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THE ASTRONOMICAL JOURNAL, 120 : 2884 х 2903, 2000 December
( 2000. The American Astronomical Society. All rights reserved. Printed in U.S.A.

MAXIMUM DISK MASS MODELS FOR SPIRAL GALAXIES POVILAS PALUNAS1,2
Laboratory for Astronomy and Solar Physics, NASA/Goddard Space Flight Center, Greenbelt, MD 20771 ; palunas=gsfc.nasa.gov

AND T. B. WILLIAMS1
Department of Physics and Astronomy, Rutgers University, Box 0849, Piscataway, NJ 08855-0849 ; williams=physics.rutgers.edu Received 1996 December 2 ; accepted 2000 September 6

ABSTRACT We present axisymmetric maximum disk mass models for a sample of 74 spiral galaxies taken from the southern sky Fabry-Perot Tully-Fisher survey by Schommer et al. The sample contains galaxies spanning a large range of morphologies and having rotation widths from 180 km s ~1 to 680 km s ~1. For each galaxy we have an I-band image and a two-dimensional Ha velocity пeld. We decompose the disk and bulge by пtting models directly to the I-band image. This method utilizes both the distinct surface brightness proпles and shapes of the projected disk and bulge in the galaxy images. The luminosity proпles and rotation curves are derived using consistent centers, position angles, and inclinations derived from the photometry and velocity maps. The distribution of mass is modeled as a sum of disk and bulge components with distinct, constant mass-to-light ratios. No dark matter halo is included in the пts. The models reproduce the overall structure of the rotation curves in the majority of galaxies, providing good пts to galaxies that exhibit pronounced structural dierences in their surface brightness proпles. Of galaxies for which the rotation curve is measured to R or beyond 75% are well пtted by 23.5 a mass-traces-light model for the entire region within R . The models for about 20% of the galaxies 23.5 do not пt well ; the failure of most of these models is traced directly to nonaxisymmetric structures, primarily bars but also strong spiral arms. The median I-band M/L of the disk plus bulge is 2.4 ^ 0.9 h in solar units, consistent with normal stellar populations. These results require either that the mass 75 of dark matter within the optical disk of spiral galaxies is small or that its distribution is very precisely coupled to the distribution of luminous matter. Key words : galaxies : fundamental parameters х galaxies : halos х galaxies : kinematics and dynamics х galaxies : photometry х galaxies : spiral х galaxies : structure
1.

INTRODUCTION

Extended H I rotation curves provide deeply compelling evidence for a dark matter component that dominates the total mass of spiral galaxies. However, the fraction of luminous to dark matter (L /D) within the optical disk (within the 25 mag arcsec~2 B-band, 23.5 mag arcsec~2 I-band isophote) is very poorly known. Typically, extended rotation curves have no distinct features that might reяect the end of the luminous disk (Bahcall & Casertano 1985). As a consequence, they provide virtually no constraint on L /D within the optical disk. Mass models in which the dark component dominates the inner mass distribution generally пt the rotation curves within the optical disk, as well as models with a negligible dark component (van Albada & Sancisi 1986 ; Lake & Feinswog 1989). The mass and luminosity of spiral galaxies are, however, very strongly connected. Spiral galaxies exhibit a very tight, one parameter relation between rotational velocity and luminosity, the Tully-Fisher (T-F) relation (Tully & Fisher 1977 ; see Jacoby et al. 1992 for a review). The total mass-tolight ratio (M/L ) within the optical disk is remarkably uniform among all spirals (Rubin 1985 ; Roberts & Haynes 1994) including low surface brightness spirals (Sprayberry et
ххххххххххххххх 1 Visiting Astronomer, Cerro Tololo Inter-American Observatory (CTIO). CTIO is operated by the Association of Universities for Research in Astronomy (AURA), Inc., under contract to the National Science Foundation (NSF). 2 NASA/NRC Resident Research Associate.

al. 1995). Indeed, the typical value of M/L [we derive here (M/L ) \ 2.4 ^ 0.9] is consistent with that expected from I normal stellar populations (Larson & Tinsley 1978 ; Bruzual & Charlot 1993 ; Worthey 1994) with no dark matter. Optical rotation curves are not, in general, яat and featureless. They span a range of shapes from linearly rising to falling with radius (Rubin 1985) and also display smaller scale "" bumps and wiggles оо (Freeman 1992). Mass models пtted to rotation curves within the optical disk rarely require dark matter halos to yield good пts (Kalnajs 1983 ; Kent 1986 ; Buchhorn 1992). These models generally reproduce the large-scale features and sometimes reproduce the smaller scale "" bumps and wiggles оо in the optical rotation curves. Dynamical arguments suggest that the inner regions of spirals cannot be dominated by dark matter. Within the optical radius, a dark matter halo with an average projected surface mass density greater than that of the disk would act to suppress common instabilities, such as bars and two-arm spiral structure (Athanassoula, Bosma, & Papaioannou 1987). The existence of lopsided modes, which appear in many disk galaxies (Rix & Zaritsky 1995), may require even lower halo mass densities. A bulge or thick disk also act to stabilize the disk, further lowering the allowed dark matter density. It is possible to construct a model disk galaxy with an almost яat rotation curve which is stable with no dark matter (Sellwood & Evans 2001). This reverses the disk stability argument Ostriker & Peebles (1973), which is often used to support the need for a massive dark matter halo within the optical radius. Of course, spherical halo mass 2884

DISK MASS MODELS outside the optical radius has no eect on disk stability. A stellar bar would also interact with a massive halo through dynamical friction. For large dark matter densities, this interaction acts to rapidly slow the pattern rotation speed of the bar (Weinberg 1985 ; Debattista & Sellwood 2000) far below that expected in real galaxies (Sellwood & Wilkinson 1993), or measured in any galaxy (Merriпeld & Kuijken 1995 ; Gerssen, Kuijken, & Merriпeld 1999). Simulations of galaxy formation through dissipationless collapse of matter yield strongly triaxial halos (Dubinski & Carlberg 1991 ; Warren et al. 1992), which produce nonaxisymmetric disks if the halo dominates the inner mass density. Adding a dissipational gaseous component to these simulations leads to rounder, but still triaxial halos (Katz & Gunn 1991 ; Dubinski 1994). The intrinsic ellipticity and noncircular motions of these disks would produce large scatter in the T-F relation (Franx & de Zeeuw 1992), which is not observed. This limits either the halo triaxiality to less than that predicted by any of the formation models or the mass of dark matter within the optical disk. The shapes of the gravitational potentials of dark matter halos for two galaxies have been determined through dynamical models of a polar ring (Sackett et al. 1994) and the яaring of H I gas (Olling 1996). Both of these studies suggest that halos have very яat mass distributions, with axis ratios between 0.1 and 0.3, and thus resemble a disk more closely than a sphere. The various pieces of evidence discussed above strongly suggest that the luminous component contains a dynamically signiпcant fraction (є 1 ) of the mass in the inner 2 regions of spiral galaxies. However, most rotation curves (see Casertano & van Gorkom 1991) for two possible counter-examples) show no signiпcant change in the transition between this inner region and the outer, dark matterх dominated region. The disk and halo must somehow conspire to hide this transition (van Albada & Sancisi 1986). Moreover, because of the low scatter in the T-F relation and in M/L s, the disk-halo conspiracy must act consistently among all spiral galaxies to couple tightly the evolutionary history and the present structure of luminous disks and dark halos. This is extremely surprising, because after the initial collapse the evolution of the halo and disk proceed on dierent scales and through dierent and complex physical processes which are linked only through a weak gravitational coupling. All of the above considerations led van Albada & Sancisi (1986) to advance the maximum disk hypothesis. Under this hypothesis, the mass of the luminous disk in a spiral galaxy is assumed to be as large as possible, consistent with the galaxyоs rotation curve. The mass contribution of the dark matter halo is therefore assumed negligible in the inner parts of spirals. A maximal disk does not eliminate the disk-halo conspiracy ; in a sense, it makes it more puzzling because it minimizes the overlap in the distribution of dark and luminous matter. However, it does make the tight correlation between mass and luminosity in the inner parts of spirals more plausible. Not all lines of evidence support the maximum disk hypothesis. Kuijken & Gilmore (1991) quote a local Galactic surface mass density, derived from the velocity dispersion of K dwarfs, which is 30% higher than that of "" identiпed оо matter. Cosmological N-body simulations of hierarchical universes yield halos which are not well approximated by isothermal spheres with cores (Dubinski & Carlberg 1991 ; Warren et al. 1992 ; Navarro, Frenk, &

