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THE ASTROPHYSICAL JOURNAL, 546 : 931 õ 951, 2001 January 10
( 2001. The American Astronomical Society. All rights reserved. Printed in U.S.A.

THE DISK AND DARK HALO MASS OF THE BARRED GALAXY NGC 4123. II. FLUID-DYNAMICAL MODELS BENJAMIN J. WEINER
Observatories of the Carnegie Institution of Washington, 813 Santa Barbara Street, Pasadena, CA 91101

AND J. A. SELLWOOD AND T. B. WILLIAMS
Department of Physics and Astronomy, Rutgers University, 136 Frelinghuysen Road, Piscataway, NJ 08854 Received 2000 April 18 ; accepted 2000 August 23

ABSTRACT We report a dynamical determination of the separate contributions of disk and dark halo masses to the rotation curve of a spiral galaxy. We use ÿuid-dynamical models of gas ÿow in the barred galaxy NGC 4123 to constrain the dynamical properties of the galaxy : disk M/L , bar pattern speed, and the central density and scale radius of the dark halo. We derive a realistic barred potential directly from the light distribution. For each model we assume a value of the stellar M/L and a bar pattern speed ) and add a dark halo to ït the rotation curve. We then compute the gas ÿow velocities with a p twodimensional gasdynamical code and compare the model ÿow patterns to a two-dimensional velocity ïeld derived from Fabry-Perot observations. The strong shocks and noncircular motions in the observed gas ÿow require a high stellar M/L and a fast-rotating bar. Models with I-band disk M/L of 2.0õ2.5 h , or 75 80%õ100% of the maximum disk value, are highly favored. The corotation radius of the bar must be ¹1.5 times the bar semimajor axis. These results contradict some recent claimed "" universal îî galaxy disk/halo relations, since NGC 4123 is of modest size (rotation curve maximum 145 km s~1 and V \ flat 130 km s~1) yet quite disk-dominated. The dark halo of NGC 4123 is less concentrated than favored by current models of dark halos based on cosmological simulations. Since some 30% of bright disk galaxies are strongly barred and have dust lanes indicating shock morphology similar to that of NGC 4123, it is likely that they also have high stellar M/L and low-density halos. We suggest that luminous matter dominates inside the optical radius R of high surface brightness disk galaxies. 25 Subject headings : dark matter õ galaxies : halos õ galaxies : kinematics and dynamics õ galaxies : structure õ hydrodynamics
1

. INTRODUCTION

Extended rotation curves of spiral galaxies provide the strongest evidence for mass discrepancies on galactic scales, requiring galaxies to be embedded in halos of dark matter. However, the relative contributions of disk and halo to the mass of galaxies are not well known. The rotation curve of an axisymmetric disk galaxy does not provide enough information to disentangle the disk and halo contributions to the mass distribution. Even a well-sampled and spatially extended rotation curve can be ïtted well by any combination of disk and halo from no disk to a "" maximum disk îî value (van Albada et al. 1985). There is no consensus yet on whether maximum disks are preferred ; we review arguments for and against in ° 2. As a result, we do not know the relative importance of disk and halo, which hampers our understanding of major problems in galaxy formation and dynamics, including the efficiency of assembly of baryons into disks, the origin of the TullyFisher relation, and the generation of bars, spiral arms, and warps. Here we show that the disk-halo degeneracy can be broken in barred galaxies. Two-dimensional velocity ïelds show that bars drive noncircular streaming motions (see Weiner et al. 2000, hereafter Paper I). The strength and location of the noncircular motions are governed by the ellipticity of the potential, hence the mass of the bar, and, to a lesser extent, by the bar pattern speed, the angular rate of ïgure rotation of the bar. In strongly barred galaxies, the bar is the dominant visible component in the inner galaxy, and the (unobservable) dark halo should be more rounded 931

than the bar. Thus, the extra information in the noncircular motions can determine the mass-to-light ratio (M/L ) of the bar and, by extension, the mass of the stellar disk. In this paper we model the noncircular ÿow pattern of gas in the disk of NGC 4123, which we obtained in Paper I from optical and radio emission-line observations. Since we observe only the line-of-sight velocity, we cannot reconstruct the full space velocity ïeld of the gas, as we can for an axisymmetric galaxy. Furthermore, shocks are observed in the gas velocity ïeld, so ÿuid-dynamical models of the gas ÿow are needed. We compare these models against the observed velocity ïeld in order to derive the disk M/L . The disk M/L determines the importance of the bar, hence the ellipticity of the potential, and thus controls the strength of noncircular motions and shocks : large shocks require a fairly noncircular potential and hence a signiïcant contribution from the stellar bar. We construct models for the galactic potential from our photometric observations and run ÿuid-dynamical simulations of the gas ÿow in a range of model potentials, for dierent disk M/L and bar pattern speeds ) . We then p compare the model velocity ïelds to that observed to ïnd the most likely values of M/L and ) . As in Paper I, we assume a distance to NGC 4123 of 22.4pMpc, which is based on a Hubble constant of 75 km s~1 Mpc~1 ; all values of M/L in this paper are implicitly followed by a factor of h . 75
2

. THE DISK-HALO DEGENERACY AND MAXIMUM DISKS

Rotation curves of axisymmetric disk galaxies that extend beyond the optical disk clearly require dark matter


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Vol. 546

but do not contain enough information to distinguish between disk-halo decompositions ranging from zero-mass to maximum disks (van Albada et al. 1985 ; van Albada & Sancisi 1986). The "" maximum disk îî hypothesis, requiring that the disk be as massive as possible, was originally invoked simply for deïniteness to place a lower bound on the dark mass. We review here some arguments for and against maximal disks (see also Bosma 1999 ; Sellwood 1999). Nonaxisymmetric structure provides some information about disk M/L . Two-armed spiral patterns are common in disks, and excessive halo mass suppresses two-armed spiral structure, favoring disks 50%õ100% of maximum mass (Athanassoula, Bosma, & Papaioannou 1987). The maximum disk or even "" no halo îî assumptions can reproduce the overall shape of the rotation curve inside the optical disk (e.g., Kalnajs 1983 ; van Albada et al. 1985 ; Kent 1986 ; Begeman 1987 ; Freeman 1992 ; Palunas 1996 ; Broeils & Courteau 1997). Small-scale "" bumps and wiggles îî in the rotation curve do not oer compelling support for maximum disks (van der Kruit 1995 ; Palunas 1996). Quillen & Frogel (1997) integrated periodic orbits in potentials for the ringed barred galaxy NGC 6782 and concluded that its disk should have an M/L of 75% ^ 15% of maximum (see also Quillen, Frogel, & Gonzalez 1994 for application to a bar). Dynamical friction from a heavy halo will slow down a bar (Weinberg 1985 ; Debattista & Sellwood 1998) ; bars are slowed down rapidly in even mildly submaximal disks. There are several reasons to believe that bars rotate quickly (see the review of Elmegreen 1996), and two bars have been observed to be fast rotators (Merriïeld & Kuijken 1995 ; Gerssen, Merriïeld, & Kuijken 1999). The bar of the Milky Way is also best modeled as a fast rotator (Weiner & Sellwood 1999 ; Fux 1999). Bottema (1993, 1997) has argued that stellar velocity dispersions imply that disks supply only about 63% of the maximum rotation velocity, so that disk M/L is 40% of maximum (see also Fuchs 1999). However, Bottemaîs analysis rests on a chain of assumptions, and the upper bound can be relaxed (Bosma 1999). While the Oort limit from the vertical stellar velocity dispersion has a long history as a measurement of disk mass (e.g., Oort 1932 ; Kuijken & Gilmore 1991), it is still unclear whether even the Milky Way disk is maximal or not (Sackett 1997 ; Binney & Merriïeld 1998). Its application to external galaxies has two major complications : (1) we cannot measure both vertical scale height and vertical velocity dispersion in any individual galaxy ; and (2) integrated-light measurements of the vertical velocity dispersion will be dominated by young stars, which have a lower dispersion in the Milky Way (see Binney & Merriïeld 1998 ; but cf. Fuchs 1999). Maller et al. (2000) used gravitational lensing by an edge-on disk galaxy at z \ 0.41 to measure its potential, concluding that a maximal disk is ruled out, though a model with a massive bulge and subdominant halo is allowed. The lack of kinematic information makes attempts at rotation curve decomposition preliminary. Courteau & Rix (1999) show that if galaxies have maximal disks, there should be a relationship between scale length and the peak of the disk rotation curve V , predict2.2 ing a correlation between scale length and residual from the Tully-Fisher relation *V : smaller disks should have a higher rotation width at 2.2given luminosity. They ïnd no a

such correlation between the residuals from the velocity widthõmagnitude and scale lengthõmagnitude relations and conclude that the disk contribution to rotation width is small. In searching for a correlation between scale length and rotation width at a ïxed luminosity, there is a surface brightness dependence : larger disks will have lower surface brightness. As Courteau & Rix (1999) note, the lack of correlation demonstrates that surface brightness is not a second parameter in the Tully-Fisher relation. Low surface brightness (LSB) disks do indeed lie on the same TullyFisher relation as high surface brightness (HSB) disks (Zwaan et al. 1995 ; Sprayberry et al. 1995), although a "" baryonic Tully-Fisher îî relation may be required (McGaugh et al. 2000 ; OîNeil, Bothun, & Schombert 2000). Lower surface brightness disks appear to be quite halodominated (e.g., de Blok & McGaugh 1996, 1997 ; Swaters, Madore, & Trewhella 2000) and should, in fact, violate the scale lengthõTully-Fisher residual relation expected for maximal disks. However, to preserve the Tully-Fisher relation, either disks are negligible in all galaxies or the relation between halo/disk ratio and surface brightness must be ïnetuned. The lack of correlation between scale length and *V indicates that there is no feature in the rotation curve 2.2 to distinguish putatively disk-dominated and halo-dominated portions of the rotation curve. This ïne-tuning is related to the well-known "" disk-halo conspiracy îî noted by Bahcall & Casertano (1985) : if disks are maximal, then disk and halo rotation curves must be of similar amplitude to maintain ÿat rotation curves (but see Casertano & van Gorkom 1991). The mystery in the end is why LSB and HSB disks lie on the same Tully-Fisher relation. The best way to resolve the disk-halo degeneracy and its attendant problems and to understand the origin of the Tully-Fisher relation is actually to measure the degree of maximality of disks of galaxies over a range of velocity width and surface brightness. We are now attempting to make this measurement through observations of a number of barred galaxies. In this paper we carry out this procedure for NGC 4123.
3.

MODELING. I. THE STELLAR MASS DISTRIBUTION

We wish to estimate the gravitational potential of the three-dimensional stellar mass distribution of the galaxy given a two-dimensional surface brightness distribution. There are three potential difficulties : spatial variations in M/L or extinction, deprojection to estimate the threedimensional luminosity density from its projection into the two-dimensional plane of the sky, and conversion of luminosity density into mass density. In fact, we do not need the full three-dimensional gravitational potential but only its derivatives in the galaxy midplane ; vertical gradients can be neglected. This is because the emission-line kinematic tracers we observe arise in the gas phase, which lies in a thin layer. (The H I disk can be warped and/or ÿaring at large radius, but these phenomena generally occur outside the optically luminous disk of the galaxy ; see the review by Binney 1991.) This requirement considerably simpliïes the task of deprojection. As argued in Paper I, color and extinction variations are mild in our I-band image, and correcting for them would be quite model-dependent ; we therefore neglect them here. Projection is of course formally degenerate and impossi-


No. 2, 2001

DISK AND HALO MASS OF NGC 4123. II.

