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THE ASTROPHYSICAL JOURNAL, 562 : 164 õ 178, 2001 November 20
( 2001. The American Astronomical Society. All rights reserved. Printed in U.S.A.

PROBING FOR DARK MATTER WITHIN SPIRAL GALAXY DISKS THILO KRANZ, ADRIANNE SLYZ,1 AND HANS-WALTER RIX
Max-Planck-Institut fur Astronomie, Konigstuhl 17, 69117 Heidelberg, Germany ; kranz=mpia.de, slyz=mpia.de, rix=mpia.de Received 2001 May 23 ; accepted 2001 July 24

ABSTRACT We explore the relative importance of the stellar mass density as compared to the inner dark halo using the observed gas kinematics throughout the disk of the spiral galaxy NGC 4254 (M99). We perform hydrodynamic simulations of the gas ÿow for a sequence of gravitational potentials in which we vary the stellar disk contribution to the total potential. This stellar portion of the potential was derived empirically from color-corrected K-band photometry reÿecting the spiral arms in the stellar mass, while the halo was modeled as an isothermal sphere. The simulated gas density and the gas velocity ïeld are then compared to the observed stellar spiral arm morphology and to the Ha gas kinematics. We ïnd that this method is a powerful tool to determine the corotation radius of the spiral pattern and that it can be used to place an upper limit on the mass of the stellar disk. For the case of the galaxy NGC 4254 we ïnd R \ 7.5 ^ 1.1 kpc, or R \ 2.1 R (K@). We also demonstrate that for a maximal disk the CR exp prominentCR spiral arms of the stellar component overpredict the noncircular gas motions unless an axisymmetric dark halo component contributes signiïcantly (Z 1 ) to the total potential inside 2.2 3 K-band exponential disk scale lengths. Subject headings : galaxies : halos õ galaxies : individual (NGC 4254) õ galaxies : kinematics and dynamics õ galaxies : spiral õ galaxies : structure
1.

INTRODUCTION

In almost all galaxy formation scenarios nonbaryonic dark matter plays an important role. Todayîs numerical simulations of cosmological structure evolution reproduce fairly well the observed distribution of galaxy properties in the universe (e.g., Kaumann et al. 1999), and attempts to model the formation of single galaxies have been made as well (Steinmetz & Muller 1995). In these simulations the baryonic matter cools and settles in the center of dark halos where it forms stars. The distribution of stars and gas in a galaxy depends strongly on the local star formation and merging history. At the same time that the stars are forming, the halos evolve and merge as well. The ïnal relative distribution of luminous and dark matter in the centers of the resulting galaxies is under debate because the mass distribution of the dark matter component is difficult to assess directly. Measuring luminous and dark matter mass proïles separately requires innovative strategies because the halo is poorly constrained, and equally good ïts to measured rotation curves can be achieved for a wide range of visible mass components (Broeils & Courteau 1997). In order to deïne a unique solution to this so-called "" disk-halo degeneracy,îî the "" maximal disk îî solution was introduced. It assumes the highest possible mass-to-light ratio (M/L) for the stellar disk (van Albada et al. 1985 ; van Albada & Sancisi 1986). A practical deïnition is given by Sackett (1997) who attributes the term "" maximal îî to a stellar disk if it accounts for 85% ^ 10% of the total rotational support of the galaxy at R \ 2.2 R . exp This approach has proven to be very successful in matching observed H I and Ha rotation curves (van Albada et al. 1985 ; Kent 1986 ; Broeils & Courteau 1997 ; Salucci & Persic 1999) and also satisïes some dynamical constraints, such as the criteria of forming m \ 2 spirals (Athanassoula,
1 Now at University of Oxford, NAPL, Keble Road, Oxford OX2 6UD, UK ; slyz=astro.ox.ac.uk.

Bosma, & Papaioannou 1987), as well as observational constraints on the structure of the Milky Way (Sackett 1997). However, modern numerical N-body simulations ïnd signiïcant central dark matter density cusps (Fukushige & Makino 1997 ; Moore et al. 1999). Even if the prediction of these strong density cusps may not be entirely correct, the simulations ïnd that the dark matter is of comparable importance in the inner parts of galaxies (Blumenthal et al. 1986 ; Moore 1994 ; Navarro, Frenk, & White 1996, 1997), and it thus has a considerable inÿuence on the kinematics. In this case a stellar disk of a galaxy would turn out to be "" submaximal.îî It is important to determine the relative proportion of dark and luminous matter in galaxies for a better understanding of the importance of the baryonic mass in the universe. This proportion also bears information on the dynamics and structure of the dark matter itself. Spiral galaxies are well suited to study dark matter distributions because their distinctly ordered kinematics provide an excellent tracer of the gravitational potential in the disk plane. Since bars in galaxies are very prominent features with distinct dynamic characteristics, they are especially well suited to evaluate the amount of luminous matter. Sophisticated studies of barred galaxies indicate that their stellar disks alone dominate the kinematics of the inner regionsõthe stellar contribution is "" maximal îî (Debattista & Sellwood 1998, 2000 ; Weiner, Sellwood, & Williams 2001a). However, studies of our own Milky Way, maybe also a barred spiral, still do not give a clear answer as to whether the disk is maximal (Sackett 1997 ; Englmaier & Gerhard 1999) or not (Kuijken 1995 ; Dehnen & Binney 1998). Bottemaîs analysis of the stellar velocity dispersion in various galactic disks led to the conclusion that disks cannot comprise most of the mass inside the radial range of a few exponential scale lengths (Bottema 1997). Aside from the dynamical analysis of single systems, other attempts to tackle this problem have been undertaken. Maller et al. (2000) used the geometry of gravitational lens systems to 164


DARK MATTER WITHIN SPIRAL GALAXY DISKS probe the potential of a lensing galaxy. They concluded that a maximum disk solution is highly unlikely. Courteau & Rix (1999) applied statistical methods to learn about the mass distribution in galaxies. In their analysis they found no dependence of the maximum rotation velocity on a galaxyîs disk size. This is considered to be a strong argument to rule out a maximum disk solution. The conÿicting ïndings of dierent studies leave the question of the relative proportion of dark and luminous matter in galaxies still open. In this paper we want to exploit the fact that the stellar mass in disk galaxies is often organized in spiral arms, thus in kinematically cold nonaxisymmetric structures. In the canonical cold dark matter (CDM) cosmology, the dark matter is collisionless and dominated by random motions. Although the introduction of weakly self-interacting dark matter was proposed to avoid current shortcomings of the CDM model (Spergel & Steinhardt 2000) it seems to raise other, comparably severe problems (Miralda-Escude 2002 ; Ostriker 2000). Hence, it seems reasonable to assume that CDM is not substantially self-interacting but dynamically hot and thus not susceptible to spiral and nonaxisymmetric structure. In light of this, the key to measuring the baryonic to dark matter fraction is to make use of the nonaxisymmetric structure that we can observe in the stellar light distribution. Using deviations from axisymmetry of stellar disks, several eorts have already been made to constrain the dark matter content of galaxies (e.g., Visser 1980 ; Quillen 1999 and references therein). Some of the most signiïcant conclusions came from studies of massive bars, which are the strongest nonaxisymmetric structures in disk galaxies. Spiral arms comprise a less prominent but still signiïcant mass concentration. Already very early theoretical calculations of gas shocking in the gravitational potential of a spiral galaxy (e.g., Roberts 1969) told us that we could expect "" velocity wiggles îî with an amplitude of 10õ30 km s ~1 while crossing massive spiral arms. For ionized gas, measurements of the velocity to this precision can be achieved with common long-slit spectrographs. The imprint of the spiral pattern in the velocity ïeld of real galaxies is indeed not very strong, as apparent in the two-dimensional velocity ïelds of M100 (Canzian & Allen 1997), of low surface brightness galaxies (Quillen & Pickering 1997), or of a sample of spiral galaxies (Sakamoto et al. 1999). There are only a few spiral galaxies without bars that show stronger wiggles in the velocity ïeld that are associated with the arms, e.g., M81 (Visser 1980 ; Adler & Westpfahl 1996) and M51 (Aalto et al. 1999). In order to still achieve the goal of measuring M/L, we need to compare the weak features that we expect in the measured velocity ïeld to detailed kinematic models. Using new high-resolution K-band photometry to map the stellar component and employing a modern hydrocode to simulate galactic gas ÿows, we are conïdent that our models show enough details and that we can eventually measure M/L. If the arms are a negligible mass concentration relative to the dark matter distribution in the galaxy, these wiggles should appear only very barely in the velocity ïeld. The main aim of this project is to ïnd out what fraction of the rotation speed comes from a mass component with spiral arms. In order to do this, we compare the strength of the wiggles in a galaxyîs observed velocity ïeld to a model of the gas velocity ïeld arising in a potential whose disk-halo fraction is known. As input for our gas dynamical simulations, we derive the stellar poten-