2885

White 1996). The mass density proпles of these simulated halos have slowly changing logarithmic slopes and continue to rise all the way into the centers. The central mass concentration of these simulated halos would seem to be inconsistent with a maximum disk. In this work we model the luminous mass distribution for a sample of 74 spiral galaxies. Our aim is to test the maximum disk hypothesis by analyzing how well features in the rotation curve are reproduced by the mass models. We therefore do not include a dark halo in the пts, but rather evaluate the quality of the пts under the strict maximum disk hypothesis. We extract surface brightness proпles for the disk and bulge from I-band images of normal spiral galaxies and derive optical Ha rotation curves from FabryPerot velocity maps with two spatial dimensions. The optical radius is deпned as the radius to the extinction corrected 23.5 mag arcsec~2 isophote in I. The Fabry-Perot maps provide high signal-to-noise rotation curves, and enable us to average over local kinematic features. We пt axisymmetric mass models to optical rotation curves assuming constant, but distinct, mass-to-light ratios for the disk and bulge. The two dimensional information provided by the photometric images and velocity maps allows us to assess the importance of nonaxisymmetric features. This will be fully explored in a future paper.
2.

THE DATA/SAMPLE

Our sample consists of 74 пeld and cluster spirals in the vicinity of the Hydra-Centaurus cluster. For each galaxy in the sample we have an I-band image and a two dimensional Ha velocity map. Sixty-one galaxies are taken from Schommer et al. (1993, hereafter SBWM). The observations and preliminary reductions of the data for these galaxies are described there. Observations and reductions for 13 additional galaxies are presented in this paper. The sample includes galaxies which are members of the clusters : Antlia (6), Hydra (12), Centaurus (16), and Klemola 27 (5) (see Table 1). I-band CCD images were taken on 1994 March 15 with the Cerro Tololo Inter-American Observatory (CTIO) 0.9 m telescope at the Cassegrain focus. The detector was a TEK 1024 with a scale of 0A 9 pixel~1. The exposure time .3 for each image was 10 minutes. Typical seeing was D1A . .5 The images were bias-subtracted and яat-пelded with twilight sky яats using IRAF.3 A large fraction of the SBWM images were taken with a TI 800 ] 800 CCD which had large, 1.5%, яat-пeld errors. The errors were large-scale and primarily near the edges of the пeld, which made precise sky estimation difficult and created signiпcant distortions at low surface brightness. An illumination correction was constructed for the TI images by combining all 40 I-band images from the run. For each image the galaxy and stars were removed and the remaining pixels were scaled by the sky level. The resultant image was smoothed over a scale of 20 pixels. The corrected images are яat to 0.2%. All the images were calibrated using Graham standards (Graham 1982). The instrumental magnitudes of the standards were determined using DAOPHOT II (Stetson 1987). DAOPHOT was also used to automatically locate cosmic rays and stars in the galaxy images. Stars and cosmic rays were distinguished using the DAOPHOT sharpness paramххххххххххххххх 3 IRAF is distributed by NOAO, which is operated by AURA, under contract to the NSF.

TABLE 1 GALAXY SAMPLE

Galaxy (1)

Cluster (2)

D (Mpc) (3) H Type (RC3) (4) F Type (5) k 0 (mag arcsec~2) (7) B/D (10) / (deg) (12) log (2v ) 0 (km s ~1) (13) M/ L D (solar) (14)

m I (mag) (6)

R 23.5 (kpc) (8)

r d (kpc) (9)

i (deg) (11)

M/ L B (solar) (15)

M/L (solar) (16)

Cen Cen

2886

Cen Cen Cen Cen Cen Cen Cen Cen Cen Cen Cen Cen Cen Cen

a1644d83 (a) ...... e215g39 (b) ........ e216g20 (c) ........ e263g14 (d) ........ e267g29 (e) ........ e267g30 (f) ........ e268g37 (g) ........ e268g44 (h) ........ e269g61 (i) ........ e317g41 (j) ........ e322g19 (k) ........ e322g36 (l) ........ e322g42 (m) ....... e322g44 (n) ........ e322g45 (o) ........ e322g48 (p) ........ e322g76 (q) ........ e322g77 (r) ........ e322g82 (s) ........ e322g87 (t) ........ e323g25 (u) ........ e323g27 (v) ........ e323g39 (w) ....... e323g42 (x) ........ e323g73 (y) ........ e374g02 (z) ........ e374g03 (aa) ...... e375g02 (ab) ...... e375g12 (ac) ....... e375g29 (ad) ...... e376g02 (ae) ....... e376g10 (af) ....... e377g11 (ag) ...... e381g05 (ah) ...... e381g51 (ai) ....... e382g06 (aj) ....... e382g58 (ak) ...... e383g02 (al) ....... e383g88 (am) ...... e435g26 (an) ...... e435g50 (ao) ...... e436g39 (ar) ....... e437g04 (aq) ...... e437g30 (ar) ....... e437g31 (as) ....... e437g34 (at) .......

Antlia Antlia Antlia Antlia

Antlia Antlia Hydra Hydra Hydra Hydra Hydra

79.62 61.29 77.85 69.83 76.23 75.92 68.50 49.95 69.40 81.17 45.23 43.51 55.99 52.88 44.19 60.91 64.28 38.19 65.84 52.13 59.76 54.90 69.90 59.73 69.63 41.42 43.22 43.75 42.86 56.12 59.44 46.21 46.01 79.56 70.72 65.44 106.20 85.40 59.51 40.32 40.60 51.25 48.10 54.24 56.17 54.83

10 5 10 3 2 3 5 3 3 2 6 4 5 5 5 3 4 3 5 3 4 5 10 10 10 3 6 3 3 5 4 8 2 10 3 10 4 5 4 5 5 4 4 4 7 3

I II II I I I I I I I I I II I I I II I I II I II I I I I I II I I II I II I II I I I I II II II II I I II

14.42 12.01 12.07 11.32 11.84 11.48 12.39 12.22 10.81 11.53 12.64 10.83 11.91 11.57 11.52 12.59 11.97 11.49 11.05 11.47 11.34 11.14 13.35 11.53 12.44 9.91 11.51 12.01 9.28 11.99 11.35 11.00 10.72 13.30 11.44 13.20 11.21 12.22 11.69 10.13 13.53 11.81 11.76 10.89 12.99 14.25

21.23 19.74 19.35 18.54 19.31 19.77 19.65 18.87 19.17 18.89 19.90 19.06 20.76 19.79 19.12 19.94 20.50 18.84 19.46 19.87 18.69 19.98 20.46 19.57 18.46 19.08 20.11 19.17 18.97 19.83 19.21 20.85 19.85 19.35 19.34 19.60 19.94 19.76 19.78 19.67 21.34 19.97 19.72 19.57 20.37 21.73

7.32 12.92 13.46 15.93 15.86 16.40 12.50 8.38 24.00 17.66 8.10 14.19 14.57 12.53 10.36 10.56 12.22 8.57 19.16 14.11 14.27 15.85 9.35 16.61 9.47 18.08 13.20 8.65 27.64 13.15 14.68 19.53 17.35 9.33 15.18 8.51 32.75 16.82 15.25 20.33 5.55 11.39 10.84 19.05 8.89 5.32

3.64 4.20 2.90 3.60 4.31 4.98 3.43 1.91 5.97 4.19 2.47 3.50 4.65 3.71 2.85 2.87 2.91 1.71 5.34 4.52 3.31 3.76 3.42 4.44 2.06 4.42 4.25 2.15 6.66 3.54 3.87 8.12 3.51 2.42 2.71 2.33 9.42 4.87 4.77 3.65 2.23 2.32 2.80 5.28 3.12 2.91

0.06 0.08 0.09 0.04 0.12 0.62 0.08 0.04 0.03 0.03 0.00 0.04 0.05 0.32 0.03 0.00 0.30 0.04 0.11 0.02 0.01 0.05 0.07 0.03 0.02 0.45 0.03 0.01 0.12 0.00 0.16 0.02 0.12 0.02 0.12 0.01 0.12 0.10 0.02 0.06 0.00 0.29 0.06 0.12 0.02 0.04