933

ble to invert uniquely. One approach is to ït a threedimensional parametric model to the data by projecting it into two dimensions. Prolate Ferrers ellipsoids are the most convenient analytical forms with easily derived barlike potentials, and several previous gasdynamical studies of barred galaxies have used them to model bars. Duval & Athanassoula (1983) used an n \ 0 (uniform density) Ferrers bar in their study of NGC 5383 ; Regan, Vogel, & Teuben (1997) used an n \ 1 Ferrers bar to model NGC 1530 ; and we have used an n \ 1 Ferrers bar to model the Milky Way (Weiner & Sellwood 1999). Lindblad, Lindblad, & Athanassoula (1996), on the other hand, used a photometric model less dependent on an analytical form to derive a potential for their gasdynamical simulations, rectifying their J-band image of NGC 1365 to face-on and decomposing it into the ïrst several even Fourier components. A disadvantage of the analytical approach is evident from the I-band image of NGC 4123 presented in Paper I. The bar in this galaxy has a complex shape that does not lend itself to modeling by simple analytic expressions. It has two components, both of which are clearly nonellipsoidal. The high surface brightness central component is distinctly rectangular, while the elongated component is also quite boxy and possibly broadens, if anything, toward the ends, where an ellipsoid would taper. We are therefore motivated to construct a model that is independent of analytic expressions for the luminosity density. Our approach is inspired by Quillen et al. (1994), who estimated the potential of a face-on barred galaxy in a model-independent way by integrating over the actual light distribution. We wish to apply the same method to an inclined galaxy, which we must ïrst deproject to face-on. 3.1. Deprojection Disk galaxies are relatively thin (e.g., van der Kruit & Searle 1982), and a simple stretch of our two-dimensional image should give us a reasonable approximation to a face-on view of the galaxy. As the galaxy has a ïnite scale height, which is probably not the same for all components, such a deprojection will introduce biases, which we address in this section. There are a number of foreground stars that must be removed before the rectiïcation process. We masked out circular apertures around each star. NGC 4123 is very close to bisymmetric, especially in the bar region, so we replaced the missing data in the masked regions with the data from the corresponding regions after a 180¡ rotation. Since the galaxy is so bisymmetric, we chose to conserve computing resources by running our simulations on a half-plane and enforcing bisymmetry (see ° 5). We constructed a strictly bisymmetric image by rotating the image 180¡ and averaging the rotated and original images. We assume the inclination i \ 45¡, derived from our kinematic data in Paper I, and simply stretch the galaxy image along the projected minor axis by the factor 1/cos i to produce a face-on image of surface brightness, while conserving total luminosity. We selected NGC 4123 for this study in part because it does not have a large spheroidal bulge. In fact, the only spheroidal component present is an unresolved pointlike source at the very center of the galaxy (Paper I), having a luminosity 1.8 ] 108 L in the I band. We ït this source _ independently and remove it before deprojection. The ïnite thicknesses of the disk and bar will introduce inaccuracies into our deprojection. The typical scale height

of a disk is a few hundred parsecs, but the vertical extent of the bar may be up to 2õ3 times greater. Bars in simulations develop pronounced "" peanut îî shapes (Combes & Sanders 1981 ; Raha et al. 1991), which have larger scale heights than the disk, and the box/peanut bulges seen in some edge-on galaxies are associated with bars (Kuijken & Merriïeld 1995 ; Bureau & Freeman 1999). It is possible that the peanuts and boxes occur at the bar center but that the bar is thinner farther out along its length. Dettmar & Barteldrees (1990) showed that some edge-on galaxies with box and peanut bulges have an additional thin component, which they suggest is a bar. NGC 4123 could be such an object, given the two-component structure of its bar (Paper I). The bias introduced by rectiïcation is signiïcant only when a structure has a scale length s along the deprojected minor axis of the galaxy that is comparable to its vertical scale height z . At an inclination of i \ 45¡, for example, 0 erroneously rectifying a sphere with s \ z would stretch it 0 along the galaxy minor axis by a factor of 1.414, a 41% error in its linear extent. However, a spheroid with a modest 2 : 1 ÿattening (s \ 2z ), once rectiïed, would be erron0 eously stretched by a factor of just 1.12, and for an object with 3 : 1 ÿattening the stretch is too large by just 5%. These factors are only marginally dierent for nonellipsoidal cross sections. As noted above, the boxy structure of NGC 4123 inside D15A radius could be thickened into a box- or peanut-type bulge. Our models may not be reliable in the very center for this reason, although the bulge is unlikely to be as thick as 1 : 1. Outside the central boxy structure, the bar is probably about as thin as the disk, and the rectiïcation bias should be minimal. The elongated part of the bar has a projected scale length of s D 16@@ on the plane of the sky, parallel to proj the galaxy minor axis (not the bar minor axis). The deprojected length s in the plane of the galaxy, parallel to the galaxy minor axis, is greater, s ^ 1.4s \ 2.4 kpc. If the proj bar has a scale height typical of disk galaxies, z D 400 pc, 0 as we have assumed, then the erroneous stretching induced by rectiïcation is very small. Rectiïcation stretches along the galaxy minor axis, not the bar minor axis, which works in our favor. 3.2. Calculating the Potential In order to calculate the gravitational ïeld from this face-on image, we must make two additional assumptions : (1) some relation between light and mass and (2) a form for the vertical structure of the disk. We follow usual practice by assuming that light is directly proportional to mass, that is, a constant M/L throughout the galaxy. This simplest possible assumption is generally reasonable in the inner parts of galaxies (see Kent 1986), as evidenced by weak color gradients and the success of maximum disk models with constant M/L . Barred galaxies generally have shallower abundance gradients than unbarred galaxies (e.g., Martin & Roy 1994 ; Zaritsky, Kennicutt, & Huchra 1994), which suggests they are well mixed and any gradient in M/L must also be fairly shallow. We considered the eect of a radially varying M/L in Paper I. Even if the outer disk M/L at 15 kpc is just half that of the bar, the eect on the disk contribution to the rotation curve is just 5% at 15 kpc. Our modeling procedure is sensitive to the M/L inside the bar, and varying the disk M/L outside the bar makes little dierence to the derived disk and halo masses.


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Vol. 546

The potential and accelerations in the midplane of a ïnite-thickness disk are weaker than those from a razorthin disk. We assume a vertical distribution of the common form o(z) P sech2(z/2z ) and a scale height of z \ 200 pc, 0 0 similar to that found in edge-on disk galaxies (van der Kruit & Searle 1982 ; de Grijs & van der Kruit 1996 ; de Grijs 1997) and slightly smaller than in the Milky Way, since NGC 4123 is smaller (Mihalas & Binney 1981). The scale height is eectively a smoothing length for the potential. With the z-distribution given, we calculate the potential and the accelerations at every point in the midplane, using a Fourier transform method to convolve a Greenîs function with the surface density distribution (Hockney 1965). The rectiïed I-band image from our Cerro Tololo InterAmerican Observatory (CTIO) 0.9 m photometry (Paper I) taken in 1A seeing has 0A 9 pixels ; in order to keep the cell .2 .3 size and number of cells in the simulation grid reasonable, we binned it 2 ] 2 to make the pixel size 0A78, or 84.7 pc. . We then used the fast Fourier transform (FFT) algorithm to generate the x- and y-acceleration components on a 1024 ] 1024 (87 kpc square) grid, although we use only a 256 ] 512 subsection of this grid for our simulations ; the large grid ensures that forces from mass outside the subsection are calculated accurately. Since the FFT grid is larger than the ïeld of the CTIO observations, we extrapolated the disk surface brightness proïle. The extrapolation contains just 2% of the total disk luminosity and is consistent with the observed surface brightness proïle from our wider ïeld Las Campanas photometry. We tested the eect of changing the assumed scale height z by factors of up to 2. The eects on the velocity jump 0 across the bar shock are fairly small ; lowering the scale height makes little dierence, while increasing the scale height to 800 pc weakens the predicted velocity jump slightly, which would require a slightly higher bar M/L to match the observations (see ° 7.4). The assumed scale height has little eect on our conclusions based on the strength of the velocity jump (since we conclude that the bar M/L is high). However, the smoothing due to the scale height does aect the models at R \ 1 kpc ; the streaming motions at the inner Lindblad resonance (ILR) weaken with increasing scale height. Therefore, the models are not robust inside the ILR (see ° 7). The method also assumes that z is constant over the bar 0 and disk. As argued above, only the inner 10Aõ15A of the bar can be substantially thicker than the disk, and the eect is small outside the innermost few grid cells ; but inside D10 grid cells, the approximate size of the ILR, the potential could be aected by the uncertainty in scale height.
4

high as 108 M (Kormendy & Richstone 1995). _ We modeled the nucleus/bulge as a spherical Gaussian distribution, o(r) P exp ([r2/r2), with a scale radius of 200 c pc. We chose this scale radius to be softer than implied by the optical observations to keep the forces from varying too strongly over the innermost few grid cells, which would cause large numerical diusion eects. The Gaussian proïle drops o rapidly, so the softening aects the forces in only the innermost cells. Since the M/L of the nucleus is uncertain, we ran sets of models with three dierent prescriptions for the M/L , setting ! /! , the ratio of nucleus/bulge to disk M/L , equal bd to either 0, 0.5, or 1.0. The nuclear mass inÿuences the gas ÿow by changing the strength or size of the inner Lindblad resonance (see Athanassoula 1992 ; Weiner & Sellwood 1999). While the dierent choices of nuclear mass do aect the ÿow in the inner few grid cells, the dynamical parameters disk M/L and bar pattern speed ) are robust against p variations in nuclear mass, as discussed further in ° 7. Our 21 cm observations of NGC 4123 (Paper I) revealed the presence of a large, extended H I disk, with a hole of lower surface density in its center. The total mass of atomic gas, including helium, is 1.06 ] 1010 M . In Figure 8 of Paper I, we show the small contribution _ the azimuthally of averaged atomic gas to the rotation curve. In this paper we add this axisymmetric contribution to that of the now nonaxisymmetric stellar disk. We do not have a map of the molecular gas distribution ; as discussed in Paper I, singledish observations show that the contribution from molecular gas to the rotation curve is quite small, and we neglect it. In Paper I we found that the observed one-dimensional rotation curve of NGC 4123 can be ïtted satisfactorily for a wide range of disk M/L by adding quite simple dark matter (DM) halo models. This property is true for many disk galaxies with extended 21 cm rotation curves and lies at the heart of the disk-halo degeneracy. The DM halos used in the two-dimensional models of the velocity ïeld considered here are identical to those determined by the ït to the 21 cm rotation curve for the corresponding axisymmetric M/L models in Paper I. As there, we study two sets of models having two dierent popular halo density proïles : 1 o pseudoisothermal , 0 1 ] (r/r )2 c o (r) \ 5 (1) h 4r3 s o NFW-type . 6 s r(r ] rs)2 The density proïle of the ïrst set has the pseudoisothermal form, while that of the second set is an "" NFW-type îî power-law proïle (Navarro, Frenk, & White 1996), discussed more fully in Paper I. For a given disk M/L , the best-ït pseudoisothermal and NFW-type halo rotation curves both ït the axisymmetric rotation curve well. Since they are ïtting the same data, the two halo contributions to the rotation curve are quite similar, and substituting one halo proïle for the other makes little dierence to the gas ÿow pattern inside the bar region. In fact, the preferred values of the main parameters, disk M/L and ) , scarcely dier for the two possible halos p (see ° 7). We have assumed spherical halos. As noted in Paper I, a ÿattened halo with a slightly dierent density distribution could produce essentially the same rotation curve. Since our

