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A INTRODUCTION T E tl mltajv 08/13/06 v. emulateapj style X nvriyo e eio 0 aeBv otes,Albuquer Northeast, Blvd Yale 800 Mexico, New of University aais niiul(11 C32 G 98 G34)–glxe:s galaxies: – NGC3344) 3938, NGC 342, IC (M101, individual galaxies: aais ieaisaddnmc aais tutr methods: – structure galaxies: – dynamics and kinematics galaxies: nvriyo otnhm nvriyPr,Ntiga,NG Nottingham, Park, University Nottingham, of University hrnE ed n ihr .Rand J. Richard and Meidt E. Sharon eateto hsc n Astronomy, and Physics of Department rf eso coe 2 2018 22, October version Draft colo hsc Astronomy, & Physics of School Dtd coe 2 2018) 22, October (Dated: ihe .Merrifield R. Michael A TR SPEEDS PATTERN ABSTRACT tutr a xedoe agrrda ag than range wave radial resulting larger the a over that such extend different linked, can in are structure patterns zones multiple radial 1997b), Tagger & arms), Masset the CR of the side convex outside uncommon. the also mostly along is lie lanes which dust would with speed generated (e.g. (i.e. spiral same (CR) a the 2008), Resonance Corsini with Corotation early-types; in its in occur found where near as not Furthermore, ends does bar 1988). speed, Sparke the & pattern ev- two-armed (Sellwood common as the general taken a initially where for ends), M83, bar idence in the be- from (as alignment emanates spiral apparent spiral and the Tagger bar galaxies, & tween barred (Masset In zones radial struc- 1997b). and distinct speeds occupy a different may span of ture waves not radius, do in general range in large propagate resonances which to the between expected OLR) Since are and waves ILR observed. Lindblad; as outer radius, and (inner disk as a of 1991). tions Kahn features, & Sellwood 1984; 1990; transient Carlberg al. & et (Sellwood Thomasson rapidly-evolving, simulations al. several et in Bertin or found compan- 1964; Shu persisting unclear or & waves Lin 1989), remains density bars (i.e. revolutions long-lived it many of over are 1979), presence spirals Norman whether the & (Kormendy to ions linked tionally nteter f“oeculn”(ynte l 1988; al. et (Sygnet coupling” “mode of theory the In frac- large over exist can waves how uncertain also is It observa- been have spirals grand-design Although itdwt ah(salse ihthe with (established each with ciated os yaayigtehigh-quality the analyzing By ions. prl r ecie ymultiple by described are spirals r pes u eetydeveloped recently Our speeds. n 98adNC34 nodrto order in 3344 NGC and 3938 C hr osbe eitrrtour interpret we possible, Where fglxe eel surprisingly a reveals galaxies of hrceitcsgaue fthe of signatures characteristic e frsnn opigo multiple of coupling resonant of atr pe eair,calling behaviors, speed pattern nl atr pe nothers. in speed pattern ingle u,N 87131 NM que, R,UK 2RD, 7 dfrtefis ie eewe Here time. first the for ed vrmr fteapparently the of more over sbet eemn directly determine to ssible A AITO IN VARIATION IAL numerical ia – piral 2 is possible for a single pattern. In this scenario, the dependence of the pattern speed, and affords straightfor- CR of an inner pattern overlaps with the ILR of an ward tests for bar-spiral and spiral-spiral relations, and outer pattern, and at this overlap energy and angu- spiral winding. lar momentum are efficiently transferred outward in the Also as in Paper II, for use as a kinematic tracer we disk. This was first demonstrated between simulated consider observations of the ISM, which have become the bars and spirals by Masset & Tagger (1997b) and later standard choice of spiral tracer for meeting the conti- by Rautiainen & Salo (1999), who find, in addition, that nuity requirement of the method. The application of other resonance overlaps, such as the coincidence of the the TW and TWR methods to CO and HI observations spiral’s inner 4:1 resonance with the bar CR, are also of spiral galaxies (e.g. Westpfahl 1998; Rand & Wallin effective. Given that the radial extent of resonant or- 2004; Merrifield, Rand & Meidt 2006) avoids significant bits can be quite large, these overlappings can be spa- problems with the stellar component in these systems, tially broad, and frequency diagrams can therein be suit- namely the faintness of the old stellar disk, and the ef- ably revealing. However, not all such resonance over- fects of star formation and obscuration by dust in spiral laps are accompanied by signs of true mode-coupling (i.e. arms by which the application of the continuity equation boosted beat modes detectable in simulation power spec- is invalidated. tra at the overlap; Masset & Tagger (1997b)). Furthermore, where the ISM is dominated by either Spiral-spiral mode coupling may also occur, typified, the molecular (traced by CO) or atomic gas phases, perhaps, by the transition commonly observed between conversion among phases can be assumed to occur at a strong two-armed spiral and more complex structure low levels on orbital timescales such that, together with at some radius. In their simulations, Rautiainen & Salo the low true efficiency of star formation in spirals, the (1999) find evidence for spiral structure in the absence of dominant component arguably obeys continuity (e.g. a bar, spiral-spiral mode coupling, and multiple pattern Zimmer, Rand & McGraw 2004; Rand & Wallin 2004). speeds without mode coupling. In addition, as an improvement on previous ISM-based To observationally investigate these possibilities, TW spiral studies, here we analyze both molecular gas we have undertaken a project to measure spi- observations from the BIMA SONG (Helfer et al. 2003; ral pattern speeds and their radial variation using maps include zero-spacing flux information) and archival the Radial Tremaine-Weinberg method developed by 21-cm emission data tracing the atomic hydrogen phase. Merrifield, Rand & Meidt (2006), subsequently tested (The flux information at the largest scales in all of the on simulations (Meidt et al. 2008a); hereafter Pa- HI cubes considered here is comparable to that achieved per I), and recently applied to a real spiral in the with single-dish observations; see references in §§2.2-2.5). grand-design galaxy M51 (Meidt et al. 2008b, here- Our consideration of both CO and the more extended after Paper II). Like its traditional counterpart–the HI here serves two main purposes. In galaxies that are TW method for measuring a single, constant pattern not molecule-dominated over the extent of the detectable speed (Tremaine & Weinberg 1984)–the TWR calcula- CO emission, the HI supplements H2 to establish a total tion yields measurements of pattern speeds using ob- particle, continuity-obeying tracer. The radial range of servationally accessible quantities based on a require- detectable pattern speeds is also increased in this way. ment of continuity. The method is direct, and so Part of our treatment, in this case, entails an investi- overcomes several of the obstacles in reliably estimat- gation (where possible) into the sensitivity of TWR so- ing pattern speeds with other, indirect methods (e.g. lutions to the CO-to-H2 conversion factor X adopted in resonance identification or modeling; Elmegreen et al. combining the data, which has been suggested to vary lin- 1989, Elmegreen et al. 1996, Rautiainen & Salo 2005, early with metallicity (e.g. Boselli et al. 2002). Follow- Garcia-Burillo et al. 1993). ing Zimmer, Rand & McGraw (2004) and Paper II, we Moreover, the TWR method provides a resolution for consider the effect of variation in X with radius (but not those applications of the TW method where pattern the possibility, for example, of arm-interarm variations). speed estimates show a clear, systematic departure from We also take into consideration distortions or warps ob- a single value (e.g. Zimmer, Rand & McGraw 2004 and servable in the outer HI disks in our sample (described in Merrifield, Rand & Meidt 2006). Particularly in disks §§2.1-2.5), which violate of one of the main TW assump- with multiple