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accepted, ApJPreprint typeset using LATEX style emulateapj v. 11/10/09

OBSERVATIONAL EVIDENCE AGAINST LONG-LIVED SPIRAL ARMS IN GALAXIES

K.FoyleMax-Planck-Institute fur Astronomie, Konigstuhl 17, D-69117, Heidelberg, Germany and

Dept. of Physics & Astronomy, McMaster University, Hamilton, Ontario, L8S 4M1, Canada

H.-W.RixMax-Planck-Institute fur Astronomie, Konigstuhl 17, D-69117, Heidelberg, Germany

C. L. DobbsMax-Planck-Institut fur extraterrestrische Physik, Giessenbachstraße, D-85748 Garching, Germany and

Universitats-Sternwarte Munchen, Scheinerstraße 1, D-81679 Munchen, Germany

A. K. LeroyNational Radio Astronomy Observatory, 520 Edgemont Rd., Charlottesville, VA 22903 USA; Hubble Fellow

F. WalterMax-Planck-Institute fur Astronomie, Konigstuhl 17, D-69117, Heidelberg, Germany

accepted, ApJ

ABSTRACT

We test whether the spiral patterns apparent in many large disk galaxies should be thought of asdynamical features that are stationary in a co-rotating frame for ! tdyn, as implied by the densitywave approach for explaining spiral arms. If such spiral arms have enhanced star formation (SF),observational tracers for di!erent stages of the SF sequence should show a spatial ordering, from up-stream to downstream in the corotating frame: dense H I, CO, tracing molecular hydrogen gas, 24 µmemission tracing enshrouded SF and UV emission tracing unobscured young stars. We argue that sucha spatial ordering should be reflected in the angular cross-correlation (CC, in polar coordinates) usingall azimuthal positions among pairs of these tracers; the peak of the CC should be o!set from zero, indi!erent directions inside and outside the corotation radius. Recent spiral SF simulations by Dobbs& Pringle, show explicitly that for the case of a stationary spiral arm potential such angular o!setsbetween gas and young stars of di!ering ages should be observable as cross-correlation o!sets. Wecalculate the angular cross-correlations for di!erent observational SF sequence tracers in 12 nearbyspiral galaxies, drawing on a data set with high quality maps of the neutral gas (H I, THINGS),molecular gas (CO, HERACLES) along with 24 µm emission (Spitzer, SINGS); we include FUV images(GALEX) and 3.6 µm emission (Spitzer, IRAC) for some galaxies, tracing aging stars and longertimescales. In none of the resulting tracer cross-correlations for this sample do we find systematicangular o!sets, which would be expected for a stationary dynamical spiral pattern of well-definedpattern speed. This result indicates that spiral density waves in their simplest form are not animportant aspect of explaining spirals in large disk galaxies.Subject headings: galaxies: general —

1. INTRODUCTION

The nature of a coherent spiral structure that extendsover a large portion of the galaxy is still not fully un-derstood. While there are many variations in theories toexplain such a structure, typically self-excited models forspiral structures can be divided into two groups based onthe longevity of the spiral pattern. A long-lived quasi-stationary spiral structure theory has been developedlargely using analytical studies. The spiral features areattributed to quasi-steady global modes of the disk (Lin& Shu 1964, 1966; Lynden-Bell & Kalnajs 1972; Bertinet al. 1989a, 1989b, etc.). The other approach, which haslargely been based on simulations, purports that spiralsare short-lived, recurrent, transient patterns that orig-

foylek@physics.mcmaster.ca

inate from recurrent gravitational instabilities (Goldre-ich & Lynden-Bell 1965; Julian & Toomre 1966; Toomre1981; Sellwood & Carlberg 1984; Sellwood 2010). It re-mains a challenge to decipher which of these descriptionsbest describes spiral structure. The interested reader canrefer to excellent reviews in Binney & Tremaine (2008)and Sellwood (2010) for more information. In these re-views it is explained that it has proven very di"cult todecipher which of these models best describes observedgalaxies and it may be that a combination of these mod-els is required. It is clear that careful comparisons be-tween observations and simulations are needed in orderto disentangle these theories.In this work, we examine simulations and observations

for evidence of a long-lived, quasi-stationary spiral struc-ture based on the predictions of such a model. As origi-

2 Foyle et al.

nally explained by Roberts (1969, hereafter R69), if thespiral pattern is a quasi-stationary, dynamical pattern ina corotating frame and SF is occurring at an enhancedrate in the arms, then there should be a temporal se-quence of events, as material streams in and out of thisspiral pattern.As gas passes through the minimum of the spiral poten-

tial, it gets compressed and molecular clouds form. Thisis followed by SF, leading initially to dust-enshroudedyoung stars and then to unobscured young stars andclusters. If this SF sequence of events were to occurin a steady state, then it would translate into a set ofspatial o!sets for tracers of di!erent stages of this SFsequence, with the size of the o!sets reflecting both thetime di!erence between these stages and the relative ve-locity between the material and the spiral pattern. Inthis way, observed spiral arms are produced by the con-tinual triggering of SF through compression in the gaspeak (Lubow et al. 1986). Even if the spiral structure isnot directly shock triggering SF (e.g., Foyle et al. 2010),provided there is an appreciable amount of SF in thearms, one still expects bright stars to be observed down-stream from the arms and this has been noted in severalcases (e.g., Tamburro et al. 2008; Egusa et al. 2009).Roberts et al. (1975) found observational evidence for