77 50 74 60 51 55 55 62 76 71 79 53 71 68 67 76 57 70 63 80 55 58 53 69 48 52 71 64 44 80 75 76 73 41 82 54 79 60 67 51 82 81 63 77 52 63

278 29 306 289 313 294 120 244 253 105 300 104 42 89 308 36 259 172 8 138 283 275 265 79 358 303 149 20 309 321 159 93 58 299 58 86 153 213 274 117 70 82 320 124 334 77

2.29 2.48 2.65 2.54 2.66 2.72 2.50 2.48 2.74 2.69 2.41 2.50 2.38 2.44 2.52 2.36 2.54 2.61 2.63 2.53 2.66 2.63 2.33 2.45 2.51 2.71 2.34 2.45 2.75 2.43 2.62 2.55 2.59 2.49 2.70 2.47 2.80 2.58 2.55 2.64 2.26 2.56 2.56 2.62 2.38 2.24

3.00 2.16 2.42 0.92 2.94 4.32 2.33 2.04 2.59 2.43 2.06 1.28 1.47 1.69 1.73 0.99 2.34 3.61 2.28 2.40 2.61 2.53 2.02 1.71 1.13 2.63 1.45 1.73 2.19 1.95 2.48 3.92 2.40 2.25 2.92 3.28 4.32 2.86 2.75 2.76 2.08 3.62 2.65 3.06 3.22 4.58

0.15 0.81 0.83 0.63 0.48 1.14 0.00 1.54 0.00 1.09 0.00 0.22 0.00 0.76 0.00 0.00 1.27 0.57 1.43 0.60 2.75 2.54 0.00 0.00 0.00 1.00 0.00 0.13 2.18 0.00 0.92 0.00 0.00 0.00 0.00 0.00 0.79 2.29 4.00 1.71 0.00 0.00 0.00 0.65 0.00 0.00

2.84 2.06 2.29 0.91 2.68 3.10 2.17 2.02 2.51 2.39 2.06 1.24 1.39 1.46 1.69 0.99 2.09 3.49 2.20 2.37 2.61 2.53 1.89 1.66 1.10 2.12 1.41 1.72 2.19 1.95 2.26 3.86 2.14 2.19 2.61 3.25 3.93 2.81 2.77 2.71 2.08 2.80 2.49 2.81 3.17 4.42

TABLE 1хContinued

Galaxy (1)

Cluster (2)

D (Mpc) (3) H Type (RC3) (4) F Type (5) k 0 (mag arcsec~2) (7) B/D (10) / (deg) (12)

m I (mag) (6)

R 23.5 (kpc) (8)

r d (kpc) (9)

i (deg) (11)

log (2v ) 0 (km s ~1) (13)

M/L D (solar) (14)

M/L B (solar) (15)

M/L (solar) (16)

Hydra

K27 K27 K27 K27

2887

K27

e437g54 e438g08 e438g15 e439g18 e439g20 e441g22 e444g21 e444g47 e444g86 e445g15 e445g19 e445g35 e445g39 e445g58 e445g81 e446g01 e446g17 e501g01 e501g11 e501g15 e501g68 e501g86 e502g02 e509g80 e509g91 e510g11 e569g17 e572g17

(au) ...... (av) ....... (aw) ...... (ax) ....... (ay) ....... (az) ....... (ba) ...... (bb) ...... (bc) ....... (bd) ...... (be) ....... (bf) ....... (bf) ....... (bh) ...... (bi) ....... (bj) ....... (bk) ...... (bl) ....... (bm) ...... (bn) ...... (bo) ...... (bp) ...... (bq) ...... (br) ....... (bs) ....... (bt) ....... (bu) ...... (bv) ......

Hydra Hydra Hydra Hydra Hydra Hydra

50.15 124.3 49.96 122.20 59.84 90.40 60.68 62.40 58.18 60.34 66.05 68.02 61.59 70.78 61.12 98.34 58.52 55.57 54.94 49.43 45.77 54.28 56.91 92.86 72.29 81.13 57.77 93.44

3 10 4 4 4 4 10 6 10 10 4 3 3 4 4 4 3 7 7 1 10 4 3 4 6 1 3 0

I I II I I II I I II II I I I I II I I I II I II II I II II I I I

12.94 12.66 11.38 12.09 11.84 11.10 12.86 12.82 11.56 12.05 11.69 11.69 11.17 11.75 11.36 12.07 11.05 12.64 12.51 10.35 11.93 11.78 11.55 11.86 12.70 11.84 11.99 11.55

20.35 18.60 21.06 19.59 19.06 20.19 21.67 19.63 19.41 20.26 19.35 18.96 18.65 19.24 19.76 19.63 20.51 21.12 20.46 19.44 21.27 20.84 18.85 20.70 21.23 19.01 18.29 18.76

8.24 14.27 15.16 20.99 12.28 27.97 10.97 9.51 12.93 11.12 15.10 13.27 15.48 15.96 15.80 18.96 18.87 11.00 10.37 19.66 12.01 13.67 12.42 20.29 13.61 17.65 8.03 19.22

2.84 3.15 2.87 5.90 3.02 7.50 6.44 2.71 3.55 2.51 4.32 3.38 3.58 4.05 3.60 5.27 6.87 5.06 2.46 5.24 3.99 3.07 2.90 4.36 3.68 3.52 1.69 4.56

0.11 0.18 0.10 0.30 0.08 0.21 0.05 0.00 0.20 0.52 0.03 0.20 0.03 0.06 0.07 0.28 0.30 0.06 0.01 0.36 0.19 0.08 0.13 0.14 0.09 0.13 0.05 0.12

83 40 71 43 65 73 84 71 78 66 67 42 61 63 79 53 54 55 81 60 70 60 63 61 79 66 45 48

49 89 215 276 281 176 64 22 252 56 70 175 64 330 3 323 148 334 94 111 18 201 73 171 131 64 171 59

2.46 2.50 2.53 2.76 2.64 2.77 2.36 2.44 2.62 2.58 2.60 2.76 2.78 2.60 2.67 2.63 2.60 2.39 2.41 2.75 2.54 2.52 2.64 2.70 2.46 2.68 2.55 2.83

5.60 0.90 2.14 4.15 3.46 4.26 4.50 1.92 2.86 4.02 2.37 4.27 3.74 2.42 2.74 2.83 3.11 4.81 2.61 3.46 4.95 2.75 2.61 3.95 2.96 3.02 1.48 3.71

0.24 0.40 2.31 0.00 0.00 1.07 0.57 0.00 0.51 1.14 0.51 3.44 1.49 0.03 0.00 1.01 0.24 0.00 0.00 2.47 3.28 0.49 1.31 2.38 0.04 1.88 0.19 1.76

5.05 0.82 2.15 3.20 3.21 3.70 4.32 1.92 2.46 3.04 2.32 4.13 3.68 2.29 2.55 2.43 2.45 4.54 2.57 3.19 4.69 2.59 2.46 3.75 2.71 2.89 1.41 3.51

NOTE.хCol. 1 : ESO-Uppsala Catalog number for galaxies in the sample. The codes for Fig. 1 are also given. Col. 2 : cluster membership. Cen \ Centaurus. Col. 3 : distance in megaparsecs, assuming Hubble яow with respect to the cosmic microwave background frame and H \ 75 km s ~1 Mpc~1. Col. 4 : Hubble type in de Vaucouleurs et al. 1991 (RC3). Col. 5 : Freeman type. Col. 0 6 : total I-band magnitude. Col. 7 : disk central surface brightness in I. Col. 8 : major axis radius of the I \ 23.5 mag arcsec~2 isophote in kpc. Col. 9 : exponential disk-scale length. We quote the scale length of the outer disk for Freeman type II galaxies. Col. 10 : bulge-to-disk luminosity ratio. Col. 11 : inclination. Col. 12 : major-axis position angle measured north through east. Col. 13 : log of the circular rotation velocity width in km s ~1. Col. 14 : I-band M/L of the disk in solar units. Col. 15 : I-band M/L of the bulge in solar units. Col. 16 : I-band M/L of the disk plus bulge in solar units.

