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Dark and Baryonic Matter in Bright Spiral Galaxies: I I. Radial Distributions for 34 Galaxies
Susan A. Kassin1,2 , Ro elof S. de Jong3 , & Benjamin J. Weiner
4

arXiv:astro-ph/0602027 v1 1 Feb 2006

ABSTRACT
We decomp ose the rotation curves of 34 bright spiral galaxies into baryonic and dark matter comp onents. Stellar mass profiles are created by applying color-M /L relations to near-infrared and optical photometry. We find that the radial profile of the baryonicto-dark-matter ratio is self-similar for all galaxies, when scaled to the radius where the contribution of the baryonic mass to the rotation curve equals that of the dark matter (RX ). We argue that this is due to the quasi-exp onential nature of disks and rotation curves that are nearly flat after an initial rise. The radius RX is found to correlate most strongly with baryonic rotation sp eed, such that galaxies with RX measurements that lie further out in their disks rotate faster. This quantity also correlates very strongly with stellar mass, Hubble typ e, and observed rotation sp eed; B -band central surface brightness is less related to RX than these other galaxy prop erties. Most of the galaxies in our sample app ear to b e close to maximal disk. For these galaxies, we find that maximum observed rotation sp eeds are tightly correlated with maximum rotation sp eeds predicted from the baryon distributions, such that one can create a Tully-Fisher relation based on surface photometry and redshifts alone. Finally, we compare our data to the NFW parameterization for dark matter profiles with and without including adiabatic contraction as it is most commonly implemented. Fits are generally p oor, and all but 2 galaxies are b etter fit if adiabatic contraction is not p erformed. In order to have b etter fits, and esp ecially to accommodate adiabatic contraction, baryons would need to contribute very little to the total mass in the inner parts of galaxies, seemingly in contrast with other observational constraints.
Subject headings: galaxies: fundamental parameters - galaxies: general - galaxies: halos - galaxies: kinematics and dynamics - galaxies: spiral - galaxies: stellar content

1.
Department of Astronomy, The Ohio State University 2 currently at: UCO/Lick Observatory, University of California, Santa Cruz, CA 95064; kassin@ucolick.org 3 Space Telescop e Science Institute, 3700 San Martin Drive, Baltimore, MD 21218 4 Department of Astronomy, University of Maryland, College Park, MD 20742
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Intro duction

The flatness of rotation curves has long b een the most direct evidence for the existence of a dominant comp onent of dark matter in spiral galaxies (Sofue & Rubin 2001). However, the distribution of dark matter is p oorly constrained, even in galaxies where the presence of dark matter is dominant (e.g., Ver1


heijen 1997; de Block & McGaugh 1998). Interestingly, the main uncertainty in the radial density distribution of dark matter, as derived from rotation curves, stems from the p oorly known stellar mass distribution (e.g., Verheijen 1997). In this pap er, we do not rely on the typical maximal disk assumption (van Albada et al. 1985; van Albada & Sancisi 1986) to derive stellar mass profiles, but use color-massto-light ratio (M /L) relations first given by Bell & de Jong (2001) and later up dated by Bell et al. (2003). These color-M /L relations were determined from the analysis of sp ectrophotometric spiral galaxy evolution models and give an upp er limit to the baryonic mass present in spirals. They are most p owerful when applied to data in the near-infrared since this wavelength regime traces the older stellar p opulations that contain most of the mass, while avoiding much of the effects of dust (extinction at the near-infrared K -band is only ab out 10% of that at the optical B band; Martin & Whittet 1990). The colorM /L relations allow us to investigate radial variation in stellar M /L, and provide us with a consistent way of scaling stellar M /L from one galaxy to the next. With the distributions of stellar mass in galaxies reasonably well determined, we can b egin to investigate those of the dark matter. The maximum disk assumption (Sackett 1997) can b e examined with the color-M /L relations. We can obtain an upp er limit to the numb er of galaxies that have maximal disks, and also compare the shap es of rotation curves derived from baryonic mass distributions in galaxies to the observed rotation curves as in Palunas & Williams (2000). There are many pieces of evidence in the literature for maximal disks; we mention a few noteworthy ones here. Possibly the strongest evidence comes from the work of Weiner, Sellwood, & Williams (2001) who used fluid2

dynamical models of gas flow to model the two-dimensional velocity field of a barred galaxy. They find that such models require 80 - 100% of the maximal disk M /LI value. Palunas & Williams (2000) modeled the baryonic mass distributions of 74 spirals and found that a mass-follows-light model could reproduce the overall structure of the optical rotation curves in the ma jority of galaxies. These authors found that 75% of the galaxies in their sample have a rotation curve out to R23.5 that is entirely accounted for by baryons. The mass models for 20% of their sample fail b ecause of nonaxysymmetric structures. For a more inclusive discussion on maximal disks, see Palunas & Williams (2000). It has b een known for nearly 20 years that the radius where dark matter b egins to contribute to the rotation curve of a galaxy is smaller for low surface brightness galaxies than it is for brighter galaxies (e.g., Persic & Salucci 1988, 1990; Broeils 1992; Persic, Salucci, & Stel 1996; de Blok & McGaugh 1997). A numb er of authors have demonstrated this by evaluating either the mass discrepancy of galaxy disks (ratio of total to baryonic mass) by adopting a stellar M /L, or a similar quantity at a chosen radius. For example, Persic, Salucci, & Stel (1996) find that spiral galaxies with V = 100 km s-1 have > 75% of their mass in dark matter within Ropt 3.2h where h is the disk scale-length, and that galaxies with V = 150 km/s have only > 40%. McGaugh & de Blok (1998) find that the total M /LB evaluated at 4h is smaller for galaxies of higher surface brightness and brighter magnitude, and does not correlate with h. These authors also find that the radius where dark matter b egins to dominate over the luminous mass is greater for higher surface brightness galaxies. Zavala et al. (2003) find that the ratio of the velocity due to the baryons to the total maximum velocity dep ends mainly on disk central surface


density such that denser galaxies have larger ratios. These authors also find that the mass discrepancy evaluated at 2.2h and 5h dep ends on central surface brightness, but puzzlingly does not dep end on h, baryonic mass, or B band luminosity. In addition, Zavala et al. (2003) find that M /LB anti-correlates significantly with B -band central surface brightness and does not correlate with B -band luminosity, in disagreement with McGaugh & de Blok (1998) and the general qualitative trend in the literature. Most recently, Pizagno et al. (2005) evaluate the total M /Li at 2.2hi and find median values of 2.4 and 4.4 for galaxies with stellar masses greater than 1010 M and 4.4 b etween 109 -1010 M , resp ectively. These authors are all generally in qualitative agreement, but there are discrepancies, and we will discuss their origin in the b ody of this pap er. A combination of resolved H or CO and H I velocity data is necessary to determine the dark matter distribution of a galaxy from its center to b eyond 5h. H data are essential to measure the steep rise of rotation curves in the inner 2h of galaxies (e.g., Palunas & Williams 2000), whereas H I data are key to prob e b eyond where H can b e measured (e.g., de Blok & McGaugh 1997). Since rotation curves in the optical p ortion of galaxies are usually affected by non-axisymmetric structures that do not trace the main galactic p otential, such as bars and spiral arms, the distributions of dark matter derived from H data can b e quite variable. Two examples of studies probing dark matter distributions that have used b oth typ es of rotation curves in a direct manner, indep endent of prior parameterizations for the dark matter, are Persic, Salucci, & Stel (1996) and McGaugh & de Blok (1998). Persic, Salucci, & Stel (1996) coadd 1100 rotation curves and find that they could b e determined by a single galactic parameter (e.g., luminosity). This formulation, however, is refuted by a numb er of authors

(e.g., Verheijen 1997; Bosma 1981). McGaugh & de Blok (1998) find a regularity b etween the mass discrepancy and acceleration of a galaxy as a function of radius. This follows directly from the Tully-Fisher relation, probably the b est observational example of selfsimilarity in the dark matter comp onent of galaxies. The mass discrepancy-acceleration relation shows that knowledge of the baryonic mass distribution of a galaxy allows one to calculate its dark matter distribution. Such a relation steps b eyond the usual Tully-Fisher relation, which predicts the dark matter content of a galaxy at a particular radius given its baryonic mass at that radius, to one that predicts radial distributions. Theories and simulations of galaxy formation should b e able to reproduce the distributions of dark matter observed in galaxies. Until now, comparisons to predictions for dark matter distributions have consisted of fits of functional formulations for dark matter halos to rotation curve data, with stellar M /L as a free parameter. Now that we have a handle on the stellar M /L in galaxies with the color-M /L relations, we re-examine these fits. In particular, we test the main incarnation of the density distribution of dark matter halos in N-b ody simulations of a CDM universe: the formulation of Navarro, Frenk, & White (1996) (NFW), which is the simplest and most p opular analytical description. This formulation predicts that dark matter is significant down to the very inner radii of galaxies, in contradiction to maximal disks. Moreover, dark matter halos are exp ected to contract due to baryon collapse. As this contraction is normally implemented (Blumenthal et al. 1986), even more dark matter is caused to move toward the centers of galaxies, making the situation worse (e.g., McGaugh & de Blok 1998). We p erform fits to the NFW formulation with and without including contraction as it is normally implemented. The main as-