or extended structures, this behavior im- tions (namely, that the disk is flat; Tremaine & Weinberg plies that the pattern speed is not constant either be- 1984). cause it varies temporally or spatially. By allowing the Based on the a priori models described in §3.1, the pattern speed to vary in the radial direction, the TWR best-fit pattern speed solutions calculated for each galaxy method explicitly allows for the possibility that a galaxy are presented in §§ 3.2-3.5. There, the resonances asso- may contain a number of distinct features at different ciated with each, which we identify through comparison radii, such as bars and spiral arms, each with their own with angular rotation curves, serve as a main informant pattern speeds. It also makes possible the detection of of our results; our findings are motivated by, and com- spiral winding and hence the estimation of the lifetime pared with, the large body of work linking resonances to of a galaxy’s current pattern. the dynamics and morphology of disks. We also discuss Our aim with this paper is to expand to a larger sam- the limitations and sensitivities of the TWR calculation ple of galaxies the TWR analysis recently applied to (described in Paper I) as applied to each galaxy and sum- the grand-design spiral galaxy M51 (Paper II). Here, as marize our results in §4. there, we apply the method according to the prescrip- tion outlined in Paper I. The calculation employs reg- 2. THE SAMPLE ularization, which smooths otherwise intrinsically noisy solutions through the use of a prior models of the radial 2.1. Selection and Overview 3
Fig. 1.— Clockwise from top left: Zeroth moment maps for M101, IC 342, NGC 3938 (made from the combined CO and HI cubes) and NGC 3344 (made from the HI cube) showing the logarithm of the column density N in units of cm−2. The horizontal bar near the bottom indicates the physical scale. In the following subsections we describe the four spi- kpc; in this case, considering the total column density is ral galaxies analyzed in this paper, M101, IC 342, NGC necessary for meeting the continuity requirement of the 3938, and NGC 3344. This small sample is not meant method. Lastly, for NGC 3344, where the ISM is domi- to be representative of variations in spiral structure with nated by the atomic phase over the majority of the disk galaxy classification type, or to embody all possible the- area (see §2.5), we analyze HI data alone, and with our ories of spiral structure. These nearby galaxies were se- TWR solution predict a pattern speed for the small bar. lected simply for their clear spiral structure and mod- In three of the four cases, a warp-like distortion is ev- erate inclination, as well as for the availability of ob- ident in the HI distribution and kinematics, often with servations (specifically, both CO and HI in three of the little further evidence of a strong spiral pattern. A warp four cases) of sufficient sensitivity and resolution to allow in the outer disk, characterized by position and/or in- structure to be resolved and the radial variation of Ωp to clination angles that differ from those values in the disk be investigated; see Paper I. interior, violates the TW assumption that the disk is flat With this sample we explore three scenarios for ap- (Tremaine & Weinberg 1984). In implementing the reg- plying the TWR method to observations of the ISM. We ularized TWR method we therefore adopt the procedure first consider both the molecular and atomic components advocated in Paper I (see the beginning of § 3.1), which of the ISM in M101, which is molecule-dominated over accommodates for the presence of information about the the detectable extent of the CO emission, and investi- warp in the TWR quadrature by excluding it from reg- gate whether there exists a relation between the speeds ularized solutions. of the multiple structures apparent throughout the disk. Zeroth and first moment maps for each galaxy are We then apply the method to two galaxies (IC 342 and shown in Figures 1 and 2. For M101, IC 342 and NGC NGC 3938) where the ISMs are dominated by neither 3938, these maps were generated from the combined CO the atomic or molecular gas phases in the central few and HI cubes, which were regridded to a uniform channel 4
Fig. 2.— Clockwise from top left: First moment maps for M101, NGC 3938, IC 342 (made from the combined CO and HI cubes) and NGC 3344 (made from the HI cube) in units of km s−1. width and co-added in units of particles cm−2. Prior to squares fits to the standard three-parameter approxima- combination, the CO cubes were smoothed to the resolu- tion (i.e. Faber & Gallagher 1979) tion of the HI and CO intensities converted to molecular Vmax(r/rmax) column densities assuming a constant CO-to-H2 conver- Vrot(r)= (1) 20 −2 −1 −1 n 3/2n sion factor X=2.0×10 cm [K km s ] , the Galactic (1/3+2/3(r/rmax) ) mean value (e.g. Hunter et al. 1997). (Subsequent vari- as shown in Figure 3. The errors bars on each mea- ations in the X-factor are considered when applicable.) sured velocity there reflect the average deviation from Tilted ring fits to the velocity field of the four galaxies this value on either the approaching or receding side, were performed using the GIPSY (van der Hulst et al. which together we find better represent the uncertainty 1992) task ROTCUR in order to derive the kinematic in the rotation curve than do the formal errors returned parameters used in the TWR calculation. For these fits, by ROTCUR. With the fitted velocities, we then gener- the systemic velocity is initially fixed to the value given ate a set of smooth curves for Ω, Ω±κ/2, and Ω±κ/4; in the literature, while the kinematic center, position an- these curves are intended to reduce the impact of non- gle (PA), and inclination are allowed to vary. (The value axisymmetric motions (e.g. spiral streaming) on our res- and the errors on Vsys were subsequently determined by onance identifications and were invoked without regard fitting with all other best-fit parameters held fixed.) The to specific mass component characterization. values we find are in good agreement with those nomi- nally adopted for each galaxy and are listed in Table 1. 2.2. M101 For the purposes of identifying resonances (corotation, This SABcd galaxy (D=7.4 Mpc; Jurcevic & Butcher inner and outer Lindblad), we also derived the circu- 2006) is an excellent candidate for analysis with the lar velocity in each ring with the best-fit kinematic pa- TWR method. The molecular gas filling the central 3’ rameters held fixed. In each galaxy these velocities are hole in the HI emission features a bar discovered by modeled in the unwarped portion of the disk with least- Kenney, Scoville & Wilson (1991). The bar is a clear 5
Fig. 3.— Clockwise from top left: Rotational velocities for M101, IC 342, NGC 3938, and NGC 3344. Values measured with tilted ring fits to the velocity fields shown in Figure 2 with all optimal parameters listed in Table 1 held fixed are shown as filled circles (crosses) in the unwarped (warped) region of the disk (as identified in §§ 3.2-3.5). The points are shown spaced at the resolution of each map with error bars (shown in dark gray) representing the average variation in the derived velocity from side to side (approaching and receding). The solid line in the plots for M101, IC 342 and NGC 3938 is the least-squares fit of the velocity model (Equation 1) to the velocities −1 measured in the unwarped portion of the disk (filled circles), with best-fit parameters: Vmax=189 km s , rmax=15.89 kpc, n=0.43 for −1 −1 M101; Vmax=167 km s , rmax=5.62 kpc, n=0.76 for IC 342; Vmax=154 km s , rmax=7.3 kpc, n=0.76 for NGC 3938; Vmax=168 km −1 s , rmax=5.8 kpc, n=0.29 for NGC 3344. (For NGC 3344, to achieve an accurate fit these parameters had to be constrained a priori to a range that best reproduces the flatness for r&3 kpc). All solid lines are used to derive the smoothed angular rotation and resonance curves presented in §§ 3.2-3.5.