2888

PALUNAS & WILLIAMS

Vol. 120

eter. The cosmic rays were removed by replacing aected pixels with the biweight of surrounding pixels (the biweight is a robust estimate of the mode of a distribution which is optimized for small samples ; see Beers, Flynn, & Gebhardt 1990). The stars were removed by яagging pixels in a circular aperture centered on each star. Ha spectroscopy for each galaxy was obtained in 1993 April with the CTIO 1.5 m telescope and the Rutgers Fabry-Perot imaging spectrophotometer. A TEK 512 CCD with a 1A 9 pixel~1 scale was used. The observations for .0 each galaxy consist of eight to 15 images with a 110 km s ~1 FWHM bandpass (2.4 A at Ha), spaced at 1 A intervals to sample the Ha emission line. Wavelengths were calibrated during the day. Calibration drifts during the night were monitored by taking exposures of a neon lamp every hour. The maximum drift rate was 0.1 A hr~1. The images were bias-subtracted and яat-пelded, with dome яats taken near the wavelength of each image, using IRAF. Typical seeing was D1A . .5 Transparency and instrumental throughput variations were measured by performing photometry on stars common to all the images for each galaxy using DAOPHOT. Cosmic rays were removed using the procedure described above. The images were convolved with a Gaussian to compensate for variations in seeing. The stellar positions were used to establish transformations and the images were shifted to a common coordinate system. The sky in each frame was determined in an annulus centered on the galaxy. There was generally no observable wavelength dependent structure in the sky over the small 6 A gradient across Fabry-Perot пeld of view at the observed wavelengths. The series of images yields, at each pixel, a short segment of the spectrum around Ha. The spectra have been -- пtted with Voigt proпles (Humlicek 1979) to yield maps of the velocity, velocity dispersion, Ha intensity, continuum intensity, and their respective uncertainties. The kinematic data extend to a median of four disk scale lengths or 1.1 in I. We used stars in each of the images to пnd R 23.5 relative astrometric corrections between the photometric and kinematic images.
3

of speciпc predictions for variable M/L , we adopt a constant M/L for our models. We assume that luminous parts of spiral galaxies are composed of two principal components : a яat disk and a rounder bulge. Each component is characterized by a distinct spatial and kinematic stellar distribution and stellar population. In addition, the disk harbors the cold gas. Our models are based on a two-component, disk and bulge, photometric decomposition. To derive the mass distribution we assume that each component has a separate, constant M/L . In real galaxies, the disk and bulge can be further divided into subcomponents such as the thin and thick disk and the nucleus ; these are not considered here. The thick and thin disks in external galaxies have been distinguished only in edge-on projections (Burstein 1979) and the nuclei are not resolved in our data. Spiral galaxies are approximately axisymmetric ; the axisymmetry is broken by bars and spiral structure. Our models assume strict axisymmetry. The surface brightness proпles and the rotation curves for each galaxy are derived with a пxed and consistent center, position angle, and inclination, which we derive from both the photometry and the kinematic data. The dierence in the projected disk and bulge axis ratios is taken into account in the disk-bulge decomposition. In this analysis there is no radial dependence of the geometric parameters. Such a dependence is often implicit in isophotal or kinematic tilted-ring analysis. Our goal is to examine how well maximum disk models reproduce the mass distribution in a large and diverse sample of spiral galaxies. Individual models with halos could be constructed but would be poorly constrained by our data. In °° 3.1 and 3.2 we present our photometric and kinematic models. The two-dimensional nature of our data set allows us to derive independent geometric parameters from both the photometric images and the kinematic maps (SBWM). In ° 3.3 we combine these results to derive a luminosity distribution and rotation curve with consistent geometric parameters. In ° 4.3 we present the mass models. 3.1. Photometric Models The goal of the photometric models is to separate the disk and bulge and deduce their radial luminosity distributions. The two primary features we use to distinguish the disk from the bulge in an image are the dierence in the radial surface brightness proпle and the axis ratio of the isophotes. The surface brightness proпles of most spiral galaxies are exponential over some fraction of the disk (Freeman 1970). Approximately 40% of spirals, however, deviate strongly from purely exponential disks. These spirals, designated as type II by Freeman, have proпles that are яat or slowly rising toward the center and have a steeper exponential outer proпle. The proпles of type I galaxies more closely follow the canonical exponential disk. The proпles of disk galaxies often also exhibit smaller scale deviations. In order to model fully the radial structure of the disk these features must be included. We use an exponential component for the disk in our disk-bulge decomposition, but the пnal model for the disk is taken as the radially binned surface brightness proпle of the galaxy after subtracting the bulge model. The bulges of spiral galaxies are three-dimensional structures. This leads to the dierence in axis ratios of the disk and bulge isophotes in the projected image of an inclined

. SPIRAL GALAXY MODELS

We derive axisymmetric mass models of spiral galaxies assuming that the radial mass distribution follows the radial luminosity distribution with constant M/L . The assumption of constant M/L requires that extinction and population gradients be small across the luminous disk, which may not be an adequate representation of real galaxies. De Jong (1995) пnds signiпcant color gradients in the proпles of a large sample of spiral galaxies. His analysis attributes these gradients primarily to changes in stellar populations, with younger, more metal poor stars at large radii. This would tend to lower the stellar M/L at large radii and, therefore, decrease the radius at which the "" missing mass оо becomes important. De Jongоs models predict that (M/L ) can I change by a factor of 1.5 from the inner to outer parts of spirals. However, his models predict a scatter of D1.5 mag in the I-band Tully-Fisher and a smaller scatter in color bands redder than I. Neither of these predictions are in agreement with observations. The large scatter is not observed, and an analysis of the T-F relation shows larger scatter both at visible color bands (Bothun & Mould 1987) and in the near infrared (Bernstein et al. 1994) than at I. These issues clearly need to be resolved, but, in the absence

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spiral. The isophotes of the bulge are typically not circular, and thus bulges are typically not spherical.
3.1.1. Disk-Bulge Decomposition

There have been two strategies for performing disk-bulge decompositions. One has been to пrst perform an isophotal analysis ; the resulting radial luminosity proпle is пtted with an exponential disk and a bulge model (Schombert & Bothun 1987), such as the R1@4 law (de Vaucouleurs 1948) or Plummer model. The second method, from Kent (1986), uses the geometric properties of the disk and bulge. The proпles derived from isophotal пts are functions of up to пve variables at each isophote : semimajor axis, ellipticity, position angle, and center coordinates. The dependence on the last four variables, while included in the пts, is often not presented. This dependence leads to distortions in the radial luminosity proпle which are not in the underlying radial light distribution. The geometric parameters can be distorted locally by bright star-forming regions. Unavoidably, the parameters have a radial dependence that follows a bar and/or spiral arms. The isophotes are not well approximated by ellipses in the region where the surface brightness changes from bulge dominated to disk dominated. An elliptical isophote пtter makes a compromise пt that biases the true radial proпle. For example, in a highly inclined galaxy with a compact nearly spherical bulge, the surface brightness of the bulge projects to large galactic radii along the minor axis. An elliptical isophote пtter will produce a proпle in which the bulge seems to extend to larger radii. This will bias bulge models, such as the R1@4 or Plummer model, which have strong tails. Byun & Freeman (1995) present a systematic study of these eects in model galaxies. The method of disk-bulge decomposition developed by Kent uses the dierent radial scaling properties of the disk and bulge proпles along the major and minor axes. The method relies on the dierent axis ratios of disk and bulge isophotes and the assumption of axial symmetry. The disadvantage of this method is that it uses information only near the major and minor axis and is, therefore, sensitive to nonaxisymmetric structure in these regions. Our method combines the best properties of these two methods by using the full two-dimensional information in the galaxy image. We пt two components, a disk and a bulge model, to the image of the galaxy. We assume that the underlying distribution of light in spiral galaxies is axisymmetric, with a thin disk and an oblate spheroidal bulge with constant axis ratio. Under this assumption the projected isophotes of the model disk are ellipses with constant axis ratio, position angle and center. The projected isophotes of the model bulge are ellipses with the same position angle and center as the disk, but a dierent, larger, axis ratio. We exploit the exponential form of the disk but do not constrain ourselves to a more speciпc functional form for the bulge. For the bulge we use a series expansion of Gaussians (Bendinelli 1991), which we have generalized to model oblate distributions. At large radii the surface brightness of this bulge model is always negligible compared with the disk. The disk and bulge are separated in three steps : (1) an exponential disk model is пtted to and subtracted from the image, (2) the bulge model is пtted to the resultant image, and (3) the bulge model is subtracted from the original image leaving an image of the disk. The exponential disk of