such o!sets from a sample of 24 galaxies and posited thatthe shock strength and spiral strength could account forthe ordering of spiral disks into Hubble type and lumi-nosity class. Figure 1 taken from R69 illustrates the po-sitions of the gas shock and young stars inside corotationof a trailing spiral pattern and illustrates the predictedo!sets. Using a rigidly rotating spiral potential in hydro-dynamical simulations, Gittins & Clarke (2004) showedthat one can find three spiral patterns each tracing adi!erent stage in this process. The first is traced bythe stellar mass density and the location of the poten-tial minima. The second spiral pattern is traced by thedust, which is known to be in locations of compressedgas and the third marks the location of the young stars.Due to the finite time for SF, the young stars should bedownstream of the dense gas in this picture. The loca-tion of the compressed gas relative to the minimum ismore complex; traditionally it is located upstream of theminimum, but the location of the shock depends on thesound speed of the gas and can be downstream of theminimum in a cold, or multiphase medium (Wada 2008;Dobbs 2007).Figure 2 shows what would be expected using a simple

toy model of a two-arm spiral pattern with a corotationradius of 2.7Rexp (i.e. Kranz et al. 2003). We consider agas and recent SF tracer with an onset time of 3 Myr be-tween the two. The upper right panel of Figure 2 showsthe radial profile of the angular o!sets between themcalculated from Eq. 6, discussed in the following section,using the rotation curve shown on the left. In the innerregions large positive o!sets are expected and beyondcorotation the o!sets become negative. The lower panelshows the position of the tracers in the spiral arms. Thevariation in the angular o!sets of the tracers with radius,produces a spiral pattern that is e!ectively more tightlywound. The o!sets are small enough that the arm pat-terns of the two tracers will likely overlap, depending onthe amount of dispersion and width of the tracers. Thiscould make by-eye determinations of o!sets challenging.

A number of observational studies have looked for qual-itative evidence for angular o!sets of SF tracers. Earlyobservations showed dust lanes on the inside part of thearms with respect to H II regions (e.g., Lynds 1970).Mathewson et al. (1972) and Rots (1975) found thatatomic hydrogen was o!set from optical arms and dustlanes in M51 and M81 respectively. A detailed studyof M33 by Humphreys & Sandage (1980), however, pre-sented a more complex picture. While two of the armsdid show dust lanes on the inner edge of the arm andbright young stars on the outer part of the arm, the thirdarm showed no such features or ordering.Ideally one would like to examine o!sets between the

gas and young stars. Due to the lack of atomic hydrogen(H I) in the inner part of the disk and the fact that themolecular gas forms giant molecular clouds out of whichstars form, it is natural to look for o!sets between COand H! or 24 µm emission (with both H! and 24 µmemission being tracers of recent SF, e.g., Calzetti et al.2007). A number of studies have looked for such o!setsand examined the streaming motions near the arms, es-pecially in M51 and M81 (e.g., Vogel et al. 1988; Rand& Kulkarni 1990; Lord & Young 1990; Garcia-Burillo etal. 1993; Rand 1995; Loinard et al. 1996; Shetty et al.2007; Egusa et al. 2009). However, all such studies usedby-eye determinations of these o!sets by selecting indi-vidual patches of a single arm. Such methods can easilyintroduce potential biases.Tamburro et al. (2008, hereafter T08) developed an al-

gorithmic technique to measure o!sets between SF trac-ers by locating the peak of the cross-correlation functionbetween H I and 24 µm emission. This also allowed fora measure of the timescale of going from H I atomic gasto dust-enshrouded massive stars (as traced by 24 µmemission). Their findings were in agreement with thesimple prescription of R69. They found corotation radiiat !2.7rexp, which is consistent with other works (e.g.,Kranz et al. 2003) and short timescales of 1-4 Myr forthe timescales of SF emerging from H I. Indeed the an-gular o!sets were found to be very small (5!) betweenthe tracers. T08’s work is the first to approach o!setmeasurements using an algorithmic method as opposedto by-eye determinations and they used the highest res-olution images to date. Our method is based on the onedeveloped by T08 and we describe it in greater detail in§2.Recently, Dobbs & Pringle (2010, hereafter DP10) have

looked in detail at the distribution of cluster ages acrossspiral arms in realistic simulations. In their sample, theyincluded a galaxy with a fixed spiral potential whichmimics a long-lived spiral with a constant pattern speed.The top left panel of Figure 3 shows the cluster age dis-tribution in this galaxy. If one compares this to the illus-tration of R69 (Figure 1), one sees that the distribution ismore complex in these simulations. However, the clustersare still ordered in the way expected. Previous studiesby Dobbs et al. (2006) have also shown that the distri-bution of the giant molecular clouds has a sharp edgeon the upstream side of the arms where fresh materialis flowing in and a dispersed, smooth distribution on thedownstream side.However, imposed stationary potentials are not realis-

tic. DP10 also included three spiral structures withoutexternal potentials, in which the spiral structure forms

Evidence Against Long-Lived Spiral Arms 3

Figure 1. Illustration of the relative location of gas shock, sharpH I peak and newly formed stars in H II regions in the gaseousspiral arms of a trailing spiral pattern (top; Figure 7 of R69).

naturally. They included a flocculent spiral as well as abarred and interacting galaxy. Unlike imposed stationarypotentials, the active potentials do not produce a spatialseparation between the gas and SF tracers. Dobbs &Bonnell (2008) have shown that the stars and gas arecoincident with the spiral potential minima, because gasaccumulates in the potential minima as the spiral struc-ture forms. The spiral arms in the flocculent model ofDP10 evolve over timescales of 100 Myr, which is typi-cal for spirals with recurrent spiral structure (Sellwood& Carlberg 1984; Fujii et al. 2011; Sellwood 2010). Thespiral patterns found with active potentials have no fixedpattern speed or corotation radius. Longer lived tran-sient spirals, where spiral arms evolve over a few hun-dreds of Myr are possible (e.g. Thomasson et al. 1990),and may also lead to o!sets similar to the static potentialcase. However such structures are not readily produced,at least in numerical simulations (Sellwood 2010), and wedid not include this scenario in our simulated galaxies.In this work, we look for evidence for a systematic

