················· NG C . . . . + . − . ····················· NG C . . . . + . − . . ± . ·················· NG C . . . . + . − . . ± . ·················· NG C . . . . + . − . . ± . . + . − . . ± . . + . − . . ± . NG C . . . . + . − . . ± . ·················· NG C . . . . + . − . . ± . . + . − . . ± . . + . − . . ± . NG C . . . . + . − . . ± . ·················· NG C . . . . + . − . . ± . ·················· NG C . . . . + . − . . ± . . + . − . . ± . . + . − . . ± . NG C . . . . + . − . . ± . . + . − . . ± . . + . − . . ± . NG C . . . . + . − . ····················· NG C . . . . + . − . . ± . ·················· NG C . . . . + . − . ····················· NG C . . . . + . − . ····················· NG C . . . . + . − . ····················· NG C A . . . . + . − . . ± . ·················· NG C . . . . + . − . . ± . . + . − . . ± . . + . − . . ± . NG C . . . . + . − . . ± . ·················· T a b l e c on ti nu e d ILLIAMS ET AL . T a b l e ( c on ti nu e d ) G a l a xy P G C D i P A B a r ? Ω c l u m p , A Q A R C R , A Ω P , MM Q MM R C R , MM Ω c l u m p , M H α Q M H α R C R , M H α M p c ◦◦ k m k p c s k p c k m k p c s k p c k m k p c s k p c NG C . . . . + . − . . ± . ·················· NG C . . . . + . − . . ± . ·················· NG C . . . . + . − . . ± . ·················· NG C . . . . + . − . . ± . ·················· NG C . . . . + . − . . ± . ·················· NG C . . . . + . − . ····················· NG C . . . . + . − . . ± . ·················· NG C . . . . + . − . . ± . ·················· NG C . . . . + . − . ····················· NG C . . . . + . − . ····················· NG C . . . . + . − . . ± . ·················· NG C . . . . + . − . . ± . ·················· NG C . . . . + . − . . ± . ·················· NG C . . . . + . − . . ± . ·················· NG C . . . . + . − . . ± . ·················· NG C . . . . + . − . . ± . . + . − . . ± . . + . − . . ± . NG C . . . . + . − . ····················· NG C . . . . + . − . . ± . ·················· NG C . . . . + . − . . ± . ·················· NG C . . . . + . − . . ± . ·················· NG C . . . . + . − . . ± . ·················· NG C . . . . + . − . . ± . ·················· NG C . . . . + . − . . ± . ·················· NG C . . . . + . − . . ± . ·················· NG C . . . . + . − . . ± . . + . − . . ± . . + . − . . ± . NG C . . . − . + . − . ····················· N O TE — F o r eac hg a l a xy , w e li s tit s NG C nu m b e r , P G C nu m b e r , d i s t a n ce , i n c li n a ti on , po s iti on a ng l e , a nd w h e t h e r it h a s a b a r( f o r y e s , f o r no ) . W e u s ea s ub s c r i p tt o r e f e r t o t h e t r ace r i nqu e s ti on– MM f o r M U S E s t e ll a r m a ss , M H α f o r M U S E - H α m ea s u r e m e n t s , a nd A f o r A L M A C O m ea s u r e m e n t s . W ea l s o s ho w t h e qu a lit y fl a g s ( i nd i ca t e d a s Q ) , a nd c o -r o t a ti on r a d ii ( R C R ) , a ndou t e r /i nn e r L i ndb l a d r a d ii ( R O / I L R ) . F o r b r e v it y , on l y t h e fi r s t c o -r o t a ti on r a d i u s i ss ho w n , a nd L i ndb l a d r a d ii a r e h i dd e n . HANGS P
ATTERN S PEEDS B. TREMAINE-WEINBERG PLOTS FOR ALL MUSEGALAXIES8 W
ILLIAMS ET AL . h m s s s s -36°05'00"30"06'00"30"07'00"30" RA (J2000) D ec ( J ) IC5332, M h m s s s s -36°07'30"00"06'30"00"05'30"00"RA (J2000) D ec ( J ) Velocity x > ( )510152025 < v > ( k m s ) P = 13.96 ± 5.65 km s kpc Figure 14.