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tial of the galaxy from color-corrected K-band photometry, while the dark matter component is modeled as an isothermal sphere with a core. Simulations are performed for a variety of potential combinations and values for the pattern speed of the spiral structure. The results from these simulations are then compared to the observed kinematics. The paper is organized as follows. In the next sections we present the galaxy sample and the observations (° 2) and the general modeling techniques (° 3). In ° 4 we present results for the galaxy NGC 4254 (M99) and discuss them in ° 5. We summarize in ° 6.
2

. THE SAMPLE DATA AND NGC 4254

We aim to perform this study on a sample of about half a dozen galaxies to account for systematic errors in the analysis of single systems. We could not identify a data set in the literature that fulïlled satisfactorily the requirements for this project. Hence, we decided to collect data on a sample of galaxies ourselves. The data acquisition procedure was carried out in the following way. First we took near-infrared (NIR) photometry of possible sample candidates to check whether their spiral structure shows up reasonably well in the K band. From that sample we selected the most promising candidates and obtained kinematic data for them on a second observing run. The requirement for the kinematic measurements was to trace the gas velocity perturbations of the spiral arms, ideally across the whole disk. The two classical methods for obtaining two-dimensional gas velocity ïelds are H I or CO radio observations and Fabry-Perot interferometry. Alternatively, long-slit spectra taken at dierent position angles can be used to map the disk. Single dish H I or CO observations are not suited for our project because they suer from relatively bad angular resolution and poor sensitivity to faint emission between spiral arms and the outer part of the disk. Sakamoto et al. (1999) recently published CO observations of a sample of spiral galaxies. In the velocity maps the signature of the spiral arms is in the majority of cases not or only barely visible. This most likely results from beam-smearing of the already weak velocity perturbations. Two-dimensional Fabry-Perot velocity ïelds provide the required angular resolution but usually give only a very patchy representation of the disk : mainly the H II regions show up in the map (e.g., Canzian, Allen, & Tilanus 1993 ; Regan et al. 1996 ; Weiner et al. 2001b). Because the coverage we can achieve by taking eight long-slit spectra across a galaxyîs disk is reasonably high, in combination with its good angular resolution and sensitivity to faint Ha emission we chose to collect our kinematic information by taking long-slit spectra. This paper deals only with the results of the ïrst galaxy from the sample : NGC 4254. 2.1. Observations and Data Reduction Since we require photometric as well as kinematic data for this study, there are some constraints that apply for the sample selection. To reasonably resolve the structures of the stellar disk we prefer galaxies with low inclinations with respect to the line of sight (LOS). However, the LOS component of the circular motion increases with inclination i. Owing to the fact that the projection of the galaxy scales with cos i and the LOS fraction of the velocity scales with sin i, galaxies in the inclination range between 30¡ and 60¡ are best suited for yielding all the photometric and spectro-


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FIG. 1.õK@-band image of NGC 4254 with a total exposure time of 20 minutes at the Calar Alto 3.5 m telescope. Bright foreground stars are masked out. The overlay shows the slit orientations of the spectrograph. We took eight long-slit spectra (angles labeled in bold font) crossing the galaxyîs center to measure the two-dimensional velocity ïeld.

the sky. The spectra span a region of 6130õ7170 A centered on Ha. The spectra were obtained during three observing runs in 1999 June and 2000 May and December. The integration time for each single spectrum varied between 600 and 1800 s owing to weather and scheduling constraints. For the data reduction we used the IRAF package and applied standard long-slit reduction procedures. We determined the LOS velocity component of the ionized gas as a function of radius from the galactic center from Doppler shifts of the Ha line. The center of the emission line was determined by ïtting separately a single Gaussian proïle to it and the brighter N II line at 6584 A. The weighted com parison of the two ïts provided us with the uncertainty of the line center position. Finally, each of the eight spectra were folded at the center of the galaxy to get the two rotation curves. To do this coherently, we determined an average wavelength distance D between a prominent sky line (at j ) and Ha at our best j guess for the galactic center sky from all eight spectra. Then, separately for each slit, the Ha rest wavelength in the center of the galaxyõwhere v \ c 0õwas assigned to be at j ] D . There the two rotation sky j curves should be separated. This provided us ïnally with 16 rotation curves at dierent position angles per galaxy, each reaching out to about 2@. All 16 rotation curves are displayed later in ° 4, Figure 6, together with the results from one simulation. 2.2. Structural Properties of NGC 4254 We selected NGC 4254 (M99) as the ïrst galaxy from our sample to be analyzed, because it shows a clear spiral structure with high arm-interarm contrast. NGC 4254 is a bright Sc I galaxy located in the Virgo Cluster with a recession velocity of 2407 km s~1 adopted from NED.3 We assume a distance of 20 Mpc toward NGC 4254, taken from the literature (Sandage & Tammann 1976 ; Pierce & Tully 1988 ; Federspiel, Tammann, & Sandage 1998). It has a total blue magnitude of B \ 10.44 and a diameter of 5@ ] 4.7 on the .4 @ T sky. At 20 Mpc, 1A is 97 parsecs in the galaxy, which translates to 38.4 pc per detector pixel. Our Ha rotation curve for NGC 4254 (the Ha rotation curve is plotted later in Fig. 4, together with modeled rotation curves) rises steeply out to D35A (3.4 kpc) and then ÿattens at a rotation velocity of D155 km s~1. This agrees well with earlier estimates (Phookun, Vogel, & Mundy 1993). From the kinematics we know that the southwestern part of the galaxy is approaching. If we assume trailing spiral arms, then the galaxy rotates clockwise when viewed from our perspective. In the K band NGC 4254 shows very prominent two-arm spiral features at most radii. The northern arm bifurcates at R B 4.5 kpc, causing a three-arm pattern in the outer disk. Furthermore, the galaxy exhibits considerable lopsidedness. NGC 4254 shows a strong arm-interarm brightness contrast, noted by Schweizer (1976) to be even stronger than the one for NGC 5194 (M51). Even in the K band the brightness contrast is rather high, approximately a factor of 2 over a wide radial range. Gonzalez & Graham (1996) argued that, combined with a usual density wave, some external mechanism is needed to invoke such high contrasts. However, NGC 4254 is well separated from any other galaxy in the cluster, and there are no obvious signs
3 The NASA/IPAC Extragalactic Database (NED) is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.

scopic information. We also chose the sample to consist mostly of nonbarred high-luminosity galaxies with strong spiral arms. All data were acquired at the Calar Alto observatoryîs 3.5 m telescope. For all sample galaxies we decided to take NIR photometry to study the luminous mass distribution and gas emission line (Ha) spectroscopy to acquire kinematic information for the systems. The NIR images were taken during two observing runs in 1999 May and 2000 March with the Omega Prime camera at the 3.5 m telescopeîs prime focus (Bizenberger et al. 1998). It provides a ïeld of view of 6@ 6 .7 ] 6.76 with a resolution of 0A3961 per pixel. We used the @ . K@ ïlter, which has a central wavelength of 2.12 km. The integration time was 20 minutes on target, which allows us to trace disks out to about 4 scale lengths. Our observation sequence included equal amounts of time on the targets and sky frames used for background subtraction. The data reduction was performed using standard procedures of the ESO-MIDAS data reduction package. In total, we collected NIR photometry data for 20 nearby spiral NGC galaxies. From their NIR appearance we selected half of the galaxies as a subsample for which kinematic data should be taken. We obtained the gas kinematics from long-slit spectroscopy measurements of the Ha emission. With the setup we used, the TWIN spectrograph achieved a spectral resolution of 0.54 A per detector pixel, which translates to 24.8 km s~1 LOS velocity resolution per pixel, allowing us to determine LOS velocities with D7 km s~1 precision. We sampled the velocity ïeld of the galaxy along eight slit position angles,2 all crossing the center of the galaxy (see Fig. 1). The slit of the TWIN spectrograph measures 4@ ] 1A on .5
2 All position angles quoted in the text are in degrees eastward from north.