ordering of cold gas and young stars, which would bepredicted by a long-lived spiral pattern with a constantpattern speed. We first examine the four simulationsof DP10 for angular o!sets between the gas and stellarclusters and then examine a sample of observed galax-ies for o!sets between various gas and SF tracers. In §2we describe the algorithmic method developed by T08to measure angular o!sets between gas and SF tracers.In §3 we use this method to measure angular o!sets inthe simulations of DP10, which cover a range of spiralstructure formation mechanisms including a stationaryspiral potential, bar potential, interacting galaxy andtransient spiral structure. In §4 we turn to observationsand present our sample of 12 nearby spiral galaxies withhigh-quality maps of cold gas and SF tracers includingH I (THINGS), CO (HERACLES), 24 µm (SINGS), UV(GALEX) and 3.6 µm (IRAC). Using the same methodemployed for the simulations, we measure angular o!setsbetween these observed tracers. In §5 we compare theobserved cross-correlation functions and angular o!setsto those in the simulations and assess the predictions ofthe model of a long-lived spiral structure with a constant

Figure 2. Toy model illustration showing the relative positionbetween the gas and a recent SF tracer with an onset time of 3Myr between the two. The rotation curve of the galaxy is modeledwith an arctan function (upper left) and the corotation of the spiralpattern is set at 2.7Rexp (dashed line). With a tSF of 3 Myr, Eq. 6determines the radial profile of the angular o!sets between the twotracers (upper right). The relative positions of the spiral patternsof each tracer are shown below.

pattern speed. In §6 we present our conclusions.

2. MEASURING ANGULAR OFFSETS

We seek to algorithmically measure angular o!sets be-tween the gas and SF tracers in both simulations andobservations, which are predicted by a long-lived, quasi-stationary, spiral structure theory. We use the method ofTamburro et al. 2008, which was developed to measureo!sets between H I and 24 µm emission. Angular o!setsbetween H I and 24 µm emission allow one to measurethe timescale between gas compression and massive SFbecause peaks of H I column density are thought to bethe sites of molecular cloud formation and the 24 µmemission traces the young dust enshrouded stars. Weuse this method to measure angular o!sets between thegas and young stellar clusters in the simulations of DP10and also extend T08’s study to measure angular o!setsbetween CO, which traces the molecular gas with 24 µmemission. This provides a more direct measure than thatof H I and 24 µm emission. While T08 also used BIMASONG CO maps (Helfer et al. 2003) to measure angularo!sets between the molecular gas and 24 µm, the low sen-sitivity of these images provided too few points to makeaccurate estimates of the timescale. We also include thecross-correlation of the gas tracers with UV images and3.6 µm images which probe longer timescales. We firstdescribe the method of T08 in detail.

2.1. Method of Tamburro et al. 2008

T08 measured angular o!sets by locating the peak ofthe cross-correlation function between H I and 24 µm

4 Foyle et al.

emission in radial annuli for 14 disk galaxies.The time between these two events, tHI"24µm, the lo-

cal circular velocity, vc(r), and a pattern speed, #p, forthe spiral arms, lead to an angular o!set between thesetwo tracers of:

$"(r) = (#(r) " #p)tHI"24µm. (1)

Thus, this method is independent of the number ofarms and shape of the spiral pattern, but it does assumea fixed pattern speed. The angular o!sets should varywith radius. When the disk rotates faster than the pat-tern (inside corotation) we expect $" > 0 and when thepattern rotates faster than the disk (beyond corotation)we expect $" < 0. At corotation #(Rcor) = #p, the signchanges and $" = 0.In order to measure $", T08 divided the images into

radial annuli and cross-correlated them to find the rel-ative o!set with the best match. The best match be-tween the two images is found by minimizing the follow-ing quantity as a function of the shift, l (lag):

#2x,y(l) = %k[xk " yk#l]

2. (2)

The sum is carried out over all N elements of x, y, wherex and y refer to the two tracers at a given radius as afunction of azimuth, ", such that:

xk = fHI("k|r) and yk#l = f24µm("k#l|r). (3)

By minimizing #2 one maximizes the following quantity:

ccx,y(l) = %k[xk # yk#l], (4)

defined as the cross-correlation coe"cient. T08 used thenormalized version such that:

ccx,y(l) =%k[(xk " x)(yk#l " y)]

!

%k(xk " x)2%k(yk " y)2, (5)

where x and y are the means of x and y respectively.In this way, perfectly identical patterns have a cross-correlation coe"cient of unity and highly dissimilar pat-terns have a value much less than one. At each an-nulus, T08 searched for a local maximum of the cc(l)around l $ 0. The location of the maximum, lmax, de-fines the angular o!set between the two tracers such that$"(r) = "lmax(r). If the value of the cc(l) peak was lessthan 0.3, the point was rejected.With the angular o!set, $"(r), and the rotation curve,

vc(r), one can write Eq. 1 as:

$"(r) =

"

vc(r)

r" #p

#

tHI"24µm. (6)

T08 used #2 fitting of the above with the measured an-gular o!sets and derived best fits for tHI"24µm and #p.This method can be used for any two tracers. We firstemploy this method on the simulations of DP10 and ex-amine which models fit the predictions of R69. We thenuse this method to repeat the work of T08 as well as lookfor o!sets between 24 µm and the higher sensitivity COimaging from HERACLES, which traces the moleculargas. We also examine UV and 3.6 µm for o!sets withrespect to the molecular gas.

Figure 3. Distribution of stellar cluster ages (shown in di!erentcolors) in the spiral arms of a long-lived spiral potential with aconstant pattern speed (top left), a barred galaxy (top right), a self-excited flocculent galaxy (bottom left), and a galaxy interactingwith a companion (bottom right; Figure 2 of DP10). While theordering of the stellar clusters is systematic in the long-lived spiral,the other cases show a much more complex distribution.