0 0
4

. MODELING. II. ADDITIONAL MASS COMPONENTS

The total gravitational ïeld must account for components of the mass distribution other than the stellar disk ; these include the nucleus, gas disk, and dark halo of NGC 4123. We removed the central pointlike source of 1.8 ] 108 L _ from the image before the rectiïcation described above. Its M/L is uncertain, since it is blue and an emission-line source and is likely to have a young stellar population or even nonstellar luminosity, both suggesting a low M/L . Conversely, there could be a black hole at the center of NGC 4123, although upper limits on the black hole massõ toõbulge mass relation make it highly unlikely that a black hole in such a modest-sized galaxy could have a mass as


No. 2, 2001

DISK AND HALO MASS OF NGC 4123. II.
TABLE 1 SIMULATION PARAMETERS AND GOODNESS OF FIT s2/N M/L I (1) 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.50 1.50 1.50 1.50 1.50 1.50 1.50 1.50 1.50 1.50 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.25 2.25 2.25 2.25 2.25 2.25 2.25 2.25 2.25 2.25 2.25 2.50 2.50 2.50 2.50 2.50 2.50 2.50 ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ) p (2) 10.0 12.0 14.0 16.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 10.0 12.0 14.0 16.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 10.0 12.0 14.0 16.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 10.0 12.0 14.0 16.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 10.0 12.0 14.0 16.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 10.0 12.0 14.0 16.0 18.0 19.0 20.0 r /a L (3) 2.38 2.04 1.76 1.53 1.34 1.25 1.16 1.07 0.92 0.85 0.80 2.48 2.14 1.86 1.63 1.47 1.36 1.28 1.20 1.11 0.88 0.83 2.51 2.16 1.89 1.67 1.49 1.40 1.31 1.23 1.14 0.85 0.78 2.52 2.18 1.91 1.69 1.51 1.43 1.33 1.24 0.86 0.79 0.71 2.53 2.18 1.93 1.70 1.53 1.46 1.35 1.26 0.85 0.77 0.68 2.54 2.20 1.96 1.73 1.55 1.48 1.39 Raw Errors (4) 102.30 55.21 14.22 12.79 14.76 16.00 17.57 19.44 21.36 22.97 24.03 84.42 41.26 21.04 10.89 10.64 11.73 13.36 15.17 16.56 17.21 17.51 68.50 36.06 10.24 7.06 8.44 9.70 10.74 10.97 10.88 11.11 11.34 74.36 22.99 7.09 6.52 5.89 5.63 5.23 5.27 5.65 5.92 6.33 70.29 22.89 12.56 7.41 6.27 4.81 3.54 3.78 4.22 4.59 4.87 86.60 89.74 40.68 20.93 6.53 8.50 5.36

935

data are restricted to the midplane of the galaxy, they do not constrain the halo ÿattening, nor does the halo ÿattening aect the results. If the halo were nonaxisymmetric, it could conceivably contribute to the observed noncircular streaming motions. However, to have a signiïcant eect on our results, the halo would have to be both quite prolate inside the bar radius, where the noncircular motions are large, and axisymmetric farther out so as not to disturb the apparently circular ÿow pattern in the outer disk, at which point it would be part of the bar, for all practical purposes.
5

8 km s~1 Dispersion (5) 29.83 14.74 4.13 3.73 4.75 5.15 5.56 6.03 6.52 6.92 7.16 24.14 11.30 6.01 3.56 3.67 3.97 4.38 4.83 5.20 5.34 5.37 19.21 10.25 3.09 2.50 2.99 3.34 3.62 3.69 3.64 3.64 3.66 20.18 6.81 2.40 2.39 2.11 1.98 1.87 1.86 1.97 2.06 2.14 19.56 7.54 4.67 2.64 2.23 1.84 1.40 1.42 1.55 1.64 1.67 26.20 25.09 12.66 7.16 2.32 3.03 2.10

. MODELING. III. FLUID-DYNAMICAL MODELS

5.1. T he Model Parameters The disk M/L is the main parameter for the gravitational potential, and the nuclear mass and dark halo proïle are subsidiary parameters. To run a simulation of the gas dynamics, we also need to know the bar pattern speed ) . p For a given potential, the value of ) also determines the p of the bar ; a faster radius r of the Lagrangian point L1 rotatingLbar corresponds to a smaller Lagrangian radius. A particle at the L1 point would rotate at the same angular speed as the bar, and so r is the corotation radius. Two L models with the same ) and dierent M/L have similar, p but not exactly equal, values of r because the value of r L depends somewhat on the shape ofLthe potential. We parametrized our models by M/L and ) and calculated r /a, the ratio of the Lagrangian radiusp to the bar L semimajor axis, for each model. The ratio r /a is commonly L used to characterize the bar pattern speed because it generalizes across galaxies of dierent rotation curve height and bar size. Bars with r /a [ 1.5 are generally considered to be "" fast rotating.îî TheLelongated stellar orbits that are generally presumed to make up the bar are conïned inside the well of the eective potential that extends to L1, so r /a [ L 1.0 is favored (e.g., Teuben & Sanders 1985 ; Binney & Tremaine 1987). See Elmegreen (1996) for a review of the issues surrounding pattern speeds and their determination in barred galaxies. A "" set îî of models includes a full range of disk M/L and bar pattern speed ) . Each set is 88 models with all comp binations of eight values of disk M/L from M1.0, 1.5, 1.75, 2.0, 2.25, 2.5, 2.75, 3.0N and 11 values of ) from M10, 12, 14, 16, 18, 19, 20, 21, 22, 23, 24N km s~1 kpc~1p . In order to check the dependence of the models on the subsidiary parameters ! /! and dark halo proïle, we ran b the six sets of models covering d three choices of ! /! M0.0, bd 0.5, 1.0N and the two halo proïles, isothermal and NFWtype. We do not expect these subsidiary parameters to aect the modelsî ït to the data nearly as strongly as the main parameters, M/L and ) . Conversely, we do not expect to p make a robust determination of ! /! and halo proïle from b run. the models. In total, 528 models were d The models of the set with isothermal halos and ! /! \ bd 0.5, which contains the best-ïtting model, are listed by M/L and ) in Table 1. Thep M/L range covers from an M/L \ 1.0 disk, which is dominated by the dark halo at nearly all radii, to a supermaximal disk at the high end of M/L \ 3.0 (see Fig. 8 of Paper I). The ) range covers from quite slow rotating bars p to superfast bars with r /a \ 1. L 5.2. T he Fluid-Dynamical Code We used a two-dimensional grid-based gasdynamical code (kindly provided to us by E. Athanassoula) to simulate


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s2/N M/L I (1) 2.50 2.50 2.50 2.50 2.75 2.75 2.75 2.75 2.75 2.75 2.75 2.75 2.75 2.75 2.75 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ) p (2) 21.0 22.0 23.0 24.0 10.0 12.0 14.0 16.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 10.0 12.0 14.0 16.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 r /a L (3) 1.31 0.87 0.80 0.71 2.54 2.22 1.99 1.77 1.58 1.51 1.46 1.36 1.28 0.85 0.78 2.57 2.25 2.04 1.84 1.63 1.55 1.49 1.43 1.34 1.27 0.85 Raw Errors (4) 5.11 5.07 4.33 4.36 129.66 138.08 62.56 36.94 17.23 7.88 9.83 7.68 6.93 6.13 5.46 196.67 186.90 80.47 48.29 37.61 22.39 14.74 10.49 11.43 10.22 7.84 8 km s~1 Dispersion (5) 1.87 1.82 1.60 1.55 36.64 38.19 19.22 11.44 6.12 2.86 3.22 2.82 2.48 2.21 1.93 52.13 51.66 23.89 14.96 11.50 7.45 5.48 3.39 3.97 3.31 2.75

NOTE.õSimulations for the set of models with isothermal halos and ! /! \ 0.5, listed by their parameters M/L in col. (1) and ) in col. (2). b (3) : the value of r /a implied by ) . Col. (4) : the reduced s2 for the I p Col. d L p modelsî ït to the velocity data using purely statistical errors (see ° 7). Col. (5) : the reduced s2 when an additional 8 km s~1 dispersion is added to the error budget.

the gas ÿow in the models of NGC 4123. The FS2 code is a second-order, ÿux-splitting, Eulerian grid code for an isothermal gas in an imposed gravitational potential, originally written by G. D. van Albada to model gas ÿow in barred galaxy potentials (van Albada 1985). Athanassoula (1992) used it to study gas ÿow patterns in a variety of model galaxy potentials. We have modiïed it to suit our approach, but the heart of the code remains the ïnitedierence scheme described by van Albada (1985). By its nature, the code approximates the interstellar medium as an Eulerian ÿuid, smooth on scales of the grid cell size. Without some idealization it is hopeless to simulate the extremely complex dynamics of the multiphase interstellar medium (ISM). Various methods including grid codes, particle hydrodynamics, and sticky-particle codes have been advocated (see Weiner & Sellwood 1999). Essentially, applying the Euler equations to the ISM simply asserts that the ISM has a pressure or sound speed deïned in a coarse-grained sense, over scales greater than the codeîs resolution. The smoothed particle hydrodynamics (SPH) models of Englmaier & Gerhard (1997), who ran a simulation under the same conditions as used by Athanassoula (1992) with the FS2 Eulerian grid code, yielded very similar results. There are some small dierences between the results of the grid and SPH codes, chieÿy because of the dierent tradeos in resolution in Eulerian and Lagrangian schemes.

However, the chief features, such as the strength of the shocks in the bar (see ° 7), are quite similar. This comparison reassures us that the simulation results are not dependent on the particular ÿuid-dynamical algorithm. The code does not include the self-gravity of the gas, although the axisymmetrized rotation curve due to the atomic gas is included as part of our model for the potential, discussed above in ° 4. The gas self-gravity is negligible for this study, since the gas surface density is considerably lower than that of the stellar disk and of the halo. This is especially so in the bar region of NGC 4123, as can be seen from the rotation curve mass models in Figure 8 of Paper I. Since we compare the gas velocities to data only within the bar, the lack of gas self-gravity has little eect on the results. We use a grid having 256 ] 512 cells, each 84.7 pc square, and enforce a 180¡ rotation symmetry so that the grid is eectively 512 ] 512 and bisymmetric. Tests show that the results are not dependent on the cell size. The barred potential rotates at a ïxed pattern speed ) , and the grid rotates p with the bar. The x-axis of the grid is along the line of nodes of the galaxy disk. This puts the major axis of the bar at an angle of 53¡ to the x-axis. (It is desirable to avoid having the bar aligned with either the x- or y-axis of the grid.) The simulation time step is variable, chosen automatically via a Courant condition, and is generally 0.1õ 0.2 Myr. We take the sound speed of the gas to be 8 km s~1 (similar to the Galactic value ; e.g., Gunn, Knapp, & Tremaine 1979), corresponding to a temperature of 104 K, a preferred temperature for the diuse interstellar medium. Varying the sound speed within reasonable limits of several kilometers per second does not materially aect the derived gas ÿow. The strength and shape of the velocity jump across the primary shocks (see ° 7.4) do not change much for simulations with sound speed from c \ 1 to D25 km s~1. For s c Z 50 km s~1, the magnitude of the velocity jump s decreases : the sound speed becomes comparable to the streaming motions, and the shocks are no longer strong since the Mach number is not much greater than 1. However, such high sound speeds for the ISM are unrealistic ; additionally, high sound speeds would require a more massive bar to produce the observed shock, and the bar we infer is already near-maximal (° 7). 5.3. Initial Conditions We begin each simulation with an axisymmetric potential so that the gas starts on circular orbits. To achieve this, we redistribute the only nonaxisymmetric part of the mass, the stellar disk and bar, by averaging it over azimuthal angle. We then cause linear growth of the bar by interpolating the gravitational ïeld between the initial axisymmetric and the ïnal observed barred shape. We allow 0.1 Gyr to reach the fully barred state, a time much longer than the time step, so that the ÿow adjusts steadily to the growth of the bar. The bar growth time is roughly one-third of an orbital period for material in the bar. We set the initial gas surface density to be constant at 10 M pc~2 inside a radius of 8 kpc ; outside that radius it falls o_ exponentially with a scale length of 13 kpc. These values were chosen by examining the H I data. (We did not use an initial density distribution with a central hole because the H I hole is probably caused by the bar once it forms and may be ïlled in by molecular gas.) Longer bar growth times and dierent initial density distributions have very little eect on the results.