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sumption b ehind these fits, namely that the NFW halos can b e fit to data that subtends only their very inner parts ( 0.1Rvirial ), is problematic. Furthermore, the simulations that produce the NFW functions have difficulty forming disk galaxies once hydrodynamics is included (e.g., Abadi et al. 2003), signaling p ossible unknown interactions b etween baryons and dark matter. To address these concerns, we turn the problem of fitting dark matter halos to data around and present a simple universal form for dark matter profiles in terms of baryonic mass profiles. This pap er is organized as follows: In ?2 we briefly discuss our galaxy sample. Radial baryonic mass distributions and their rotation curves are derived in ?3. In ?4 these baryonic rotation curves are compared with rotation curves from the literature to derive rotation curves due to dark matter. We discuss maximal disks in ?5. In ?6 we define quantities to describ e the baryonic and dark matter distributions and show how they correlate with general galaxy prop erties. We investigate the radial b ehavior of dark matter in ?7, and mass profiles are fit with the NFW profile in ?8. In ?9, we summarize our conclusions. Throughout this pap er we adopt a Hubble constant (Ho ) of 70 km s-1 Mp c-1 . When distancedep endent quantities have b een derived from the literature, we have reduced them to this value of Ho and always quote the converted values. 2. The Galaxy Sample

the remaining 4 galaxies can b e found in Verheijen (1997). More details on the sample selection are given in Pap er I. The profiles were calculated in elliptical annuli of increasing distance from the centers of the galaxies and have b een corrected for Galactic extinction. The galaxies NGC 1090, NGC 2841, and NGC 3198, which have images from the Sloan Digital Sky Survey (SDSS) Second Data Release (Abaza jian et al. 2004), have images in which almost half of the galaxy was off of the detector. This is also the case, but to a much lesser extent for NGC 3521. This effect can b e observed in the g-band images in Figure 1 of Pap er I; these galaxies are flagged in the following analysis. All the analyses in this pap er have b een p erformed for b oth the approaching and receding sides of a handful of galaxies with complete imaging. No significant change (to within the zero-p oint uncertainties) was found b etween the results for each side of these galaxies. Physical parameters (i.e., Hubble typ e, distance, R25 , integrated magnitudes) and bulge-disk decomp ositions can also b e found in Pap er I. 3. Baryonic Matter

Our data consist of surface brightness profiles, physical parameters, and rotation curves for 34 bright spiral galaxies. These galaxies have inclinations in the range 30-65 degrees in order to reduce the effects of dust, while still b eing able to obtain accurate kinematical information. Surface brightness profiles for 30 galaxies were presented in Kassin, de Jong, & Pogge (2005) (hereafter, Pap er I); those for 4

In this section, we derive the baryonic mass surface density profiles of the galaxies, and compute the comp onent of the galaxies' observed rotation that is due to the baryons. These comp onents are called "baryonic rotation curves," to distinguish them from observed rotation curves. In ?4, we will introduce the term "dark matter rotation curves," to denote the contribution from dark matter to the observed rotation curves. 3.1. Radial Baryonic Surface MassDensity Distributions

To determine a galaxy's radial stellar surface mass-density distribution, we apply a color-M /L relation to its surface brightness


Fig. 1.-- Color-M /L relation for B - R color and
(M /L) at K from Bell et al. (2003) (solid line). The galaxies in the sample are plotted as filled circles, and a reddening vector is plotted for for correction to face-on for a Milky Way-type galaxy viewed at an inclination of 80 with the Tully et al. 1998 formalism.

profiles. These relations were derived from sp ectrophotometric spiral galaxy evolution models in Bell & de Jong (2001), and were up dated in Bell et al. (2003). The color-M /L relations show a relatively tight correlation ( 0.1 dex spread for the color and metallicity range where our galaxies lie) b etween the optical color of a galaxy and its stellar massto-light ratio, (M /L) . Color-M /L relations are most useful when applied to (M /L) in the near-infrared since they are the least affected by dust obscuration, as discussed in Bell & de Jong (2001). Sp ecifically, we choose to use the relation b etween B - R color and (M /L) in the K -band. This relation is comp osed of the optical color with the largest wavelength baseline and the reddest near-infrared band. It is reproduced in Figure 1. If imaging is unavailable at R, we use the relation for B - V , and if imaging is unavailable at K we use H . For those galaxies without a significant 5

bulge contribution (such that inclusion of a bulge comp onent does not change the rotation curve due to baryons b eyond the uncertainties of the color-M /L relations), we apply a color-M /L relation directly to the azimuthally averaged radial B - R color profiles to derive radial (M /L) profiles at K . The LK profile for each galaxy is then multiplied by the galaxy's (M /L) profile to derive a radial stellar surface mass-density profile. For these galaxies, using a radial B - R color profile is consistent with using an ap erture B - R color since this color-M /L relation has a shallow slop e. For those galaxies with a significant bulge comp onent, a bulge-disk decomp osition is p erformed. A characteristic B - R color is adopted for each comp onent based on the average colors of the bulge and disk, and the LK profiles are multiplied by the resulting (M /L) . Extinction corrections are discussed in ?3.4. We also extend the exp onential disk surface brightness profiles until approximately (defined here as r = 10000 ).1 When available from the literature, we use radial H I 21 cm measurements to determine the contribution of interstellar gas to the radial baryonic surface mass-densities. There are gas measurements available for NGC 1090 from Gentile et al. (2004), NGC 3198 from Begeman (1987), and NGC 3949, NGC 3953, and NGC 3992 from Verheijen (1997). The neutral gas comp onent is included by scaling the H I surface mass-density by a factor of 1.32 to account for the abundance of helium. For those galaxies without gas mass measurements, we assume that gas does not contribute to its baryonic surface massdensity. This is probably not a bad assumption for high surface brightness galaxies since a galaxy with a Hubble typ e b etween Sa and Scd has a gas mass fraction that is typically
1

Extended galactic stellar disks have b een discovered for some galaxies via very deep imaging (see Ibata et al. 2005, and references within).


Fig. 2.-- Rotation curve for NGC 1090 due to the
stellar mass component (solid line) is compared with the rotation curve due to stars and H I gas from Gentile et al. 2004 (dashed line). The observed H and H I rotation curves are plotted as open triangles and circles, respectively. The upper and lower bounds for the stellar mass rotation curve (dotted lines) are due to the 0.1 dex uncertainty in the color-M /L relations.

rotmod task in the GIPSY software package (van der Hulst et al. 1992), which calculates rotation curves for galaxies comp osed of a truncated exp onential disk (Casertano 1983) and a spherical bulge when applicable. Exp onential disks are assumed to have a scaleheight of 0.3 kp c, which is typical of bright spirals. The difference b etween using a different scale-height for each galaxy of 0.1hI R kp c, as in Sparke & Gallagher (2000), and a global value of 0.3 kp c is negligible. The results also remain unchanged if we used the relation b etween the central surface brightness and the ratio of vertical scale-height to vertical scalelength given by Bizyaev & Mitronova (2002). 3.3. Uncertainties in Baryonic Rotation Curves

MH I /Mgas+stars 0.03 (Rob erts & Haynes 1994), and hence does not greatly affect its baryonic mass. The average gas mass fraction for the 5 galaxies in our sample that have gas mass measurements is 0.04, consistent with this measurement. Figure 2 demonstrates the effect of including the interstellar gas comp onent in the baryonic rotation curve of NGC 1090, which is generally within the uncertainty of the stellar mass rotation curve. Therefore, by not including interstellar medium contributions in the analysis, only a minor systematic effect is introduced. 3.2. Baryonic Rotation Curves

A baryonic rotation curve is calculated for each galaxy from its baryonic surface massdensity profile and is plotted in Figure 3. The rotation curve calculation is done with the

The largest source of uncertainty for the baryonic rotation curves lies in the normalization of the color-M /L relation, which is mainly determined by the stellar IMF at lowmass. Since the faint end of the IMF is relatively unconstrained, there may exist many low-mass, low-luminosity stars that can contribute significantly to the mass budget of a stellar p opulation without creating a detectable increase in luminosity or change in color. For their derivation of the color-M /L relations, Bell & de Jong (2001) and Bell et al. (2003) adopted a truncated Salp eter IMF which derives from the constraint that baryonic rotation curves should not over-predict observed rotation curves for spiral galaxies in Ursa Ma jor (the Verheijen 1997 sample). With this constraint, they predict fewer low mass stars than a Salp eter IMF. These relations thus give an upper limit to the stellar mass present; a lower normalization of the color-M /L relation cannot b e excluded. We re-derive this constraint on the upp er limit to the IMF by re-calculating maximum disk fits for galaxies in the Verheijen (1997) sample without including a dark halo, as was 6


NGC 157

NGC 908

NGC 1241

NGC 289

NGC 1087

NGC 1385

NGC 488

NGC 1090

NGC 1559

Fig. 3a.-- Observed rotation curves (dots for H I, triangles for H or N II, thin dashed lines for models),
baryonic rotation curves (solid lines), and dark matter rotation curves where applicable (thick dashed lines). For many of the H and N II rotation curves, the error bars are smaller than the points at the resolution of the plots. The effects of the Á0.1 dex uncertainty in the color-M /L relations on the baryonic rotation curves are plotted as dotted lines. On the x-axis, the radii R25 and R = 2.2hI R are marked with thick and thin bars, respectively.

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NGC 1832

NGC 2280

NGC 3198

NGC 2090

NGC 2841

NGC 3198 (zoom)

NGC 2139

NGC 2841 (zoom)

NGC 3223

Fig. 3b.-- See Figure 3a.

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NGC 3319

NGC 3893

NGC 3992

NGC 3521

NGC 3949

NGC 4051

NGC 3726

NGC 3953

NGC 4062

Fig. 3c.-- See Figure 3a.

9


NGC 4138

NGC 5371

NGC 7083

NGC 4651

NGC 5806

NGC 7217

NGC 4698

NGC 6300

NGC 7606

Fig. 3d.-- See Figure 3a.

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done in the original Verheijen (1997) fits. For these fits, we define maximal disk to b e the greatest p ossible contribution of the bulge and disk to the observed rotation curve of the galaxy. To do this, we scale the K -band derived stellar mass rotation curves as high as p ossible without over-predicting the observed rotation curves b eyond the very inner parts. In order to b est match the overall shap es of the observed rotation curves, we allow overpredictions of the inner 2 p oints of H I rotation curves due to b eam-smearing and overpredictions of the inner parts of H rotation curves. We also re-scale the stellar and gas mass distributions from the distance used in Verheijen (1997) to 20.7 Mp c b efore fitting the maximal disks, unlike the approximate rescaling done in Bell & de Jong (2001). Gas distributions derived from H I measurements are multiplied by 1.32 to account for helium, as in Verheijen (1997). In Figure 4, the resulting maximum disk (M /L) are plotted against the B - R colors of the galaxies corrected for extinction with the Tully et al. (1998) formalism. In this figure, we also plot maximum disk (M /L) for those galaxies presented in this pap er that have B RK photometry. All the p oints in Figure 4 are upp er limits to the (M /L) ; galaxies cannot have values greater than those defined by the lower envelop e in this plot without over-predicting their rotation curves b eyond the very inner parts. It is apparent from Figure 4 that the Bell et al. (2003) color-M /L relation is consistent with all the galaxies to within 0.1 dex, which is the uncertainty in the relations. Other than the IMF, there are five primary sources of uncertainty in the determination of the baryonic surface mass-density distributions: uncertainties in the distances to galaxies2 , 0.1 dex spread in the color-M /L
2

Fig. 4.-- Maximum disk stellar (M /LK ) versus
reddening-corrected B - R colors for galaxies in the Verheijen (1997) sample (Xs) and in this paper (triangles). Plotted for reference are the colorM /L relations of Bell & de Jong 2001 (dashed line) and Bell et al. 2003 (solid line).