step 1 is пtted in a region well away from the bulge and extrapolated into the central bulge dominated region. The extrapolation is carried out in dierent ways for type I and type II disks (see below). The purpose of пtting the exponential disk model is to derive a global inclination, position angle, and scale length and to provide a reasonable extrapolation of the disk into the central bulgeхdominated region for the purpose of isolating the bulge light. For type I disks the galaxy image is divided into two regions ; the disk-dominated region and the central bulgeх dominated region. The disk-dominated region is deпned as an elliptical annulus centered on the galaxy : the inner edge of this annulus has axis ratio of the bulge isophotes and the outer edge has the axis ratio of the disk isophotes. The semimajor axis of the inner edge of the annulus is set sufficiently far from the center that the bulge is negligible compared with the exponential disk. The bulge dominated region is deпned as an elliptical disk centered on the galaxy with the axis ratio of the bulge isophotes. The semimajor axis of each division is determined from a plot of the surface brightness binned in elliptical annuli with the approximate axis ratio of the disk. We пt a projected exponential disk to the image of the galaxy in the disk dominated region : (1) k (a ) \ k e~aD@rD , DD D0 where k is central surface brightness, r is the disk scale D D0 length, and a is the length of the semimajor axis of a ellipse D of constant surface brightness for the disk. In general, a2\ R2[1 ] f 2 sin2 (/ [ / )] , (2) 0 where f 2\ (b/a)~2 [ 1 is deпned in terms of the axis ratio, b/a, R is the distance from the galaxy center as measured on the sky, and / is the position angle of the major axis. For a 0 яat disk the inclination is given by i \ cos~1 (b/a). The exponential disk is subtracted from the galaxy image and a bulge model is пtted in the bulge-dominated region. The bulge model is a series of Gaussians : nc k e~a2@r2k , (3) k (a ) \ ; BB BB nr2 k/1 Bk where c is the total light in each component, r is the scale k k length of the kth Gaussian component, and a Bis the length B of the semimajor axis of an ellipse of constant surface brightness for the bulge. Each Gaussian has a common center and axis ratio. The position angle is пxed at the value derived for the disk, as required by the assumption of axisymmetry. Bulges are пtted with one to six Gaussian components. We use the maximum number of components that gives a unique and stable пt. If we add additional components we пnd either that : multiple components converge to the same scale length, with each of these components contributing a fraction of the intensity, or that the scale length of an additional component diverges and the intensity is reduced to the point that the component contributes a negligible constant oset. We have found that many of the standard пtting laws for the bulges of spiral galaxies such as de Vaucouleurs or Plummer models are inadequate representations of real bulge images. The bulge model is subtracted from the original galaxy image, and the residual image is binned in elliptical annuli to arrive at the пnal disk model. To avoid truncation features and to integrate total magnitudes, we extrapolate the disk model to large radii using the parameters of the exponential disk.

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For type II galaxies we divide the galaxy into three regions : the outer disk region, the inner disk region, and the central bulge region. We пt an exponential disk in the outer region to derive the ellipticity and position angle. We then пt an exponential disk in the inner disk region with its own intensity and scale length, but with the ellipticity and position angle пxed at the value derived from the outer disk. Because the surface brightness is nearly constant in this region, the shapes of the isophotes are not well deпned and the light distribution contains no information about the position angle or inclination of the galaxy. We subtract the inner disk model from the image and the bulge model is пtted in the central region as described above. The bulge model is subtracted from the original galaxy image and the residual image is binned in elliptical annuli to arrive at the пnal disk model. The disk model is extrapolated to large radii using the parameter of the exponential disk derived in the outer region. The disk-bulge decomposition is not generally iterated except to adjust the border of division between the disk and bulge regions. The disk model near the center is a reasonable, but arbitrary, extrapolation. Attempts to make small improvements in the solution through iteration depend on the details of the assumed disk extrapolation. The advantages of our method are that it does not start with built-in biases of the isophotal surface brightness proпle, and it sets additional, geometric, constraints on the disk-bulge decomposition and therefore exploits the available twodimensional information.
3.1.2. T hree-dimensional L uminosity Distribution

extinction is uniform over the luminous disk. The problem, however, is considerably more complex. The actual distribution of dust in galaxies is not well known and scattering may be as important as absorption in the I band. Furthermore, multicolor photometry of spiral galaxies by de Jong (1995) suggests that radial color gradients may be due primarily to population gradients. We adopt A \[1.0 log (b/a) given by Giovanelli et al. int (1994). Similar results are found by Han (1992), Bernstein et al. (1994), and Willick et al. (1995). Galactic extinction, A ext in the B band, is taken from Burstein & Heiles (1978). Reddening between B and I is assumed to be E(B[I) \ 0.45. The median value of Galactic extinction for this sample is 0.12 mag. 3.2. Kinematic Models We derive rotation curves from two-dimensional FabryPerot Ha radial velocity пelds. We assume that the Haemitting gas is in an axisymmetric rotating thin disk. In polar coordinates the model, projected on the sky, is given by cos i cos (/ [ / ) 0 v(r, /) \ v ] v (r) sin i , sys circ J1 [ sin2 i cos2 (/ [ / ) 0 (7) where i is the inclination, / is the position angle of the 0 projected major axis, v (r) is the circular velocity proпle, circ velocity. The disk center is an and v is the systemic sys implicit pair of parameters in the model. The term in brackets is equal to the cosine of azimuthal angle in the plane of the galaxy measured from the major axis. The parameters of the kinematic model are derived by пtting to the two-dimensional data in concentric elliptical annuli using a Levenburg-Marquardt s2 minimization technique (Press et al. 1992). The covariance matrix at the s2 minimum is used to estimate the errors in the parameters. The errors in the kinematic center are generally larger that those of the photometric center. The primary reason for this is that the kinematic center is poorly constrained along the minor axis and couples to the systemic velocity along the major axis. The center was therefore пxed by the centroid of the continuum distribution. In each annulus we пtted v , circ / , and i. The global kinematic position angle and inclina0 are the average, weighted by the estimated errors, of tion these parameters from each ellipsoid. The пnal rotation curve is extracted with all of the geometric parameters пxed. The rotation velocity is estimated independently on each side of the minor axis. 3.3. Geometric Parameters The photometric and kinematic models yield independent estimates of the major axis position angle and the inclination. We merge these results and rederive the models using consistent parameters. The position angle is generally better constrained by the kinematic models. A distinct line of nodes delineates the position angle in the velocity map, while the photometric position angle depends on the average distribution of luminosity around an annulus. The photometric position angle is therefore more easily biased by global nonaxisymmetric features such as spiral arms.