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The gas response can never reach a completely steady state, since the gas inside corotation will always lose angular momentum to the bar ; however, after the bar growth time, the evolution of the gas ÿow pattern is quite slow.1 We continue our simulations to 0.2 Gyr to allow the gas ÿow to settle after the bar has grown and to 0.3 Gyr to check that the ÿow has stabilized. There are some small changes from 0.1 to 0.2 Gyr in the gas density and velocity ïelds in the bar region, as the gas response to the fully grown bar stabilizes. After 0.2 Gyr, gas continues to fall to the center, but there is little change in the velocity ïeld in the bar region from 0.2 to 0.3 Gyr. We
1 Gas builds up in the center as a result of torque can be signiïcant if the code is run for many rotation gigayears. We turned o gas recycling in the code : mass build-up, it can provoke numerical instability wood 1999). from the bar, which periods, e.g., several though it decreases (see Weiner & Sell-

tested a bar growth time of 0.2 Gyr and found no signiïcant dierence after the gas response had settled. In order to compare the simulations to the data, for each simulation we projected the gas velocity ïeld at 0.2 Gyr into line-of-sight velocities and scaled and rotated it to match the orientation of the observed Fabry-Perot velocity ïeld of the bar region as shown in Paper I. 5.4. T he Best-Fit Model and V ariants Figure 1 shows the gas density ïeld viewed at 0.2 Gyr run time in four representative simulations. Figure 1a has M/ L \ 2.25 and ) \ 20.0, ! /! \ 0.5, and an isothermal p bd halo. This is the best-ïtting simulation discussed in ° 7. Figures 1bõ1d show variations on this model. The density ïelds have been reprojected and rotated to match NGC 4123îs orientation on the plane of the sky, for purposes of comparison. Shocks in the bar, visible as regions of high gas density, are associated with jumps in the model gas velocity

FIG. 1.õGas density ïeld in four representative simulations, at 0.2 Gyr simulation time, projected to the galaxyîs orientation on the plane of the sky ; north is up and west is to the right. The gray scale runs from 0 to 40 M pc~2 in face-on gas surface density. The bar runs east-west, and the oset regions of _ high gas density along the bar are the loci of shocks. The white box indicates the bar region plotted in Figs. 2 and 3. (a) Heavy-disk, fast-bar model with M/L \ 2.25, ) \ 20 km s~1 kpc~1.(b) Heavy-disk, slow-bar model with M/L \ 2.25, ) \ 10 km s~1 kpc~1.(c) Light-disk, fast-bar model with M/L \ 1.0, p p ) \ 20 km s~1 kpc~1.(d) Light-disk, slow-bar model with M/L \ 1.0, ) \ 10 km s~1 kpc~1. p p


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ïeld. There are multiple regions of high gas density in the bar, which correspond to a very strong primary shock and some smaller secondary shocks. The straight features of high gas density traceable from the center are the locations of the primary shocks and correspond to the dust lanes visible in the color map (Fig. 2 of Paper I). The natural explanation for the cause of the shock is the elongation of the streamlines along the bar (see Prendergast 1983). Gas falling down along the bar potential reaches high velocities, and as the gas climbs away from the center, up the potential, it decelerates. Eventually a pile-up occurs and the shock is formed. The strength and location of the gradient and shock are strongly inÿuenced by the ellipticity and rotation rate of the potential. A massive bar causes a strong spiral pattern in the gas disk outside the bar region, as can be seen in Figure 1. The spiral response is enhanced by the spiral structure and mild ellipticity of the stellar disk outside the bar, discussed in Paper I ; our simulation method causes the outer disk structure to rotate at the same angular speed as the bar. An additional simulation of a model in which we forced the disk outside the bar to be axisymmetric still generated a weaker spiral arm pattern. The spiral pattern continues to evolve from 0.2 to 0.3 Gyr, propagating outward and winding up to tighter pitch angles. We ran another test simulation for a long time (over 1 Gyr) and determined that the spiral continues to wind slowly. The spiral pattern is expected to evolve on a much longer timescale than the gas ÿow in the bar since the orbital timescale (the rotation period) is considerably longer at the larger radii where the spiral pattern occurs, and forcing from the bar is greatly reduced outside the bar radius. The steadiness of the gas ÿow pattern inside the bar allows us to compare a snapshot of the simulation to the observed velocity ïeld inside the bar radius. The evolution of the spiral pattern renders a snapshot comparison to observations outside the bar radius unreliable ; fortunately, most noncircular motions are inside the bar radius, as discussed in Paper I.
6

. FEATURES OF THE SIMULATED VELOCITY FIELDS

The models consist of two-dimensional velocity ïelds over the two-dimensional (M/L , ) ) parameter space with the two subsidiary dimensions of !p /! and halo type. It is impossible to show anything more b d a small fraction of than the models without overwhelming the reader. We present four representative models in Figures 1, 2, and 3 and discuss some of their features. These models are drawn from the set with ! /! \ 0.5 and isothermal halos and represent both bd low- and high-mass disks and low and high pattern speeds. In ° 7.4 we also discuss the systematic changes of the model velocity ïelds with M/L and ) , plotting one-dimensional p subsets of the models against the observations. 6.1. Low- versus High-Mass Disks Figures 2a and 2c show the eect of varying the disk mass. These are two simulated velocity ïelds that dier only in disk M/L . The ïeld in Figure 2a is for an approximately maximum disk (M/L \ 2.25), while the ïeld in Figure 2c is for a low-mass disk, with M/L \ 1.0, or about 40% of the maximum disk. Both are for fast-rotating bars, with ) \ p 20, corresponding to r /a \ 1.16 and 1.35, respectively. The L reprojected into the plane of the velocity ïelds have been

sky and rotated to match the observations shown in Paper I. The projected major axis of the galaxy is at 57¡ north of west, while the major axis of the bar runs nearly east-west. The shocks in the bar are the roughly horizontal (eastwest) isovels in the inner 60A of each simulation. Across these isovels, perpendicular to the bar, there is a velocity gradient, particularly large in Figures 2a and 2b. The lowõM/L simulations produce less of a shock in the bar than the highõM/L simulations, as can be seen by examining the bar region of the simulated velocity ïelds in Figures 2c and 2a, respectively. Strongly nonaxisymmetric motion in the low-mass disk (Fig. 2c) is restricted to the inner 15A radius (D1.6 kpc), and the magnitude of the velocity jump across the shock is not large except very close to the bar center. By contrast, the heavy disk produces oset shocks parallel to the bar that extend to 2 or 3 times farther out and have larger velocity jumps. The residual velocity ïelds in Figures 3a and 3c show the importance of the change in shock strength with M/L . The fast-bar, highõM/L simulation in Figure 3a matches the observed velocities fairly well outside the ILR. By contrast, the fast-bar, lowõM/L simulation in Figure 3c produces large systematic residuals up to 50 km s~1, north and south of the bar, because the velocity jump across the shock is not large enough, as discussed further in ° 7.4. Inside the ILR, all models show large residuals, which are not meaningful (see ° 7.1) ; both data and models are aected by limited resolution, and the structure of the ILR is more sensitive to ! /! than global M/L . bAsd discussed above in ° 5, both simulations produce a spiral pattern driven by the bar ; the pattern is stronger in the highõM/L simulation, as is to be expected. The curving shocks to the southeast and northwest are along the spiral arms. The spiral arms in the highõM/L velocity ïeld of Figure 3a are in approximately the same position as the actual spiral arms observed in NGC 4123 just outside the bar. These spiral arms are also visible in the Fabry-Perot Ha velocity ïeld of NGC 4123 (Fig. 4 of Paper I) as chains of large bright H II regions. However, the spiral arms are not a particularly good argument for or against any model because the spiral pattern evolves over time, as mentioned in ° 5. Furthermore, the strength of the spiral arm density contrast and the associated shock are quite strong early in their evolution, stronger than is observed, and the observed spiral pattern actually appears to have a slower pattern speed than the bar, as discussed in Paper I. These factors suggest that the observed spiral pattern in the outer disk is decoupled or at best in resonance with the bar, rather than directly driven by it (see Sellwood & Sparke 1988). 6.2. Fast versus Slow Bars Figures 2a and 2b show the eect of varying the bar rotation speed while holding the M/L ïxed at 2.25, close to the maximum disk value. Figure 2a is for a fast-rotating bar, with ) \ 20 km s~1 kpc~1 and r /a \ 1.35, while Figure p L 2b is for a slow bar, with ) \ 10 and r /a \ 2.53. p The simulation with a fast-rotatingL bar produces oset shocks that lie almost parallel to the bar approximately at the location of the observed shocks and their associated dust lanes. The slow-rotating bar also produces shocks, but the magnitude and extent of the shocks are larger, and the location of the shocks shifts somewhat. Both eects occur because the gas overtakes the slow-rotating bar at a much higher relative speed (see ° 7.4).


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FIG. 2.õFour simulated gas velocity ïelds in the bar region (inset boxes in Fig. 1), reprojected onto the plane of the sky to match the observations. North is up and west is to the right. The projected major axis of the galaxy is at 57¡ northwest. The color scale shows the line-of-sight velocity ïeld. The bar runs approximately east-west, and the shocks are the large velocity gradients perpendicular to the bar. The area between the ellipses is that used for comparison to the data in ° 7. Panels are as in Fig. 1 : (a) heavy disk, M/L \ 2.25, fast bar ; (b) heavy disk, slow bar ; (c) light disk, M/L \ 1.0, fast bar ; (d) light disk, slow bar.

Figures 3a and 3b show the velocity residuals for these models. While the fast-bar, highõM/L model does fairly well (outside the ILR), the slow-bar model in Figure 3b has very large residuals. The velocity jump across the shock is too large and too sharp, causing substantial residuals of Z50 km s~1 (see also ° 7.4). Finally, Figure 2d shows the velocity ïeld for a light disk, M/L \ 1.0, and slow bar, ) \ 10 (r /a \ 2.38). There is a L fairly strong shock in this pmodel, despite the low M/L , because the gas streams through the bar at higher relative speed. However, the shock is displaced very far upstream (in the direction of galactic rotation) compared with the other models ; it is so far from the bar major axis that it is practically out of the bar, in strong disagreement with the observed location. This disagreement is reÿected in the very large residuals, D100 km s~1, seen in Figure 3d.

7.