As Fig. 1, but for IC5332. For this galaxy , Q = 3 . HANGS P
ATTERN S PEEDS h m s s s s D ec ( J ) NGC0628, M h m s s s s D ec ( J ) Velocity x > ( )246810 < v > ( k m s ) P = 31.37 ± 2.70 km s kpc Figure 15.
As Fig. 1, but for NGC0628. For this galaxy , Q = 1 . ILLIAMS ET AL . h m s s s s -0°28'30"29'00"30"30'00"30"31'00" RA (J2000) D ec ( J ) NGC1087, M h m s s s s -0°31'00"30'30"00"29'30"00"28'30"RA (J2000) D ec ( J ) Velocity x > ( )1520253035 < v > ( k m s ) P = 36.10 ± 2.27 km s kpc Figure 16.
As Fig. 1, but for NGC1087. For this galaxy , Q = 1 . HANGS P
ATTERN S PEEDS h m s s s s -19°23'30"24'00"30"25'00"30"26'00" RA (J2000) D ec ( J ) NGC1300, M h m s s s s -19°26'00"25'30"00"24'30"00"23'30"RA (J2000) D ec ( J ) Velocity x > ( )304050607080 < v > ( k m s ) P = 18.38 ± 3.62 km s kpc Figure 17.
As Fig. 1, but for NGC1300. For this galaxy , Q = 3 . ILLIAMS ET AL . h m s s s s -36°07'00"30"08'00"30"09'00"30" RA (J2000) D ec ( J ) NGC1365, M h m s s s s -36°09'30"00"08'30"00"07'30"00"RA (J2000) D ec ( J ) Velocity x > ( )200204060 < v > ( k m s ) P = 51.81 ± 3.16 km s kpc Figure 18.
As Fig. 1, but for NGC1365. For this galaxy , Q = 3 . HANGS P
ATTERN S PEEDS h m s s s -24°29'00"30"30'00"30"31'00"30" RA (J2000) D ec ( J ) NGC1385, M h m s s s -24°31'30"00"30'30"00"29'30"00"RA (J2000) D ec ( J ) Velocity x > ( )1020304050 < v > ( k m s ) P = 40.09 ± 7.80 km s kpc Figure 19.
As Fig. 1, but for NGC1385. For this galaxy , Q = 3 . ILLIAMS ET AL . h m s s s m s s -47°12'00"30"13'00"30"14'00"30" RA (J2000) D ec ( J ) NGC1433, M h m s s s m s s -47°14'30"00"13'30"00"12'30"00"RA (J2000) D ec ( J ) Velocity x > ( )202224262830 < v > ( k m s ) P = 19.87 ± 1.61 km s kpc Figure 20.
As Fig. 1, but for NGC1433. For this galaxy , Q = 1 . HANGS P
ATTERN S PEEDS h m s m s s s -43°19'30"20'00"30"21'00"30"22'00" RA (J2000) D ec ( J ) NGC1512, M h m s m s s s -43°22'00"21'30"00"20'30"00"19'30"RA (J2000) D ec ( J ) Velocity x > ( )102030405060 < v > ( k m s ) P = 22.08 ± 0.21 km s kpc Figure 21.
As Fig. 1, but for NGC1512. For this galaxy , Q = 1 . ILLIAMS ET AL . h m s s s m s -54°54'55'56'57' RA (J2000) D ec ( J ) NGC1566, M h m s s s m s -54°57'56'55'54'RA (J2000) D ec ( J ) Velocity x > ( )25.027.530.032.535.037.5 < v > ( k m s ) P = 22.18 ± 1.73 km s kpc Figure 22.