C

D

The three-dimensional luminosity distribution of the disk is trivially related to the projected distribution. Assuming no internal extinction o (r) \ k (r) cos i d(z), where r is the D distance from the center of the D galaxy and d(z) is the Dirac delta function. The surface brightness decreases by a factor of cos i in the deprojection. Under our assumptions the two-dimensional elliptical bulge surface brightness distribution can be uniquely deprojected to the three-dimensional spheroidal luminosity distribution via Abelоs integral equations (Stark 1977). A two-dimensional elliptical Gaussian distribution deprojects to a three-dimensional spheroidal Gaussian distribution. n c k (4) o (a8 ) \ ; e~a8B@rB , BB k/1 Jn3(1 [ e2)r3 BB 8 where a is the deprojected semimajor axis of the spheroid, B 1 8 a \ x2] y2] z2 . (5) B 1 [ e2 B The ellipticity of the spheroid, f2 1 B e\ , (6) B 1 [ f 2 sin2 i B follows directly from the galaxy inclination and the axis ratio of the bulge projected on the sky.
3.1.3. Photometric Corrections

The luminosity proпles are corrected for internal extinction, A , and Galactic extinction, A . Corrections for ext internalint extinction by dust generally give the total fraction of light absorbed within the galaxy. The surface brightnesses of galaxies in this sample are derived by assuming the

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The inclination, however is more poorly constrained by the kinematic model for many galaxies. For rising rotation curves the slope of the rotation curve and the inclination are degenerate parameters. In the extreme case of a rigid rotator the degeneracy between the rotation proпle and inclination is complete, and there is no independent kinematic information on the inclination. SBWM found some galaxies with large deviations between the photometric and kinematic inclinations. The deviations are primarily for galaxies with rising rotation curves. We therefore determine the position angle from the kinematic model and the inclination from the photometric model. The models are then iterated with the position angle пxed in the photometric model and the inclination пxed in the kinematic model. The пnal values of the geometric parameters are пxed and used consistently in both the photometric and kinematic models. The values are tabulated in Table 1. 3.4. Galaxy Parameters In Table 1 we list the photometric and kinematic parameters for galaxies in the sample. After пxing the geometric parameters of the models, we derive values of the central surface brightness (k ) and scale length (r ) for the disk. Dust extinction and 0 projection eects are d included in the models. We quote r in the outer parts of the disk. For d Freeman type II galaxies, therefore, the central surface brightness and scale length are not directly related quantities. The radius of the 23.5 mag arcsec~2 isophote (R ) is measured directly from the photometric proпles 23.5 and, therefore, is also corrected for dust extinction (assuming the dust acts uniformly over the disk) and projection eects. The magnitudes for each galaxy are derived by integrating the disk proпle and adding the total bulge luminosity. We quote total magnitudes integrated to inпnity. We calculate the ratio of the total luminosity bulge to that of the disk (B/D). We deпne the velocity width, used in the Tully-Fisher relation, to be twice the rotation speed, measured by a weighted average of the rotation curve points where the rotation curve becomes яat. For rotation curves that are still rising we deпne the width to be twice the maximum rotation speed. 3.5. Mass Models We assume that the mass distribution of a spiral galaxy follows the deprojected luminosity distribution with constant M/L s for each component. For the disk, we use a Fourier transform method for computing the rotation curve of a яat axisymmetric mass distribution given by Kalnajs (1965) : 1 v2 (u) \ circ 2n where u \ ln r and B(p) \ 2nGA(p) and A(p) \

For a spheroidal mass distribution the rotation curve in the symmetry plane is given by Binney & Tremaine (1987) : . (11) 0 Jr2[ e2 a8 2 B For a Gaussian distribution this integral reduces to a degenerate hypergeometric series in two variables. No dark halo is included. We also do not include a gas component. H I may contribute approximately 10% of the mass within the optical radius of late-type spirals (Broeils & van Woerden 1994) and thus further lower the allowed mass of the luminous component. H I gas disks in spiral galaxies have longer scale lengths than the stellar disks and therefore contribute to the mass distribution primarily at larger radii, where the rotation velocity due to the stellar disk begins to fall o.
4

v2 (r, z \ 0) \ 4nGJ1 [ e2 circ B

P

r o(a8 2)a8 2da8

. RESULTS

The model rotation curves of the bulge and disk are пtted to the kinematic data by adjusting the M/L of each component ; the best пt is derived by minimizing s2. For most galaxies this results in a пt that nowhere signiпcantly exceeds the data. For galaxies that would require a halo to get an acceptable пt, we limit the radius of the пt so that the model rotation curve does not exceed the rotation curve data and evaluate the quality of the пt out to this radius. In Figure 1 we present the models. The top panel for each galaxy shows the face-on surface brightness proпle. The contribution of the bulge is indicated by the dashed lines. The central surface brightness of the disk is marked with a diamond. The lower panel shows the rotation curve and the пtted mass model. The rotation speed for the side receding with respect to the center is marked with crosses and that for the approaching side is marked with open circles. The rotation curves for the model bulge and disk are traced with dashed lines. The full model rotation curve is equal to the disk and bulge models summed in quadrature and is indicated by an unbroken line. In addition, the radii of prominent features, such as bars or rings, are marked with a vertical dot-dashed line. A scale bar in arcseconds and the inclination are given to gauge the eect of seeing. 4.1. Morphology of Surface Brightness Proпles In addition to the relative brightness of the disk and bulge, the surface brightness proпles of spirals can be classiпed by the morphology of their disks. Two common assumptions are that spiral galaxies have an exponential disk and that they have a universal constant central surface brightness (Freeman 1970). The discovery of low surface brightness galaxies (Bothun, Impey, & Malin 1991 ; Schombert et al. 1992 ; McGaugh, Bothun, & Schombert 1995) most clearly indicates that spiral galaxies fail the second assumption. The surface brightness proпles of disks are also not strictly exponential. The most prominent distinction in disk proпles is between Freeman types I and II, described in ° 3.1. Galaxy disks also frequently exhibit less extreme deviations from a pure exponential, such as a point of inяection where the scale length changes. One statement of Freemanоs law is that the ratio of the optical disk radius to the exponential scale length, R /r , 23.5 d is a constant. In Figure 2 we plot R /r versus the central 23.5 d surface brightness. The tight correlation for exponential (i.e., type I) disks follows trivially from the deпnition of the optical radius and the wide distribution of central surface

P

`= dpB(p)ejpu , ~=

(8)

!(1 ] jp/2)!(1 [ jp/2) !(1 [ jp/2)!(1 ] jp/2)

(9)

`= dueuk(u)e~jpu , ~= where ! is the Gamma function and j \ J [1.

P

(10)

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FIG. 1.хMass models. T op panels : I-band surface brightness proпle. The bulge proпle is given by the dashed line. The diamond indicates the disk central surface brightness. Bottom panels : Maximum disk пts to the rotation curves. The dashed lines indicate the bulge and disk contributions. See text for a full description.

brightnesses in our sample. More signiпcantly, R /r correlates with absolute magnitude (Fig. 3). We test 23.5 signiпthe d cance of this correlation with the Spearman rank-order correlation coefficient (Press et al. 1992). The correlation coefficient is [0.29. The probability that there is no correlation is 0.01. The low-luminosity galaxies have lower central surface brightness and relatively яatter surface brightness proпles over the optical disk as indicated by a

larger R /r . This leads to model disk rotation curves that rise 23.5 dslowly for low-luminosity galaxies than those more for high-luminosity galaxies. 4.2. Morphology of Rotation Curves Within the optical radius, spiral galaxy rotation curves are not generally яat but span a range of morphologies, from rising linearly to falling with radius (Rubin 1985 ;

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FIG. 1.хContinued

Persic, Salucci, & Stel 1996). These authors also show that the shape of the rotation curves correlates with luminosity ; low-luminosity, small rotation velocity galaxies have rotation curves that are rising, while high-luminosity, large rotation velocity galaxies have falling rotation curves. An examination of the rotation curves in this sample suggests such a trend, but a large range of rotation curve shapes is found at every scale of rotational velocity. At high maximum rotation velocities (D300 km s ~1) : ESO 374G02 (Fig. 1z) is falling, ESO 375G12 (Fig. 1ac) is яat, ESO 269G61 (Fig. 1i) rises over D0.4R before яattening, and 23.5

ESO 381G51 (Fig. 1ai) rises over the entire optical radius. At low maximum rotation velocities (D100 km s ~1) : ESO 374G03 (Fig. 1aa) is яat past D0.25R , ESO 322G19 23.5 (Fig. 1k) is яat past D0.5R , Abell 1644d83 (Fig. 1a) rises 23.5 over the entire optical radius. The rotation curve of ESO 441G21 (Fig. 1ba) also rises to about 100 km s ~1 over the optical radius but does so almost linearly. 4.3. Model Fits In spite of the variety of surface brightness proпle and rotation curve morphologies, maximum disk models with

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constant M/L s provide good пts to optical rotation curves for a majority of the galaxies in our sample. The overall structure of the rotation curves is reproduced by the models. The models пt the data for rotation curves that are rising linearly or with a curve, that are яat or falling, or that have strong inяection points. The galaxies with good пts span a range of velocity widths (2v ) from 180 km s ~1 for circ Abell 1644d83 (Fig. 1a) to 680 km s ~1 for ESO 572G17 (Fig. 1bv). For most of the galaxies no halo is required for the models to пt the data within the optical radius or to the last measured point.