Of the 34 galaxies in our sample, 4 have distances estimated from Cepheid variables for which the uncertainties are much less than other typ es of distance

relations, uncertainties in the zero-p oints of these relations, uncertainties in the photometric zero-p oint calibrations, and uncertainties in the determination of a galaxy's inclination to the line of sight. Secondary sources of uncertainty generally have a small effect on the baryonic rotation curves. They include secondary photometric ("b ootstrap") calibrations, the neglect of interstellar gas content, and p osition angle uncertainties. Dust reddening also plays a role in the uncertainty of the baryonic rotation curves; this will b e discussed in ?4.3. Note that the transformation of Two Micron All Sky Survey (2MASS; Jarrett et al. 2000; Cutrie et al. 2000; Jarrett et al. 2003) and SDSS photometry to the KronCousins system introduces negligible uncertainties, as shown in Pap er I. As an illustration, Figure 5a-d shows the
measurements, and are typically 10%.

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effects of three of the primary sources of uncertainty on the baryonic rotation curve of NGC 157. In all the plots in Figure 5, we plot |vb |/vb where vb is the difference b etween the velocity derived from the color-M /L relations and this velocity affected by the named uncertainty. Figure 5a shows the effect of an uncertainty of Á 20% in the distance. The distance uncertainty causes a maximum change in |vb |/vb of 0.24 which corresp onds to a change in the velocity of nearly 40 km s-1 . The average change is |vb |/vb = 0.09 (|vb | 13 km s-1 ). Figure 5b shows the effect of a systematic change of Á0.1 dex in the color-M /L relations. This uncertainty causes a maximum change in |vb |/vb of 0.12 (|vb | 25 km s-1 ), and an average change of 0.12 (|vb | 20 km s-1 ). In Figure 5c,d, the effects of introducing systematic errors in the photometric zero-p oint calibrations are shown. In Figure 5c, the effect of the actual zero-p oint uncertainties for NGC 157 is shown (B = 0.03, R = 0.03, K = 0.04). This causes an average change in |vb |/vb of 0.10 (|vb | 17 km s-1 ). While most of our optical photometric zero-p oints are good to 5%, the worst uncertainty in an optical zero-p oint calibration for any of the surface brightness profiles that we use in this analysis is Á15%, as given in Table 3 of Pap er I. All of the uncertainties on the near-infrared zero-p oints are 4%. To show the effect of a photometric calibration that is not as good as that of NGC 157, in Figure 5d we show what would happ en to the rotation curve if photometric zero-p oint errors were Á15% in the optical and Á4% in the near-infrared. Such uncertainties cause an average change in |vb |/vb of 0.13 (|vb | 22 km s-1 ). In Figure 5e,f, we show the effects on the baryonic rotation curve of NGC 157 of changing the inclination and p osition angle in the derivation of its surface brightness profiles. Typical uncertainties in p osition angles and

Fig. 5.-- Effects of uncertainties on the baryonic
rotation curve of NGC 157. For one side of the uncertainty (e.g., +0.1 dex), the |vb |/vb curve is plotted as a dotted line, and for the other side (e.g., -0.1 dex) the curve is plotted as a solid line. The effects of the following uncertainties are plotted: (a) Á20% uncertainty in the distance to the galaxy, (b) Á0.1 dex scatter in the color-M /L relation, (c) actual photometric zero-point uncertainties for NGC 157, (d) photometric zero-point uncertainties for the galaxy with the poorest zeropoint, (e) a change in the inclination of 5 , and (f ) a change in the position angle of 5 . The radii 2.2hK and R25 are marked as thin and thick lines, respectively, on the x-axis in panel a.

inclinations are b oth Á5 (Pap er I). In Figure 5e, we plot the effect of a change in the p osition angle of Á5 , and in Figure 5f we show the effect of a change in the inclination of Á5 . Uncertainties in the inclination have a larger effect (on average |vb | 0.03; |vb | 5 km s-1 ) than those in the p osition angle (on average |vb | 0.007; |vb | 1 km s-1 ). However, b oth these uncertainties are small when compared to other sources of error discussed in this section. In Figure 6, to compare the color-M /L re12


baryonic rotation curves. 3.4. Effects Due to Dust

Fig. 6.-- Baryonic rotation curves derived from
the color-M /L relations (plotted as dotted lines for the Á0.1 dex scatter in the relations) are compared with baryonic rotation curves derived from constant (M /LK ) of 0.75 (thin solid line) and 1.0 (thick solid line). The thin dashed line plotted for NGC 7606 is the model observed rotation curve of Courteau 1997. All other features of the plot are the same as Figure 3.

lations to constant (M /LK ) , baryonic rotation curves created with the color-M /L relations and with (M /LK ) of 0.75 and 1.0 are plotted for 4 example galaxies. The rotation curves created with (M /LK ) of 0.75 and 1.0 are approximately consistent with the Á0.1 dex uncertainty of the color-M /L relations. This is similarly the case for all galaxies in our sample, and is due to the shallow slop es of the color-M /L relations ( 0.14-0.18 in log10 M /L). To summarize, other than the IMF, of the other sources of uncertainty in the determination of the baryonic rotation curves, only the distance uncertainty, the Á0.1 dex scatter in the color-M /L relations, and zero-p oint uncertainties can produce non-negligible effects. These three sources of uncertainty vary from galaxy to galaxy and introduce scatter in the 13

Since the reddening vector in Figure 1 lies nearly parallel to the color-M /L relation, to first order, errors in foreground dust reddening estimates should not strongly affect the final relative derived masses of the stellar p opulations, as foreground dust will systematically b oth redden and extinguish galactic light. In this section, we discuss the p ossible effects of dust reddening and extinction on absolute derived stellar masses. We examine the empirical inclinationdep endent extinction correction of Tully et al. (1998), which should describ e to first order the dust content of galaxies. We correct to face-on the total integrated colors and magnitudes of galaxies in our sample. In doing this, we ignore radial dust gradients in the disk and the fact that dust reddening is likely to b e large for the inner parts of galaxies. While the K -band magnitudes barely change when extinction corrected (0.05 mag on average), the B - R colors do (0.09 mag on average). This, however, should not affect the final derived stellar masses, since the color-M /L relations that we use have a very shallow slop e in B - R color. To examine this effect further, for 4 galaxies that span the range of inclinations in our sample, we apply the Tully et al. (1998) extinction correction to their radial K -band profiles, and use their extinctioncorrected B - R colors to derive (M /LK ) . From the resulting surface mass-density profiles, we derive baryonic mass rotation curves. In Figure 7, we plot for the 4 galaxies the difference b etween the extinction-corrected baryonic rotation curve and the uncorrected one. For NGC 157, NGC 3726, NGC 7217, and NGC 7606, the rotation curves on average differ by 1.3, 1.1, 2.7, and 5.0 km s-1 , resp ectively. The greatest difference is for NGC 7606 which has the largest inclination


the baryonic rotation curve of each galaxy uniquely. 4. Dark Matter

For each galaxy, we compare its baryonic rotation curve derived in ?3.2 with its observed rotation curve taken from the literature to derive a "dark matter rotation curve." 4.1. Observed Rotation Curves

7.-- Difference between the extinctioncorrected (V ) and uncorrected (Vo ) baryonic rotation curves. The extinction-corrected baryonic rotation curves are derived from colors and magnitudes corrected for extinction with the formalism of Tully et al. 1998. of the four galaxies, 63.9 . These differences in velocity are all less than the uncertainties of the baryonic rotation curves themselves, as discussed in ?3.3. Galaxy disks have b een observed to have a relatively constant face-on dust opacity of 0.5 magnitude in the I -band (Holwerda et al. 2005). Correcting for this, AI would increase the baryonic velocity by 20 km s-1 . However, since the maximal disk limits in Figure 4 would also need to b e corrected, and the color-M /L relations re-scaled accordingly, the net effect would more or less cancel out. However, it is very likely that more complex dust models where the effects of dust dep end on star/dust geometry are necessary, such as those by Disney (1989) and Gordon et al. (2001). We choose to leave this approach to a future analysis, since the detailed radiative transfer codes are not publicly available and are difficult to apply to real galaxies. Moreover, different galaxies may have various amounts of dust and different star/dust geometries which would cause dust to affect

Fig.

Observed rotation curves are plotted in Figure 3; Table 1 lists the tracer and literature reference for each. Some galaxies have two observed rotation curves, one from H or N II observations which traces the inner parts, and another from H I observations which traces the outer parts. If errors for the observed rotation curves were given in the original table or plot that they were taken from, then these are used in this pap er. If no errors are given, then we estimate them to b e the difference in rotation velocity b etween the approaching and receding sides of the galaxy; rotation curves for which we estimate the errors in this manner are noted in Table 1. This estimate will generally give errors larger than the true measurement errors since it will b e more affected by non-axisymmetric features such as spiral arms and slight warps in the gas distributions. Note that the 2-3 innermost p oints of the H I rotation curves and the outer few p oints of the H and N II rotation curves have a greater uncertainty than other p oints due to b eam smearing and low signal-to-noise, resp ectively. For many of the rotation curves we have obtained data from the authors, but for a handful we could not. For those few galaxies, rotation curves are extracted from plots in the literature with the DataThief program (Tummers 2000); these galaxies are noted in Table 1. Errors inherent to the extraction of a rotation curve vary from plot to plot, but tend to b e 5 km s-1 .