COMPARISON OF SIMULATED VELOCITY FIELDS TO DATA

We have simulated gas velocity ïelds for six sets of 88 combinations of mass-to-light ratio and pattern speed. We reproject and rotate the velocity ïeld of each simulation into the orientation of the galaxy on the plane of the sky and estimate how well it resembles our observed velocity ïeld. As our goodness-of-ït tests yield remarkably tight constraints on our main parameters, M/L and ) , we also p oer simple physical explanations for the systematic changes, which we illustrate by taking one-dimensional cuts through the velocity ïelds for several ranges of models. 7.1. T he Region Used for the Comparison For a meaningful comparison of models to data, it is


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FIG. 3.õResidual velocity ïelds in the bar region (inset boxes in Fig. 1), for four representative models. The residuals are the Fabry-Perot velocity ïeld from Paper I, minus each of the four models shown in Fig. 2. The orientation is as in Fig. 2. The area between the ellipses is that used for comparison to the data in ° 7. All models have large residuals inside the inner ellipse (8A radius), where both data and models are unreliable as a result of limited resolution ; see ° 7.1. Outside that radius, model (a) has signiïcantly smaller residuals than the others. Panels are as in Fig. 1 : (a) heavy disk, M/L \ 2.25, fast bar ; (b) heavy disk, slow bar ; (c) light disk, M/L \ 1.0, fast bar ; (d) light disk, slow bar.

necessary to use only regions where models and data are primarily sensitive to the parameters being varied and not to other systematic eects. We compare the models and data only in the area between the two ellipses on the sky indicated in Figures 2 and 3, as well as in Figure 4 of Paper I. The outer ellipse has diameters 60@@ ] 48@@, and the inner ellipse has diameters 24@@ ] 16@@. The outer ellipse is chosen to include nearly all the emission within the bar but exclude the large H II regions at the end of the bar, near the beginning of the spiral arms. The inner ellipse is chosen to exclude the knot of emission at the very center of the bar. We chose this region of the data for several reasons. The region of most interest is inside the bar radius, where the nonaxisymmetric motions are strong. Outside the bar radius, there are plenty of velocities with small errors derived from strong Ha emission, but including them in the comparison would place a large statistical weight on

regions where the bar-induced motions are weak and therefore carry little useful information. Furthermore, the spiral arm patterns, which are the main source of noncircular motions outside the bar, evolve over time (° 5). While the galaxy itself does have spiral arms and associated velocity perturbations, comparing them to the spirals in simulations that have not settled would be misguided. We also exclude data at the very center of the galaxy. The velocity gradients are extremely high in this region and are probably unresolved in our data. Also, there are several reasons why our models are most unreliable in this region : our model for the nucleus is probably too soft, deprojection errors from the thicker part of the bar are likely to be the most signiïcant there, our assumption of uniform zthickness is likely to be violated, and the resolution of the gasdynamical code is also worst in the center, in the sense of having relatively few grid cells across the scale of a stream-


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line. None of these problems are troublesome outside R D 10@@, but together they force us to exclude data within that radius from the comparison. The distribution of Ha emission, and hence velocity data, is rather sparse within our comparison region. However, there is still a substantial amount of data. In particular, there are emission regions within and on either side of the bar, which allow us to measure the strength and the location of the main bar shock. As seen in Figure 3, all of the emission regions within the bar (but outside the ILR) provide power to discriminate between models. 7.2. Overall Comparison and L ikelihood Within the comparison area (including the bar but excluding the central region and the disk outside the bar) we can simply use our observed velocities and their associated errors to compute the s2 statistic as a measure of how well the models ït the data. Since our velocity data are oversampled, on 0A 6 pixels but with 1A seeing, we binned .3 .3 down the velocity with a 2 ] 2 bin, to provide 0A 2 sam.7 pling as a compromise between preserving resolution in the velocity ïeld and attaining statistical independence for adjacent pixels. This sampling approximately matches the models, which we do not need to smooth to equalize their resolution. Other choices, such as no binning, or a 3 ] 3 bin (which undersamples the seeing), produce dierent values of reduced s2, but the relative quality of the models is similar to that presented below. Overlarge binning makes s2 artiïcially low, since it smooths both data and model heavily. After the 2 ] 2 binning, we have N \ 471 velocity measurements within the region of interest, with errors ranging from 2.5 to 12 km s~1 ; the median error is 7.4 km s~1. These errors are strictly statistical errors from the Voigt proïle ïtting to the Fabry-Perot data (see Paper I). For each simulation, we calculated s2 of the model, after computing the best-ït galaxy systemic velocity and subtracting it from the observations. For the set with isothermal halos and ! /! \ 0.5, the reduced value s2/N is tabulated with bd the models in column (4) of Table 1. Examining the results in Table 1 and Figure 4, it is clear that heavy-disk, fast-bar models with M/L º 2.0 and ) º 18 are the best match to the data. The best ït is for M/p \ 2.25 and ) \ 20. Light L p disks and slow bars yield miserable ïts. The best models are not perfect ïts to the data, in the sense of attaining s2/N ¹ 1.0. Given the number of idealizations inherent in the modeling procedure, this is not surprising, and we cannot expect the models to reproduce every feature of the data. Another issue is that the purely statistical errors on the velocity data are not completely realistic. Many pixels in bright H II regions have errors much less than 8 km s~1, which are highly weighted in the computation of s2, probably excessively so. A given line of sight likely passes through one to a few H II regions, each of which may have a velocity dierent from the bulk matter-weighted velocity ; our measurement is emission weighted and biased toward dense regions, since Ha emission measure is proportional to n2 l. Hence, our Ha measurements may not sample the e mean velocity ïeld fairly, and the velocity uncertainties should be adjusted for the rms oset of an individual H II region from the mean. In order to get a sense of the importance of this eect, we added an additional ad hoc dispersion component of 8 km s~1 in quadrature to the initial error estimate. This is equal

FIG. 4.õQuality of s2/N over the ) õM/L halos and ! /! p\ 0.5, bd The contours are *(s2/ N \ 471, surrounding ) \ 20. p

modelsî ït to the velocity data. A contour plot of plane for the 88 models in the set with isothermal calculated using strictly statistical velocity errors. N) \ 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, and 100, with the minimum : s2/N \ 3.54, at M/L \ 2.25 and

to the gas sound speed we assumed in our models and a reasonable estimate for the velocity dispersion among H II regions. The eect is to reduce the weight of very low error velocity measurements somewhat, giving the "" better îî values of s2/N tabulated in column (5) of Table 1. As a trade-o, some discriminatory power is lost : the dierence in goodness of ït between models is somewhat lessened. However, the conclusions are unchanged ; light disks and slow bars are still grossly worse ïts to the data. In practice, the purely statistical velocity errors are certainly underestimates, since no systematic eects have been budgeted for, while the boosted error bars probably make the models look too good. The two values of s2/N in columns (4) and (5) of Table 1 then bracket the amount of conïdence one should have in a given model. Figures 4 and 5 show contour plots of s2/N distributed on the ) õM/L plane. Figure 4 shows the values of s2/N p calculated using the strictly statistical velocity error bars, and Figure 5 shows s2/N calculated with the additional 8 km s~1 dispersion added to the error budget. These ïgures show graphically (in both senses of the word) that only heavy-disk, fast-bar models work. LowõM/L disks and slow bars are ruled out. The models that ït the data reasonably well range from M/L \ 2.0 to 2.5 and from ) \ 18 to 24 km s~1 kpc~1 p (r /a \ 1.5 õ 0.8). The disk of NGC 4123 is between 80% and L 100% of the maximum disk mass, and the bar is fast rotating. If instead of adding an additional 8 km s~1 dispersion error component we add 10.8 km s~1 dispersion, then the best model, with M/L \ 2.25 and ) \ 20, has s2/N \ 1.00. p We can then use the values of s2 obtained using this additional 10.8 km s~1 error to determine conïdence limits. Formally, all models but the best few are ruled out. At the 99.99% conïdence level, the models allowed must have *s2 \ 20, or *(s2/N) \ 0.04. The only models allowed are the two with M/L \ 2.25 and ) \ 20 and 21. Even if we p


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FIG. 5.õQuality of modelsî ït to the velocity data. A contour plot of s2/N over the ) õM/L plane for the 88 models in the set with isothermal halos and p ! /! \ 0.5, calculated using the statistical velocity errors and an additional 8 km s~1 dispersion. The contours are *(s2/N) \ 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, and b with N \ 471, surrounding the minimum : s2/N \ 1.40, at M/L \ 2.25 and ) \ 20. 50, d p

attempt to compensate for the correlation in measurements introduced by the seeing, assuming the number of independent measurements is only one-third of N \ 471, the only models allowed at 99.99% conïdence are those with M/ L \ 2.25 and ) \ 20õ24 and the model with M/L \ 2.5 and ) \ 23. p p In practice, these restrictive limits are partly due to the assumption that all the sources of error are Gaussian. The level of statistical signiïcance is nonetheless impressive : models in Figures 4 and 5 with somewhat worse values of s2/N are in fact much poorer matches to the data. 7.3. Results for Other Model Sets We show in Figure 6 the results for all six sets of models, again plotting *(s2/N) over the ) õM/L plane for each set. Although the variation of s2/N pfor dierent sets diers slightly in the details, the primary result is the same. The best model always has M/L of 2 or 2.25 and ) in the range 18õ23 km s~1 kpc~1. Models with M/L \ 2 op ) \ 18 are r p always signiïcantly worse. Figure 6 shows that the subsidiary parameters of halo shape and ! /! have little eect on bd the distribution of *(s2/N). We also computed the results for all sets of models when an additional 8 km s~1 dispersion is added to the statistical errors on the observed velocities, as described above. This procedure weights the observed velocities slightly dierently and, in down-weighting velocity measurements with extremely small errors, is probably a more realistic basis for choosing the best-ït model. When we compute values of

s2/N with the added 8 km s~1 dispersion, the best model, with M/L \ 2.25, ) \ 20, ! /! \ 0.5, and isothermal bd halo, remains the bestp; the model with the same parameters but ! /! \ 0 is almost as good. For each set of ! /! and bd b) halo type, the best model always has M/L \ 2.25 and d in p the range 20õ23 km s~1 kpc~1. Models with M/L \ 2 and ) \ 18 are always ruled out at the 99.99% conïdence p level. No matter how we model the nucleus and the halo or treat the errors on the observed velocities, Figure 6 shows that the best models are always in the same region of M/L and ) , requiring a heavy disk and fast bar. The six sets of p models have best values of s2/N that dier somewhat : there is some preference for a low M/L for the nucleus, which is plausible since it is blue and an emission-line source. There is not a strong discrimination between NFW and isothermal halo proïles. Overall, the best value for disk M/L is I 2.25, and M/L \ 2.0 is a strong lower limit. I compared the models to velocity data only Since we have within the bar radius, technically we require only that M/L must be high within the bar radius. However, in Paper I we showed that an M/L that begins to decline outside the bar radius does not cause a signiïcant decrease in the disk contribution to the rotation curve. Therefore, the high M/L indicates that the disk contribution is at or near maximum. 7.4. A Slit Cut through the Data and Models The previous comparisons of our models with twodimensional data indicate very tight constraints on our two main parameters. In order to gain a physical understanding