3. ANGULAR OFFSETS IN SIMULATIONS

DP10 have recently simulated four di!erent models ofspiral structure and compared the distribution of stel-lar clusters of di!erent ages. Their simulations includeda stationary spiral potential which mimics a long-livedspiral structure with a constant pattern speed, a barredgalaxy, an interacting galaxy like NGC 5194 and a tran-sient spiral produced using the spiral potential from themodels of Sellwood & Carlberg (1984). As we saw in§1, the long-lived spiral structure was simulated usinga stationary potential and indeed the ordering of thestellar cluster ages fit the predictions of R69 and others(see Figure 1). Since the long-lived structure requiresa stationary potential in order to ensure its longevity,the other three cases may be more realistic (see Figure 3for all four cases). In these cases the ordering is muchmore complex. For the flocculent galaxy, each di!erentsegment of the spiral arms contains clusters of di!erentages. The interacting galaxy has an almost incoherentdistribution.We have used the same simulations presented in DP10

and analyzed them for angular o!sets between the gasand stellar clusters of di!erent ages in the way designedby T08 for observations. For di!erent snapshots we con-vert the gas and stellar cluster onto an R-" grid and crosscorrelate the annuli and locate the peak of the cross-correlation function as described in §2. We show herethe results of the cross-correlation between the gas and100 Myr clusters in all four simulations. The gas andthe 100 Myr clusters represent timescales close to thoseobserved between H2 and the UV, which we will examinein §5.

3.1. Stationary Spiral Potential

Evidence Against Long-Lived Spiral Arms 5

Figure 4. Cross correlation function between the gas and the100 Myr clusters for the simulated stationary spiral potential (toppanel), bar potential (second from top), interacting spiral (thirdfrom top) and flocculent spiral (bottom) at a fiducial radius of 7kpc in each case. The cross-correlation function of the stationarypotential shows four equally spaced peaks due to the self-similarityof the four-armed spiral structure. In the other cases, the centralpeaks are less well-defined and show very small, if any, o!sets.

The stationary spiral potential is an imposed potentialwith a constant pattern speed. Thus, we expect an or-dering of the clusters and gas to be in the way predicted.The pattern has a corotation radius at 11 kpc. The up-per panel Figure 4 shows the cross-correlation functionbetween the gas and the expected distribution of the 100Myr stellar clusters at a fiducial radius of 7 kpc. Thegalaxy has four arms, so there are four peaks in the cross-correlation function due to the self-similarity of the pat-tern. At each radius, we fit the central peak with a fourdegree polynomial, as was done by T08. The peak of thepolynomial locates the angular o!set. Figure 5 showsthe chosen angular o!set between the gas and 100 Myrclusters at a series of radial annuli. As one moves out inradius the o!sets decrease in value as we saw with the toymodel in §1. Unfortunately, the stellar clusters do not ex-tend beyond corotation so we are unable to confirm thatthe sign of the o!sets changes beyond corotation.Since the arms are well-defined in the stationary po-

tential we fit straight lines to the gas arms in logR-" anddefined a ridge line to the arm. At each radial positionthe arm is at a position "ridge. We then measure the dis-tance of the clusters with respect to the arm. Figure 6

Figure 5. Radial profile of the angular o!sets based on the peak ofthe cross-correlation function between the gas and 100 Myr clusterparticles for the stationary spiral potential. The o!sets show asystematic trend going from high to low values. The gas and starparticles in this simulation do not extend beyond the corotation ofthe spiral potential. Thus, we are unable to currently test whetherthe o!sets cross zero at this point.

shows stellar clusters at 2, 50, and 100 Myr and their an-gular distance to the ridge line. We have only fit one armso the self-repeating pattern is due to the other arms.One notes that the older clusters have drifted furtherdownstream from the arms (i.e. ("ridge " ") increasesfor clusters of older ages). Both these findings and thesystematic ordering of the angular o!sets fit the predic-tions of a long-lived, stable, spiral structure. This is notsurprising because in these simulations such a structureis imposed. This exercise shows that if such a structureexists in nature, we should be able to detect o!sets be-tween gas and SF tracers using our analysis.We now turn to the other simulations, where the spiral

structure has formed naturally in the simulations and isnot fixed with a stationary potential. In these cases,however, we do not expect a constant pattern speed forthe spiral pattern, which is one of the assumptions in theR69 model.

3.2. Barred Galaxy

The second panel from the top of Figure 4 shows thecross-correlation functions of the gas and the expecteddistribution of the 100 Myr stellar clusters for the barredpotential at a fiducial radius of 7 kpc. Here we findtwo cross-correlation peaks due to the two arms in thisgalaxy, but the cross-correlation peaks are small and, inmany cases, are below the 0.3 cuto! value used by T08and which we also adopt for the observed galaxies. De-spite these low cc(l) values, the peak is well-defined sowe are able to determine the o!set angles. We find thatthe o!sets are very small and, at all radii, are less than adegree. One concern is that by 100 Myr the ordering ofthe particles has been disrupted. In comparison to theother simulated galaxies, the bar primarily extends oversmaller radii, and the orbital times are shorter. By 100Myr, the particles have made at least one and possiblytwo passages around the galaxy, disrupting the order-

6 Foyle et al.

Figure 6. Angular distance of clusters of varying ages to thegaseous spiral arms in the stationary spiral potential simulation.By fitting a ridge line to a spiral arm in the gas (!ridge), we mea-sure the angular distance to each age of stellar cluster (2 Myr:black; 50 Myr: blue; 100 Myr: red). As the clusters age they driftfurther downstream with respect to the arm, as expected in thestationary, spiral potential model.

ing of the gas and star particles. Due to this, we alsoexamined star particles at early times, before completepassages. For the 50 Myr clusters the o!sets were easierto detect and larger, but still quite small with the largestbeing ! 5!. A further challenge in the barred galaxy isthe highly elliptical orbits of the particles, which makethe transitions more di"cult to detect using the cross-correlation method, which relies on the azimuthal angleof the particles. In general, we find that o!sets can bedetected for the barred galaxy, but that they are quitesmall and the ordering of the particles can be easily dis-rupted.