TABLE 1 Parameters used in the TWR calculation. Parameter M101 IC 342 NGC 3938 NGC 3344 Dynamical Center RA (α) (J2000) 14h3m13s.13 3h46m48s.4 11h52m49s.8 10h43m31s.5 Dynamical Center DEC (δ) (J2000) 54◦20’56” 68◦5’47”.8 44◦7’11”.7 24◦55’18”.3 Distance (Mpc) 7.4 2.0 11.3 6.9 −1 Systemic Velocity (Vsys, km s ) 244±8 30±2 809±3 586±3 Position Angle(◦) 42±3 42±3 21±2 155±2 Inclination (◦) 21±6 31±5 14±3 25±4 Note. — Entries for the dynamical center, inclination and position angle for each galaxy were derived with a tilted ring analysis of the first moment of the data cube using the GIPSY task ROTCUR. Optimal values for the distance originate with references cited in §§ 2.2-2.5. candidate-driver of the spiral structure that appears to radial dependence of the pattern speed throughout the emanate from the bar region. At the outer radii, tidal disk. We aim to derive the bar and spiral pattern speeds, interaction with companion galaxies (e.g. NGC 5474 and identify multiple spiral modes and spiral winding, if and NGC 5477; Huchtmeier & Witzel 1979, Waller et al. present. 1997) is thought to be responsible for the lop-sidedness As considered in the upcoming discussion (§ 3.2), we and distortion in the HI distribution, and may also be the note that the rotational velocities between r∼7 and 14 source of the spiral (Waller et al. 1997). The high resolu- kpc and at radii r&19 kpc in the top right of Figure 3 are tion (∼7”) of the total H2+HI maps constructed from the not well fit here. Also, while the rise in the latter zone BIMA SONG CO data and THINGS HI data (with RO- appears on both the approaching and receding sides, a BUST weighting scheme; Walter et al. 2008) allows us to ∼75 km s−1 asymmetry exists therein. apply the TWR method with exceptional leverage on the In addition, as demonstrated by Wong & Blitz (2002) 6
20 −2 −1 −1 assuming X=1.8×10 cm [K km s ] (slightly lower H2 column density (Helfer et al. 2003) there exceeds the than our adopted value), and using the BIMA SONG HI, we do not consider the contribution of the molecular data we analyze here, the ISM is molecule-dominated gas here. over the extent of the CO emission. However, the disk 3. RESULTS is known to sustain a metallicity gradient, which, for a linear dependence of X on metallicity (e.g. Boselli et al. 3.1. Applying the TWR Method 2002), could imply variation in X with radius. So, while We apply the regularized TWR method as in Paper I. a constant offset should have no implications for TWR For each galaxy we consider several smoothed, testable solutions, we also consider the effect of variation in X models for Ωp(r). These models vary as polynomials (or- outward from the center. der n.2) designated into at most three distinct radial zones. Where a priori evidence suggests that there is 2.3. IC 342 little information from a strong pattern beyond a cer- For this Seyfert SABcd galaxy (D=2.0 Mpc; tain radius, or that the TW assumptions are otherwise Crosthwaite et al. 2000), we base our analysis on the violated by the presence of a warp, for example, our mod- BIMA SONG CO data cube and the 38” VLA HI obser- els also include the parameterization of a cut radius, rc, vations first published by Crosthwaite et al. (2000). As beyond which all bins are calculated without regulariza- in M101, the CO fills the central hole in HI distribution, tion (i.e. the functional form is unconstrained). These but here the atomic gas makes a significant contribution models, with rc marking the end of the dominant struc- to the total gas density before the edge of the CO emis- ture, have been demonstrated to sufficiently separate the sion. A thorough discussion of the HI-CO overlap and compromised zones in the disk from regions where infor- the features (a bar, a two-armed spiral, and a four-armed mation about patterns can be reliably extracted in the spiral) in the two gaseous components can be found in TWR calculation (Papers I and II). All transitions rt be- the study of Crosthwaite et al. (2001). tween distinct zones and all cut radii, where present, are treated as free parameters. 2.4. NGC 3938 For each model, the two numerical solutions for Ωp(r) from either side of the galaxy (y>0 and y<0; NGC 3938 (D=11.3 Mpc; Jimenez-Vicente et al. 1999) Merrifield, Rand & Meidt 2006) are averaged to con- is a nearly face-on, late type (SAc) galaxy exhibiting a struct a single, global model solution. Each model solu- central two-armed spiral that branches into multi-armed tion is then judged based on a simple reduced χ2 statis- structure in the optical. The two strong spiral arms are 2 tic, with the best-fit solution corresponding to the χν - evident in the molecular gas, which reaches a surface den- minimum in the full parameter space. sity comparable to that of the HI inside the edge of the As in Paper II, the random, measurement errors used CO emission. But where the stellar disk exhibits floc- in the regularized calculation, and with which we judge culent, but clear, spiral structure (with as many as six the best-fit solution, reflect uncertainty arising with the arms; Elmegreen et al. 1992), the HI disk is characterized chosen flux cut-off in the first moment maps. The sys- by less well-organized, irregular structure. The combina- tematic errors on each measurement represent uncer- tion of archival WSRT HI data with BIMA SONG CO tainty in the PA, which is the dominant source of er- observations establishes a total H2+HI kinematic tracer ror in TW and TWR estimates (Debattista 2003; Pa- at roughly 20” resolution that extends to just beyond the ◦ per I), roughly 20% for δPA=3 . But, here we report optical extent of this galaxy (van der Kruit & Shostak these as a dispersion on each measured value, rather than 1982). present individual solutions for each PA; this is possible here since, unlike in M51 (Paper II), we find no mean- 2.5. NGC 3344 ingful evidence that the form of the model associated The application of the model-independent TWR with the best-fit solution for any of the galaxies in the method to this HI-dominated, isolated SABbc galaxy current sample varies from PA to PA. Also, unless oth- (D=7.4 Mpc; Verdes-Montenegro et al. 2000) should, erwise noted, errors due to uncertainty in, e.g., the incli- in principle, clearly establish the relation of the ring- nation angle are generally smaller and are not reported; like morphological features at r=1 kpc and r=7 kpc, these prove to be of little consequence to the accurate identified by Verdes-Montenegro et al. (2000) in opti- placement of radial bins defined in the quadrature (as cal images, to the large-scale patterns present in the suggested in Paper I). The additional change introduced disk (and their resonances). The outer ring, in par- in the measurements Ωp through a change in sin i is fur- ticular, although thought unlikely to be related to the thermore shared by Ω and κ, and so our resonance iden- small bar (not observed in the gas) with aB∼0.7 kpc tifications, in particular, should not be effected by error (Verdes-Montenegro et al. 2000), has not been otherwise in the inclination to first order. A thorough account of conclusively related to the spiral. In search, as well, of our methodology can be found in Paper I (and references spiral winding and multiple spiral modes, here we an- therein). alyze 20” resolution archival WSRT HI data where the dominant two-armed spiral and the outer ring are clear, 3.2. M101 in addition to the outer lopsided region exhibiting a twist We apply the TWR calculation with radial bin width in the isovels (van der Kruit & Shostak 1982). The zone ∆r=7”=0.27 kpc (D=7.4 Mpc), the resolution of the of the bar and the inner ring, which falls within the cen- combined cube. Together with the position of the outer- tral 26” where there is little 21-cm emission, corresponds most slice on each side, |y|=30.4 cos i kpc, this establishes to less than two resolution elements. So although the the extent of integration along each slice, equipping solu- ring is resolved in CO (Regan et al. 2002), and the peak tions with 113 bins in total. The best-fit solution given a 7