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FIG. 2.õDierent results from the point-by-point correction of the M/L in the K@ image. (a) The original azimuthally averaged brightness proïle together with the corrected proïle. The outward blueing of the galaxy leads to a stellar disk scale length that is smaller than in the uncorrected K@-band light. (b) The azimuthal brightness proïle at a radius of 39A ; azimuth 0 is to the north. The proïle shows a clear two-arm structure that is preserved after the correction. .2 Note, however, that the shape of the proïle changes.

for recent interaction. Phookun et al. (1993) reported in a recent paper the detection of high-velocity H I clouds outside the diskîs H I emission. The authors argue that infalling H I gas may be responsible for the unusual "" onearmed îî outer structure of the spiral. In that case the spiral structure of NGC 4254 may have been recently reorganized, enhancing the southern arm or the arm-interarm mass density distribution. Recent interactions could in principle corrupt the projectîs assumption of a steady state spiral pattern. But as we will ïnd from our full analysis, the steady state assumption is seemingly not far o on timescales of a few dynamical periods. NGC 4254 harbors a small barlike structure at its center with a major axis position angle of B40¡. From both ends of the bar two major arms emerge with a third arm splitting o the northern arm. By analyzing a g[K color map of NGC 4254 one learns that this third arm is signiïcantly bluer than the other regions of the galaxy and thus consists of a younger stellar population. In the K band the disk of NGC 4254 is well approximated by an exponential with a scale length of R B 36@@, exp corresponding to B3.5 kpc if a brightness average of the arm and interarm regions is considered and bright H II regions are removed from the image. Except for the very center, the whole surface brightness proïle is well ïtted by a double exponential model with an inner "" bulge îî scale length of B0.6 kpc (see Fig. 2a)
3.

3.1. T he Stellar Potential
3.1.1. K-Band L ight as T racer of the Stellar Mass

MODELING

In the forthcoming sections we describe the modeling needed to connect our photometric and kinematic observations. In this pilot study, we restrict our analysis and description to the spiral galaxy NGC 4254 (M99). The data modeling involves four discrete steps. First we derive a stellar gravitational potential from the K-band images. Second, for each stellar M/L, we ïnd a dark matter halo proïle to match the total rotation curve. Third, we perform hydrodynamic simulations of the gas within the combined stellar and dark matter potential. Finally, we compare the predicted gas velocity ïeld to our Ha observations.

We would like to derive the stellar potential directly from the deprojected K-band stellar luminosity density of the galaxy. To this end, it is important to understand how well the NIR surface brightness traces the stellar mass density. There are two major concerns which complicate the direct translation : both dust extinction and spatial population differences could introduce K-band M/L variations. Observing at D 2 km reduces the eect of the dust extinction signiïcantly. Since we look at most galaxies from a nearly face-on perspective, we expect the optical depths to be D0.5 in the K band, leading to local ÿux attenuations of less than 10% (Rix & Rieke 1993 ; Rix & Zaritsky 1995). Signiïcant global M/L variations may arise from red supergiants ; they emit most of their light in the NIR and are fairly numerous and thus could considerably aect the total light distribution in K-band images. Supergiants evolve very rapidly and therefore are mainly found within the spiral arms where they were born. This would tend to increase the arminterarm contrast, leading to stronger inferred spiral perturbations in the stellar potential. We used the approach of Bell & de Jong (2001) to correct for local M/L dierences, which may still be present in the K-band images. From spiral galaxy evolution models, assuming a universal initial mass function, Bell & de Jong (2001) ïnd that stellar M/L correlate tightly with galaxy colors. They provide color-dependent correction factors, which we can use to adjust our K-band image to constant M/L. For NGC 4254 we used a g-ïlter image taken from Z. Freiîs galaxy catalog (Frei et al. 1996). The g ïlter (Thuan & Gunn 1976) has a passband that is very similar to the Johnson V ïlter, so we can use the color correction given for V [K while using g[K colors. The M/L is given by log (M/L ) \[1.087 ] 0.314(V [ K) . (1) This relative correction was performed on a pixel-by-pixel basis by scaling each ÿux by the corresponding M/L (see Fig. 3b). NGC 4254 has g[K of 2.8, which results, according to (1), in a K-band maximal-disk stellar M/L, ! , of *


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FIG. 3.õDierent stages of processing of the K-band image prior to the calculation of the potential. (a) The original frame ; (b) image of the galaxy after the correction for constant M/L was applied ; (c) ïnal image after removal of compact star-forming regions by means of azimuthal Fourier decomposition. The images are not yet deprojected to face-on.

0.62. If we estimate a maximal-disk stellar M/L from the disk potential we use for the simulations, we ïnd ! B 0.64. * These values are in good agreement, considering the uncertainties that contribute to the estimation of ! . In our * approach even small changes in NGC 4254îs distance or inclination result in errors of B20%. The galaxy becomes signiïcantly bluer from the center to the outer disk regions, with the bluest regions being the spiral arms themselves. As a result the mass density proïle of the corrected disk gets steeper : the disk scale length (after deprojection) changes from 3.5 to 2.7 kpc (see Fig. 2a). In general the correction for constant M/L does not particularly change the arminterarm contrast of NGC 4254 and thus does not reduce the nonaxisymmetric component of the stellar potential. In the force ïeld of the galaxy, the relative strength of the tangential component makes up D6% of the radial force component before and after the correction. Nevertheless, the correction modiïes the nonaxisymmetric force component on average by D30% (see Fig. 2b). This has an evident eect on the data and the models, which we think is favorable. The correction seems to work ïne for most of the disk. However, there are some thin dust lanes at the inner edges of the spiral arms in the green image of NGC 4254 that are optically thick. Because of the quite poor resolution of the green image, the dust lanes are not distinctly visible in the image. For these the correction leads to some overcorrection because virtually all the green light is absorbed by the dust lane. But since these optically thick dust lanes are narrow and still at the location of the NIR spiral arms, we assume that their inÿuence on the total potential is negligible. To reduce the inÿuence of star-forming H II regions and OB associations, which are distinctly visible in the K-band data, we ïltered the image by means of a Fourier decomposition. From the azimuthal decomposition we discarded the Fourier terms higher than N \ 8 and subtracted the residual from the K-band image of NGC 4254. This correction does not depend signiïcantly on the number of Fourier terms included in the ït. This procedure cleaned the image of the patchy small scale structure and left us with the smooth global spiral pattern (see Fig. 3c).
3.1.2. Deprojection

a global s2 ït of an axisymmetric rotation curve model to the 16 observed rotation curves. The model is based on a combination of stellar and dark halo rotation curves that were scaled to an average rotation curve from the six slit positions closest to the major axis (see Fig. 4). Deriving i and P.A. is an iterative process. The ïnal values get determined only after the ïrst full hydrodynamic simulations, when actual nonaxisymmetric model rotation curves can be ïtted to the observed kinematics. As it turns out, in the case of NGC 4254 the dierence in the resulting inclination can be considerable. An axisymmetric model yields a disk inclination of i \ 30¡ , while the simulations suggest an inclina.8 tion of i \ 41¡ . We use the latter as the inclination of the .2 galaxy. However, spatial deprojection eects are not particularly large in the inclination range of 30¡õ 40¡. The major axis position angle is not sensitive to the details of the rotation curve ; we adopt P.A. \ 67¡ . .5
3.1.3. Calculation of the Gravitational Potential