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FIG. 6.õQuality of modelsî ït to the velocity data, for all models. The value of s2/N is shown over the ) õM/L plane, for all six sets of models, calculated using strictly statistical velocity errors. Each set contains 88 models. The left panels are for isothermal halos,pthe right panels for pseudo-NFW halos. The top panels have ! /! \ 1, the middle panels ! /! \ 0.5, the bottom panels ! /! \ 0.0. The contours are *(s2/N) \ 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, and 100. d bd (a) Isothermalb halos, ! /! \ 1.0, best s2/N \ 4.90. (b) NFW halos, ! /! b \d1.0, best s2/N \ 5.02. (c) Isothermal halos, ! /! \ 0.5, best s2/N \ 3.54. bd d b (d) NFW halos, ! /! \ 0.5, best s2/N \ 4.08. (e) Isothermal halos, ! /! b\ 0.0, best s2/N \ 3.74. ( f) NFW halos, ! /! \ 0.0, bestds2/N \ 3.94. bd bd bd

of these constraints, we examine a subset of the data in more detail. As we emphasized in Paper I, the strongest signatures of the nonaxisymmetric motions caused by the bar are the oset shocks along the bar, which are also the locations of the dust lanes. We extract a slice nearly perpendicular to the bar from the Fabry-Perot data in order to show the velocity jump in the shock, which we compare with the

corresponding data from the models. We make this comparison for two subsets of the two-parameter M/Lõ) space that include our best-ïtting model, for which M/L p 2.25 \ and ) \ 20. Thep position of our pseudoslit through the Fabry-Perot data is indicated by the vertical line in Figure 4 of Paper I. It is at 17A (2 kpc distance in the disk) east of the galaxy center,


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runs north-south, nearly perpendicular to the bar, and is 2 pixels, or 0A72, wide. We selected it to pass through several . regions of Ha emission, including one that straddles the velocity jump associated with the shock and two regions of Ha emission on either side of the shock. The spatial resolutions of the data and models shown here are similar, about 1A for both. .4 The velocity data are the points with error bars plotted in Figures 7 and 8, while the predictions from sets of models are drawn as smooth curves. The x-axis is distance from the bar major axis in arcseconds, in the direction of galaxy rotation, so that gas overtakes the bar moving from left to right on this plot. The y-axis is projected line-of-sight velocity. The strong gradient in the central part of the data is a combination of the changing projection of the gas streamlines and the shock at the leading edge of the bar. The gradient is clearly spatially resolved. The two H II regions away from the shock do not have signiïcant internal velocity gradients. The change in velocity from one side of the bar to the other is 140 km s~1 in line-of-sight velocity (uncorrected for inclination), a substantial fraction of the velocity width of the galaxy. This gradient arises from the changing projection of the noncircular streamlines and the shock in the bar (see the velocity ïeld in Paper I). The shock is caused by elongation of the streamlines along the bar, so its strength is closely dependent on the bar mass, which controls the ellipticity of the potential.
7.4.1. V arying the Disk Mass

FIG. 8.õCut through the velocity data and models of varying r /a. The L Fabry-Perot velocity data ( ïlled circles) and associated errors from a cut through the data, nearly perpendicular to the bar. The x-axis is distance from the bar in arcseconds. The y-axis is projected line-of-sight velocity. The lines are the velocity ïelds of models ; all have M/L \ 2.25. The ) p values are 10.0 (dotted line), 14.0 (short-dashed line), 20.0 (solid line), and 24.0 (long-dashed line).

In Figure 7 we keep ) ïxed at 20 and plot four models p with M/L \ 1.0, 1.75, 2.25, and 3.0. At M/L \ 1.0, the simulation predicts a rather small velocity change, less than half that observed, but the velocity change increases and becomes steeper as M/L rises. Furthermore, there is a sys-

tematic shift in the position of the strong shock, where the gradient is steepest ; the much smaller velocity wiggles are due to secondary shocks. Of these four models, only that with M/L \ 2.25 shows a shock of about the right strength at the location of the observed velocity jump. The constraint on M/L is strikingly tight. Nonaxisymmetric forces become weaker as M/L decreases, as the inÿuence of the bar declines and that of the round halo grows. Models with M/L \ 2.0 cannot produce a strong enough shock, even at 2 kpc from the galaxy center. The presence of shocks with a large velocity jump along the length of the bar requires that the streamlines be quite elliptical, which requires a massive bar. If the dark matter is dominant in the inner disk, the potential is rounder and cannot produce the shocks.
7.4.2. V arying the Pattern Speed

FIG. 7.õCut through the velocity data and models of varying M/L . The Fabry-Perot velocity data ( ïlled circles) and associated errors from a pseudoslit cut through the data, nearly perpendicular to the bar. The x-axis is distance from the bar in arcseconds. The y-axis is line-of-sight velocity. The lines are the velocity ïelds of models ; all have ) \ 20. The p M/L values and line styles are 1.0 (dotted line), 1.75 (short-dashed line), 2.25 (solid line), and 3.0 (long-dashed line).

In Figure 8 we keep M/L ïxed at 2.25 and plot models of varying ) . The models plotted have ) \ 10, 14, 20, and 24 p p km s~1 kpc~1, corresponding to r /a \ 2.53, 1.93, 1.35, and L 0.68. As the pattern speed is decreased and the Lagrangian radius is increased, the shock moves upstream, in the direction of rotation of the galaxy and of the bar. This behavior occurs because at slower bar pattern speeds the gas overtakes the bar more quickly and climbs farther out of the bar potential before shocking. The higher velocity of the gas in the corotating frame also causes the shock to steepen for slower bars, as noted in ° 6. At ) \ 10, the pattern speed is p so slow that the velocity gradient of the shock is extremely steep. This model is so extreme that the global ÿow pattern changes and the relation between pattern speed and shock location breaks down. The behavior of models in this pseudoslit cut through the data shows clearly why the observations favor models with


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high M/L and high ) . This is also generally true of the p models not shown in either Figure 7 or Figure 8. There is some covariance between the parameters M/L and ) , but p their eects on the location and gradient of the shock are rather dierent. The best models have high-mass disks and fast bars, and it is not possible to rescue low-mass disks with M/L \ 2.0 by appealing to slow bars.
8.

DISCUSSION. I. PROPERTIES OF THE DARK HALO

Our measurement of the disk M/L of NGC 4123 determines the disk mass distribution and, since we have an extended H I rotation curve, the mass distribution of the dark halo. We ïnd that the disk dominates the mass inside the optical radius R (see the rotation curve decomposi25 tions in Paper I for M/L \ 2.25). We have used only data inside the bar radius, but the result is robust : the disk dominates the mass even if its M/L were to decline outside the bar radius (° 6.3 of Paper I). Dark matter halos with either isothermal proïles with a constant-density core or power-law proïles with an inner cusp are both consistent with the data, as discussed in Paper I. However, the halo rotation curve V (R) and equivalently the enclosed mass halo M (\R) are fairly well constrained, especially at large R. halo we have detailed knowledge of the radial distribuHence, tion of luminous and dark mass in this galaxy. In this section we use the mass distributions to test models of the relation between disks and their dark halos and of the process of disk formation by dissipative collapse within the halo. 8.1. Comparison to "" Universal îî Relations The tight correlation between galaxy luminosity and rotation width, the Tully-Fisher relation, suggests that there is some kind of universal relation between galaxy disks and halos, especially since galaxy rotation curves are approximately ÿat out past the optical radius. Recently there have been a number of suggestions for universal relations that govern the disk-to-halo ratios of galaxies, which can be tested against NGC 4123. Additionally, models of structure formation yield predictions for dark matter halo parameters. The "" universal rotation curve îî claimed by Persic, Salucci, & Stel (1996) gives a relation of disk and halo mass with rotation velocity and suggests that galaxies of smaller rotation width are more dark matterõdominated. For NGC 4123, their relation predicts that dark matter should become dynamically detectable at 0.6R (6.7 kpc) and that 25 the mass within this radius is 40% dark. In contrast, we ïnd that the galaxy is at most 15% dark matter within that radius. Several studies motivated by cosmological simulations of structure formation have examined the structure of dark matter halos. Such simulations produce centrally concentrated dark halos, generally with a broken power law density proïle, with a central cusp slope of r~1 or r~1.5 (e.g., Navarro et al. 1996 ; Syer & White 1998 ; Kravtsov et al. 1998 ; Moore et al. 1999 ; Klypin et al. 2000). However, these studies do agree that the central concentration of dark halos is dependent on the mass of the halo and on the cosmological model. Such halos follow one-parameter families : the halo scale density and radius are linked. It is tempting to identify a one-parameter halo family with the Tully-Fisher relation and derive a relation between halo scale and rotation width. Navarro et al. (1996) pro-

posed a relation of halo to galaxy that produces galaxies that are very dark matterõdominated : a galaxy with a maximum rotation of 200 km s~1 is 90% dark matter within the optical radius, and a galaxy with maximum rotation of 100 km s~1 is 96% dark matter within the optical radius. The ) \ 0.3 model of Navarro (1998a), where there is less dark matter to go around, still produces very dark matterõ dominated galaxies. Figure 2 of Navarro (1998a) proposes a disk-halo decomposition of the well-studied galaxy NGC 3198, whose rotation width is about 150 km s~1, similar to that of NGC 4123. In the model of Navarro (1998a), NGC 3198 is 75% dark matter inside the optical radius, the dark halo dominates the rotation curve even at 2 kpc from the galaxy center, and the disk M/L is ^0.8 h . This model is 75 clearly inconsistent with our result for NGC 4123. In Figure 9 we plot the (preõbaryonic collapse) halo scale density and radius predicted for a galaxy of circular velocity V \ 130 km s~1, from the models of Navarro (1998b), and c the conïdence ellipses for our NFW-type halo ïts to the H I rotation curve of NGC 4123, from Paper I. In all cases, for disk M/L from 2.0 to 2.5 and for all three cosmologies, the halos determined by our modeling have lower scale density or radius (or both) than the cold dark matter (CDM) halo predictions. The implication is that either the dark halo is of similar size to the prediction but less dense by a factor of D3 or the dark halo has a smaller break radius than predicted. Since the inner part of an NFW-type halo has o D r~1, a smaller scale radius means a lower halo density at a given physical radius in kiloparsecs. In terms of the NFW halo concentration parameter c, we ïnd that the halo of NGC 4123 has c \ 2.9, while Navarro (1998b) predicted c ^ 5 for the "CDM cosmology with ) \ 0.3. M

FIG. 9.õParameters of the dark halo vs. predictions. The scale density o and radius r of the NFW-type dark halos allowed by our models s compared to thesprecollapse halos predicted by the simulations of Navarro (1998b). The 1 p conïdence ellipses in o [ r space determined from our s s rotation curve ïts in Paper I are shown for the allowed disk M/L from 2.0 to 2.5. The halo parameters predicted for a galaxy of NGC 4123îs rotation width are indicated for three cosmologies.