3.3. Interacting Galaxy

The second panel from the bottom of Figure 4 showsthe cross-correlation functions for the gas and the ex-pected distribution of the 100 Myr stellar clusters of theinteracting galaxy, which is very similar to NGC 5194,at a fiducial radius of 7 kpc. In this case, we find only asingle, broad, uneven cross-correlation peak. The mea-sured angular o!sets show no trend and in most casesthe angular o!sets are close to zero and thus we do notshow them graphically. Since this simulated galaxy isvery similar to NGC 5194 (see DP10 for further details),it is interesting to compare the cross-correlation functionwith the observations, which we will do in the followingsection.

3.4. Transient, Flocculent Spiral

Finally we also examine the cross-correlation functionof the transient, more flocculent spiral structure that

used the spiral potential from the models of Sellwood& Carlberg (1984). The bottom panel of Figure 4 showsthe cross-correlation function between the gas and theexpected distribution of the 100 Myr stellar clusters at afiducial radius of 7 kpc. At this radius and all other onesno obvious cross-correlation peak is found, suggesting nopreferred position between the gas and clusters. We wereunable to properly fit a peak and measure angular o!setsand thus, we do not show the radial variation here.

3.5. Summary of Numerical Models

If a long-lived spiral structure with a constant pat-tern speed accurately describes spiral structure, than weshould be able to detect angular o!sets between gas andSF tracers as we have seen with the imposed stationarypotential of DP10. However, if a stationary potential isnot imposed, simulations of spiral structure do not showa systematic ordering of angular o!sets between the gasand stellar clusters in fitting with the model predictionsof a long-lived, stable, spiral structure. In the case ofthe barred galaxy, the cross-correlation peak could befit, but the values were very low and the angular o!setsmeasured, were small. In the other two cases, the o!setsshowed no radial variation and were close to zero. Wenow turn to our sample of observed galaxies and compareour findings.

4. ANGULAR OFFSETS IN OBSERVED GALAXIES

We chose 12 galaxies in common with the sample ofT08 that have coverage in THINGS (Walter et al. 2008)and SINGS (Kennicutt et al. 2003). Eight of these galax-ies also have coverage in HERACLES (Leroy et al. 2009).For three galaxies we also included FUV images fromGALEX (Gil de Paz et al. 2007) and 3.6 µm images fromIRAC (Kennicutt et al. 2003). The sample presents amixture of both grand design and more flocculent spi-rals, which are similar to the four simulated galaxies ofDP10.The galaxies were deprojected according to the values

of T08, with the exception of NGC 2841, NGC 3521 andNGC 5194 (see Table 1). For NGC 5194, we used the val-ues adopted by Leroy et al. (2008). NGC 2841 and NGC3521 were deprojected using GALFIT (Peng et al. 2002)and are roughly equivalent to those found in HyperLedaExtragalactic Database. Figure 7 shows deprojected 3.6µm images for our sample as well as the direction ofrotation, which is defined by assuming a trailing spiralpattern.The images were aligned to the THINGS astrometric

grid and were degraded to a common resolution of either6$$ for our analysis of H I and 24 µm images or 13$$ for ouranalysis of CO, 24 µm, UV and 3.6 µm images. Foyle etal. (2010) describes in detail the processing of the images.After initial processing, the images are translated from

rectangular to polar coordinates, (r,"), and divided intoannuli of 5$$ width as was done by T08. At our convolvedresolution, this allows us to Nyquist sample the data. Asan example, Figure 8 shows the H I emission (bottompanel) and H2 map (upper panel) in polar coordinates forNGC 5194 with the contours of 24 µm emission overlaid.One notes that in both cases the arm patterns in thesetracers are nearly coincident. Thus, any o!sets betweenthese tracers must be small.

Evidence Against Long-Lived Spiral Arms 7

Table 1Properties of 12 Sample Galaxies

Name Inclination P.A. D rexp Vmax

[!] [!] [Mpc] ["] [km s#1]NGC 628 7 20 7.3 1.1 220NGC 2403 63 124 3.22 1.3 128NGC 2841 63 148 14.1 0.92 331NGC 3031 59 330 3.63 3.62 256NGC 3351 41 192 9.33 0.86 210NGC 3521 65 162 10.05 0.74 242NGC 3621 65 345 6.64 0.8 144NGC 3627 62 173 9.25 0.95 204NGC 5055 59 102 7.82 1.16 209NGC 5194 20 172 7.77 1.39 242NGC 6946 33 242 5.5 1.73 201NGC 7793 50 290 3.82 1.16 109

Note. — Sample properties: the inclination and position an-gles used to deproject the galaxies, their adopted distance, scalelength and maximum amplitude of their rotation velocity. Withthe exception of NGC 2841, NGC 3521 and NGC 5194, all valuesare those from T08.

Figure 7. Sample of 12 galaxies (numbers correspond to NGCnumbers). We show the deprojected 3.6 µm images with arrowsdenoting the direction of rotation based on the assumption of atrailing spiral pattern.

The cross-correlation coe"cients were calculated forangular o!sets from -180! to +180! in increments of !0.1!. We restrict the search for the peak between ±30!,because the o!sets must be at least this small based onvisual inspection (see Figure 8). We fit the region ofthe maxima with a four degree polynomial and calculateits peak value. The angular position of the peak is theangular o!set, $"(r), between the tracers at each radius.Any local maxima with a cross-correlation coe"cient lessthan 0.3 is rejected, in following T08. The sign of theangular o!sets is corrected for the direction of rotationof the galaxy.