The smoothed image is then used to calculate the maximal disk stellar potential by numerical summation over the whole mass distribution of the galaxy, which is deïned by the surface mass density &(r) we get from the K-band images. &(r ) j ' (r ) \[G! ; *i * or [r o jEi i j with o r [ r o \ J*x2] *y2] v2 . (3) i j The indices i and j denote dierent radius vectors, specifying here the pixels of the image array. Beyond the size of the detector array, the surface mass density is assumed to be zero. The stellar M/L ! is assumed to be constant, after the corrections described*in ° 3.1.1. The factor v in equation (3) accounts for the ïnite thickness of galactic disks following (Gnedin, Goodman, & Frei 1995). This softening factor was chosen to be compatible with a vertical exponential density scale height of h \ 400 pc, independent of radius. z This assumption most probably holds for NGC 4254, which is a late-type spiral and its central spheroid, which appears barred and extends only to about a disk scale length. In addition, small bulges may not be spherical and thus might be well described by the constant scale height of 400 pc assumed for most of the simulations. Moreover, our derived potential does not depend signiïcantly on the choice of h . z (2)

To deproject the K-band images to face-on, we need to derive the disk inclination, i, and the major axis position angle (P.A.) from the observed gas kinematics. We perform


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FIG. 4.õDierent decompositions of the total gravitational potential into stellar and dark halo components, illustrated for the example of NGC 4254. The measured rotation curve (data points with error bars)õan average of six single rotation curves closest to the major axisõis well ïtted by all decompositions (solid line). Displayed are three cases for the contribution of the stellar disk f \ 20%, 60%, and 100% (dash-dotted line). The halo d contribution (dashed line) was adjusted to yield a good ït to the measurements in combination with the preselected stellar contribution. The ït parameters for the three cases are listed in Table 1.

Varying h by 10% leads to relative changes of the resulting forces by z D 5.1 ] 10~3 on average. Only for the most massive stellar disk models we performed an explicit diskbulge decomposition and substantially increased the softening factor for the bulge to 760 pc. This helps to achieve a better ït at the very inner part of the observed rotation curve. We also calculate a potential ' from an equivalent axisymmetric density distribution *,ax to the measured ïtted K-band luminosity proïle. The rotation curve is evaluated from the stellar potential by v2 (R) \ R *,ax d' *,ax dr (r)

Thus, v and R uniquely specify all properties of the halo. = c For any fraction f of the maximal stellar mass we can d obtain the halo parameters from the best ït of the combined stellar and halo rotation curve to the observed kinematics. In the ïnal potential the two components are combined in the following way : ' (R, f ) \ f ' (R) ] ' (R, f ) , (9) tot d d* halo d with f ranging from 0 to 1, and ' denoting the stellar d * potential with maximal ! . The contributions to the circu*: lar velocity add quadratically v2 (R, f ) \ f v2 (R) ] v2 (R, f ) . (10) tot d d* halo d For every f we are able to ïnd a dark matter proïle, which d complements the luminous matter rotation curve. s2 ïts to the observed data are similar (see Table 1), necessitating this study. The spiral structure in galaxies is tightly related to the density wave theory that is used to describe the formation of these structures in spiral disks (for a review see Athanassoula 1984). If the spiral pattern evolves more slowly than the orbital timescale, resonances occur at certain radii in the galactic disk. We assume that the spiral pattern remains constant in a particular corotating inertial frame, speciïed by the pattern speed ) . At the corotation radius R CR () \ V /R , where V is p circular rotation speed) everythe p CR
TABLE 1 DARK HALO PARAMETERS f d (M /M ) D D max 0.2 ............ 0.4444 ........ 0.6 ............ 0.85 .......... 1.0 ............ R c (kpc) 1.08 2.00 3.03 5.68 7.30 v = (km s~1 ) 155 150 150 155 155 s2/N 2.45 3.42 3.30 2.00 1.84

K

.

(4)

R The maximal stellar M/L in equation (2) is determined by ïtting the rotation curve emerging from the axisymmetric potential to the inner part of the observed rotation curve with the highest possible ! . Later, when combining stellar * and dark halo potentials, we take fractions f of this stellar d "" maximum disk îî ït, to explore various luminousõtoõdark matter ratios.
3.1.4. T he Dark Halo Potential

3.1.5. T he Pattern Speed ) and Corotation p

For the present analysis we use the radial density proïle of a pseudoisothermal sphere as the dark matter component in our model. Its radial density proïle, characterized by a core of radius R and a central density o , is given by c c R 2 ~1 . (5) o(R) \ o 1 ] c R c The corresponding rotation curve is

C A BD

R R v2 (R) \ v2 1 [ c arctan halo = R R c (Kent 1986), and the potential is ' (R) \ halo

C

A BD

(6)

R v2 (r) halo dr . (7) r 0 The asymptotic circular velocity in this (inïnite mass) halo v is related to the two other free parameters R and o by = c c v \ J4nGo R2 . (8) = cc

P

NOTE.õDark halo parameters used to generate the potentials used for simulations. The s2/N values refer to an axisymmetric model.


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thing corotates with the spiral pattern. In addition to this corotation resonance there are more radii, at which the radial oscillation frequency i and the circular frequency ) are commensurate. In linear density wave and modal theories, these so-called inner and outer Lindblad resonances should enclose the radial range for which a m \ 2 spiral pattern may develop and persist over a longer period of time (Lin & Shu 1964). On the other hand, nonlinear orbital models for open galaxies indicate that the symmetric, strong part of the stellar spiral might end at the inner 4 : 1 resonance (Patsis, Contopoulos, & Grosbòl 1991). Finding the radial locations of these resonances in real galaxies is a delicate endeavor, and there are no widely agreed upon ways to determine them. For our analysis the pattern speed ) is crucial since it p modiïes the forces acting in a nonrotating inertial frame by adding in coriolis and centrifugal force terms. Therefore, we need to perform a set of hydrodynamic simulations at dierent ) , to ïnd the best-matching value and thereby to also p determine the corotation radius. As it turns out, the simulations are well suited to ïnd these values reliably. However, to reproduce all the gravitationally induced features found in the observed kinematics, we suspect that the assumption of a constant pattern speed for the whole disk is only a crude approximation of the dynamical processes occurring in the disk of NGC 4254. The pattern rotation speed most probably depends slightly on the radius. A radiusdependent pattern speed would eventually drive evolution of the spiral pattern. Although spiral structure is most likely a transient feature in the galactic disk, it persists for several dynamical timescales (Donner & Thomasson 1994 ; Patsis & Kaufmann 1999).4 But, since for now we have no means to reliably determine the radial dependence of the pattern rotation speed, we choose to ignore the time evolution of the spiral pattern and assume a constant pattern speed for the simulations. 3.2. Hydrodynamic Simulations The next step is to calculate the expected kinematics for a cold gas orbiting in ' with a certain ) . We model the gas p ÿow with the BGK tot hydrocode (Prendergast & Xu 1993 ; Slyz & Prendergast 1999). This is an Eulerian, grid-based hydrodynamics code that is derived from gas kinetic theory. The simulations are challenging because cold gas in galactic disks rotates with such high Mach numbers that if the ÿow is diverted from a circular orbit by nonaxisymmetric forces, the consequence is etched in the gas in patterns of shocks and rareïed regions. BGK is well suited for these simulations because, as veriïed by extensive tests on standard one-dimensional and two-dimensional test cases of discontinuous nonequilibrium ÿow (for a review, see Xu 1998), at shocks and contact discontinuities BGK behaves as well as the best high-resolution codes, and it gives better results at rarefaction waves because it naturally satisïes the entropy condition. To model the two-dimensional gas surface densities and velocity ïelds for NGC 4254, we carried out a set of simulations on a 201 ] 201 evenly spaced Cartesian grid. Our data for NGC 4254 extend out to a radius of D11.6 kpc ;
4 In light of our project it is interesting to note that in both N-body simulations referenced here, submaximal disks were adopted in the initial conditions.