As Fig. 1, but for NGC1566. For this galaxy , Q = 3 . HANGS P
ATTERN S PEEDS h m s s s s s -59°14'00"30"15'00"30" RA (J2000) D ec ( J ) NGC1672, M h m s s s s s -59°15'30"00"14'30"00"RA (J2000) D ec ( J ) Velocity x > ( )10203040 < v > ( k m s ) P = 22.52 ± 0.44 km s kpc Figure 23.
As Fig. 1, but for NGC1672. For this galaxy , Q = 1 . ILLIAMS ET AL . h m s s s s -22°20'00"30"21'00"30"22'00"30" RA (J2000) D ec ( J ) NGC2835, M h m s s s s -22°22'30"00"21'30"00"20'30"00"RA (J2000) D ec ( J ) Velocity x > ( )10.012.515.017.520.022.525.0 < v > ( k m s ) P = 46.33 ± 2.82 km s kpc Figure 24.
As Fig. 1, but for NGC2835. For this galaxy , Q = 1 . HANGS P
ATTERN S PEEDS h m s s m s s D ec ( J ) NGC3351, M h m s s m s s D ec ( J ) Velocity x > ( )051015 < v > ( k m s ) P = 42.90 ± 0.89 km s kpc Figure 25.
As Fig. 1, but for NGC3351. For this galaxy , Q = 1 . ILLIAMS ET AL . h m s s s D ec ( J ) NGC3627, M h m s s s D ec ( J ) Velocity x > ( )0102030 < v > ( k m s ) P = 29.15 ± 2.36 km s kpc Figure 26.
As Fig. 1, but for NGC3627. For this galaxy , Q = 2 . HANGS P
ATTERN S PEEDS h m s s s s s D ec ( J ) NGC4254, M h m s s s s s D ec ( J ) Velocity x > ( )1020304050 < v > ( k m s ) P = 52.20 ± 2.45 km s kpc Figure 27.
As Fig. 1, but for NGC4254. For this galaxy , Q = 3 . ILLIAMS ET AL . h m s s s D ec ( J ) NGC4303, M h m s s s D ec ( J ) Velocity x > ( )2.50.02.55.07.510.012.515.0 < v > ( k m s ) P = 54.07 ± 1.67 km s kpc Figure 28.
As Fig. 1, but for NGC4303. For this galaxy , Q = 1 . HANGS P
ATTERN S PEEDS h m s s s D ec ( J ) NGC4321, M h m s s s D ec ( J ) Velocity x > ( )2.55.07.510.012.515.017.5 < v > ( k m s ) P = 118.52 ± 7.40 km s kpc Figure 29.
As Fig. 1, but for NGC4321. For this galaxy , Q = 4 . ILLIAMS ET AL . h m s s s D ec ( J ) NGC4535, M h m s s s D ec ( J ) Velocity x > ( )5.07.510.012.515.017.5 < v > ( k m s ) P = 18.80 ± 1.08 km s kpc Figure 30.
As Fig. 1, but for NGC4535. For this galaxy , Q = 3 . HANGS P
ATTERN S PEEDS h m s s s -21°01'02'03'04' RA (J2000) D ec ( J ) NGC5068, M h m s s s -21°04'03'02'01'RA (J2000) D ec ( J ) Velocity x > ( )012345 < v > ( k m s ) P = 6.77 ± 3.22 km s kpc Figure 31.
As Fig. 1, but for NGC5068. For this galaxy , Q = 3 . ILLIAMS ET AL . h m s s s s -43°24'30"25'00"30"26'00"30"27'00" RA (J2000) D ec ( J ) NGC7496, M h m s s s s -43°27'00"26'30"00"25'30"00"24'30"RA (J2000) D ec ( J ) Velocity x > ( )12.515.017.520.022.525.027.530.0 < v > ( k m s ) P = 16.90 ± 0.70 km s kpc Figure 32.
As Fig. 1, but for NGC7496. For this galaxy , Q = 1= 1