4.1. Angular O!sets Between H I and 24 µm Images

Figure 9 shows a sample cross-correlation function be-tween H I and 24 µm at a radial annulus of 80$$ for NGC5194. One notes two broad peaks due to the two spiralarms and the self-similarity of the structure. The zoomedbox on the right shows the peak and the polynomial fit(gray curve), the maximum of which locates the angularo!set at that radius (vertical black line). The o!sets aremeasured in a similar fashion at all radii.Figure 10 shows the angular o!sets (black points) ver-

sus radius for all 12 galaxies. The colors reflect the valueof the cross-correlation with low values (% 0.3) beingblue and high values (% 0.8) being red in the neighbor-hood of the chosen o!set corresponding to the peak value(±20!). Around the peak value, there is a broad regionwhere cc(l) is high. One expects to find positive angularo!sets smoothly decreasing with radius and crossing zeroat corotation. The radial profiles of the angular o!sets donot show any smooth trend of going from positive to neg-ative values and show considerable scatter. The profilesdo not correspond in any way to the model predictions.Our profiles also do not agree with the profiles found byT08, who found o!sets that agreed with the picture ofR69. In §4.3 we examine this in detail. However, if thereader is not interested in this discussion, the §4.3 canbe skipped.

4.2. Angular O!sets Between H2 and 24 µm, UV and3.6 µm Images

The H I distribution is not as concentrated to the spiralarms as H2 (e.g., Foyle et al. 2010). Since the H I emis-sion is more evenly distributed than the 24 µm emission,this may make the measurements of angular o!sets moredi"cult. Thus, we cross-correlate H2 maps with otherSF tracers to measure angular o!sets.Eight of the galaxies had CO coverage from HERA-

CLES at 13$$ resolution. As described in Foyle et al.(2010), we process and convert those maps to producedeprojected maps of H2 using an XCO conversion factor.We then transform the images to polar coordinates. Aswe did with H I and 24 µm emission, we measure angu-lar o!sets between the H2 and 24 µm emission by locat-ing the peak of the cross-correlation function. Figure 11shows the radial profile of the angular o!sets betweenthese two tracers. In a similar fashion to Figure 10, wefind little evidence for systematic o!sets, in contrast toR69 and T08. There is some evidence for a sequence ofo!sets in NGC 6946 and NGC 5194. Thus, we examinethese galaxies as well as NGC 628 in greater detail andwith other tracers for possible angular o!sets.

8 Foyle et al.

Figure 8. H I (bottom) and CO (top) emission of NGC 5194 in polar coordinates with contours of the 24 µm emission overlaid. It isclear from these overlays that any angular o!sets between both sets of tracers must be small.

Figure 9. Example of a cross-correlation function, cc(l), for H Iand 24 µm as a function of azimuthal o!set for NGC 5194 at a ra-dius of 80”. Two broad peaks are present due to the self-similarityof the two spiral arms. The box on the right shows the regionzoomed in (±30!). The fourth degree polynomial fit overlaid ingray. The location of the maximum of the polynomial (black ver-tical line) gives the angular o!set between the two tracers. Thereis negligible o!set between these tracers at this radius (2.7!). Thisexample is not atypical for di!erent galaxies, and radii.

For three galaxies with prominent spiral structure weadded FUV images from GALEX (Gil de Paz et al. 2007)which trace a longer timescale and should show the great-est o!sets. We also included 3.6 µm emission maps whichtrace the underlying old stellar population. We mea-sured angular o!sets between the H2 and the UV emis-sion as well as H2 and the 3.6 µm emission using thecross-correlation method described above. The timescalebetween the H2 and the UV is similar to the timescalebetween the gas and the 100 Myr clusters in the simula-tions of DP10 (see §3).

Figure 12 shows the angular o!set profiles for cross-correlations of H2 with the UV emission on the left andthe H2 with the 3.6 µm emission on the right. Of thesegalaxies, only NGC 628 presents any trend suggestiveof the picture of R69. However, it is important to notethat many of these o!sets are well below the resolutionof the images, especially for the case of H2, which onlyhas 13$$ resolution. We also tried other combinations ofthese images and divided the images into quadrants toisolate individual arms. Our findings were similar in allcases.

4.3. Comparison with Tamburro et al. (2008)

We have found no evidence for a systematic orderingof angular o!sets in the way found by T08. We examinethe cause of this discrepancy in greater detail here. Fig-ure 13 shows the cross-correlation functions of H I with24 µm emission at several radial annuli for NGC 5194.The left panels show the full range from -180! to +180!.We also zoom in on the peak of the function in the boxedregions and display these on the right. The zoomed re-gions reflect the range of -30! to +30!. The grey curveshows the fourth degree polynomial fit.The top panel of Figure 13 exhibits what one would

hope to find when seeking the maximum of cc(l). At thisradius a single, narrow peak is found. The other panels,show, however, that at most radii, the cc(l) is more com-plex, with a very broad peak and uneven features. Thepolynomial fit is sensitive to the range over which one fitsand selecting this range can be subjective. Due to this,we also tried selecting the maximum cc(l) value, ratherthan fitting the peak. Even in this case, no systematicvariation of o!sets were found. We tried a number ofvariations of T08’s method, including smoothing and fil-tering of the images, variations of the cross-correlation

Evidence Against Long-Lived Spiral Arms 9

Figure 10. Radial profiles of the angular o!sets based on thelocation of the peak of the cross-correlation function between H Iand 24 µm vs. radius (black dots). The colors reflect the valueof the cross-correlation coe"cients from high (red) to low (blue)in the range of ±20!. Values below 0.3, our chosen cuto!, arenot shown. The angular o!sets are corrected for the direction ofrotation. None of the galaxies show the trend predicted by the R69model.