hence, on a 201 ] 201 grid this gives a resolution of about 116 pc per side of a grid cell, which is considerably higher than the eective force smoothing of 400 pc. Given that the gas surface mass density of the modeled galaxies is much lower than the density of the stellar disk and halo, we neglect the self-gravity of the gas, and compute its response to a ïxed nonaxisymmetric gravitational potential derived from the corrected NIR image of the galaxy with a threefold reduced resolution (see ° 3.1). We approximate the interstellar medium as a monatomic gas. The gas temperature proïle is taken to be uniform, and by imposing an isothermal equation of state throughout the simulation we assume that the gas instantaneously cools to its initial temperature during each updating time step. The initial density proïle of the gas is exponential with a scale length equal to a third of the disk radius, namely, 3.86 kpc. We begin each simulation with the gas initially in inviscid centrifugal equilibrium in the axisymmetric potential given by ' and ' (see ° 3.1). Following the initialization of *,a halo the gas xin centrifugal balance in an axisymmetric potential, we slowly turn on the nonaxisymmetric potential at a linear rate computed by interpolating between the ïnal nonaxisymmetric potential and the initial axisymmetric potential so that the potential is fully turned on by the time 40 sound crossing times of the code have passed. Here the sound crossing time is deïned as the time it takes to traverse the length of the diagonal of a grid cell at sound speed. For an isothermal simulation with a sound speed of 10 km s ~1, the sound crossing time of a cell in our simulation is about 16 Myr, so that by 40 sound crossing times, the gas has evolved for 640 Myr, which is about 1.4 times the dynamical time of the galaxy measured at a radius of 11.6 kpc. After the nonaxisymmetric part of the gravitational potential has been fully turned on, we continue to run the simulation for about another dynamical time. A thorough description of the technical details of the simulations is given in a companion paper (Slyz et al. 2001, in preparation). We ran a large set of simulations both to understand the power and limitation of our modeling in general and to match the observations. Simulations were performed for a total of ïve dierent fractions f of the stellar disk : disk only, i.e., f \ 1, and f \ 0.85, d 0.6, 0.4444, and 0.2, or d accordingly, f given in d percent from 20% to 100%. In all d the cases the core radius and the asymptotic velocity of the pseudoisothermal halo were adjusted to best match the averaged rotation curve, as summarized in Table 1. The variations in s2/N between the low-mass disks and the massive ones are mainly caused by the attempt to keep R c and v at physically reasonable values. The bump at 20A to = 4) is better ïtted for the high-mass disks, which 40A (Fig. reduces the overall s2 compared to the low-mass disks. We have no secure prior knowledge of the spiral pattern speed ) . We determine it by assuming dierent values for p ) and then comparing a simulation to the data. For every p simulation with a dierent stellar/dark halo combination, we get slightly dierent values for the best-matching ) or equivalently for the corotation radius R . We coveredpthe complete range of reasonable R , i.e., CR from about a disk CR scale length to well outside the disk. We even made simulations for the case of no spiral pattern rotation, R ] O. CR To test how the amplitude of the velocity perturbations depends on the responsiveness of the gas, we ran simulations at a variety of temperatures corresponding to sound


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speeds c , of 10, 20, 30, and 40 km s~1. In the following s sections we discuss some of the results from the simulations.
4.

RESULTS FOR NGC 4254

4.1. Simulated Gas Density Figure 5 shows eight views of the simulated gas density for dierent pattern speeds overplotted as contours over the unsmoothed, deprojected, color-corrected K-band image of NGC 4254. The simulated gas density follows an overall exponential proïle with a scale length of B4.2 kpc, comparable to the one of the disk itself. The contours in the ïgure are chosen to highlight the density enhancements and locations of the gas shocks caused by the spiral arms. For almost all simulated cases the strong part of the galaxyîs spiral structure lies well inside corotation, where the circular velocity is larger than the spiral pattern speed. The gas will thus enter the spiral arms from their inward facing side, producing the strongest shocks there. For a well-matched simulation, we expect the shocks to be near the OB associations that trace the spiral arms.

It is remarkable how well the overall morphology of NGC 4254 can be matched by the gas density simulations. Not only are the two major spiral arms clearly identiïable in most simulations, but the less prominent northern arm and the locations where the arms bifurcate are reproduced in some cases well. For fast pattern speeds (R \ 5.4 õ7.58 CR kpc in Fig. 5) we ïnd a strong shock in the northern part of the galaxy that cannot be correlated with any mass feature. We believe it develops because the potential close to the upper boundaries of the computational grid is quite nonaxisymmetric, and this leads to a spurious enhancement of a shock. The shock does not propagate into regions inside the corotation radius, and therefore we refrain from smoothing the potential. It is important to note that the simulations lead to a very stable gas density distribution that does not change much after the nonaxisymmetric potential is fully turned on. When the contribution of the disk is increased in the combined potential, all spiral features get enhanced in the gas density, but the galaxy morphology is essentially unchanged. For a more detailed discussion of these issues,

FIG. 5.õSimulation results of the gas density distribution overplotted in red contours on the deprojected K@-band image of NGC 4254 (center). From the image an axisymmetric radial brightness proïle has been subtracted to enhance the contrast of the spiral arms. The results are displayed for eight dierent assumptions for the pattern speed ) or, respectively, the corotation radius R . The green circle marks corotation. The range goes from fast rotation P CR R \ 5.4 kpc to slow pattern rotation R \ 15 kpc, lying mostly outside the image frame and ïnally for no pattern rotation. CR CR


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please refer to the accompanying paper by Slyz et al. (2001, in preparation). With increasing pattern rotation (smaller corotation radii in Fig. 5), we ïnd that the predicted spiral arms become more and more tightly wound. For a comparison to the stellar spiral morphology we need to deïne some criteria to pick the right model. If the situation in NGC 4254 is similar to NGC 4321, whose gas and dust distributions and their connection to star-forming regions have been discussed in detail by Knapen & Beckman (1996), then a good matching gas morphology is one where for radii smaller than the corotation radius, the shocks in the gas density lie on the inside of the stellar spirals. Shortly downstream from there, many star-forming H II regions, triggered by gas compressions, should show up as patches in the arms. According to these criteria, the best-matching morphology can be unambiguously identiïed to be produced by a simulation with R B 7.6 kpc (Fig. 5, upper right panel), CR corresponding to a pattern speed of ) B 20 km s~1kpc ~1. p R \ 6.4 and 8.3 kpc enclose the range of possible values. CR corresponds to an uncertainty of D15% in the value of This R. CR Our results were compared to values of R for NGC CR 4254 from the literature, which were determined by dierent means. The results R D 8.45 kpc (Elmegreen, Elmegreen, & Montenegro 1992)CR and R \ 10.2 ^ 0.8 kpc (Gonzalez CR & Graham 1996), scaled to our distance assumptions, provide larger estimates than our ïndings. However, given the picture of nonlinear orbital models, where one expects the strong part of the stellar spiral to end inside the inner 4 : 1 resonance (Patsis et al. 1991) we ïnd that our result of R B 7.6 kpc is consistent with the galaxyîs morphology. CR In short, these simulated gas densities provide us with an excellent tool to determine the pattern rotation speed of the galaxy. Apart from requiring a constant global pattern rotation, our approach is independent of an underlying spiral density wave model. The overall very good representation of the whole spiral structure by the simulated gas density makes us rather conïdent that the simulations render realistic processes aecting the gas. 4.2. Simulated Gas V elocity Fields As another output of our simulations we get the twodimensional velocity ïeld of the gas. As is evident from a comparison to the gas density distribution, the velocity jumps areõas expectedõat the locations where the density map shows the shocks. They show up as areas of lower local circular velocity compared to the otherwise rather smoothly varying gas velocity ïeld. The velocity wiggles, as well as the density shocks themselves, have very tight proïles and thus are even more narrow than the physical extent of the stellar arms. They have to be compared to the observed kinematics.
4.2.1. T he Observed Kinematics

nent trace of the bar occurs at its minor axis at the slit positions of 135¡/157¡5 and 315¡/337¡ . . .5 The trace of kinematic features in the outer disk is not conspicuous in subsequent slits. The eastern part of the disk (slit positions 45¡õ135¡) displays a quite smooth velocity ïeld, while the western part shows some large scale variations. Aside from the interarm region between the inner disk and the southern arm where a B0@ wide depression is .5 moving outward in subsequent slits (positions º247¡5), no . signiïcant features are apparent in the outer disk. Unfortunately, in this interarm region the signal-to-noise ratio (S/N) is not so good. Does that mean we do not see the trace of the arms in the velocity ïeld, or is the single slit representation of the two-dimensional velocity ïeld misleading and does not allow us to identify coherent features in adjacent slits ? Clearly, the wiggles associated with spiral arms in NGC 4254 are not nearly as strong as in M81 (Visser 1980) ; thus, their identiïcation is harder. A CO map of NGC 4254îs center (Sakamoto et al. 1999) shows also no coherent wiggles across spiral arms, and we doubt that it would be much dierent on a Fabry-Perot image. Rather than being confused by the one-dimensional nature of our rotation curve slices we believe that the spiral features in the velocity ïeld are intrinsically weak.
4.2.2. Projection and Alignment