Fig. 4.— Best-fit regularized solution for M101 with ◦ ◦ rc=21.9±0.43 kpc for PA=42 ±3 . For this solution, bins exterior to rC (not shown) have been calculated without regularization. Dashed red lines represent the dispersion in solutions derived with ◦ ◦ a three-pattern speed model at PA=39 and 45 . Horizontal error Fig. 6.— Total H2+HI surface density in M101 highlighting the bars represent the dispersion in rt,1, rt,2 and rc from PA to PA. structure inside ∼18 kpc. The transitions rt,1 and rt,2 marking the −1 −1 The innermost speed corresponds to Ωp,1=47±10 km s kpc extents of the first and second pattern speeds in our solution are −1 −1 out to rt,1=6.7±0.25 kpc, followed by Ωp,2∼18±1 kms kpc shown as dashed white circles. The corotation radius rCR=3.5 of −1 −1 the innermost speed is shown as a solid white circle. The horizontal out to r 2=13.8±0.58 kpc and Ω 3=5±3 kms kpc out to t, p, bar near the bottom left indicates the physical scale. rC . Curves for Ω, Ω±κ/2 and Ω±κ/4 (see text) are shown in black, cyan and blue. significantly from the nominal values established in rings interior. (We note that this strong variation in the PA is found to start at a much larger radius than identified by Rownd, Dickey & Helou 1994.) Inclination variation, in particular, is a violation of the TW/TWR assumptions and so we argue that, by excluding the bins covering the outer disk from models for Ωp(r), the remaining, inner regularized bins are better equipped to reproduce the true pattern speed. (Note that with this cut radius a possible m=1 mode describing the outer, lop-sided por- tion of the disk is ignored). In this case, the most conspicuous aspect of the so- lution is the pair of transitions at rt,1=6.7±0.25 kpc and rt,2=13.8±0.58 kpc (marked in Figure 6) between −1 −1 three distinct pattern speeds, Ωp,1=47±10 km s kpc −1 −1 −1 −1 Ωp,2=18±1 kms kpc and Ωp,3=5±3 kms kpc out to rc. As discussed below, the innermost, con- Fig. 5.— Fourier power spectrum of the zeroth moment map stant speed is nearly identical to the value Ωp=48±8 shown in the top left of Figure 1. Modes up to m=4 are plotted as −1 −1 a function of radius with lines for m=1 in black dash-dot-dot-dot, km s kpc measured from the CO maps, alone (see m=2 in red solid, m=3 in green dash-dot, and m=4 in blue dash. § 3.2.1), while the second pattern speed is consistent with the value generally upheld in the literature (Ωp∼19 ◦ −1 −1 PA uncertainty ±3 is plotted in Figure 4. There shown, km s kpc for rCR∼12 kpc, e.g. Waller et al. 1997; also, are the rotation and resonance curves derived as but see the end of this section). described in §2.1. It should be noted that a ∼75 km The quality of our measurement is especially clear in s−1 asymmetry in the rotational velocities from the ap- the precision with which the transitions between the proaching and receding sides exists at radii r≥20 kpc three pattern speeds are determined. The transition rt,1, (e.g. Kamphuis 1993; Jog 2002). for example, shows less than 5% variation from PA to The outermost portion of the disk is effectively re- PA. This transition coincides with a marked decrease in moved from the solution with the parameterization of m=2 power (Figure 5) and occurs well past the edge of a cut radius rc=21.9±0.43 kpc. This radius, identified the CO emission. We can therefore recognize that the in- at the minimum of the χ2, is comparable to the loca- ner speed describes the molecular bar and the two-armed tion where the disk becomes visibly distorted. We find spiral manifest in the CO and weakly traceable in HI, that the outer distortion/lop-sidedness clear in the sur- which Elmegreen et al. (1992) identify in the B-band out face density is well characterized by the predominance to r∼6 kpc (estimated from their placement of the inner of an m=1 asymmetry beyond r∼20 kpc in the Fourier 4:1 circle in Figure 1, Plate 14, rather than with the dis- decomposition (Figure 5) and also matched by a warp in crepant value listed in Table 3 there). (Yet, we also note the outer velocity field; in addition to the rotation curve a four-armed appearance to the structure where spur-like asymmetry, with our ROTCUR analysis we find that the features associated with either arm emerge, clear in CO PA and inclination of fitted rings beyond this radius vary emission and in the B-band). 8
This result, namely that structure in HI shares the and fall in velocities between r=8 and 14 kpc in Fig- same pattern speed as the structure interior (exhibited ure 3 are more closely (yet coarsely) fit, although the solely by the molecular gas), admits a much stronger con- end of Ωp,2 more nearly approaches Ω-κ/2). The four- clusion than can be drawn from the CO, alone. Our rota- armed spiral in ESO 566-24 likewise extends between its tion and resonance curves indicate that the inner pattern inner and outer 4:1 resonances (Rautiainen et al. 2004). ends near its OLR (r∼7 kpc; or possibly the outer 4:1 Our finding is also roughly consistent with the identifica- resonance, within the errors) which would not be evident tion made by Elmegreen et al. (1992; in Figure 1, Panel from the application of the TWR method to CO data 14 there), namely that the zone dominated by the four- covering r.3.5 kpc (corresponding to the central zone armed spiral is bounded by the inner 4:1 circle with r∼6 in Figures 1 and 6 with column density &1020.6 cm−2). kpc and the CR circle with r∼12 kpc. (We would argue, This finding is consistent with the theoretical expectation however, that the end of this zone occurs past the CR, for the forbidden propagation of spiral density waves be- which we find near r=10 kpc, depending on the rotation yond this resonance, which observations corroborate (see curve.) Elmegreen et al. 1989). This is a scenario moreover favored by our TWR Likewise, corotation for this speed occurs at r=3.5 kpc, solution in that the transition to the four-armed spi- very near the transition between molecular and atomic ral occurs at a resonance overlap; where Ωp,1 ends at gas (see Figure 6). With its lack of clear structure OLR, within the uncertainties, Ωp,2 begins near its in- and relatively low surface density, this location in the ner 4:1 resonance at r∼7 kpc. Such an alignment of gaseous disk seems consistent with an expected depopu- resonances, although not conclusively associated with lation near CR owing to opposing torques (whereby gas mode-coupling (i.e. as opposed to CR-ILR overlaps is driven inward between ILR and CR and outward be- identified by Masset & Tagger 1997b), has been identi- tween CR and OLR). (This impression, however, may be fied in the simulations of Rautiainen & Salo (1999) and subject to the sensitivity of the CO map and the assumed Debattista et al. (2006), as well as in the grand-design X-factor.) spiral M51 through application of the TWR method (Pa- Interior to corotation, our determination of the disk per II). Presumably, this overlap is characteristic of a angular rotation becomes less certain at a disadvan- physical mechanism by which spiral structure can be sus- tage to definitive resonance identifications. Neverthe- tained over a large span in radius (Rautiainen & Salo less, the two-armed spiral exhibited near r∼2 kpc by 1999). the molecular gas–and which appears almost ring-like– The transition rt,2 between Ωp,2 and Ωp,3 may also be is arguably located near the inner 4:1 resonance kpc, accompanied by resonance overlap: the outermost speed, or UHR, where gas can accumulate on its path inward which spans a radial domain well-matched to that of the from corotation to ILR (Buta & Combes 1996). In ad- two-armed spiral within 12.r.20 kpc (marked by the in- dition, as will be discussed in more detail in §3.2.1, the crease in m=2 power in Figure 5), could begin at either central concentration of gas in the form of the molecu- the ILR or inner 4:1 resonance. Large fractional errors lar bar (with length ∼1 kpc) appears to lie well within on Ωp,3, of course, make this identification particularly the CR. This is also true of the stellar bar with ab∼2 vague. Likewise, the end of this pattern, as designated by kpc (Kenney, Scoville & Wilson 1991), which suggests the cut radius, can be only indefinitely related to realistic a scenario quite different from