14


Galaxies marked with the reference "Mathewson & Courteau" in Table 1 are rotation curves that were originally presented in Mathewson, Ford, & Buchhorn (1992) and were later modeled by Courteau (1997). For these galaxies, we plot b oth the actual and model rotation curves in Figure 3. Since we are primarily interested in large-scale trends, we adopt the model rotation curves in the following analysis in order to avoid much of the fine structure inherent to the actual data. 4.2. Dark Matter Rotation Curves

At those radii where the observed rotation sp eed of a galaxy is greater than that of its baryons, the additional gravitational comp onent is assumed to b e due to dark matter. A dark matter rotation curve is derived as the square root of the difference of the squares of the observed rotation velocity and the baryonic rotation curve velocity at each radius (Binney & Tremaine 1987). In doing this, it is assumed that the halos of galaxies are axially symmetric, the disk and halo are aligned, and the observed gas is in circular orbits. Dark matter rotation curves for the galaxies are plotted in Figure 3. For 10 galaxies, the baryonic rotation curves over-predict the observed rotation curves for some range in radius3 . These galaxies are NGC 157, NGC 1559, NGC 2139, NGC 2841, NGC 3198, NGC 4138, NGC 4698, NGC 5371, NGC 6300, and NGC 7083. The baryonic rotation curve for the -0.1 dex scatter in the color-M /L relations also over-predicts for the first six galaxies listed ab ove. However, for these six galaxies, the over-prediction oc3

We do not include in this count baryonic rotation curves that over-predict the inner parts of H I rotation curves, since they are affected by b eam-smearing in this region. Also, Figure 4 suggests that this numb er is less than 10, but we ignore the very inner parts of the galaxies when calculating maximal (M /LK ) s in ?3.1.

curs only in the very inner parts where the observed rotation curve is affected by structures such as rings, bars, inner windings of spiral arms, and/or irregular morphology. Baryonic rotation curves are derived under the assumption of circular motion and need not trace such structures. For NGC 6300 and NGC 7083, the baryonic rotation curves for -0.1 dex do not over-predict the observed rotation curves. The galaxy NGC 6300 has a bar and a ring in the region where the over-prediction occurs, but the image of NGC 7083 shows no sign that its baryonic rotation curve should over-predict. This is somewhat acceptable, though, since the baryonic rotation curve for NGC 7083 for -0.1 dex does not over-predict. Two galaxies, NGC 4698 and NGC 5371, have baryonic rotation curves that overpredict their observed rotation curves, even for the -0.1 dex scatter in the color-M /L relations. For b oth these galaxies, the overprediction cannot only b e explained by bars and/or rings in the galaxy images. To examine things further, we create baryonic rotation curves using the p osition angle and inclination that were used to derive their observed rotation curves; the resulting curves also over-predict. Next, we create baryonic rotation curves for the -20% uncertainty in the galaxies' distances and b oth the distance and color-M /L uncertainties; they are plotted in Figure 8. For NGC 4698, the baryonic rotation curve created by taking into account b oth these effects still over-predicts. However, for this galaxy, its distance was calculated with a Virgocentric infall calculation and was found to b e triple-valued (Pap er I). We derive a baryonic rotation curve for it with the distance solution which is closer than the one chosen in Pap er I (9.7 Mp c, as opp osed to the chosen solution of 19.1 Mp c). This baryonic rotation curve is plotted in Figure 8 and overpredicts only the inner p ortion ( 1.5 kp c) of the observed rotation curve. For NGC 5371,

15


4.3.

Uncertainties in Dark Matter Rotation Curves

Fig. 8.-- Effects on the baryonic rotation curves
of the -0.1 dex scatter in the color-M /L relations and the -20% uncertainty in distance for galaxies where the baryons over-predict the observed rotation curve. Original baryonic rotation curves (thick solid lines), those derived by taking into account the distance uncertainty (dashed lines), and those derived by taking into account both the distance uncertainty and the color-M /L scatter (dotted lines) are triangles for H and small points for H I. For NGC 4698, we also plot the baryonic rotation curve derived from the adoption of a different distance to the galaxy (thin solid line). Observed rotation curves are plotted as points. The radius 2.2hI R is marked as a thick solid line on the x-axis.

Uncertainties in the dark matter rotation curves arise from a numb er of effects. The most significant are those inherent to the determination of the baryonic mass comp onent, as discussed in ?3.3. Other uncertainties include: non-circular motions that p erturb the underlying p otential (i.e., spiral arms, bars, substructure), statistical errors from the measurement of velocities in radial bins, systematic errors in measuring the velocity (i.e., b eam smearing and slit p osition angle error), and uncertainties in the measurement of the dynamical centers of the galaxies. There also may b e differences b etween the centers of galaxies determined from photometry and those determined from the observed rotation curves. However, center measurements are not exp ected to differ much since photometric centers are always chosen to b e the brightest pixel in the nucleus which coincides with the dynamical center of most galaxies. 5. Maximal Disks

the baryonic rotation curve created by taking into account the distance uncertainty overpredicts, while that created by taking into account b oth distance and color-M /L uncertainties does not. The sort of over-prediction that is observed for NGC 4698 and NGC 5371 can also b e due to such factors as a mismeasurement of the observed rotation curves and/or the effects of dust. A stellar p opulation affected by dust may app ear redder (and hence heavier and have a faster rotation) than it is intrinsically since the reddening effect of dust is slightly greater than its extinction effect. These galaxies could also signal a need to lower the normalization of the color-M /L relations. If we did this, then many of the other galaxies in the sample would b e submaximal, but the qualitative results of this pap er would not change.

The radius R = 2.2h is where the rotation curve of a self-gravitating exp onential disk reaches its p eak (Freeman 1970). A commonly used definition of a maximal disk is given by Sackett (1997) where the galaxy disk provides 85% Á 10% of the total rotational supp ort of the galaxy at 2.2hR . In the following analysis, we define a galaxy to have a maximal disk if it has a baryonic mass (disk and bulge) contribution to the observed rotation curve of > 90% at 2.2h, or similarly, a dark matter contribution to the observed rotation curve of < 10% at 2.2h. We choose to adopt this definition of maximal disk over that of Sackett (1997) since we p erform detailed bulgedisk decomp ositions, and are therefore able to model the combined bulge plus disk baryonic rotation curves. Furthermore, we use h

16


measured at K , and when K -band imaging is unavailable, we use H . Since near-infrared bands trace most of the mass of the stellar p opulations, a near-infrared scale-length is more analogous to the stellar mass scalelength of a disk. Disk scale-lengths measured at K are typically 1.2 times shorter than those measured at B (de Jong 1996). In Figure 3, there are 4 galaxies that have submaximal disks, even if the -0.1 dex scatter in the color-M /L relations is taken into account. Since the normalization of the colorM /L relations is an upp er limit, the numb er of galaxies in our sample that do not have maximal disks can in principle b e much greater than 4. However, for those galaxies that we observe to have maximal disks, the overall shap es4 of the inner parts of their observed rotation curves (within R25 ) are generally matched by the shap es of their baryonic rotation curves. This is evidence that many of the disks have a significant baryonic comp onent in their inner parts. These 4 galaxies are NGC 33195 , NGC 3992, NGC 4062, and NGC 7606; they have baryonic mass contributions to their total masses at 2.2hI R of 22%, 50%, 57%, and 68%, resp ectively. In Figure 9 we plot for these 4 galaxies the baryonic rotation curves with and without uncertainties in distances and the color-M /L relations taken into account; these baryonic rotation curves still under-predict the observed rotation curves for NGC 3319, NGC 3992, and NGC 4062. However, for
4

5

This is distinguished from the "bumps and wiggles" of the observed rotation curves that are likely due to small-scale features such as bars and spiral arm p erturbations (Palunas & Williams 2000; Kranz, Slyz, & Rix 2003; Slyz, Kranz, & Rix 2003). For NGC 3319, we do not take into account its innermost 2 observed rotation curve p oints since they are measured from H I and are likely affected by b eam-smearing. Also, the distance measurement for NGC 3319 is from Cepheid variable stars and has an uncertainty of only 10%.

NGC 7606, the baryonic rotation curve that takes into account b oth these effects follows the observed rotation curve fairly well. For NGC 4062, its distance was calculated using a Virgocentric infall calculation and was found to b e triple-valued (Pap er I). We create baryonic rotation curves for NGC 4062 with the other 2 distance solutions (17.6 and 24.4 Mp c), b oth of which are greater than the one chosen in Pap er I; they are plotted in Figure 9. These baryonic rotation curves have a strange b ehavior since they underpredict the observed rotation curve in the inner parts, but match it in the outer regions. This may b e the effect of such factors as an underestimate of the stellar mass in the inner parts of the galaxy due to blue spiral arms in the B -band image (see image in Figure 1 of Pap er I), a p oor bulge/disk decomp osition, and/or an underestimate of the uncertainties of the observed rotation curve. In summary, while NGC 4062 and NGC 7606 may not have submaximal disks, NGC 3319 and NGC 3992 likely do. There are a few galaxies that have marginally submaximal disks. For these galaxies, the baryonic rotation curves for the +0.1 dex scatter in the color-M /L relations result in maximal disks. These galaxies are NGC 1241, NGC 2139, and NGC 2280, and they have baryonic mass contributions to their total rotational supp ort at 2.2hI R of 75%, 73%, and 72%, resp ectively. One striking example of a galaxy that is likely close to maximum disk is NGC 157. This galaxy has H and H I rotation curves that have a sudden steep decline at 8kp c ( 3hK ) and flatten afterward, which results in a hump-like structure. While this p eculiar b ehavior cannot b e absolutely confirmed by Ryder et al. (1998), there are strong lines of evidence presented in their pap er that p oint to this hump-like structure as physical. This same structure is also found in the baryonic 17


NGC 3319

NGC 3992

NGC 4062

NGC 7606

Fig. 9.-- Same as Figure 8, but for galaxies where
the baryons under-predict the observed rotation curves, and for the +0.1 dex scatter in the colorM /L relations and the +20% distance uncertainty. For NGC 4062, baryonic rotation curves derived for 2 different distance determinations are plotted as thin solid lines.