To compare the simulated gas velocities to the observed data, we need to project the modeled velocity ïeld according to the real orientation of the galaxy so that we can obtain the LOS velocity along locations corresponding to the slit positions taken with the spectrograph. For this procedure we take the velocity components and transform them into velocity components parallel and orthogonal to NGC 4254îs major axis. Since the simulations yield truly two-dimensional velocity ïelds, the component parallel to the major axis does not contribute to the LOS velocity as it reÿects only tangential motion. The orthogonal component is multiplied by the sine of the inclination of the disk to account for the LOS fraction of the velocities. We read the velocities along slits that correspond to the slit positions along which we took measurements with the spectrograph. The angular width of the grid cells of the simulation (1A19) is comparable to the slit width of the . spectrograph we used (1A5), which was also the average . seeing conditions. The angles between the spectrograph slit orientations must be translated to angles in the plane of the galaxy to actually compare the same parts of the velocity ïelds. This translation makes use of the following relationship : tan (r [ P.A.) cos i \ tan (r [ P.A.) (11) int app where r is the apparent angle of the spectrograph slit app across the galaxy on the sky and r is the corresponding intrinsic angle within the plane of int galaxy. P.A. is the the position angle of the galaxyîs major axis on the sky. All angles are in degrees measured eastward from north. Solving for r gives int tan (r [ P.A.) app ] P.A . (12) r \ arctan int cos i

The rotation curves at the 16 slit positions from the observations are shown in Figure 6 as data points. It is apparent that the long-slit spectra allow a good velocity coverage along the slits. Almost all rotation curves show contiguous data points out to a radius of Z1@ . The spectra .5 show a lot of wiggles on a small spatial scale. Jumps of [30 km s~1 on a scale of D5A are common. The very prominent jumps that we observed in slit positions 22¡ and 225¡ .5 clearly exceed the average wiggle sizes. Inside of about 0@ .3 the small bar inÿuences the velocity ïeld. The most promi-

C

D

The foreshortening of the radial proportions owing to projection in the direction of the minor axis is determined by R proj \ RJcos2 (r ) ] [sin (r ) cos i]2 . int int (13)


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FIG. 6.õSimulation results of the gas velocity ïeld in comparison to the observed rotation curves. Displayed are the measured rotation curves (data points), the axisymmetric model (thin line) and the rotation curves from the hydrodynamic simulations (thick line) for all 16 slit position angles. The parameters for the simulation were f of 60% and R \ 7.58 kpc. There are no error bars plotted for the data, but the errors can be estimated from the d CR point-to-point scatter of the data.

Finally, the detector pixel sizes of the TWIN, where the observed velocities come from, and the Omega Prime camera, whose pixel scale is the reference grid for the simulations, must be adjusted to perfectly align with each other.
4.2.3. Overall Fit Quality

Figure 6 shows the 16 separate rotation curves with a corresponding simulated velocity ïeld overplotted. The simulation used here for comparison is the one for which the gas density distribution best ïtted the K-band image (displayed in Fig. 5). It has a corotation radius of R \ CR 7.58 kpcõcorresponding to a pattern speed of ) \ 20 km p s~1kpc ~1õand a stellar disk mass fraction f of 60%. A gas d sound speed of c \ 10 km s~1 was assumed here. s quality gets determined by a global s2An overall ït comparison of the simulated velocity ïeld to the actual

observed velocity ïeld along the 16 measured slit positions. Since with the errors from the observations, we ïnd s2/ N [ 1.0, we added an overall 9.5 km s ~1 to provide an average ït with s2/N B 1.0. The s2 ïtting excludes the very central region hosting the small bar because the modeling is not intended to ït the central bar, which might have a dierent pattern speed. The total number of data points included in the s2 ïtting is 1077. The general ït quality is governed by the eect that the projection of the simulated velocity ïeld introduces. The good overall match indicates that we quite reliably found the right position angle and inclination for the galaxy. The simulated velocities align very well with the measured data points. In addition to the good overall match, the general rising or falling shape of the separate curves is also excellently reproduced by the simulations. The lopsidedness of


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FIG. 7.õComparison of four simulations for dierent fractions f of the stellar mass component. The simulation was done with a corotation radius of 7.58 d kpc and a gas sound speed of 10 km s ~1. Displayed is one of the 16 slit positions. Clearly, the predicted "" wiggles îî in the rotation curve grow much stronger for higher disk mass fractions.

the galaxy is reÿected in the shape of the rotation curves on the receding and approaching side of the disk. At the receding side (67¡ ) the rotation curve rises steeply and continues .5 to rise out to 2@, while the approaching side (247¡ ) rises less .5 steeply but ÿattens out or even drops beyond 0@ . These .7 characteristics are closely reproduced by the models. A close inspection of the two proïles shows, however, that the overlap in the match of the simulated velocities with the measurements is not always satisfactory. The agreement of local features in the simulations and the measured data is sometimes very good and even occurs in subsequent slit positions. However, there are also many locations where the match is poor. This is particularly the case in the inner region of the galaxy, where the small bar dominates the kinematics. Both proïles show strong wiggles where the slit crosses the bar, especially close to the minor axis of NGC 4254îs velocity ïeld, which is also close to the minor axis of the bar itself. While the simulations show a rather symmetric imprint, the measurements exhibit a signature dierent from that, leading to a signiïcant mismatch at several slit positions, e.g., 292¡5 and 337¡ . This . .5 might be caused by the bar, having a slightly dierent pattern speed. In the outer parts of the rotation curve we also ïnd several wiggles in the observed data that have no correspondence to the wiggles in the simulations and vice versa. It is important to note that we do not expect to reproduce all the wiggles in the galaxyîs rotation curves, since we are only modeling those created by the nonaxisymmetric gravitational potential. The wiggles originating for another reasonõlike expanding supernova gas shellsõare not considered by the simulations and thus do not show up in the resulting velocity ïeld. One very interesting thing to mention is the fact that even for moderate dierences in the ït quality of the axisymmetric disk model (see Table 1), the ït quality of the nonaxisymmetric simulated velocity ïeld is rather independent of the initial axisymmetric model. And, moreover, the formally preferred axisymmetric maximum-disk decomposition (see Table 1) turns out to be the most unfavored model, once the simulations were performed. This implies that even if an axisymmetric model proïle provides a better ït to a measured rotation curve, it does not necessarily mean that this combination provides the best ït when one considers the two-dimensional nonaxisymmetric gas evolution.
4.2.4. V arying the StellarõtoõDark Matter Ratio

Table 1. Since the nonaxisymmetric perturbations are induced in the potential by the stellar contribution, we expect the amplitude of the wiggles in the modeled rotation curves to depend signiïcantly on the nonaxisymmetric contribution of the stellar potential, whereas we expect the radial distribution of the wiggles to be rather independent of the stellar mass fraction. As expected, in the simulations with the lightest disk, the wiggles look like modulations on the axisymmetric rotation curve. In the case of the maximum disk, the rotation curves are strongly nonaxisymmetric (see Fig. 7). To describe this characteristic more quantitatively, we learn from Figure 8 that the amplitude of the deviations from axisymmetry increases linearly with the mass fraction of the stellar disk, which proves the general validity of the concept with which we try to approach this problem. The strongest velocity wiggles arising in the modeled velocity ïelds are the ones connected to the central barlike feature. However, since here we are not interested in modeling the dynamics of the bar, we exclude this inner part from the analysis. If we do a formal s2 comparison of the models with the observed data, we ïnd that for most of the sub-