observational findings resonances (e.g. CR; at least with our current determi- in favor almost exclusively of fast stellar bars (with nation of the angular rotation), although it is just be- 1.rCR/ab.1.4; Corsini 2008), at least in early-type yond the end of the outer two-armed spiral identified by barred galaxies. Instead, the stellar bar here seems Elmegreen et al. (1992) near r=19 kpc (again, according reminiscent of the slow bars found in the simulations to their Figure 1, Panel 14, rather than Table 3). In ad- of Combes & Elmegreen (1993) and Rautiainen et al. dition, whether or not the resemblance to Ω-κ/2, Ω-κ/4 (2008). or Ω suggested in Figure 4 is significant remains unclear As for the pair of pattern speeds Ωp,2 and Ωp,3 out- at this point, especially in light of the nearly linear rise side rt,1=6.7±0.25 kpc, we can similarly use the curves (and the asymmetry) in the rotation velocities outside in Figure 4 to interpret their radial domains in terms of r∼19 kpc (not well modeled here; see Figure 3) suggest- resonances. This is less straightforward, however, pri- ing that the angular rotation curve may flatten out near marily because of our uncertainty in the poorly-fit ro- this radius. The values of Ω-κ/2 and Ω-κ/4 would be tation curve here, as well as the complex nature of the lower (and Ω+κ/2 and Ω+κ/4 higher) than in Figure 4, spiral structure visible in the HI surface density, which in which case Ωp,3 may coincide with Ω-κ/4. is characterized by power in several Fourier modes (see Despite our uncertainty in the outermost speed, over- Figure 5 and optically, Elmegreen et al. 1992). Notably, all, the TWR solution presents compelling evidence for between r∼6 kpc and 13 kpc, the structure visible in HI extensive spiral structure described by multiple pattern appears four-armed (see the top left panel in Figure 1 speeds that are moreover related by their overlap at and Figure 5). Meanwhile, near r=13 kpc a two-armed resonance. Given that this galaxy is tidally interact- pattern starts to predominate again, as indicated in Fig- ing (e.g. as argued by Ideta 2002), our pattern speed ure 1 and by Figure 5 where the power in the m=4 mode solution may be suggestive of the scenario explored by decreases relative to the m=2 mode. Salo & Laurikainen (2000) in simulations of M51. Those In this case, the transition rt,1 in Figure 4 would seem authors find that waves of higher and higher pattern to associate the second speed Ωp,2 with the four-armed speeds are excited as tidally-induced waves near Ω-κ/2 spiral, here found to extend between the inner and outer propagate inward. But considering that resonance over- 4:1 resonances at r∼6.8 kpc and r∼14 kpc. (This is un- lap is not a feature in those simulations, our finding of a changed even with rotation curve models where the rise link between the speeds in M101 (and in M51; Paper II)– 9 similar to the overlaps so far demonstrated between bars and spirals in simulation (e.g. Masset & Tagger 1997b; Rautiainen & Salo 1999)–may imply that at least the in- ner disk is dominated by internal, rather than external, effects. The speeds and structure within r 3.2.1. The inner pattern speed in M101 As stated previously, the inner disk appears to sustain only a single constant pattern speed, Ωp,1=47±10 − − km s 1kpc 1. This may be surprising, given the distinct Fig. 7.— bar pattern visible in the CO emission, which we might Plot of solution-reproduced (dots and crosses) and actual (open circles) integrals Fig. 8.— Best-fit regularized solution for IC 342 with ◦ ◦ rc=9.8±0.19 kpc for PA=42 ±3 . For this solution, bins exte- rior to rc (not shown) have been calculated without regulariza- tion. Dashed red lines and green horizontal error bars represent the dispersion in the pattern speeds and in rt and rc in solu- Fig. 9.— Total H2+HI surface density in IC 342 highlighting the tions derived with a two-pattern speed model at the three PAs. structure inside ∼9.8 kpc. The transition rt marking the extent The values in the zone of the bright spiral structure correspond to −1 −1 of the inner pattern speed in our solution is shown as a dashed Ωp,1=38±7km s kpc out to rt=5.7±0.71 kpc and Ωp,2=11±6 black circle. The horizontal bar near the bottom left indicates the −1 −1 km s kpc out to rc. Curves for Ω, Ω±κ/2 and Ω±κ/4 are physical scale. shown in black, cyan and blue. Crosthwaite et al. (2001) estimate based on their deter- This scenario has been suggested by mination rCR=4.2±0.7 kpc from inspection of the field of Kenney, Scoville & Wilson (1991) who, in first re- ◦ velocity residuals. According to the solution in Figure 8, porting on the 25 offset between the molecular and as parameterized by rt this pattern terminates at either stellar bar position angles, noted a resemblance to the corotation (here at r∼5 kpc; conditional on the rotation hydrodynamical simulations of Combes & Gerin (1985). curve, which is different from that of Crosthwaite et al. Such a decoupled central gas concentration is found 2001) or OLR (within the uncertainties) in favor of the when the stellar bar is slow and drives spirals that −1 −1 second, lower speed Ωp,2=11±6km s kpc . develop inside corotation, as we suggest here. The transition between the two speeds occurs very near A slow bar, ending well inside corotation, also seems to where the spiral structure becomes visibly four-armed be reconciled with central DM content in M101 implied (see Figure 9). This seems to establish that the outer in the multi-component simulations of Petitpas et al. arms are, in fact, best described by a speed that is dis- (2004), which best reproduce the observed velocity field tinct from that of the spiral interior, as previously sug- with a minimum disk. Where the DM contribution to the gested by Crosthwaite et al. (2000). Moreover, Figure 8 gravitational potential is large, Debattista & Sellwood suggests a possible resonant link between the two speeds (2000) find that interaction between a bar and the DM in IC 342, in that the lower speed begins near its ILR halo through dynamical friction decelerates the bar (or possibly its inner 4:1 resonance). The low speed (Weinberg 1985; Debattista & Sellwood 2000), such also seems to end at the 4:1 resonance, and possibly CR that it grows in length disproportional to the greater (within the uncertainties). However, the cut radius rc increase in rCR. It should be noted, however, that the in Figure 8 seems less an accurate bound on the four- slowness of the bar implied in this case is a matter of armed spiral, which continues out to r∼11 kpc (see the debate (cf. Sellwood 2008, Dubinski et al. (2009) and zeroth-moment map in Figure 1), than an indication of Weinberg & Katz 2007 who argue that slow bars are an where the warp begins. In this case, whether the warp artifact of the resolution of the N-body simulations used and four-armed spiral are related or share a common ori- to model the interaction). gin is unclear. Although we might also infer from Figure 8 that the 3.3. IC 342 inner speed lacks an ILR, our uncertainty in the rota- The TWR calculation for this galaxy proceeds with the tion curve at the center is high, given the low resolu- use of a radial bin width ∆r=0.41 kpc (as established by tion of the maps. Without more confidence in the be- the resolution of the combined maps) using slices out to havior of the Ω-κ/2 curve, in particular, it remains un- |y|=13.3 kpc. The best-fit solution, plotted in Figure clear whether this speed, and the structure it describes, 8 together with angular rotation and resonance curves, has an inner bound. Currently, then, both the spiral measures two distinct pattern speeds inside the cut ra- structure traced by CO and the distinct bar with length dius rc=9.8±0.19 kpc. As in M101, we find that this cut aB∼1.5 kpc (Crosthwaite et al. 2000) appear to rotate −1 −1 radius accurately reflects the location where the distor- with speed Ωp∼40 km s kpc . As in M101, this no- tion/warp in the outer disk begins, e.g. as identified in tably implies that the bar ends well inside its corotation the HI surface density/velocity field (Crosthwaite et al. radius, a circumstance which may also be evidenced by 2000; see Figure 1). the ∼9◦ offset between the molecular and stellar bar ma- −1 −1 Furthermore, the speed Ωp,1=38±7 kms kpc in- jor axes (Crosthwaite et al. 2001), as similarly discussed side rt=5.7±0.7 kpc is nearly identical to the value in §3.2.1. 