rotation curve for NGC 157. Moreover, a dark matter halo in the shap e of a NFW model is not consistent with this structure unless baryons make a significant contribution to the inner 5 scale-lengths of NGC 157. In summary, although a lower normalization of the IMF cannot b e ruled out, it seems unlikely that at least NGC 157 is strongly submaximal. 6. Dark and Baryonic Matter Scaling Relations

mass distributions, we also tabulate the differences in their values for renormalizations of the color-M /L relations by Á0.1 and -0.3 dex. We choose to create the quantity Vb,max b ecause it can b e derived from imaging alone and obviates the need for much more exp ensive and time-consuming line width observations needed to obtain Vtot,max . For bright spirals, Vb,max should not differ from Vtot,max by very much, and due to the flat and usually noisier nature of the observed rotation curves, the measurement of Vb,max is more straightforward than that of Vtot,max . This quantity will b e discussed further in ?6.2. We derive two quantities from the dark matter rotation curves calculated in ?4.2: R10 , the radius where dark matter contributes 10% to the velocity of the observed rotation curve, and RX , the radius where the dark matter contribution equals that of the baryons (the "cross-over radius"). The quantity R10 is similar to RIBD of Salucci (2001) and Rt of Persic et al. (1996). The radius RX is analogous to R2:1 of McGaugh & de Blok (1998). The radii R10 and RX are listed in Table 2 along with the difference in values for renormalizations of the color-M /L relations by Á0.1 and -0.3 dex. Note that for some galaxies we cannot measure R10 or RX ; this is in general b ecause their rotation curves do not extend far enough in radius. 6.1. Baryonic Scaling Relations

In this section, we examine scaling relations for dark and baryonic matter. We use quantities from Tables 1 and 4 of Pap er I, and derive others: Vtot,max the maximum observed rotation curve velocity, Vb,max the maximum baryonic rotation curve velocity, R(Vb,max ) the radius at which V = Vb,max , and Mb the baryonic mass. These quantities are listed in Table 2, and if they are derived from the baryonic

In Figure 10, basic physical parameters of galaxies are plotted versus Mb : total absolute B and K -band magnitudes, Hubble Ttyp e, R25 , and B and K -band central surface brightnesses (Åo,B , Åo,K ). They have the following correlation coefficients: 0.91 (B ), 1.00 (K ), 0.40 (Hubble T-typ e), 0.78 (R25 ), 0.29 (Åo,B ), and 0.58 (Åo,K ). Integrated magnitudes, sizes, Åo,K , and maximum rotation velocities correlate very well with Mb , such that galaxies with greater Mb are brighter, larger, 18


Fig. 10.-- Basic physical parameters of galaxies versus baryonic mass. Open circles represent galaxies from the SDSS that have partial imaging. A typical error bar for Mb is plotted in panel a; error bars for the other quantities are the size of the data points at the resolution of the figure. To make the plot in panel c such that points do not overlap, we add fractions with values < 1 to the integer Hubble T-types.

Fig. 11.-- Relations for Vtot,max and Vb,max , which trace the total and baryonic mass component of the galactic potential, respectively. Open circles represent galaxies from the SDSS that have partial imaging. Error bars for T-type, LB , and R25 are the size of the data points at the resolution of the figure. ple have. In addition, a tight correlation indicates that there is not a wide spread in the degree of maximality. The quantity log10 Vtot,max also correlates with Hubble T-typ e and R25 such that galaxies that rotate faster have earlier T-typ es and are larger. The quantity R(Vb,max ) correlates with log10 Mb with a correlation coefficient of 0.66, but correlates relatively weakly with log10 Vb,max , log10 Vtot,max , and Åo,K with correlation coefficients of 0.08, 0.35, and 0.16, resp ectively. As discussed ab ove, the maximum rotation sp eeds predicted from the baryon distributions are tightly correlated with the observed maximum rotation sp eeds. To examine this relation further, we plot the ratio of Vb,max to Vtot,max versus Åo,K in Figure 12, and a find root mean square (rms) deviation from unity of only 0.18. Using this result, one can create a Tully-Fisher relation from two passband surface photometry and a redshift alone (e.g., 19

and have brighter Åo,K . Morphology and Åo,B also correlate with Mb , but to a lesser extent. In Figure 11, relations are plotted for log10 Vb,max and log10 Vtot,max , which trace the baryonic and total mass comp onents of the galactic p otential, resp ectively. In Figure 11a, along the lines of Rob erts & Haynes (1994), we plot log10 Vtot,max versus log10 LB and find good agreement with their results. In Figure 11b, we plot log10 Vb,max versus log10 Vtot,max ; they are strongly correlated with a correlation coefficient of 0.82. These velocities should b e equivalent for maximal disks, which many of the galaxies in our sam-


a

b

c

d

Fig. 12.-- Ratio of maximum baryonic velocity to maximum observed velocity versus Åo,K . SDSS galaxies with partial imaging are plotted as open circles. Outliers are labeled: NGC 4062 and NGC 3992 have submaximal disks, NGC 4138 has evidence of kinematic disturbance, and NGC 3319 has both a submaximal disk and a kinematic disturbance.

Fig. 13.-- Baryonic Tully-Fisher relations for
Vtot,max and Vb,max , and the size-baryonic mass relation are plotted in panels a-c. Weighted bisector fits with and without NGC 3319 are plotted as black and gray lines, respectively. In panel d, the Vtot,max residual of the Tully-Fisher relation is plotted versus the size residual from the sizebaryonic mass relation. Galaxies from the SDSS that have partial imaging are plotted as open circles.

by using the SDSS and the color-M /L relations for g - r and (M /Lr ) ). The 4 outliers in Figure 12 are galaxies that have submaximal disks and/or are kinematically disturb ed. The relation b etween Vb,max and Vtot,max is not exp ected to b e as tight for less luminous spirals, but could p ossibly b e calibrated for such a p opulation. In Figure 13a,b we plot the baryonic Tully-Fisher relations for Vtot,max and Vb,max . Weighted bisector fits are given in the form: Mb = M200 V where M200 is the stellar mass in units of 1010 M for a galaxy on the TullyFisher relation with V = 200 km s-1 . They are: M200 = 5.40 Á 0.03 and = 3.4 Á 0.3 with a rms residual of 0.20 for Vtot,max , and M200 = 6.98 Á 0.03 and = 2.8 Á 0.3 with a rms residual of 0.22 for Vb,max . If we 20

do not include the outlier in these two relations, NGC 3319, we obtain the following fits: M200 = 5.36 Á 0.03 and = 3.3 Á 0.3 with a rms residual of 0.19 for Vtot,max , and M200 = 6.75 Á 0.03 and = 3.1 Á 0.3 with a rms residual of 0.21 for Vb,max . The galaxy NGC 3319 is b oth kinematically disturb ed on one side (Moore & Gottesman 1998), and has the smallest stellar mass in our sample. For most of the galaxies in our sample, we do not have H I or molecular hydrogen measurements. If gas masses were included, the TullyFisher slop e of Mb on Vtot,max would flatten, as lower mass galaxies (e.g., NGC 3319) have a larger gas content than more massive galaxies (e.g., Verheijen 1997).


We refrain from comparing the slop es found here and in the literature (e.g., Bell & de Jong 2001; McGaugh 2005; Pizagno et al. 2005) for the following reasons. Such fits are sensitive, among other things, to the velocity measurements used (i.e., Vtot,max , Vb,max , Vf lat , V2.2 , W20 ), the range in stellar mass of the samples, and the galaxy samples used (range in Hubble typ e, field versus cluster environments). In particular, Gurovich et al. (2005) find a break in the stellar mass and baryonic Tully-Fisher relations for less massive galaxies (107.6 M 109.6 M ). This break is such that the Tully-Fisher relation for less massive galaxies is steep er than that for galaxies with stellar masses from 109.6 -1011.2 M . Gurovich et al. (2005) find slop es for the lower and upp er mass ranges of their H -band sample, using W20 as a velocity measurement, of 4.4 Á 0.3 and 3.3 Á 0.3, resp ectively. This break/curvature makes a comparison b etween slop es of different samples risky at b est, esp ecially when different selection criteria are used. A 2D Kolomogrov-Smirnov test would b e ideal, but b eyond the scop e of this work considering the modest stellar mass range of our sample. The size-baryonic mass relation, where size is parameterized by hI R , is plotted in Figure 13c. We p erform a weighted bisector fit and find hI R = 0.67 Á 2.31 (Mb /1010 M )0.60Á0.08 kp c with a rms of 0.16. We compare the Vtot,max residuals of the Tully-Fisher relation with the size residuals of the size-mass relation in Figure 13d, and find a scatter plot. If galaxy disks are maximal and the dark matter halos of galaxies of different surface brightness are identical, then it is predicted by McGaugh & de Blok (1998) and Courteau & Rix (1999) that there should b e a correlation b etween these residuals. Courteau & Rix (1999) argue that the lack of residual correlation, essentially the surface brightness indep endence of the Tully-Fisher relation, implies that all

disks are submaximal. This argument, however, is not entirely straightforward, since in the Pizagno et al. (2005) model for disk collapse within a halo, variations in dark matter halo parameters can create enough scatter in the Tully-Fisher relation to hide the predicted correlation. Furthermore, McGaugh & de Blok (1998) and Sellwood (1999) argue that submaximal disks do not solve the problem, and one is left with a fine-tuning problem: either disk M /Ls correlate with surface brightness, halo contributions vary, or Newtonian dynamics falters. Figure 12 illustrates this fine-tuning problem by showing that Vb,max /Vtot,max is indep endent of surface brightness, so that the total velocity "knows ab out" the baryonic contribution. Changing the normalization of the color-M /L relations re-scales the y-axis of the plot, but the ratio remains indep endent of surface brightness. 6.2. Dark Matter Scaling Relations

In Figures 14a-d and 15, the baryon/dark matter equality radius in units of the disk scale length, RX /hI R , is plotted versus a numb er of galactic parameters: Vtot,max , Vb,max , Hubble T-typ e, Mb , B , K , Åo,B , and Åo,K . In Figure 14e, we plot RX versus hI R . There are fewer p oints in these figures than in previous ones since not all galaxies have a dark matter rotation curve that allows us to determine RX . The correlation coefficients for these relations and for R(Vb,max ) with RX are listed in Table 4. The radius RX is found to correlate most strongly with Vb,max and very strongly with Mb , Hubble T-typ e, and Vtot,max . There are no changes in the relative strengths of the relations if the color-M /L relations are renormalized by Á0.1 or -0.3 dex. The trends with RX that we find are qualitatively in agreement with those in the literature (e.g., de Blok & McGaugh 1997; McGaugh & de Blok 1998; Pizagno et al. 2005), with some exceptions that are discussed here.