Figure 4 shows the distribution of the ratio of the maximum radius out to which the mass models provide a good пt to the optical radius ; R /R . The histogram is shaded for models that provide fit good пt out to the last a 23.5 measured point of the rotation curve. Of galaxies for which the rotation curve is measured to R or beyond, 75% are 23.5 well пtted by a mass-traces-light model for the entire region within R . For 21% of the galaxies the models provide poor пts 23.5 to strong bars or spiral arms ; these cases are due assigned R /R \ 0 in the histogram. The existence of fit 23.5 strong nonaxisymmetric structures suggests that there

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FIG. 1.хContinued

should be less dark matter in these galaxies. These cases are discussed below in ° 4.4. For Freeman type I spiral galaxies, a thin exponential disk mass distribution has a rotation curve which reaches maximum at 2.15 disk scale lengths and falls slowly thereafter, dropping 10% by 3.75 disk scale lengths. Despite the apparent restriction of this shape, a wide range of optical rotation curve shapes can be successfully modeled because of the variation of the number of disk scale lengths within the optical radius among galaxies and the addition of a bulge component.

Few galaxies have the canonical яat rotation curve across the entire optical disk. ESO 375G12 (Fig. 1ac) and ESO 376G02 (Fig. 1ae) are two examples of galaxies with good пts that do. The optical radii in these galaxies span 4.2 and 3.8 disk scale lengths, respectively. The model rotation curve for ESO 375G12 begins to fall at the optical radius, but unfortunately the rotation curve data for ESO 375G12 extend to only 3.3 disk scale lengths. ESO 376G02 shows slight evidence of dark matter, but only near the optical radius. The rotation curve for ESO 374G02 (Fig. 1z) has a gentle linear fallo over most of the optical radius, and the

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model provides an excellent пt over this entire range. The optical radius of ESO 374G02 extends over 4.1 disk scale lengths. These galaxies have prominent bulges. The яatness of the rotation curves is achieved though a combination of the bulge and disk rotation curves. ESO 383G88 (Fig. 1am) and ESO 323g42 (Fig. 1x) have less prominent bulges. The rotation curves of these galaxies rise less dramatically than those for the яat rotation curves above, but they do яatten at larger radii. The optical radius

of these spirals extends to 3.2 and 3.7 disk scale lengths, respectively, and the models reproduce the turnover in the rotation curves. The optical disks of ESO 376G10 (Fig. 1af ) and ESO 501g01 (Fig. 1bl) extend to only 2.0 and 2.2 disk scale lengths, respectively. These galaxies also have small bulges. The rotation curves reяect this morphology ; they rise with a curve over most of the optical disk, reaching maximum near the optical radius. The rotation curve for ESO 501g01

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FIG. 1.хContinued

extends signiпcantly past the optical disk to 4.1 scale lengths. The model rotation curve does not fall signiпcantly below the data over this entire range. Freeman type II disks, which are distinguished by a яat inner surface brightness proпle, are modeled with a constant inner mass density. This leads to rotation curves that are linearly rising over the constant density region. The radius of the turnover varies from 20% to 60% of the optical radius for the galaxies in our sample. Examples include : ESO 445G81, 0.22R (Fig. 1bi) ; ESO 375G02, 23.5

0.23R (Fig. 1ae) ; ESO 216G20, 0.31R (Fig. 1c) ; ESO 23.5 501G86, 0.55R (Fig. 1bp) ; and ESO 23.5 509G91, 0.62R 23.5 (Fig. 1bs). The 23.5 of the linearly rising region in the rotasize tion curve varies accordingly. ESO 381G51 (Fig. 1ai), ESO 435G26 (Fig. 1an), ESO 438G15 (Fig. 1aw), and ESO 501g11 (Fig. 1bm) also have яat surface brightness proпles near their centers ; however in these cases the proпles roll o slowly approaching an exponential asymptotically near the edge of the optical disk. The rotation curves in these cases rise slowly, with a curve

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matched by those of the models. ESO 435G26 has a strong bar and the пt within the bar radius is not good. The proпles of some spirals have inяections which are less prominent : the slope of the proпle changes but does not become яat. This feature is reяected in the rotation curves of ESO 317G41 (Fig. 1j) and ESO 322G82 (Fig. 1s). 4.4. Nonaxisymmetric Structure A fraction, about 20%, of the пts fail in the inner regions ; major structures in the model and/or rotation curve do not

match up. These bad пts occur well within the optical disk and are not likely due to a dominant dark matter component. The galaxies which have the poorest пtting mass models often have strong nonaxisymmetric structures in the form of bars or strong spiral arm structure. These structures aect both the surface brightness proпle and the rotation curve. The strong nonaxisymmetric gas motions induced by a bar distort the measured rotation curve within the bar radius as noted for ESO 435G26. Strong spiral structure can aect the shape of the surface brightness proпles and

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FIG. 1.хContinued

induce large nonaxisymmetric motions, as well as bias the determination of inclination and major-axis position angle. ESO 268g37 (Fig. 1g) and ESO 374g03 (Fig. 1aa) both have bars along the major axis and very strong spiral structure. ESO 323g39 (Fig. 1w) has a bar along the minor axis and strong spiral ams. ESO 323g25 (Fig. 1u) has very strong grand design spiral arms. The inclination of this galaxy along with the pitch angle and position angle of the spiral arms conspire in such a way that the arms closely follow the

ellipticity of the disk from 5 to 10 kpc. The large structure seen in the model at these radii is due to this chance alignment. This eect is also seen in ESO 322g36 (Fig. 1l) and ESO 569G17 (Fig. 1bu), which also have strong spiral structure. A detailed analysis of the eect of strong nonaxisymmetric structures will be presented in future paper. However, if a dark matter halo dominates the mass within the optical radius of these galaxies, it should act to stabilize

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FIG. 1.хContinued

the disk against these nonaxisymmetric modes. It is difficult, therefore, to attribute the poor пt of these models to a large dark-to-luminous mass fraction. 4.5. Small-Scale Structure Mass models for spiral galaxies sometimes, but not always, reveal a correlation between small-scale "" bumps and wiggles оо in the surface brightness proпle and the rotation curve (Kent 1986 ; Freeman 1992). We пnd that in the two-dimensional maps the residuals in the photometry and the residual kinematic motions are highly correlated. How these structures show up in one-dimensional, radial, surface brightness proпles and rotation curves depends strongly on how the data is sampled. The bumps and wiggles correlation is therefore most probably due to local perturbations, such as spiral arms and spiral arm streaming motions, rather than the global mass distribution.

4.6. Mass-to-L ight Ratios The median I-band M/L (disk plus bulge) for our sample of galaxies is 2.4 h in solar units with an rms scatter of 0.9. This accords well 75 the M/L predicted from stellar popuwith lation synthesis models. If we exclude the galaxies with the worst пts the median M/L rises to 2.7 ^ 0.8. Worthey (1994) estimates that a normal stellar population that forms in a single burst will have an initial I-band M/L of about 1 in solar units and will increase to about 5 over a time of 15 Gyr. His models have an M/L of 2.4 at an age of 6 Gyr and an M/L of 2.7 at 8 Gyr. Spiral galaxies are composed of stellar populations that span a range of ages, and their M/L depends on the number of stars formed in each generation. A typical spiral might be about 10 Gyr old and form most of its stars over the пrst 4 Gyr. The theoretical M/L s are highly dependent on the assumed mass function for the stellar population and the details of the star formation history. Thus the agreement of our M/L s with theory, although noteworthy, is not in itself

FIG. 2.хDisk central surface brightness vs. R /r . The solid symbols 23.5 d are for exponential, Freeman type I disks and the open symbols are for Freeman type II disks with a яat central disk proпle.

FIG. 3.хAbsolute magnitude vs. R /r . The symbols are the same as 23.5 d in Fig. 2. The slope of the best-пt least-squares line is [0.37.