11 Fig. 10.— Best-fit regularized solutions for NGC 3938 with ◦ ◦ A PA=21 ±2 . Dashed red lines for both solutions, Ωp (thick line) B and Ωb (thin line) represent the dispersion in the pattern speeds at the three PAs. The green horizontal error bar shows the disper- sion from PA to PA in rt in solutions derived with a two-pattern Fig. 11.— Total H2+HI surface density in NGC 3938 highlighting speed model. In the two-speed solution, the value in the inner zone the structure inside ∼15 kpc. The transition rt marking the extent B −1 −1 B corresponds to Ωp,1=53±9.2 km s kpc out to rt=3.4±0.7 kpc of the inner pattern speed in solution Ωp is shown as a dashed B −1 −1 black circle. The horizontal bar near the bottom left indicates the followed by Ωp,2=7.7±1.6 km s kpc , while the single, constant A −1 −1 physical scale. speed solution measures Ωp =6.6±2.1 km s kpc . Curves for Ω and Ω±κ/2 are shown in black and cyan. slices |y|≤13.1 kpc contain only 8 radial bins. This, to- On the other hand, the molecular bar and inner spi- gether with the low inclination of the disk–which com- ral could have different pattern speeds measurable, in promises the signatures of departures from axisymmetry principle, but for the width of the radial bins; informa- in the velocity field–limits how well radial variation in tion from the zone of the molecular gas, which covers a the pattern speed can be detected, if present. Under relatively small extent, is limited to only a minor con- this scenario, we find that two TWR solutions calcu- tribution in solutions. Although the two-pattern speed lated with regularization over all radial bins (i.e. with solution in Figure 8 yields a significantly better fit to no rc) fit the data equally well. The first ascribes a sin- A −1 −1 the data, a higher pattern speed can, in fact, be recog- gle, constant pattern speed Ωp =6.6±2.1 km s kpc nized in solutions modeled with three distinct pattern to structure throughout the disk, while the second in- 2 ◦ speeds: the lowest χν solution of this type at PA=42 corporates the measurement of a distinct, higher pattern i −1 −1 favors a transition at rt=1.6±0.45 kpc from Ωp=34±9 B − − − − speed Ωp,1=53±9.2 km s kpc inside rt=3.4 kpc (in 1 1 i 1 1 − − km s kpc to a higher Ωp=70±12 km s kpc , with addition to the lower ΩB =7.7±1.6 km s 1kpc 1). −1 −1 p,2 Ωp=12±4 kms kpc between rt=5.3±0.64 kpc and The measurement of any pattern speed at all may be rc=9.8±0.19 kpc. (Errors represent the dispersion in surprising, given the low inclination angle and the ap- three-speed solutions where the inner-most transition, parent lack of strong, organized structure in both the HI found best at the optimal PA, is held fixed from PA to and (smoothed) CO emission analyzed here. Compar- PA). This speed is particularly compelling since the bar ison with the rotation curves in Figure 10 greatly aids would end very near CR, and at this radius overlap with our interpretation of the measurements, although any the lower, spiral pattern’s inner 4:1 resonance. conclusions that might be drawn are tenuous at best; for However, we emphasize that this identification is in- this low inclination, rotation curves are highly uncertain conclusive at this point; the zone of the higher pattern (though similar to those throughout the literature, e.g. speed is covered by only four radial bins, or less than Jimenez-Vicente et al. 1999 and Combes & Becquaert 11% of the disk. (Note that the CO–which has a much 1997). higher resolution than the HI–alone cannot be used as With this in mind, we find it noteworthy that the low a continuity-obeying kinematic tracer, unlike in M101.) A B speed Ωp (or Ωp,2) is very near Ω-κ/2 in the outer disk Sensitivity to variation in the CO-to-H2 conversion factor (Figure 10; and possibly near Ω-κ/4, as well). This sim- (e.g. given the metallicity gradient in this galaxy mea- ilarity seems reminiscent of a pattern formed by closed, sured by Vila-Costas & Edmunds 1992) is also currently precessing elliptical orbits guided by the near-constancy untestable; with such a large radial bin width any varia- of Ω-κ/2. But since there is no dominant m=2 Fourier tion in X that we might accommodate would be crude at component visible in the 21 cm emission or in the NIR (as best and, with only four bins covering the CO emission, observed by Castro-Rodriguez & Garzon 2003) at these would likely yield an insignificant result. radii, it is unclear that this speed is associated with spi- ral structure, at all. A lack of spiral streaming motions 3.4. NGC 3938 in the atomic gas (as also observed in the ionized com- Given the resolution of the HI data (to which the ponent of the ISM; Jimenez-Vicente et al. 1999), would BIMA CO cube had been smoothed prior to the com- furthermore lead us to expect that, for any asymmetry bination) and, as a consequence, the rather large bin in the surface density, a value much closer to the disk width ∆r=1.7 kpc, our TWR solutions calculated in angular rotation frequency Ω would be measured. 12 At radii r.6 kpc, on the other hand, clear spiral struc- pered by the low inclination of the disk, the value of this ture is identifiable in the NIR (see the K-band image analysis lies in the prospect of renewed perspective on in Castro-Rodriguez & Garzon 2003). At the very least, the nature of the structure in this galaxy. Although in this seems to suggest that the two-pattern speed solution no way definitive (based on the TWR method, alone), B A B Ωp (with rt as marked in Figure 11) may be more realis- the lower measurement near Ω-κ/2 (either Ωp or Ωp,2), A tic than Ωp ; the faster inner speed seems more compati- in particular, may be compatible with several pieces of −1 −1 evidence that point to the influence of the DM halo first ble with structure at these radii than Ωp∼7km s kpc since, even with uncertainty in Ω at the inner radii and contemplated by van der Kruit & Shostak (1982). the near-constancy of Ω-κ/2 at the outer radii, the low As investigated by Frenk et al. (1988) or by Jog (1997), pattern speed’s ILR would lie at r&5 kpc. for example, the DM halo may be responsible for the ap- B pearance of structure in this isolated galaxy, which other- Nonetheless, whether Ωp,1 is a reliable measurement of the expected higher pattern speed is difficult to assert. wise seems difficult to reconcile with the finding that the gas disk is everywhere subcritical to gravitational insta- The transition rt (see Figure 11) lies just inside the edge of the CO emission tracing a two-armed spiral (clear in bility (as remarked upon by Combes & Becquaert 1997, the unsmoothed BIMA SONG map; Helfer et al. 2003), and aside from an alternative increase in instability pos- and so the measurement presumably reflects the speed of sible through the coupling between multiple components this pattern. However, even though this speed appears in the system, e.g. Jog 1992). to cover a quarter of the disk, in general we expect that Specifically, where the Toomre stability parameter in two radial bins would not supply sufficient leverage on the gas is high, Bureau et al. (1999) and Frenk et al. the measurement. (This is arguably the reason why the (1988) suggest that the torque due to a triaxial halo two TWR solutions are indistinguishable). Indeed, with (predicted by Cold Dark Matter simulations of hierar- so few bins in this zone we are prevented from diagnosing chical structure formation; e.g. Dubinski & Carlberg radial variation of order higher than zero. Furthermore, 1991) with slow figure rotation can drive structure in the accuracy of the measurement, if real, depends on how extended HI disks. Material (though not exclusive) to accurately information about the two speeds can be sepa- this scenario, the disk of NGC 3938 exhibits intrin- rated according to the parameterization of the transition sic ellipticity: as similarly diagnosed in other eccen- r =3.4 kpc, which here may be severely limited