calculation and restrictions of narrow regions around thearms in the case of non-axisymmetric distributions ofstars and gas and were unable to reproduce the findingsof T08.We carried out a careful comparison of NGC 628 also

studied by T08. While we were able to reproduce someof their o!sets, the o!sets measured were sensitive tosmall changes in the polynomial fit and the way in whichthe cross-correlation function is calculated (i.e. use ofmean or median for x and y in Eq. 5). Figure 14 showsthe o!sets found by T08 (black) and two of our attemptsusing di!erent fitting ranges for the polynomial (blue andred). While T08’s o!sets show a clear progression fromhigh values to low values, which cross zero, our pointsare much more scattered. In part, we can attribute thisto the fact that T08 selected the peak to fit and chosethe range over which they fit by-eye. We also tried tochoose specific fitting ranges at each radius, but foundthe choice to be subjective.Beyond the sensitivity of the peak position to the range

of the polynomial fit, resolution may be a cause for thelack of agreement with T08. The angular resolution

Figure 11. Same as Figure 10, but showing the location of thepeak of the cross-correlation function between H2 and 24 µm emis-sion.

varies with radius and the thin black line in Figure 14shows that all of the o!sets found by T08 are below theresolution of the image. The common image resolutionis 6$$ and at a radius of 30$$ this corresponds to an an-gle of 11!. Most of the o!sets found by T08 were lessthan 5! except in the very inner regions, where resolu-tion concerns are even greater. However, given that thecross-correlation function is calculated considering all az-imuthal positions, it is possible that one can probe belowthe resolution limit to measure the o!sets. The fact thatT08’s o!sets are not randomly scattered points to thispossibility.It is not clear how resolution might a!ect the cross-

correlation function and the determination of the peak.We first examine how resolution a!ects the autocorrela-tion function of two H I maps at a resolution of 6$$ and13$$ using NGC 5194. The autocorrelation function iscalculated like the cc(l) (Eq. 5), but in this case both xand y represent the same H I map. For both resolutions,we expect the peak of the autocorrelation function to becentered at zero with a maximum of unity. Figure 15shows the autocorrelation function for a series of annuli.Since the azimuthal resolution varies with radius, we listthe azimuthal resolution for each pair of images in theupper left of the panels. We see that, as expected, theautocorrelation functions have a peak at zero and max-ima of unity. However, one notes that as the resolutionsize increases, the autocorrelation function broadens andhas a higher coe"cient over a greater range of angles. Wedivided each annulus into units smaller than the resolu-

10 Foyle et al.

Figure 12. Same as Figure 10, but showing the location of thepeak of the cross-correlation function between H2 and UV emission(left) and H2 and 3.6 µm emission (right).

tion. Thus, as the resolution size increases, the fractionof pixels with identical values increases. This provides abetter agreement in the cross-correlation, which broad-ens and increases the peak.In order to examine how the peak position might vary

with resolution, we consider the cross-correlation of H I

and 24 µm at two di!erent resolutions in Figure 16.We use pairs of images at our common resolution of 6$$

(black), and 13$$ (gray). Like the autocorrelation func-tion, as the resolution size increases, the cc(l) broadensand has a higher value. The functions are very similarin shape, but as one can see in the zoomed in regions,when the resolution size is large (small radii), the shapeof the peak can change substantially. This a!ects thepolynomial fit and ultimately the position of the peakcenter. In the inner regions, where the resolution e!ectsare greatest, we found that the peak positions varied byas much as 8!. However, at larger radii the di!erence wasless than a degree and the median deviation was 0.5!.Thus, while the peak position remains roughly the same,it is important to consider the uncertainties introducedby resolution in determining its precise position.In summary, we can attribute our lack of agreement

with T08 to a combination of three causes: 1) the o!setvalue is very sensitive to the polynomial fitting of thepeak; 2) the o!set value is very sensitive to small changesin the way the cc(l) function is calculated; and 3) almostall of the o!sets are below the resolution limit of theimages and we have found that changes in the resolutionof the image do a!ect the shape of the cc(l) functionparticularly at small radii.

Figure 13. Cross-correlation functions cc(l) between H I and 24µm as a function of azimuth o!set for NGC 5194. Each panelrepresents neighboring annuli from 60"" (bottom) to 80"" (top). Theright column shows the boxed region zoomed in (±30!). The fourthdegree polynomial fits are overlaid in gray. The peak is often quitebroad and bumpy and the polynomial peak depends heavily on therange of points selected.

Figure 14. Comparison of the o!sets found by T08 (black) andthe o!sets measured in this study using two slightly di!erent fittingranges of the cc(l) (blue and red). The o!sets measured by T08are all below the angular resolution limit, which varies with radius(thin black line).

5. COMPARISON OF SIMULATIONS AND OBSERVATIONS

For all galaxies in the observed sample and betweenall tracers we found a clear peak in the cross-correlationfunction (see Fig 9). The location of the peak marksthe angular o!set between the two tracers considered.However, the measured angular o!sets showed no sys-tematic variation as would be expected from a long-lived,quasi-stationary spiral structure. Among the simulatedgalaxies only the one with an imposed stationary po-

Evidence Against Long-Lived Spiral Arms 11

Figure 15. E!ect of spatial resolution on the autocorrelationfunction of H I maps with 6"" (black) and 13"" (gray) resolutionfor NGC 5194 at a series of fiducial radii. Since the azimuthal res-olution varies with radius, it is listed in the upper left of each panel.As one moves out in radius (from bottom to top) the azimuthalresolution size increases. As in Figure 13 we zoom in on the peakbetween ± 30!. We see that as the size of azimuthal resolutionincreases, the peak of cc(l) broadens and increases in value.