21


and RX . In panels a-e, triangles represent galaxies from SDSS that only have partial imaging. Error bars represent the values of the quantities derived by renormalizing the color-M /L relations by Á0.1 dex; upper and lower limits to these values are plotted in gray. To make the points in panel c such that they do not overlap, we add fractions with values < 1 to the integer Hubble types. In panel f, galaxies are plotted with different symbols according to their hI R : those with hI R < 1 kpc as open circles, those with 1 kpc hIR < 3 kpc as filled circles, and those with 3 kpc hIR as open triangles. The galaxies NGC 2841, NGC 3319, and NGC 4138 are labeled in all panels; they have evidence of kinematical disturbances despite their normal appearances. The three outliers in Figures 14 and 15 are NGC 2841, NGC 3319, and NGC 4138. Despite their normal optical morphologies, these galaxies have signatures of kinematic disturbances. The galaxy NGC 2841 has a warp in its outer H I disk (Bosma 1981) and an indication of a counter-rotating stellar component for 5 r 12 (SilÄhenko, Vlasyuk, & Burenkov c 1997). Similarly, NGC 4138 has both a counter-rotating disk and a significant warp in its outer H I disk (Jore, Broeils, & Haynes 1996). This warp may be the cause of the decline of its H I rotation curve, and hence what makes this galaxy an outlier. The third outlier, NGC 3319, is discussed in ?6.1.
IR

Fig. 14.-- Relations for RX /h

22


and integrated magnitudes and central surface brightnesses. Symbols and error bars are the same as in Figure 14. The galaxy NGC 3319 has a Åo,K measurement, but not a K measurement. Error bars along the x-axis are approximately the point size at the resolution of the plot.
IR

Fig.

15.-- Relations for RX /h

McGaugh & de Blok (1998) and de Blok & McGaugh (1997) find that Åo,B plays a ma jor role in relations with galaxy prop erties, including with R2:1 , which is analogous to our RX parameter. We find that RX /hI R is correlated with Åo,B , but that it is even more correlated with Åo,K . We do not find a correlation b etween the mass discrepancy and Åo,K (in Figure 12 we plot the inverse mass discrepancy), but McGaugh & de Blok (1998) do find a relation b etween total M /LB at 4h and Åo,B . Also, we find that hI R correlates strongly with RX , once outliers are removed, which is not in qualitative agreement with other studies: McGaugh & de Blok (1998) and Zavala et al. (2003) find that h does not correlate with total M /LB evaluated at 4h or the mass discrepancy evaluated at maximum rotation velocity, resp ectively. However, the results of Pizagno et al. (2005) for relations with h are consistent

with ours; they find that more compact galaxies have a larger mass discrepancy (measured at 2.2hi ) than larger galaxies. Zavala et al. (2003) find a numb er of relations involving a quantity similar to the mass discrepancy that are in contradiction to those presented in this pap er and in the literature. Here we discuss p ossible causes of these discrepancies. Our differences with McGaugh & de Blok (1998) may b e traced to the quantities used in the analyses: they use M /LB and hB , while we use RX and hI R . Luminosity in the B -band should have more scatter in relations with galaxy prop erties than baryonic mass derived from a combination of optical and near-infrared data. This is b ecause the B -band is more affected by star formation and extinction, while the K -band is a b etter tracer of stellar mass. In addition, a near-infrared scale-length is more analogous to the baryonic mass scale-length of a disk, and hence should have less scatter in its relations with galaxy prop erties than an optical scale-length. Adding some credence to the hyp othesis that different quantities are the root of the discrepancy, McGaugh & de Blok (1998) find that relations b etween R2:1 and b oth Åo,B and MB are such that brighter galaxies have larger values of R2:1 , consistent with our results. That McGaugh & de Blok (1998) find a correlation b etween M /LB evaluated at 4h and Åo,B is likely also due to the larger range in surface brightness in their sample. In Figure 14f, R10 is plotted versus RX , and galaxies are plotted as different symb ols according to their value of hI R . Although there is some scatter, as hI R increases, these two radii move further out in the disks in tandem. That there is such a tight relation b etween R10 and RX tells us that dark matter contributions to the observed rotation curves must increase in a characteristic way b etween these two radii for all galaxies. This is likely due to the combined effects of quasi-exp onential 23


disks and observed rotation curves that are nearly flat. If this is correct, then R10 should have more scatter in its relations with galaxy prop erties than RX b ecause the observed rotation curves should not yet b e flat in the region where R10 is measured, which is found to b e the case. 7. Radial Behavior of Dark Matter

In Figure 16, the dimensionless parameter (r ) Mb (r )/Mtot (r ), where Mb is the baryonic mass, and its inverse are plotted for galaxies with an appreciable dark matter contribution. It measures the fractional contribution of baryons to the gravitational p otential as a function of radius in a galaxy, and is akin to the parameter defined by Salucci (2001) and similar parameters used in many other pap ers, except here it is evaluated at all radii. Where (r ) = 1, the baryonic mass of a galaxy accounts for its observed rotation curve, (r ) = 0.5 at r = RX , and (r ) = 0 where the dark matter accounts for its observed rotation curve. If the baryonic model over-predicts the observed rotation curve, then (r ) > 1. In Figure 16a, (r ) is plotted versus radius in terms of hI R . For many galaxies, the observed rotation curves are entirely accounted for by baryons in the inner parts. Beyond this region, baryonic mass falls off as dark matter b egins to dominate. Other galaxies are dark matter dominated throughout. Curves in Figure 16a have different line typ es that corresp ond to ranges of Vb,max . For most of the galaxies, as Vb,max increases, so does the prop ortion of baryonic to dark matter at all radii such that the fastest rotators are observed to b e dominated by baryons until quite far out into their disks. However, there is clearly much scatter ab out this trend; this conclusion can also b e inferred from Figure 14f. In Figure 16b, (r ) is plotted versus radius in units of RX . This choice of radial coordi24

Fig. 16.-- (r) Mb (r)/Mtot (r) and its inverse for galaxies with an appreciable dark matter contribution. (r) is plotted versus r/hI R and r/RX in parts a and b, respectively; its inverse is plotted versus r/RX in panel c. Different line types correspond to the Vb,max of the galaxies: Vb,max > 250 km s-1 (solid line), 201 < Vb,max 250 (dotted line), 120 < Vb,max 201 (dashed line), and Vb,max 120 (dot-dash line line). nate causes the curves to overlap at r = RX (where (r ) = 0.5), and thus allows for a b etter comparison of their radial b ehavior. In Figure 16c, we plot the somewhat less intuitive function -1 (r ) since it increases linearly with r /RX . There is little variation in the b ehavior of the curves in Figures 16b,c. Such regularity can b e explained in terms of quasiexp onential disks and flat rotation curves as follows: The rotation curve of a galaxy is nearly flat b eyond r 2hI R , or it is at least a slow function of r and varies less than Vb (r ) in this region. In addition, much of the bary-


onic mass of a galaxy is enclosed at r 2hI R , which causes the baryonic rotation curves to b e roughly Keplerian (Vb (r ) 1/r ) b eyond this radius. Therefore, for all galaxies, b eyond 2hI R , it should b e the case that (r ) Vb2 /Vt2 t 1/r , which is what we observe. All o the (r ) curves overlap nicely when plotted versus radius normalized to RX since this radius is located b eyond 2hI R , and is generally in the falling part of the baryonic rotation curves. We parametrize the trend of -1 with r /RX in a simple universal manner that holds for all galaxies: -1 = 1.71(r /RX ) + 0.021. This relation is consistent with the prediction of Palunas & Williams (2000) that either the contribution of dark matter within the optical radius of galaxies is small or that the distribution of dark matter is coupled to that of the luminous matter. This relation is also similar in spirit to the parameterization of the mass discrepancy-acceleration relation of McGaugh (2004). Since -1 is the mass discrepancy, and a = V 2 /r 1/r for flat rotation curves, a natural mass discrepancyacceleration relation arises. 8. Comparison With Theories of Halo Density Distributions

We compare the derived dark matter profiles with an analytical function designed to parametrize the density profile of dark matter halos in N-b ody simulations. In particular, we compare our data to the NFW formulation for dark matter halo density profiles (Navarro, Frenk, & White 1996) with and without taking into account adiabatic contraction of the halos. To do this, for each galaxy, we fit its observed rotation curve with a total mass rotation curve created from the addition of its baryonic rotation curve to a grid of halo models, with a reduced 2 statistic. Uncertainties in the observed rotation curves are taken to b e 10 km s-1 ; results do not differ significantly if the error bars plotted in Figure 3 are 25

used. To test the most common implementation of dark matter contraction (Blumenthal et al. 1986), we adiabatically contract the grid of halo models according to the radial density distribution of baryons in the galaxies following the formalism of Dutton et al. (2005), and p erform the fits again. The NFW fitting formula has 2 free parameters that we fit for using a grid covering: 0.5 c200 20 and 0.5 Vtot,max V200 2.5 Vtot,max . In Figure 17, we plot the b est-fit halo models for 8 example galaxies, and in Table 3 we list the b est-fit parameters along with the reduced 2 of all the fits. Most of the fits are very p oor; the average reduced 2 values with and without adiabatic contraction are: 7.1 and 4.2 for the original color-M /L relations, 14.0 and 7.3 for the +0.1 dex renormalization of the color-M /L relations, 4.8 and 3.4 for -0.1 dex, and 3.9 and 4.3 for -0.3 dex. We do not p erform fits for galaxies when most of the observed rotation curve is accounted for by the baryonic rotation curve. For baryonic rotation curves derived from the original color-M /L relations, nearly all fits to the NFW models without contraction have a smaller reduced 2 than those where contraction was p erformed. The exceptions are NGC 4062 and NGC 3992, which likely have submaximal disks. For galaxies that have a large baryonic contribution to the inner parts of their rotation curves, adiabatic contraction over-contracts the inner parts of their dark matter halos such that the resulting total mass rotation curves over-predict the measured rotation curves, as in Weiner, Sellwood, & Williams (2001), for example. For fits to the NFW models where the baryonic rotation curves were derived from color-M /L relations that were renormalized by +0.1 dex, the situation is exactly the same. For -0.1 dex, 7 galaxies have b etter fits when contraction is taken into account, and for -0.3 dex, 15 galaxies have a b etter fit. This implementation of dark mat-


NGC 157

NGC 1241

NGC 289

NGC 2139

NGC 488

NGC 2280

NGC 1090

NGC 2841

Fig. 17.-- NFW halo fits for 8 example galaxies without (left) and with (right) adiabatic contraction.
Best-fit NFW models are plotted as dashed lines, baryonic rotation curves as thick solid lines, total rotation curves (sums of the best-fit NFW models and the baryonic rotation curves) as thin solid lines, and observed rotation curves as points. For those galaxies with a rotation curve from Mathewson et al. 1992 that is modeled by Courteau 1997, we only plot the model as a dotted line for clarity.