by the tric nearby spirals by Andersen et al. (2001), the pho- t tometric and kinematic position angles of NGC 3938 size of the radial bin width. (Note that, as in IC 342, the ◦ unsmoothed 6” resolution CO alone cannot be used as are offset by nearly 50 (Daigle et al. 2006). In fact, a continuity-obeying kinematic tracer, since ISM is not Castro-Rodriguez & Garzon 2003 measure an ellipticity molecule-dominated over the extent of the CO emission. ǫJ =0.11 in the J-band. So while the ellipticity may have arisen with an asymmetric accretion of matter, for ex- Tests for sensitivity to variation in the CO-to-H2 conver- sion factor are also not well-accommodated with so few, ample, it could also reflect an m=2 potential pertur- large radial bins). bation occasioned by triaxiality in the DM halo (see, These shortcomings notwithstanding, the distinct mea- e.g., Andersen et al. 2001 and references therein). In surement inside r∼3 kpc seems credible if only that this case, the TWR solution may represent a measure of it is consistent with values indicated by two indepen- the rotation speed of such a halo, if the structure in the dent approaches using different tracers of the spiral outer disk (which, extends further in HI than in the opti- structure. Korchagin et al. (2005) find with a global cal; van der Kruit & Shostak 1982) arises in the manner modal approach that the dominant m=3 and m=4 considered by Frenk et al. (1988). Currently, however, it modes in their multi-wavelength observation-based simu- is not clear that the structure in NGC 3938 is compatible lations are well-described with a pattern speed near ∼55 with the strong two-armed patterns found in the simula- − − tions of Frenk et al. (1988). km s 1kpc 1. Martinez-Garcia et al. (2009) propose a −1 −1 Alternatively, the overall instability of the disk might similar pattern speed, ∼47 km s kpc (adopted from be increased by a global mass asymmetry resulting from Martinez-Garcia et al. 2009 for D=11.3 Mpc), based on the m=1 perturbation to the halo potential considered the azimuthal age gradient across the spiral arms calcu- by Jog (1997, 2000 and 2002), which is also a possible lated from optical images. source of the ellipticity. This latter scenario seems to From Figure 10 we might also argue that, within the B be compatible with other observable characteristics of uncertainties, Ωp,1 ends near corotation at rCR=2.5 kpc this galaxy: like Tully et al. (1996) and Bournaud et al. and so represents a physically realistic scenario, also re- (2005), we find the HI disk to be slightly lop-sided toward cently identified in M51 (Paper II). However, this speed the north, and this m=1 asymmetry appears in the HI might just as reasonably end at OLR, depending on the velocity field, as well. (As previously demonstrated with rotation curve, the determination of rt, and the measure- the Hα observations of Daigle et al. 2006, we measure a ment itself. Also, while this pattern appears to lack an ∼10-20 km s−1 difference between the approaching and ILR, this is uncertain given that the angular velocities are receding sides beyond r∼11 kpc.) In this case, the TWR the least well-determined at the inner radii. At this point measurement may reflect the response of the gas (and there also does not seem to be a clear relation between stars) to this perturbation if the elliptical orbits calcu- the two pattern speeds (e.g. overlapping CR and ILR; lated by Jog (1997) precess with frequency Ω-κ/2. Masset & Tagger 1997b), especially since resonances for Again, our TWR measurement does not confirm, or B the outer speed Ωp,2 are difficult to establish, given the distinguish between, these possibilities. In either of the flatness of the resonance curves. two scenarios it is also not clear if, or how, the inner, While our confidence in the TWR measurement is tem- higher speed may relate to the lower speed. Higher reso- 13 TWR calculation). The model-independent TWR method, more- over, resolves the ambiguity in the study by Verdes-Montenegro et al. (2000). There, the uncer- tainty in the rotation curve precluded the reliable association of the inner and outer rings with res- onances, and the estimation of a pattern speed (Verdes-Montenegro et al. 2000). With the identifica- tion above we can confirm that the outer pseudoring at r∼7 kpc originates with the spiral, as opposed to the bar, which Verdes-Montenegro et al. (2000) argue is too small to have an OLR at this radius. The inner ring, on the other hand, seem more likely related to the bar (which it surrounds; Verdes-Montenegro et al. 2000), given the proxim- ity of these two features. That we find the spiral’s ILR Fig. 12.— Best-fit regularized solution for NGC 3344 with ◦ ◦ rc=6.8±0.43 kpc for PA=155 ±2 . For this solution, bins exte- nearly coincident with the location of the inner ring rior to rc (not shown) have been calculated without regularization. might then be a manifestation of resonance overlap Dashed red lines and the green horizontal error bar represent the between the speeds of the bar and spiral. If the spiral dispersion in the pattern speeds and rc in solutions derived at the three PAs. The value in the zone of the bright spiral structure with lower speed is driven at resonance, for example, our −1 −1 corresponds to Ωp,1=44±4 km s kpc out to rc. Curves for Ω, TWR solution may suggest a reasonable speed for the Ω±κ/2 and Ω±κ/4 are shown in black, cyan and blue. bar, which we are prevented from measuring here. In lution observations are necessary to first establish which particular, if we associate the inner, ILR ring with inner A B extent of the spiral (inside of which the HI density falls of the two solutions, Ωp or Ωp , is the most appropriate for this galaxy. Only then will it be possible to more off), such a scenario could be realistically achieved via CR-ILR mode-coupling, where the bar rotates with pat- rigorously explore the relation between the two pattern −1 −1 B tern speed Ωp&150 km s kpc for rCR∼1.0 kpc. The speeds suggested by the solution Ωp and perhaps then recognize the true nature of the low speed. bar, with length aB∼0.7 kpc (Verdes-Montenegro et al. 2000), in this case would be reasonably fast, with rCR/aB∼1.4. Alternatively, we might expect a bar 3.5. NGC 3344 − − pattern speed Ω &80 km s 1kpc 1, in the event of the The best-fit solution calculated in slices out to p CR-inner 4:1 resonance overlap discussed in §3.2, in |y|=19.4 kpc with radial bin width ∆r=0.75 kpc (cor- which case r /a ∼3.5. In the former (latter) scenario, responding to the ∼20” resolution of the HI cube) is CR B the inner ring could be located near the bar’s CR (inner shown in Figure 12 along with rotation and resonance 4:1 resonance), at least with the rotation and resonance curves. Immediately we notice that, according to our curves derived here. determination of the cut radius r =6.8±0.43 kpc–near c Our TWR solution also seems to endorse the sce- the location where the warp has been identified to begin nario speculated upon by Verdes-Montenegro et al. (r∼7 kpc; Verdes-Montenegro et al. 2000)–this spiral (2000) for the origin of the warp in this galaxy, pattern ends at OLR, within the uncertainties. The namely through the non-linear coupling proposed by inner bound for the single, constant speed Ωp=44±4 −1 −1 Masset & Tagger (1997a). In this scheme, the warp km s kpc in Figure 12, on the other hand, is less (manifest in the lop-sided part of the HI distribution, clearly well established here. The paucity of 21 cm which extends 34% further toward the SE than NW; emission inside r∼1 kpc not only leads us to suspect Verdes-Montenegro et al. 2000) would originate through that the measurement may not be valid all the way to a coupling at OLR between a spiral density wave and the center, as suggested by our solution in Figure 12 two “warp waves”. Since we have established that the (calculated with bins covering all radii r≥0), but it also spiral structure ends at OLR, in the absence of tidal prevents us from constraining the angular rotation curve interaction with a companion galaxy (Thilker et al. there (and the position of the ILR) with confidence. 