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The study of Bullock et al. (1999) uses a large number of simulated halos to determine the c [ M relation and scatter. Although it does not directly propose disk-halo decompositions, we can ïnd the concentration expected for the halo of NGC 4123. In a "CDM cosmology, the halo has a mass of M \ 3 ] 1011 M for an NFW-type proïle. vir _ Using a modiïed deïnition of the concentration, c , vir Bullock et al. (1999) predict that such a halo will have c \ vir by 16.6 in a "CDM model, and the scatter is given log c \ 1.22 ^ 0.14. The halo of NGC 4123 has c \ 4.4, vir vir or log c \ 0.64, a 4 p deviation, requiring a much less vir dense halo than predicted. We emphasize that the dark halo we determine by ïtting the rotation curve is seen in its ïnal state, after the baryonic disk has collapsed inside it. The dissipative baryonic collapse must draw the halo inward so that the ïnal halo has a higher density and smaller radius than the initial state. The halo parameter predictions from Navarro (1998b) and Bullock et al. (1999) are precollapse halos, which increases the discrepancy between the predictions and our measurements of halo concentration. In the next section we describe an attempt to quantify the eects of baryonic collapse. 8.2. Baryonic Collapse and the Initial Halo Density Proïle The dominant paradigm of disk galaxy formation requires that the baryonic disk collapse within the dark matter potential, drawing the halo inward (White & Rees 1978 ; Fall & Efstathiou 1980 ; Faber 1981). The ïnal disk and halo mass distributions, which we have measured in the form of V (R) and V (R), can in principle be used to infer disk halo the initial, precollapse halo density proïle, which is clearly of great interest. We show here that using the standard prescription to approximate the collapse process leads to an unrealistic initial proïle. A more sophisticated treatment of the collapse will be needed to determine the initial density proïle of the dark halo from rotation curve information. It is generally assumed that collapse into a galaxy is a two-stage process : ïrst the halo (dark matter and baryons) forms, then the baryonic component cools within the dark matter halo, contracting until it is halted by angular momentum (Fall & Efstathiou 1980). The dissipative baryonic collapse compresses the dark halo. The formalism often used to model this collapse is known as the adiabatic compression or circular orbit approximation (see, e.g., Faber 1981 ; Blumenthal et al. 1986). This formalism is an integral part of many recent treatments of the process of disk formation (Dalcanton, Spergel, & Summers 1997 ; Mo, Mao, & White 1998 ; van den Bosch 1998, 2000 ; Navarro & Steinmetz 2000), which compute the ïnal state of the disk from its angular momentum distribution and use the adiabatic compression formalism to compute the postcollapse dark halo mass proïle. The formalism assumes that (1) the disk can be approximated by a sphere with the same enclosed-mass proïle, (2) the collapse of the disk inside the halo is roughly adiabatic, (3) detailed conservation of angular momentum applies, and (4) "" mass shells do not cross,îî i.e., dark halo particles behave as if they were all on circular orbits. Under these assumptions, the initial and ïnal halos are related by rM (r ) \ r (M (r ) ] M (r )) , (2) i i, total i f f, disk f f, halo f where M(r) indicates the mass interior to r and subscripts i and f indicate initial and ïnal quantities. The halo mass shell originating at r ends up at r . This equation assumes i f

the conservation of angular action, which holds for individual dark matter particles, but in order to apply it to mass shells one has to assume that mass shells have a well-deïned radius and that they do not cross. For this reason we prefer to call the formalism a circular orbit approximation, although many authors simply refer to it as "" adiabatic compression îî of the halo by the disk. Our determination of the disk M/L and of the halo parameters from the H I rotation curve gives us the ïnal mass proïles of both the disk and the halo. From this it is easy to use equation (2) to ïnd the radius r corresponding i to each r (i.e., the collapse factor at each radius) and the f initial mass proïle M (r ), the decompressed halo. We ii carried this out for models with an M/L \ 2.25 stellar disk and the corresponding isothermal and NFW-type ïnal dark halo proïles M (r ). , halo be The ïnal proïles fmust f more centrally concentrated than the initial proïles, so we also ïtted a very centrally concentrated ïnal halo to the rotation curve and decompressed that model, using a "" Moore-type îî halo proïle with a central density cusp of o P r~1.5 (Moore et al. 1999). The parameters of the ïtted ïnal dark halos are given in Table 2. The very centrally concentrated Moore-type halo can make an acceptable ït to the data, but only at the price of a quite large scale radius. The assumption that mass shells do not cross implies that M (r ) \ M (r ). Then the halo can be i, halo i halo f "" decompressed îî by f,solving equation (2) for the initial radius. Using assumption (1) above, we approximate the disk by a sphere and convert from M to V : f, disk d V 2(r ) r i \ (1 [ f ) 1 ] d f , (3) b V 2(r ) r hf f where f is the baryon fraction, i.e., M \f M , and b i, disk i, total V includes the stellar and H I disks. The eect bof the spherd ical assumption is not critical (Barnes 1987 ; Sellwood 1999). We assumed a baryon fraction of 0.1 ; dierent values of f b have little eect on the results, since they just change the scale of r . Figurei 10 shows the initial radius as a function of ïnal radius for the isothermal, NFW-type, and Moore-type halos. The predominant feature is a break at r D 9 kpc, f which is the location of the peak in the disk rotation curve (see Paper I). Inside R \ 9 kpc, V increases steeply, while d outside 9 kpc, V drops as V rises gently. Outside 9 kpc, the d h identical. results for all models are nearly At small radii in the isothermal halo model, r is not a f monotonic function of r , which means that the assumption i cross is not self-consistent. Under that mass shells do not the assumption of the adiabatic circular orbit approximation, there is no initial halo proïle that can yield a ïnal isothermal halo proïle for NGC 4123. In the NFW- and Moore-type halo models, r (r ) does fi not strongly violate monotonicity. There are a few bumps,

C

D

TABLE 2 HALO MODELS : BEST-FIT PARAMETERS FOR M/L \ 2.25 I Halo Type Isothermal ....... NFW type ....... Moore type ...... Scale Density (10~3 M pc~3) _ 4.68 0.127 0.040 Scale Radius (kpc) 6.33 38.2 61.0


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FIG. 10.õHalo compression in the circular orbit approximation. The initial, precollapse radius r of a halo mass shell, obtained using the circular i orbit formalism to uncompress the halo, plotted against its ïnal radius r . f The three models have ïnal halo proïles of isothermal (dotted line), NFW type (dashed line), and Moore type (solid line).

at r \ 3, 9, and 20 kpc, which are clearly related to smalli scale features in the stellar]gas rotation curve (see Fig. 8 of Paper I) and do not represent real features in the collapse proïle of the dark halo. The upper panel of Figure 11 shows the ïnal enclosedmass proïles of the three halo models, determined by ïtting the rotation curve. Since all three models are determined by the same data, they are very similar, except in the central 10 kpc, where the rotation curve is dominated by the stellar disk and the halo is not well constrained. The lower panel of Figure 11 shows the enclosed-mass proïles of the initial, uncompressed halos derived from the three halo models. The initial halos are much less concentrated than the ïnal halos. For the ïnal isothermal halo, the initial mass proïle is unphysical because r (r ) is not monotif onic. For the ïnal NFW- and Moore-type proïles, the initial proïles are essentially physical, with small deviations due to the wiggles in r (r ) caused by structure in the disk]gas rotation curve. i f Outside r D 47 kpc, the mass proïles of all three models are similar, i even for the ïnal isothermal halo. Figure 10 shows that this radius corresponds to r \ 9 õ10 kpc, the f location of the peak of the stellar]gas rotation curve. Inside this radius, the halo proïle is not strongly constrained, and the uncompression is inÿuenced by features in the stellar distribution. Outside this radius, the halo proïle is fairly well constrained by the need to ït the H I rotation curve ; the ïtted ïnal mass proïles for isothermal, NFW-, and Moore-type proïles are similar ; and the initial mass proïles in the three models are also similar. The derived initial mass proïle outside r \ 50 kpc is well i described by a power law : M P r3.3 , implying an initial i, halo i out to r ^ 70 kpc. density proïle of o P r0.3 , extending i i Inside 50 kpc, this power law also ïts the initiali proïle for the NFW model ; the Moore model is shallower, while the isothermal model becomes unphysical.

FIG. 11.õInitial and ïnal halo mass proïles in the circular orbit approximation. Upper panel : Final halo enclosed-mass proïles log M (r ) determined by ïtting the rotation curve, for three model f, halo : halo proïlesf isothermal (dotted line), NFW type (dashed line), and Moore type (solid line). L ower panel : Initial halo enclosed-mass proïles log M (r ) obtained using the circular orbit uncompression formalism. i, halo i Line styles as above. The isothermal model is unphysical inside 35 kpc (see text).

A o P r0.3 density proïle for the pregalactic initial overdensity is extremely unattractive. The problem exists in all three halo models ; it is simply a consequence of the relative disk and halo rotation curves and equation (3). The initial mass proïle is actually best determined at large radius, while a roughly constant density would only be believable in, e.g., the inner core of the pregalactic halo. We believe that the problem is in the circular orbit formalism, which can exaggerate the amount of halo collapse caused by a high-mass disk (Barnes 1987 ; Sellwood 1999). The assumption most likely to be violated is that mass shells do not cross, since a real halo will have some fraction of radial orbits ; radial orbits can provide extra stiness against compression (Barnes 1987). It is also possible that detailed conservation of angular momentum is violated, e.g., through angular momentum transfer within or between disk and halo, or through mass loss. However, transfer of angular momentum from disk to halo would exacerbate the overcooling problem found in CDM]gas simulations, which results in disks that are too small (e.g., Navarro & Steinmetz 1997, 2000). In order to produce the relatively low density and largescale radius ïnal halos that we obtained from ïtting the rotation curve, the circular orbit approximation forces an unrealistically diuse initial condition. The use of the circular orbit formalism to calculate properties of observed galaxies could be misleading. It should be possible in future work to derive the initial halo properties from measured ïnal disk and halo mass proïles, through either N-body simulation or analytic models with more complex dynamics (e.g., a model incorporating a halo distribution function and in which both angular and radial actions are conserved during compression).


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. DISCUSSION. II. IS NGC 4123 TYPICAL ?

The comparisons between the observed velocity ïeld of the bar in NGC 4123 and the gasdynamical models show conclusively that (1) the stellar disk of the galaxy is high mass, in the sense of being close to maximum disk, implying that the disk dominates the rotation curve until well outside the optical radius of the galaxy ; and (2) the bar rotates quickly. These conditions are required to match the strength and location of the shock observed in the bar. We found that NGC 4123 has a disk I-band mass-to-light ratio of 2.0õ2.5 h and is dominated by the stellar disk within the optical75 radius. The best value for disk M/L is I 2.25, and M/L \ 2.0 is a hard lower limit. Examining the I mass models in Figure 8 of Paper I shows that for a disk M/L \ 2.25, the stellar disk contribution reaches 87% of the total rotation velocity at the optical radius R of 11.1 25 kpc, or 76% of the centripetal acceleration at that radius. The disk fraction of the mass inside R is slightly lower, 25 since disk ÿattening boosts its contribution to the centripetal acceleration : the stellar disk is 72% of the mass inside R . The gas disk is responsible for another 5% of the mass 25 inside R . The dark halo, when assumed to be spherical, is 23% of 25 mass inside a sphere of radius R . Table 3 the 25 summarizes the mass fractions for models with M/L \ 2.0õ2.5. In all, the stellar disk is quite dominant inside R I . 25 We note that an error in the distance to NGC 4123 would aect our numerical value of M/L but would not aect our estimate of the degree of maximality of its disk. A larger distance D reduces the disk M/L , with M/L P D~1, but maintains the disk contribution to the rotation curve since L goes up. The result that NGC 4123 has a high-mass stellar disk and a low-concentration dark halo is interesting in itself, since it strongly violates a number of nominally universal predictions for the properties of dark halos of galaxies, as described in the previous section. However, it is clearly important to know whether the result applies to disk galaxies in general. 9.1. T he T ully-Fisher Relation Perhaps the strongest argument that NGC 4123 is typical is its place on the Tully-Fisher relation. As described in Paper I, NGC 4123 is 0.2 mag more luminous than the mean I-band Tully-Fisher relation for galaxies of type Sbc and later, which has an intrinsic scatter of 0.3 mag (Giovanelli et al. 1997). One could interpret its overluminoTABLE 3 DISK AND HALO FRACTIONS WITHIN R 25 FRACTION DISK M/L I All ......... 2.00 ........ 2.25 ........ 2.50 ........ COMPONENT Gas disk Stellar disk Dark halo Stellar disk Dark halo Stellar disk Dark halo V2 rot 0.06 0.66 0.28 0.76 0.18 0.83 0.11 Mass 0.05 0.63 0.31 0.72 0.23 0.79 0.16