Figure 16. E!ect of spatial resolution on the cc(l) function cal-culated using H I and 24 µm maps with 6"" (black) and 13"" (gray)resolution for NGC 5194 at a series of fiducial radii. Since the az-imuthal resolution varies with radius, it is listed in the upper leftof each panel. As one moves out in radius (from bottom to top)the azimuthal resolution size increases. As in Figure 13 we zoomin on the peak between ± 30!. While the cc(l) is roughly similar,features on the peak change substantially, which may introducedi!erences in the polynomial fit and the position of the maxima.

tential showed clear o!sets with a systematic variation.The barred, interacting and flocculent transient spiralsall showed very small o!sets (less than a few degrees) andthe interacting and flocculent spirals showed no clear sys-tematic variation. This agrees well with what we found inthe observations for the various tracers of gas and recentSF. The absence of observed o!sets not only suggeststhat these galaxies do not have a static spiral pattern,but also that their structure is unlikely to be the resultof a single mode transient spiral that persists for longertimes.However, upon comparing the shape of the cross-

correlation functions of the simulated galaxies, withthose observed, we do not find a simulated model thatagrees in detail. The observations tend to show at leasttwo broad peaks, depending on the nature and numberof arms in the spiral pattern (see Fig 9) . The simulatedbarred galaxy, did show two clear peaks, but the cross-correlation coe"cients were much smaller than those seenin observations. In the case of the flocculent and interact-ing spiral simulations, we did not find such clear features.It is interesting to compare the cross-correlation functionshown in Fig 9 with that of the interacting, simulated spi-ral galaxy in Fig 4. The interacting spiral is structurallysimilar to NGC 5194. While the cross-correlation func-tion of NGC 5194 showed two broad peaks, the simulatedgalaxy showed only one. One possible cause of the dis-crepancy in the shape cross-correlation function in thesimulations and observations is because we only have arelatively small number of discrete clusters in the simula-tions, rather than the more continuous map of emissionin the observation.

6. CONCLUSIONS

We have used simulations from DP10 and high res-olution observations for a sample of 12 spiral galaxiesto look for evidence for a model of spiral structure thatis long-lived, quasi-stationary with a constant patternspeed. This model predicts that SF tracers should beo!set from one another reflecting the relative velocitybetween the disk and pattern and the onset time for SF.We have used an algorithmic technique developed by

T08 to measure angular o!sets between the gas and stel-lar clusters in four simulated galaxies with di!erent spi-ral structures including a galaxy with a stationary spiralpotential, a barred galaxy, an interacting galaxy and amore flocculent, transient spiral structure. Only when astationary spiral potential was imposed did we find clearangular o!sets ordered in the way predicted by a long-lived, stable structure. For the barred galaxy, though wefound angular o!sets for younger clusters, and some in-dication of an age transition, the o!sets were very small(<5!), and thus inconclusive. For the interacting andtransient spirals, there were no clear cross correlationpeaks and no measurable o!sets.We then used the same method to measure angular o!-

sets between the gas and SF tracers in a sample of 12 ob-served galaxies. In all cases we did not have a systematicordering in the way predicted by the simple prescriptionof R69. We specifically measured o!sets between H I and24 µm emission in order to directly compare with thework of T08 and are unable to reproduce their results(see §4.3 for a discussion). This study has also measuredo!sets between CO, which traces the molecular gas with

12 Foyle et al.

the 24 µm emission. Even in this case, which directlyprobes the time between molecular cloud formation andSF, we find no systematic ordering of angular o!sets. Forthree galaxies we also examined the cross-correlation ofother tracer combinations including UV and 3.6 µm andfound similar results.Our results contradict previous by-eye estimates of

such o!sets. While there may be some patches of SFtracers that show by-eye o!sets, an algorithmic analy-sis shows that such patches are isolated and there is nooverall trend which supports the prediction of a modelwhere the spiral structure is long-lived with a constantpattern speed. Our technique made use of the highestquality images to date and, provided the timescales ofSF are at least a few Myr, angular o!sets should havebeen detectable. There are number of possible reasonswhy such a systematic ordering was not detected. Itmay be the case that there are no angular o!sets be-tween the tracers. This would be the case if the spiralstructure were simply a result of sheared patches of starforming regions or if the structure was rapidly dissolvingand reforming. However, it is also possible that o!setsare present, but that structure is more complex than asingle pattern with a constant pattern speed. Indeed, re-cent studies have measured multiple pattern speeds forseveral spiral galaxies(Meidt et al. 2009). Furthermore,there is substantial emission from SF tracers in the in-terarm regions (at least 30%) (Foyle et al. 2010). Thismay be obfuscating the o!sets near the spiral arms dueto our cross-correlation method which uses all azimuthalpositions. The arms may also have slightly di!erent o!-sets and the stars may have highly elliptical orbits, bothof which would introduce uncertainties when fitting overthe whole galaxy as our method relies on a polar coordi-nate system. Thus, there is still room for the possibilitythat systematic o!sets between SF tracers exists in thearms, but we may not be able to quantitatively measurethem with this technique. Furthermore, the fact a cross-correlation peak could be found at most radii, suggeststhat angular o!set measurements could still be used tounderstand star-forming sequences if the local dynamicsare known.The fact that there is little evidence for angular o!sets

between SF tracers as predicted by the model of a long-lived spiral structure with a constant pattern speed lendssupport to a model with a transient spiral structure thatreorganizes the interstellar medium (e.g., Elmegreen &Elmegreen 1986; Dobbs & Bonnell 2008; Sellwood 2010)or models that have spiral arms forming due to the shear-ing of gas and stars by di!erential rotation (e.g., Sei-den & Gerola 1979 and Elmegreen et al. 2003). Sim-ulations largely support such pictures and other obser-vational studies have shown that most spiral structuresare quite complex with multiple pattern speeds (e.g.,Meidt et al. 2009). We caution, however, that the cross-correlation functions of the other simulated spiral struc-tures in this study, did not agree in detail with the obser-vations. Thus, continued detailed comparisons betweenobservations and simulations will be required in order touncover the nature and persistence of spiral structure.

K.Foyle acknowledges generous support from the MaxPlanck Society, International Max Planck Research

School for Astronomy and Cosmic Physics at the Uni-versity of Heidelberg. We thank both Karl Schuster andthe referee, for their comments, which greatly improvedthis work.

Evidence Against Long-Lived Spiral Arms 13

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