2007), the spiral pattern itself could be capable of Even so, according to our estimate rILR∼1.0 kpc, the generating the warp in the outer disk. spiral could reasonably begin at the inner ring, as iden- tified in the optical by Verdes-Montenegro et al. (2000). 4. SUMMARY AND CONCLUSIONS In this case, with the single pattern speed implied by Direct pattern speed measurement affords an observa- our TWR solution, both the inner and outer rings, at tional resolution for several fundamental issues in the na- r=1 kpc and r=7 kpc (Verdes-Montenegro et al. 2000), ture and origin of spirals. The relation between bar and are arguably associated with spiral’s inner and outer spiral pattern speeds, the number and radial domains of Lindblad resonances. We note that this depends in no patterns speeds that can be sustained in the disk, and small part on the TWR method; the separation achieved whether spiral structure is steady or winding, for ex- by the cut radius between the strong spiral structure ample, can all be established with knowledge of spiral and the m=1 distortion grants a measurement for the speeds and their radial variation. The TWR method, a former constant speed which is significantly different technique for measuring radially varying pattern speeds, from that implied by the traditional TW method, supplies us with the first such measurements to address −1 −1 Ωp=26±6km s kpc (calculated with the same slices, these issues. kinematic parameters, and limits of integration as in the In this paper we have applied the TWR method to ob- 14 servations of CO and HI in four spiral galaxies. For this models and calculated without regularization. work, we have expanded the number of spiral galaxies to Even though in this case we cannot characterize the which TW-type calculations can be applied by consider- patterns that may distinguish the warped regions of the ing, for the first time, the combination of both molecular disks in our sample, accurate measurements for the pat- and atomic gas. Together, the two ISM phases better tern speeds of the structure interior can, themselves, rec- meet the continuity requirement of the method and also ommend different mechanisms by which warps are ex- afford greater insight into whether (and how) multiple pected to originate and evolve. The galaxies in this spiral pattern speeds extend over a large range of radii. sample are consistent with various existing explanations. This has notably increased the sample size of galaxies Where the spiral structure ends at OLR in a warped, analyzed with the TWR method so far. isolated galaxy (e.g. NGC 3344), for example, spirals Our TWR solutions for M101 (§3.2), which, of all the themselves may be capable of exciting the warp through solutions presented here, are equipped with the small- the non-linear coupling proposed by Masset & Tagger est radial bins, very clearly show radial variation in the (1997a). In the absence of both a companion and this pattern speed across the disk. Within the inner 3’, we resonance boundary, but where the disk is intrinsically el- find convincing evidence that the bar and spiral have the liptical, as in NGC 3938, the ellipticity, the warp and/or same pattern speed (distinct from an exterior speed), and the instabilities in the disk may arise with a DM halo or that both lie inside corotation. This represents a scenario the asymmetric accretion of matter (discussed in § 3.4). quite different from recent findings in favor almost ex- An obvious companion, on the other hand, is a clear, clusively of fast bars (with 1.rCR/ab.1.4; Corsini 2008) potential source of tidal perturbation to the disk (e.g. albeit in mostly early-type barred galaxies. M101 and IC 342). Although an accompanying warp or In M101–as well as in IC 342 and possibly NGC 3938– distortion may not necessarily relate to the spiral struc- we also find that the extensive spiral structure there is ture at resonance, where the speed of outer structure is best described with more than a single pattern speed. near the Ω-κ/2 curve, for example, this structure may Furthermore, in both M101 and IC 342 we find evidence be resonantly excited (at ILR) by the external perturber that the transition between two speeds coincides with (see e.g. Salo & Laurikainen 2000). resonance overlap. In the former, the transition between These issues, and the existence of extensive spiral the two- and four-armed patterns occurs near the overlap structure, are best developed with a larger sample of of the inner’s OLR and the outer’s inner 4:1 resonance. galaxies, where meaningful trends will be more clearly In the latter case, the inner pattern transitions to a four- established. For greatest effect, the TWR calculation armed spiral at an OLR-ILR overlap. should be applied to higher resolution HI observations Although the specific resonance overlaps identified than we analyze here. In all of the galaxies in our sam- in M101 and IC 342 have not been conclusively ple (excepting M101) the resolution of the HI data limits demonstrated as true instances of mode-coupling (e.g. our confidence in how well radial variation is constrained. Sygnet et al. 1988, Masset & Tagger 1997b), together It also inhibits our leverage on information extracted in with the CR-inner 4:1 resonance overlap in M51 (Pa- the zone traced by CO emission (which we smooth to the per II), this work suggests that there exist several pos- resolution of the HI in combining the two observations). sibilities dictated by resonance overlap by which exten- A large radial bin width likewise prevents us from sat- sive spiral structure can be sustained. Our findings are isfactorily testing the effect of variation in the CO-to- qualitatively similar to the barred spiral simulations of H2 conversion factor on TWR pattern speed estimates. Rautiainen & Salo (1999), but we note that the spatially Future studies with higher resolution observations (and coincident existence of multiple modes with different pat- therefore more radial bins across the CO-emitting disk) tern speeds, as often found there beyond the bar, is may be able to better test for the effects of a gradient untested with the TWR calculation here. Even so, the or arm-interarm variations in X. We note, though, that outer four-armed spirals in both M101 and IC 342 are these types of variations were found to have a negligi- arguably excited at resonance by the patterns interior. ble effect on TW estimates (Zimmer, Rand & McGraw On the other hand, in our sample we also find exten- 2004), and here we find that low-level variation in the sive spiral structure in the absence of clear resonance X-factor produces very little change in our TWR solu- overlap. The speed characteristic of the outer, flocculent tions for M101. structure in NGC 3938, for instance, does not seem to be In general, these types of TWR measurements in clearly related to that of the inner, two-armed spiral, if nearby galaxies are invaluable for interpreting observa- the two speeds are distinct. In addition, the tight spiral tional studies of the evolution of bar and disk parameters structure throughout the HI emitting disk in NGC 3344 now possible with HST. They also promise to be signifi- is best described with a single, constant pattern speed. cant for studies at intermediate redshift, and future stud- In no case do we find that pattern speed is a smoothly ies with JWST, which will extend structural parameter decreasing function of radius, as might be expected for a measurement to larger redshifts and allow smaller bars winding spiral. Our cut procedure, however, may intro- and the earlier evolution of disks to be studied. duce a bias against such spirals if they exist preferentially We would like to thank Pertti Rautiainen for his con- in the outer regions of the disks in our sample; the ra- structive comments on this manuscript. This material is dial bins covering the outer portion of three of the four based on work partially supported by the National Sci- galaxies where a warp is evident are excluded from our ence Foundation under grant AST 03-06958 to R. J. R. REFERENCES Andersen, D. R., Bershady, M. A., Sparke, L. S., Gallager, J. 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