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FIG. 4.хHistogram of R /R . The shaded regions indicate galaxies that have good пts out fit the last measured point. Cases where to 23.5 R /R \ 0 indicate galaxies which have poor пts due to bars or strong fit arms. spiral 23.5

FIG. 5.хM/L vs. Hubble type. The symbols are the same as in Fig. 2. I The Hubble types, classiпed in the RC3, are coded : Sa \ 1, Sb \ 3, Sc \ 5, Sd \ 7, untyped \ 10.

overwhelming evidence for the maximum disk hypothesis. The light comes primarily from a small population of bright, high-mass stars, while the most of the mass is in a large population of faint, low-mass stars. Current star formation rates vary with Hubble type, with lower rates in early-type spirals. M/L s increase over time as the stellar population fades, and therefore M/L s should also vary with type, with higher M/L s in early types. Such a correlation was found by Rubin (1985) and Kent (1986) for M/L s in the V band. In the I band this correlation is expected to be considerably weaker, and indeed it is not strong in our data (Fig. 5). The stellar population models predict a rate of evolution of M/L s in I band that is a factor of 2 less than that in V band for a given generation of stars. Also, the scatter in our M/L s for a given type is large. The plot of M/L versus axial ratio (Fig. 6) is яat, which gives us conпdence that the assumed internal extinction law is correct and conпrms the work of Han (1992), Giovanelli et al. (1994), Bernstein et al. (1994), and Willick et al. (1995). The good agreement of the values of M/L with the stellar population models also indicates that internal extinction cannot be grossly higher, as claimed by Valentijn (1990). The scatter in the M/L s in our sample is 37%. One of the primary sources of error is uncertainty in the distance ; measured M/L s are inversely proportional to the assumed distance. The sample is concentrated in the great attractor region (Dressler et al. 1987), which is dominated by the Hydra-Centaurus supercluster. The clusters are likely to have large peculiar velocities. Galaxies in the vicinity of the Centaurus cluster have a median M/L of 1.8 ^ 0.6 and galaxies near Hydra have a median M/L of 3.4 ^ 1.0. The Centaurus cluster is composed of two major subclusters (Lucey, Currie, & Dickens 1986), and the Hydra cluster is also thought to have substructure (Fitchett & Merritt 1988). The galaxies that are members of these clusters may therefore have especially large peculiar velocities. Figure 7 gives

the distribution of M/L s for the sample. Smooth Hubble яow is, unfortunately, not a precise approximation for estimating the distances to the galaxies in this sample. The large-scale peculiar motions may contribute 25% or more to the error budget (Bothun et al. 1992 ; Mathewson, Ford, & Buckhorn 1992 ; da Costa et al. 1996). This can also be seen in the T-F relation for this sample. Figure 8 shows the T-F relation for the galaxies in our sample assuming Hubble яow distances. We plot the total magnitude with closed symbols for Freeman type I and open symbols for type II galaxies. We also indicate the R isophotal magnitude at the end of the line connected to23.5 symbol. The length of the line indicates the extrapoeach lation from the isophotal to total magnitude. The fainter

FIG. 6.хM/L vs. axis ratio. The symbols are the same as in Fig. 2. I

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the slope of the T-F relation is assumed to be 10 and we consider isophotal magnitudes (SBWM ; Peletier & Willner 1993) then the scatter about the T-F relation is 0.75 mag, which implies deviations from the Hubble яow distances of 40%.
5

. CONCLUSIONS

FIG. 7.хHistogram of M/L . The hatched regions give the distribution I of M/L in Centaurus and Hydra clusters.

galaxies clearly require a larger extrapolation ; this is because the fainter galaxies have lower central surface brightness, as noted above. If the slope of the T-F relation is assumed to be near 6 and we consider the total magnitudes (Mathewson et al. 1992 ; Bernstein et al. 1994), then the scatter around the T-F line is 0.46 mag. The zero point of the relation is пxed so that the average deviation from the T-F line is zero (this assumes no bulk яow for the sample). If the deviations from the T-F line are attributed solely to peculiar motions, the error in the Hubble яow distances has a scatter of 24%. If

FIG. 8.хTully-Fisher relation. The symbols are the same as in Fig. 2. The symbols mark the total extrapolated magnitude and the lower points of the vertical lines mark the R isophotal magnitudes. The dashed line is the T-F relation assuming a 23.5 of 10, and the dotted line is the T-F slope relation assuming a slope of 6.

We пnd that spiral galaxy mass models that assume the maximum disk hypothesis yield good rotation curve пts within the optical radius for a variety of spirals with distinct surface brightness and rotation curve morphologies. Of galaxies for which the rotation curve is measured to R or 23.5 beyond, 75% are well пtted by a mass-traces-light model for the entire region within R . It is particularly striking that 23.5 spirals with very dierent disk surface brightness proпles, generically distinguished by Freeman types I and II, are well modeled under the maximum disk hypothesis. Freeman type II galaxies constitute a signiпcant fraction of spiral galaxies which fail to meet the canonical assumption that all spirals have an exponential disk. This has certainly contributed to the large discrepancy in disk scale lengths published by dierent authors for the same galaxies (Knapen & van der Kruit 1992). Type II galaxies can be further characterized by the size of the inner яat region and turnover rate. They span a large range of velocity widths and show no correlation with type. They generally break the correlation of rotation curve shape to absolute magnitude found by Rubin et al. (1985) and Persic et al. (1996). Galaxies for which our models fail to give good пts in the inner regions, D20% of the sample, generally have strong features, particularly bars but also strong spiral arms, which break the assumption of axisymmetry. Smaller scale deviations in the surface brightness proпles and rotation curves, "" bumps and wiggles,оо can also often be traced to nonaxisymmetric features. These results show that, within the optical regions of most spiral galaxies, the radial mass distribution is tightly coupled to the luminosity distribution. This is a much stronger constraint than that due to global correlations such as the T-F relation. It implies that either the mass of dark matter must be small within the optical radius or that the distribution of dark matter must be precisely coupled to the distribution of luminous matter. A dark halo which is independent of and unresponsive to the luminous disk cannot dominate the mass within the optical radius. A fraction of the derived luminous mass could be traded for halo mass, but the luminous mass traces the overall features of so many and various rotation curves that this fraction could not reasonably be too large. Persic et al. (1996) have constructed synthetic rotation curves by averaging over 500 optical rotation curves from Mathewson et al. (1992). They пt each synthetic rotation curve with model rotation curves for an isothermal halo and an exponential disk in which the scale length is пxed relative to the optical radius. They conclude that spiral galaxies with rotation velocities of 150 km s ~1 are over 40% dark matter within the optical radius, and that galaxies with rotation velocities of 100 km s ~1 are over 75% dark matter. The assumptions which lead to this conclusion are that all spirals galaxies have the same R /r and that all spirals of 23.5 d a given luminosity have the same rotation curve shape. Our results strongly suggest that both of these assumptions are too simplifying and call their results into question. We пnd that R /r is larger in low rotation velocity galaxies. This 23.5 d

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leads to more slowly rising rotation curves. Relaxing only this assumption considerably weakens their dark matter constraints. Cosmological N-body simulations of hierarchical universes suggest a universal halo proпle for mass scales ranging from dwarf galaxies to rich clusters of galaxies (Navarro et al. 1996). The halo proпles are centrally concentrated and for spiral galaxies they dominate the mass distribution at all radii. In the cold dark matter models of Navarro et al., a galaxy with a maximum rotation velocity of 300 km s ~1 is 72% dark matter within the optical radius, a galaxy with a maximum rotation velocity of 200 km s ~1 is 90% dark matter, and a galaxy with a maximum rotation velocity of 100 km s ~1 is 96% dark matter. Our results show that optical rotation curves in real galaxies exhibit a variety of shapes and that these shapes are well modeled by the luminous distribution of matter. It is difficult to reconcile our results with a universal rotation curve in which the central attraction is primarily due to a dark matter halo.

The success of the maximum disk hypothesis in modeling the mass distribution in the inner parts of spiral galaxies implies that either the mass of dark matter has to be small or that its projected distribution must follow precisely that of the luminous matter out to nearly the optical radius. We thank Ben Weiner, Jerry Sellwood, and Liz Moore for critical readings of this paper. We also warmly acknowledge the excellent support of the CTIO observing sta. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. The Rutgers Fabry-Perot (RFP) instrument was built with support from Rutgers University and from the National Science Foundation grant AST 83-19344. The RFP is operated at CTIO under a cooperative agreement between Rutgers University and CTIO.

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