As already mentioned in ° 3.2, we performed simulations for ïve stellar disk and dark halo combinations, as listed in

FIG. 8.õDeviation of the simulation from axisymmetry. Displayed is the average deviation of each radial simulation bin from the axisymmetric rotation curve. It rises linearly with the stellar disk mass contribution f . d


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maximal disks the ït is considerably better than for heavy stellar disks. By formal we mean that we use all data points for the s2 ït, regardless of whether a certain part matches well or not with two exceptions : we exclude the very central part with the bar, and we correct for the outer strong shock appearing in the fast rotating models (see Fig. 5). The result from this s2 ït is presented in Figure 9. In all cases, f \ 100% gives the worst ït to the observed rotation curves.d For the lower mass disks it is very hard to decide whether a particular disk mass is preferred. For R \ 6.4 CR kpc we ïnd about the same s2 for all nonmaximal disk models. Since we cannot reject data on a physical basis, we can state only a trend at this point. Thus, our conclusion from this part of the analysis is that the disk is most likely less than 85% maximal.
4.2.5. V arying the Gas T emperature

of 7250 K. To probe the eect of the gas temperature, we performed simulations for four dierent gas sound speeds, c , of 15, 20, 30 and 40 km s~1. With a simulation performed s at higher sound speed, we tested if our results at lower sound speeds were caused by overly cold gas being too strongly aected by the nonaxisymmetric part of the potential. Increasing the sound speed slightly broadened and shifted the shock fronts in our simulations but not enough to change the conclusions of this paper. A more detailed technical discussion on the simulated gas dynamics in NGC 4254 will be presented in a forthcoming paper by Slyz et al.
5

. DISCUSSION

This kind of analysis may be very sensitive to the temperature of the gas that is assumed for the modeling. Higher gas temperature corresponds to a higher cloud velocity dispersion c and thus to a reduction in the response of the gas s to any feature in the gravitational potential. Within the Milky Way, c varies from B6 km s~1 in the solar neighs borhood to B25 km s~1 in the Galactic center (Englmaier & Gerhard 1999). For simulations of a galactic disk with an isothermal equation of state, the most commonly used values for c are 8õ10 km s ~1, corresponding to less than s 104 K in gas temperature (Englmaier & Gerhard 1999 ; Weiner et al. 2001a). In these simulations the authors make the statement that within the reasonable limits of c \ 5 õ30 s km s~1, the modeled gas ÿows across the primary shocks are not considerably aected. Only when modeling strong bars in galaxies the simulations might be dependent on the choice of the gas sound speed (Englmaier & Gerhard 1997). For the main set of simulations presented in this paper, we chose c \ 10 km s~1, corresponding to a gas temperature s

To learn about the relative role of stellar and dark matter in the inner parts of spiral galaxies, we have extended the hydrodynamic simulations of galactic features from bars (Weiner et al. 2001a) to the regime of spiral arms. In this paper we have used color-corrected K-band images to derive the stellar portion of the potential of NGC 4254 for the analysis. From the simulation of the gas density we ïnd a very good morphological match of the gas shocks to the spiral arms in the galaxy ; this aspect of the simulations determines the pattern speed ) of the galaxy to about 15% precision. A comparison ofpthe observed and simulated kinematics has turned out to be challenging. Although the overall shapes of the dierent rotation curves were very well reproduced by the simulations, some small scale structure remains unmatched. The formal comparison of the gas velocity ïeld to the observed Ha kinematics favors simulations with small disk mass fractions f (see Fig. 9) and corred spondingly small values for the stellar M/L. With the K-band M/L discussed in ° 3.1.1 our results yield an overall stellar M/L of ! [ 0.5. We can estimate the relative mass * fractions from their contributions to the total rotation velocity. At a radius of 2.2 uncorrected K@-band exponential disk scale lengths (2.2 R B 79A or 7.7 kpc), the individual exp rotational support of the stellar and dark halo components for f \ 85% are v \ 125 km s~1 and v \ 86 km s~1. If d * halo a total mass is estimated via M(2.2R )\ exp v2(2.2R ) ] 2.2R exp exp , G (14)

we ïnd that M B 0.47 M at R \ 2.2 R , or accordhalo * exp ingly Z 1 of the total mass inside R is dark. Since our 3 limits are not very tight, we cannot use them to exp conïdence test other authorsî ïndings in detail. Projects yielding results in favor of a submaximal stellar disk usually ïnd a disk mass fraction less than our upper limit estimate. Bottema (1997) as well as Courteau & Rix (1999) conclude that the contribution of the stellar disk to the total rotation is v D 0.6 v , which translates to M D 0.6 M . At the * tot halo tot current state of the project we cannot exclude or conïrm these ïndings. This issue is going to be discussed more thoroughly as soon as we have a few more examples analyzed. In the following sections we will discuss the details that could cause deviations from a perfect match between the simulations and the measurements.
FIG. 9.õFormal s2 ït of the gas velocity simulations for dierent stellar mass contributions f , normalized by the s2 ït for the axisymmetric model d rotation curve. Displayed are cases for four dierent values of R . From CR the simulations we can rule out the cases with the highest disk mass.

5.1. Is the Concept Reasonable ? If the self-gravity of the stellar mass in the disks of spiral galaxies plays an important role, then undoubtedly the potential becomes nonaxisymmetric. The trajectory of any


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kinematic tracer in the galaxy, such as the gas, should be aected by these potential modulations. But is the H II component of the gas the best choice for tracing the galaxyîs potential ? Analytic calculations of gas shocks in the gravitational potential of a spiral galaxy (e.g., Roberts 1969) tell us that we should expect velocity wiggles with an amplitude of 10õ30 km s~1 while crossing massive spiral arms. However, kinematic feedback to the gas from regions of massive star formation, from expanding gas shells produced by supernova explosions and from other sources of turbulence, introduces small-scale random noise in the velocity ïelds. These ÿuctuations typically lie in the range of 10õ15 km s~1 (Beauvais & Bothun 1999) and seem even higher in the case of NGC 4254. The kinematic small-scale noise could be increased if the dynamics of the brightest H II regions is kinematically decoupled from the global ionized gas distribution. To check this we overplotted the Ha intensity on the rotation curves to see if the star formation regions coincide with the strongest wiggles in the rotation curves. There is, however, no discernible relation between the amplitude of a wiggle and the intensity of the H II region, indicating that the deviations are not conïned to compact H II regions. As an alternative to observing the ionized phase of the hydrogen gas, one could consider using H I radio observations. However, available H I data are limited by the larger size of the radio beam that smears out kinematic small-scale structures in the gas. In principle, stellar absorption spectra could also provide relevant kinematic information, but this approach has two disadvantages compared to an approach using gas kinematics. First, the acquisition of stellar absorption spectra with sufficient S/N would take a prohibitively large amount of telescope time. Second, stellar kinematics cannot be uniquely mapped to a given potential ; there are dierent sets of orbits resulting in the same observed surface mass distribution and kinematics. Thus, despite their apparent shortcomings, H II measurements seem to be the most promising method with which to approach the problem. Could the discrepancies between modeled and observed kinematics be related to taking the NIR K-band image of a galaxy as the basis to calculate its stellar potential ? As already discussed in ° 3.1 there are several factors that throw into question whether the K-band image is a good constant M/L map of the stellar mass distribution despite the color correction we applied. However, the two major factorsõdust extinction and the population of red supergiant starsõtend to aect the arm-interarm contrast, rather than the location of the spiral arms in the K band. Dust lanes lie preferentially inside the m \ 2 component of a galaxyîs spiral (Grosbòl, Block, & Patsis 2000) absorbing interarm light, while the red supergiants may actually have their highest density in the spiral arms directly, where they emerged from the fastest evolving OB stars. This eect should become apparent in the simulations as slightly wrong amplitudes of the gas wiggles, leaving their radial position mostly unchanged. So even if the K-band images might actually include unaccounted M/L variations, they most probably introduce only small errors, which should not result in an overall mismatch of the models with the data. The color-corrected K-band images should therefore reÿect the stellar mass accurately enough for our analysis. Certainly, there appears to be no better practical mass estimate that we could use for our analysis.