26


batic contraction, the derived concentrations for these halos are low compared to those measured from simulations. The concentrations for -0.3 dex are in slightly b etter agreement with simulations, but are still quite low. In concert with this, McGaugh, Barker, & de Blok (2003) and McGaugh (2004) found concentrations for NFW halos that are too low for a standard CDM universe. In addition, Alam, Bullock, & Weinb erg (2002) examined low surface brightness galaxies and found their dark matter halos to b e underconcentrated compared to what is exp ected from a standard CDM universe, even if they assumed the galaxies to b e dark matterdominated. 9. Summary

Fig. 18.-- Best-fit parameters for NFW fits with
and without adiabatic contraction (filled and open symbols, respectively). The median and the 68 per cent of the c200 values measured in numerical Nbody simulations of dark matter halos are plotted as solid and dashed lines, respectively. When the best-fit halo is at the edge of the parameter space searched, it is plotted as a triangle, otherwise it is plotted as a circle. In part a, best-fit parameters for the original and Á0.1 dex renormalizations of the color-M /L relations are plotted; in part b, best-fit parameters for the -0.3 dex renormalization are plotted.

We have decomp osed the rotation curves of 34 nearby bright spiral galaxies into baryonic and dark matter comp onents by applying color-M /L relations to near-infrared and optical photometry, and find the following: Ç The dark-to-luminous matter distributions are self-similar once scaled by RX , the radius where the baryonic and dark matter contributions to the rotation curve are equivalent. This b ehavior is parameterized by a simple function whose form is due to the quasiexp onential nature of galaxy disks and rotation curves that are nearly flat after an initial rise. This result is indep endent of the normalization of the colorM /L relations. Ç The radii R10 and RX , where dark matter contributes 10% and 50% to the rotation of galaxies, resp ectively, correlate with galaxy prop erties. The strongest correlation with RX is with the maximum baryonic rotation sp eed such that galaxies with RX measurements that lie further out in their disks 27

ter contraction works b est when baryons do not account for all of the inner parts of the observed rotation curve. In Figure 18, we plot the b est-fit values of V200 and c200 and the range of these parameters found in N-b ody simulations (Bullock et al. 2001; Eke, Navarro, & Steinmetz 2001). Even under the assumption of no adia-


rotate faster. The next strongest correlations are equivalently baryonic mass, observed rotation sp eed, and Hubble Ttyp e. Contrary to what is exp ected from previous studies (de Block & McGaugh 1998; Zavala et al. 2003), B -band central surface brightness is not found to b e the main driving force in relations with RX for this sample of bright galaxies. The radii RX and R10 move out in galaxy disks in tandem, consistent with a self-similar dark-to-luminous matter distribution among galaxies. Ç We confirm the normalization of the color-M /L relations given in Bell et al. (2003), which is analogous to an upp er limit on the IMF based on the dynamics of disk galaxies. A more careful analysis of the data used in b oth Bell & de Jong (2001) and Bell et al. (2003) is p erformed, and data from this pap er are included in the sample. Ç All but 4 of the 34 galaxies in our sample are close to maximal disk. Two of these 4 galaxies have maximal disks within uncertainties. A prime example of a galaxy that cannot have a submaximal disk given the current formulations of dark matter halos is NGC 157 which has a pronounced hump-like structure in its observed and baryonic rotation curves. Ç Maximum rotation velocities predicted from the baryon distributions of galaxies (Vb,max ) are tightly correlated with the observed maximum rotation sp eeds (Vtot,max ). Using this, a baryonic TullyFisher relation can b e created based on two-passband surface photometry and a redshift alone (e.g., with the SDSS and color-M /L relations for g - r and (M /Lr ) , or for more distant redshift surveys with Hubble Space Telescop e imaging). Such a relation could p ossibly 28

b e calibrated for lower luminosity galaxies. Ç We find generally p oor fits for the NFW parameterization for dark matter halos due to the significant baryonic contributions found in the inner parts of most galaxies. The concentrations of the b est-fit NFW halos are therefore much lower than what is exp ected for galaxies in a standard CDM universe. This is even the case when the colorM /L relations are renormalized by -0.3 dex. In order to have b etter fits, a normalization even lower than -0.3 dex, where baryons contribute very little to the total mass in the inner parts of rotation curves, would have to b e implemented. Adiabatic contraction, as it is normally implemented, makes these fits worse by adding more dark matter to the inner parts of galaxies.

We would like to thank Richard Pogge, James Pizagno, James Bullock, Jay Frogel, Chris Kochanek, and David Weinb erg for valuable discussions. The referee, Stacy McGaugh, is thanked for constructive comments that greatly improved the quality of this pap er. We are grateful to the following authors who provided rotation curves in tabular form via email: Kor Begeman, Gianfranco Gentile, Thilo Kranz, Povilas Palunas, Stuart Ryder, Michele Thornley, and Wilfred Walsh. We are grateful to Marc Verheijen who made available his high-quality imaging and rotation curves of galaxies in the Ursa Ma jor cluster. SAK would like to acknowledge financial supp ort from The Space Telescop e Science Institute Director's Discretionary Research Fund (DDRF). We thank the CTIO TAC for generous allocation of time for The Ohio State University Galaxy Survey and the many p eople over the years who help ed collect these


observations. Funding for The Ohio State University Bright Spiral Galaxy Survey was provided by grants from The National Science Foundation (grants AST-9217716 and AST-9617006), with additional funding by The Ohio State University. This pap er makes use of data from b oth the Sloan Digital Sky Survey and the Two Micron All Sky Survey. The Two Micron All Sky Survey is a joint pro ject of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. Funding for the creation and distribution of the SDSS Archive has b een provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Aeronautics and Space Administration, the National Science Foundation, the U.S. Department of Energy, the Japanese Monbukagakusho, and the Max Planck Society. The SDSS Web site is http://www.sdss.org/. The SDSS is managed by the Astrophysical Research Consortium (ARC) for the Participating Institutions. The Participating Institutions are The University of Chicago, Fermilab, the Institute for Advanced Study, the Japan Participation Group, The Johns Hopkins University, Los Alamos National Lab oratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, University of Pittsburgh, Princeton University, the United States Naval Observatory, and the University of Washington. This research has made use of NASA's Astrophysics Data System, the NASA/IPAC Extragalactic Database (NED), and the Hyp erLeda database and the VizieR catalog access tool. NED that is op erated by the Jet Propulsion Lab oratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. 29

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This 2-column preprint was prepared with the AAS A L TEX macros v5.2.

31


Table 1
Galaxy NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC
a b c

Tracer FP H, HI H H H H, H I H H H H H H H FP H, FP H, H HI HI HI H, H I HI HI HI HI H N I I, H H H HI H FP H, H H H

a

Reference(s) Fridman et al. 2001, Ryder et al. 1998b Walsh, Staveley-Smith, & Oosterloo 1997b Peterson 1980b,c Mathewson, Ford, & Buchhorn 1992 Rubin et al. 1985 Courteau 1997, Gentile et al. 2004 Mathewson, Ford, & Buchhorn 1992 Mathewson, Ford, & Buchhorn 1992 Mathewson, Ford, & Buchhorn 1992 Mathewson, Ford, & Buchhorn 1992 Mathewson, Ford, & Buchhorn 1992 Mathewson, Ford, & Buchhorn 1992 Mathewson, Ford, & Buchhorn 1992 Blais-Ouellette et al. 2004, Giraud 1998c Corradi et al. 1991b , van Albada et al. 1985 Mathewson, Ford, & Buchhorn 1992 Moore & Gottesman 1998b,c Sanders 1996c Verheijen 1997b Kranz 2002, Verheijen 1997b Verheijen 1997b Verheijen 1997b Verheijen 1997b Verheijen 1997b Rubin et al. 1985 Jore, Broeils, & Haynes 1996c ,Verheijen 1997 Rubin, Waterman, & Kenney 1999 Rubin, Waterman, & Kenney 1999 Begeman 1987 Courteau 1997 Buta et al. 2001, Ryder et al. 1996 Mathewson, Ford, & Buchhorn 1992 Rubin et al. 1985 Mathewson, Ford, & Buchhorn 1992

1 2 4 9 1 1 1 1 1 1 2 2 2 2 3 3 3 3 3 3 3 3 3 4 4 4 4 4 5 5 6 7 7 7

5 8 8 0 0 0 2 3 5 8 0 1 2 8 1 2 3 5 7 8 9 9 9 0 0 1 6 6 3 8 3 0 2 6

7 9 8 8 8 9 4 8 5 3 9 3 8 4 9 2 1 2 2 9 4 5 9 5 6 3 5 9 7 0 0 8 1 0

HI

7 0 1 5 9 2 0 9 0 1 8 3 9 1 6 3 9 3 2 1 2 8 1 8 1 6 0 3 7 6

HI HI

I

b

HI

The notation "FP H" is used for rotation curves derived from Fabry-Perot measurements of H. Errors are taken as the difference in velocity between the approaching and receding sides.

The rotation curve has been extracted electronically from a plot in the referenced paper.