sity as a sign that its M/L is 20% o the mean, or that it has 20% more luminous matter than the mean, or as a 10% overestimate of its distance. None of these possibilities bring NGC 4123 signiïcantly closer to the theoretical models. In any case the overluminosity is not statistically signiïcant. Scenarios in which most galaxies have a substantial amount of dark matter inside the optical radius and NGC 4123 is an outlier are unappetizing. A galaxy at the same place on the Tully-Fisher relation but with D50% dark matter inside the optical radius would have to have a stellar M/L dierent by nearly a factor of 2. If stellar M/L varied by such a large amount, the Tully-Fisher relation would have a much larger scatter than is observed. 9.2. Barred Galaxies Is NGC 4123 typical of barred galaxies ? Strongly barred galaxies generally show straight, oset dust lanes like those in NGC 4123 (see Athanassoula 1992) ; many examples can be found in the standard atlases (e.g., Sandage 1961 ; Sandage & Bedke 1988). Spectroscopic studies of strongly barred galaxies show large velocity jumps associated with these dust lanes, as in Figure 5 of Paper I, NGC 1365 (Lindblad & Jorsater 1987 ; Lindblad et al. 1996) and NGC 1530 (Regan et al. 1997). The dust lanes indicate the location of shocks, as discussed above and in Paper I. The position of the dust lanes favors fairly fast rotating bars ; in slow bars the shocks move too far ahead of the bar, as seen in the simulations presented here and the simulations of the Milky Way presented by Weiner & Sellwood (1999). The most secure dynamical determinations of bar pattern speeds in other galaxies also indicate that NGC 936 and NGC 4596 have fast-rotating bars, with corotation just outside the end of the bar (Merriïeld & Kuijken 1995 ; Gerssen et al. 1999 ; using the method of Tremaine & Weinberg 1984). The presence of shocks indicates that the disk M/L is likely to be high ; if the dark matter halo dominates the rotation curve, the potential is too round to produce strong shocks, as can be seen from our M/L \ 1.0 simulation (Figs. 2 and 7). Furthermore, a high disk M/L is also implied by fast-rotating bars, since otherwise the bar that forms in a heavy halo is quickly slowed by dynamical friction, as discussed in ° 2 (Weinberg 1985 ; Debattista & Sellwood 1998). 9.3. W hat about Unbarred Galaxies ? NGC 4123 is strongly barred. Since bars can form by instabilities in massive disks but the instability can be deterred by a massive halo, it has been argued that unbarred galaxies require massive halos (e.g., Ostriker & Peebles 1973 ; Efstathiou, Lake, & Negroponte 1982). Conversely, one could argue that barred galaxies are preferentially disk dominated. However, it takes a lot of halo to deter bar formation : as noted by Kalnajs (1987), central bulges are more efficient than extended dark halos at stabilizing disks. Adding more halo mass in the form of spherical shells at large radius has zero eect on the dynamics of the matter interior to those shells, so stabilizing a disk with a halo requires cranking up the central density. Additionally, bars can form in disks with massive halos (see, e.g., Combes & Sanders 1981 ; Debattista & Sellwood 1998). But is NGC 4123 a self-selected exception by virtue of being strongly barred ?

NOTE.õFractions of rotation velocity squared and of mass within a sphere of radius R (11.1 kpc). The mass fractions for the dark 25 halo assume that it is spherical.


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First, relations that claim universality should clearly apply to unexceptional galaxies, and NGC 4123 is quite ordinary, especially given its modest rotation width. Some 30% of galaxies in major catalogs are classiïed as barred, plus some percentage of intermediate types (Sellwood & Wilkinson 1993), and an even larger bar fraction is found from infrared imaging (Eskridge et al. 2000). It seems unsatisfying to put forward galaxy formation scenarios that exclude such a large percentage of observed galaxies. There are no dierences in the overall H I properties of barred and unbarred galaxies (Bosma 1996). Palunas (1996) ïnds that maximum disk ïts to galaxy rotation curves yield a median I-band M/L of (2.4 ^ 0.9) h , very close to our 75 value for NGC 4123. The distributions of maximum disk M/L for barred and unbarred galaxies in the Palunas (1996) sample are indistinguishable (Sellwood 1999). We are unaware of any evidence that barred and unbarred galaxies fall on dierent Tully-Fisher relations. Many Tully-Fisher surveys tend to avoid selecting barred galaxies. However, Figure 1 of Syer, Mao, & Mo (1999) plots a measure of total (disk]halo) mass-to-light ratio versus central surface brightness (a variant of the TullyFisher relation) for the 2446 galaxies for which Mathewson & Ford (1996) obtained rotation curves and I-band photometry. A total of 175 of these galaxies were classiïed as barred. There is no dierence in the distribution of barred versus unbarred galaxies, implying that the barred and unbarred galaxies do have similar luminosities at the same velocity width (see also Sellwood 1999). Syer et al. (1999) nevertheless argued that disk galaxies should have lowmass disks, M/L [ 1.4 h , in order to satisfy a stability 75 criterion designed to deter bar formation. It is extremely difficult to understand how barred and unbarred galaxies could lie on the same Tully-Fischer relation and have similar maximum disk M/L and H I properties yet have true disk mass-to-light ratios varying by a factor of 2õ3. In fact, Syer et al. (1999) themselves do not favor such an approach : they argued that perhaps disks have a critical value of M/L , so that bars do not form spontaneously but can form when induced by, e.g., encounters between galaxies. This argument is not particularly compelling, especially in light of the high disk M/L of NGC 4123 and the dynamical friction argument for high disk M/L in most barred galaxies (Debattista & Sellwood 1998). Relying on extremely dominant halos to deter bar formation leads to a division between barred and unbarred galaxies for which there is no evidence and in which the existence of intermediate types is hard to understand. Bar formation can be deterred by other means than a heavy halo ; Kalnajs (1987) emphasized the importance of bulges in stabilizing disks (see also Sellwood & Evans 2000). Bars can be deterred by a central mass dense enough to cause an inner Lindblad resonance (Toomre 1981) or destroyed by a relatively small mass accumulation at the center of the bar, which can arise naturally as the bar torque drives gas to the center (Pfenniger & Norman 1990 ; Hasan, Pfenniger, & Norman 1993 ; Friedli & Benz 1993 ; Norman, Sellwood, & Hasan 1996). Many unbarred galaxies with dense centers may have once had bars, and whether a high surface brightness galaxy is barred or unbarred today may depend more on its formation history than on its disk/halo ratio (Sellwood & Moore 1999). The high disk M/L of NGC 4123, with its modest rotation width, suggests that the common picture in which

lower rotation width galaxies are more dark matterõ dominated may not be correct for HSB galaxies. LSB galaxies appear to be quite dark matterõdominated (de Blok & McGaugh 1996, 1997 ; Swaters et al. 2000), and surface brightness, rather than rotation width, is likely to be the sequence along which dark matter content varies. Detailed models of galaxy formation predict that speciïc angular momentum, not mass, is the parameter that controls the baryon collapse factor, so that a galaxy with larger angular momentum is both more spread out (lower surface brightness) and has less baryonic mass within a given number of scale lengths (more dark matterõdominated and higher total M/L ) (Dalcanton et al. 1997).
10.

CONCLUSIONS

Comparing the observed noncircular motions in NGC 4123 to simulations of gas ÿow in model potentials, derived directly from the light distribution, shows that the galaxy has a high-mass disk and a fast-rotating bar. The best match between simulations and data is for an I-band disk M/L of 2.25 and a bar pattern speed ) of 20 km s~1 kpc~1, p implying a Lagrangian radius r of 1.35 times the bar semiL major axis a. The acceptable range of the I-band disk M/L is 2.0õ2.5. The maximum disk value is 2.5 for a disk in an isothermal halo, or 2.25 for a disk in a power-law halo similar to the NFW proïle, so the galaxy is 80%õ100% of the maximal disk. The acceptable range of ) is from 18 to p 24, implying that r /a \ 1.5. L Lighter disks can be excluded because they do not produce shocks as strong as those observed. Slower bars can be excluded because the shock occurs in a location that does not match the observations : slower bars produce a shock that is farther ahead of the bar, in the direction of galactic rotation. The observed shocks occur at the location of straight oset dust lanes of the type commonly seen in strong bars (Athanassoula 1992). The ubiquity of these dust lanes suggests that strong bars generally rotate quickly, otherwise the shocks would occur in the wrong place. Fast-rotating bars imply near-maximal disks, since in halo-dominated disks the bar is slowed by dynamical friction (Weinberg 1985 ; Debattista & Sellwood 1998). Low-mass disks also do not produce shocks that extend the length of the bar, unless the pattern speed is very slow, in which case the shocks are again too far ahead of the bar. These results suggest that strongly barred galaxies have high-mass disks and fastrotating bars. The near-maximal disk of NGC 4123 does not agree with a number of predictions that suggest that galaxies with rotation curve height ^130 km s~1 should be dominated by dark matter inside the optical radius (e.g., Persic et al. 1996 ; Navarro 1998b), nor with predictions that disk galaxies must have low M/L to satisfy stability criteria (e.g., Efstathiou et al. 1982). The high disk M/L can be consistent with a power-law NFW-type halo, but the halo must be much less dense than predicted by the relations between halo concentration and mass found in CDM and "CDM simulations (Navarro et al. 1996 ; Bullock et al. 1999). When we attempt to derive the initial state of the halo from the adiabatic compression and circular orbit formalism commonly used to model disk collapse within a dark halo (Faber 1981 ; Blumenthal et al. 1986), the inferred initial density proïle goes as r0.3. This unrealistic halo proïle indicates a failure of the circular orbit approximation.


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Though NGC 4123 is barred, there is no sign that barred galaxies are more disk dominated than unbarred galaxies, at least for HSB galaxies ; their Tully-Fisher relations and total mass-to-light ratios are similar. Requiring heavy halos to stabilize disks is not necessary, since bars can also be deterred or destroyed by central bulges. The ultimate question of why some galaxies are barred and some unbarred remains unsettled, but dark halo mass is not the controlling factor. We have shown that the noncircular motions in a barred galaxy can be used to determine the M/L of its stellar component. Together with H I observations, these can be used to constrain the radial density proïle of the galaxyîs dark matter halo. While not every barred galaxy is suitable for the modeling process outlined here, observations and modeling of more galaxies will eventually lead us to determi-

nations of stellar M/L and dark halo parameters for galaxies as a function of their rotation width, surface brightness, and position on the Hubble sequence. We thank Jacqueline van Gorkom for her contributions to and support of this project, Dick van Albada and Lia Athanassoula for providing the ÿuid-dynamical code and for comments on its use, Povilas Palunas and Julianne Dalcanton for many helpful conversations and encouragement, and Jason Prochaska and the anonymous referee for comments that improved the manuscript. This research was supported in part by NSF grant AST 96-17088 and NASA LTSA grant NAG 5-6037 to J. A. S. and by NSF grant AST 96-19510 to T. B. W. In addition, B. J. W. has been supported by a Barbara McClintock postdoctoral fellowship from the Carnegie Institution of Washington.

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