5.2. Are there Systematic Errors in the Modeling ? The most critical part of this study are the several modeling steps required to predict the gas velocity ïeld for the comparison with the data. We apply a spatial ïlter to the K-band image before calculating the disk potential to reduce the signiïcance of the clumpy H II regions (as described in ° 3.1). The residual map of the discarded component shows all the bright H II regions in the disk, giving us conïdence that we have excluded much of the structures that reÿect small-scale M/L variations. This correction removes roughly 3.5% of the total K-band light and does not depend strongly on the number of Fourier components used for the ït. For two extreme decompositions employing 6 and 16 Fourier components, the mean resulting relative discrepancy in the derived potentials is only B10~6. One remaining concern in calculating the potential might be that the mass density map cuts at the border of the image. However, the galaxy completely ïts in the frame, fading into noise before the image border. Moreover, we do not perform the simulations for the complete galaxy, but only for the inner 11.6 kpc in radius. So the edge cuto eect is very small and aects the part of the potential we are looking at even less. For the dark matter component we chose an isothermal halo with a core because of its ÿexibility in ïtting rotation curves. The functional form of the dark halo proïle has only a second-order eect on the results of the simulations compared to the variations owing to its two basic parameters R c and v . We decided not to distinguish between dierent = proïles for the present analysis. dark halo The most complex step surely is the hydrodynamic simulation of the gas ÿow in the galaxyîs potential. Beyond the tests of the code discussed in ° 3.2, it is the excellent morphological agreement between the simulated gas density proïles and the observed spiral arms that gives credence to the results of the code. The choice of the grid size was mainly motivated by the desire to achieve reasonable computing times, and not to exceed the seeing resolution. To check the grid cell sizeîs eect on the results, for selected cases we also ran simulations on grids with 2 times higher resolution as well as on grids with 2 times lower resolution. These simulations showed some minor dierences. Some gas shocks had a slightly larger amplitude in the density distribution but their locations were essentially unchanged. The morphology and the velocity ïeld of the central bar was most susceptible to the grid cell size. It results in small changes in the bar position angle. However, we have excluded the central region containing the bar from our analysis. The ïnal s2[ ït deviation between simulations with dierent grid cells ranges at about 6% and does not change the conclusions of this paper. Accordingly, we consider it safe to perform the simulations on the medium resolution grid we used. In a real galaxy the assumption of a global constant pattern rotation speed may not be fulïlled. In particular, we should expect the central bar to have a dierent pattern speed. Also, the pattern might be winding slowly, rather than being ïxed in a corotating frame. Furthermore, the gas may not have a uniform temperature. If these simpliïcations were relaxed, the location of the spiral arms in the hydrosimulations would change and eventually lead to a dierent overall ït quality. Lacking any solid basis to constrain these parameters, we are unable to implement these


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eects into the modeling procedure. Finally, the code does not include gas self-gravity. The eect of gas self-gravity is difficult to quantify without actually performing simulations, but from the literature we know that it tends to amplify the gas response. Gas self-gravity also suppresses the tendency of the gas to shock (Lubow, Balbus, & Cowie 1986). Since we are interested in the strength of the gas response to the gravitational potential and we already ïnd that for high-disk M/L the response is too strong, we assume that our upper limit holds also if gas self-gravity was included. Given the good morphological match, we may conïdently assume that all our approximations are not far o. This leaves nongravitationally induced gas motions as the main contamination in the comparison of the models to the observed kinematics. The inÿuence of nongravitational gas motion is in our case increased by the circumstance that we observe NGC 4254 from a perspective that is not far from face-on, so any motion along the z-direction is expected to appear clearly in the spectra. Eliminating these wiggles from the rotation curve is not possible since we have no means to reliably identify them. Any method we apply for excluding parts of the data will be aected by some kind of bias. 5.3. Is the Galaxy Suited for this Analysis ? Finally, we must consider the possibility that the galaxy we picked for our analysis might not be as suited as it appeared to be. NGC 4254 is not the prototype of a classical grand design spiral galaxy in optical wavelengths. However, as it can be seen in the central frame of Figure 5, the galaxy exhibits in the M/L corrected K@-band image mainly an m \ 2 spiral pattern with a strong symmetric part that ends at B5.5 kpc and fainter outer extensions. With its large angular size and moderate inclination, NGC 4254 seems to be one of the most promising candidates for this kind of study in our sample. However, as discussed earlier in ° 2.2, there are indications that this galaxy might not be as isolated and undisturbed as one might expect. In fact, the morphology itself implies some perturbative event in its recent evolution history : NGC 4254 shows a clear m \ 1 mode and a lopsided disk. In this respect it was argued earlier that infalling H I gas clumps, which are visible in radio data and do not emit in Ha, might be responsible for triggering deviations from pure grand design structure (Phookun et al. 1993). So have we reason to believe that NGC 4254 is far from equilibrium ? This is hard to tell, because on the other hand we ïnd plenty of arguments that a stable propagating density wave in NGC 4254 is responsible for its morphology. This galaxy shows many similarities to the spiral galaxy NGC 5247, whose morphological and dynamical properties were discussed by Patsis, Grosbòl, & Hiotelis (1997), based on smoothed particle hydrodynamics (SPH) simulations. We note that in our best model, the strong bisymmetric part of the K@ spiral, ends well inside the corotation radius, although fainter extensions reach out to it. This picture is in agreement with the 4 : 1 SPH models of Patsis et al. (1997). Assuming corotation

close to the characteristic bifurcation of the arms at B5 kpc on the other hand, we do not obtain satisfactory results (Fig. 5, upper left frame). On this basis it seems appropriate to conclude that NGC 4254 is at least close to an equilibrium state and suited for a case study.
6.

CONCLUSIONS

We performed hydrodynamic simulations to predict the gas velocity ïeld in a variety of potentials for the spiral galaxy NGC 4254 and compared them to observations. These potentials consisted of dierent combinations of luminous (nonaxisymmetric spiral) and dark matter (axisymmetric) components. The resulting gas spiral morphology reÿects very accurately the morphology of the galaxy and allows us to specify the corotation radius or the pattern speed of the spiral structure quite precisely. It is noteworthy that within the error range given, the bestmatching pattern speed does not depend on the mass fraction of the stellar disk relative to the dark halo. For NGC 4254 we ïnd that the corotation lies at 7.5 ^ 1.1 kpc, or at about 2.1 exponential K@-band disk scale lengths. From the kinematics of the gas simulations we could rule out a maximal disk solution for NGC 4254. Within the half-light radius the dark matter halo still has a nonnegligible inÿuence on the dynamics of NGC 4254 : speciïcally, our fraction f [ 0.85 implies that Z 1 of the total mass within 2.2 d 3 K-band disk scale lengths is dark. However, the comparison of the simulated gas velocity ïeld to the observed rotation curves turned out to be a delicate matter. The observed rotation curves show a signiïcant number of bumps and wiggles, presumably resulting from nongravitational gas eects, that complicate the identiïcation of wiggles induced by the massive spiral arms. Therefore, beyond concluding that the disk is less than 85% of maximal, we were unable to specify a particular value for the disk mass or to test the results from Bottema (1997) or Courteau & Rix (1999). But already with this statement we dier from the conclusions of Debattista & Sellwood (2000) and Weiner et al. (2001a), who argue that their conclusions for maximal disks of barred galaxies also hold for nonbarred spirals. Since we only analyzed one galaxy so far, it is inappropriate to state here that the centers of unbarred spirals, what we still consider NGC 4254 to be despite its small barlike structure in the very center, are generally governed by dark matter. In the near future we will extend this analysis to more galaxies from our sample. This will put us into the position to decide whether single galaxies dier very much in their dark matter content, or if the bulk of the spirals show similar characteristics. We thank Panos Patsis and Julien Devriendt for their valuable comments and helpful conversations. We also thank the anonymous referee for a careful reading of the manuscript and comments which improved the paper. T. K. wishes to thank Greg Rudnick, Nicolas Cretton, and Marc Sarzi for sharing tips, tricks, and thoughts.


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