32


Table 2
Observed Vtot,max Baryonic Matter R(Vb,max ) Mb +0.1dex -0.1dex -0.3dex (kpc) (1010 M ) 6.2 7.1 +1.8 -1.4 -3.5 4.1 +1.0 -0.8 -2.0 31.1 +7.4 -5.9 -14.3 7.2 +1.8 -1.5 -3.5 2.8 +0.7 -0.6 -1.4 5.4 +1.4 -1.1 -2.7 19.2 +4.8 -3.8 -9.2 2.6 +0.6 -0.5 -1.2 2.3 +0.6 -0.5 -1.1 Dark Matter R10 RX +0.1dex +0.1dex -0.1dex -0.1dex -0.3dex -0.3dex (kpc) (kpc) ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 9.1 +1.7 -1.8 -4.0 ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 16.4 +4.4 -5.3 -12.5 12.4 +4.4d -5.6 -12.4d ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 11.3 +4.2 -2.7 -4.7 ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 23.2 +7.8c -4.9 -18.2 ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ

Galaxy NGC 157

(km/s) 205

Vb,max +0.1dex -0.1dex -0.3dex (km/s) 205 +25 -2 2 -6 0 189 +23 -2 0 -5 5 296 +36 -3 2 -8 6 201 +24 -2 2 -5 9 141 +17 -1 5 -4 1 165 +20 -1 8 -4 9 250 +30 -2 7 -7 3 133 +16 -1 4 -3 9 136 +17 -1 5 -4 0

NGC 289

182

3.6

NGC 488

350

10.1

NGC 908

200

7.3

NGC 1087

136

5.9

NGC 1090

170

7.3

NGC 1241

300

7.9

NGC 1385

140

4.0

NGC 1559

150

4.5

33


Table 2--Continued
Observed Vtot,max Vb,max +0.1dex -0.1dex -0.3dex (km/s) 209 +25 -2 3 -6 1 153 +19 -1 6 -4 5 121 +15 -1 3 -3 5 177 +5 -1 9 -3 2 284 +49 -2 0 -3 5 120 +14 -1 3 -3 5 314 +38 -3 4 -9 2 50 +7 -5 -1 4 263 +32 -2 9 -3 4 Baryonic Matter R(Vb,max ) Mb +0.1dex -0.1dex -0.3dex (kpc) (1010 M ) 3.8 5.4 +1.4 -1.1 -2.7 1.6 +0.4 -0.3 -0.8 1.9 +0.5 -0.4 -0.9 6.8 +1.8 -1.4 -3.4 13.0 +3.3 -2.6 -6.4 2.3 +0.6 -0.5 -1.1 30.8 +7.6 -6.1 -14.7 0.7 +0.1 -0.1 -0.3 8.0 +1.9 -1.5 -3.6 Dark Matter R10 RX +0.1dex +0.1dex -0.1dex -0.1dex -0.3dex -0.3dex (kpc) (kpc) ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 8.7 +1.8 -3.5 -5.4 9.0 +3.4 -0.6 -6.2 6.8 +3.1 -2.1 -4.6 ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 2.6 +0.7 -0.6 -2.6d 10.3 +2.3 -1.8 -5.8d ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 12.6 +5.8 -2.9 -3.7 10.5 +2.6 -2.5 -7.3 ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 3.5 +1.1 -0.8 -3.5d 13.3 +4.2 -2.4 -6.3

Galaxy NGC 1832

(km/s) 200

NGC 2090

a

160

1.6

NGC 2139

140

4.5

NGC 2280

a

210

6.9

NGC 2841

325

4.7

NGC 3198

152

5.6

NGC 3223

320

10.4

NGC 3319

b

132

9

NGC 3521

221

3.1

34


Table 2--Continued
Observed Vtot,max Vb,max +0.1dex -0.1dex -0.3dex (km/s) 144 +22 -1 2 -4 2 186 +23 -2 0 -5 4 158 +18 -1 7 -4 6 227 +28 -2 5 -6 6 188 +21 -2 1 -5 6 167 +20 -1 8 -4 9 110 +14 -1 1 -3 2 272 +33 -3 0 -7 9 173 +32 -1 0 -4 5 Baryonic Matter R(Vb,max ) Mb +0.1dex -0.1dex -0.3dex (kpc) (1010 M ) 10.8 6.0 +1.5 -1.2 -2.9 6.8 +1.6 -1.3 -3.2 2.7 +0.6 -0.5 -1.2 12.0 +3.1 -2.4 -5.9 11.2 +2.8 -2.3 -5.5 2.4 +0.6 -0.5 -1.1 1.1 +0.3 -0.2 -0.6 4.2 +1.0 -0.8 -2.0 3.7 +0.9 -0.8 -1.8 Dark Matter R10 RX +0.1dex +0.1dex -0.1dex -0.1dex -0.3dex -0.3dex (kpc) (kpc) 20.4 +3.2 -10.6 -20.4d 13.3 +4.5 -5.5 -10.3 7.4 +0.7d -1.6 -7.4d 17.8 +0.3c -4.2 -14.0 ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 5.8 +2.1 -1.3 -5.8d 0d +1 0d 0d 7.5 +8.9 -1.4 -7.5d ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 25.3 +3.5 -2.2 -16.3 19.4 +2.6c -4.7 -13.4 ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 3.9 +2.4 -2.0 -3.9d 17.8 +3.1 -5.8 -12.6 ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ

Galaxy NGC 3726

(km/s) 169

NGC 3893

210

5.3

NGC 3949

169

3.9

NGC 3953

225

7.7

NGC 3992

272

13.1

NGC 4051

170

0.3

NGC 4062

162

3.4

NGC 4138

195

1.0

NGC 4651

210

2.6

35


Table 2--Continued
Observed Vtot,max Vb,max +0.1dex -0.1dex -0.3dex (km/s) 223 +14 -3 5 -6 8 289 +21 -4 3 -8 7 190 +27 -1 7 -2 7 220 +12 -3 5 -6 7 245 +30 -2 7 -7 2 283 +35 -3 1 -8 4 228 +28 -2 5 -6 7 Baryonic Matter R(Vb,max ) Mb +0.1dex -0.1dex -0.3dex (kpc) (1010 M ) 3.8 6.5 +1.6 -1.3 -3.2 33.5 +8.5 -6.7 -16.4 5.3 +1.3 -1.1 -2.6 6.7 +1.7 -1.4 -3.3 13.1 +3.3 -2.7 -6.4 7.9 +2.0 -1.6 -3.9 12.9 +3.2 -2.6 -6.2 Dark Matter R10 RX +0.1dex +0.1dex -0.1dex -0.1dex -0.3dex -0.3dex (kpc) (kpc) ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 13.7 +5.8 -1.3 -8.8 ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 2.2 +0.8c -0.7 -2.2d ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 20.4 +2.6 -5.5 -13.5 ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ

Galaxy NGC 4698

(km/s) 220

NGC 5371

242

13.9

NGC 5806

200

1.1

NGC 6300

208

4.3

NGC 7083

210

6.5

NGC 7217

284

2.0

NGC 7606

280

9.8

a b c

Quantities are calculated from B - V instead of B - R Quantities are calculated from (M /L)
,H

instead of (M /L)

,K

Upper limit Lower limit

d

36


Table 3
origonal color-M /L c V200 2 (km/s) 0.5 0.5 8.0 3.0 16.5 3.0 ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 1.0 0.5 1.5 0.5 ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 0.5 0.5 ÇÇÇ ÇÇÇ 2.5 0.5 2.0 0.5 8.5 0.5 6.0 0.5 0.5 0.5 2.0 1.0 12.5 0.5 1.0 0.5 111 61a 143 168 199a 204 ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 400 105 745 150a ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 89a 89a ÇÇÇ ÇÇÇ 301 71a 470 105a 242 582 140 345 733 158a 269 304 121 281 334 159 9.5 13.8 0.7 1.2 9.1 5.3 ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 6.4 11.4 4.7 7.3 ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 8.8 19.1 ÇÇÇ ÇÇÇ 1.5 2.7 1.2 3.9 13.4 23.3 1.7 4.0 5.3 10.1 0.3 0.9 3.1 4.1 1.0 5.9 +0.1 dex V200 (km/s) 61a 61a 148 238 204 199a ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 400 80a 150a 150a ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 71a 71a 475 105a 402 452a 160 285 158a 158a 314 314 136 171 379 94 -0.1 dex V200 (km/s) 96a 61a 138 133 214a 199a ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 160 215 745 354 ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 394 394 77a 77a 336 156 495 265 217 647 130 235 178 233 233 259 116 146 394 239 -0.3 dex V200 (km/s) 96 131 138 143 279 199a 116 96a 68a 68a 125 360 180 585 328 143 350a 350a 94 389a 122 77a 261 336 375 525 207 337 120 145 178 158 189 294 121 116 154 414

c



2

c



2

c



2

Galaxy NGC 157 NGC 289 NGC 488 NGC 908 NGC 1087 NGC 1090 NGC 1241 NGC 1385 NGC 1559 NGC 1832 NGC 2090 NGC 2139 NGC 2280 NGC 2841 NGC 3198 NGC 3223 NGC 3319 NGC 3521 NGC 3726

0 0 6 1 8 0 Ç Ç Ç Ç 0 0 0 0 Ç Ç Ç Ç Ç Ç Ç Ç 0 0 1 0 3 0 4 0 0 0 1 1 7 0 0 0

.5 .5 .5 .0 .0 .5 ÇÇ ÇÇ ÇÇ ÇÇ .5 .5 .5 .5 ÇÇ ÇÇ ÇÇ ÇÇ ÇÇ ÇÇ ÇÇ ÇÇ .5 .5 .5 .5 .0 .5 .0 .5 .5 .5 .5 .5 .0 .5 .5 .5

19.9 28.2 0.8 1.7 6.5 6.8 ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 9.7 15.6 9.3 22.2 ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 2.2 5.6 0.5 2.4 27.6 53.9 2.8 6.5 14.8 39.9 0.3 1.3 4.3 6.2 3.6 10.2

4.0 0.5 10.5 5.0 20.0 8.0 ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 5.0 0.5 3.0 0.5 ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 0.5 0.5 20.0 2.5 4.0 0.5 3.0 0.5 12.5 1.0 8.0 1.5 9.5 0.5 2.5 1.5 18.5 4.5 1.0 0.5

6.7 8.2 0.8 0.8 12.6 5.2 ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 5.3 8.7 3.4 4.0 ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 4.2 6.9 0.8 0.4 1.1 2.5 0.2 2.7 11.0 18.6 1.1 2.8 5.0 3.7 0.2 0.7 2.6 3.5 0.3 2.9

16.0 2.0 11.5 7.5 20.0 18.5 20.0 14.0 16.5 4.0 12.0 1.0 20.0 1.5 4.0 0.5 3.5 0.5 20.0 0.5 20.0 13.5 7.5 2.0 5.0 1.0 15.0 3.5 11.5 5.0 20.0 11.5 3.5 1.5 20.0 13.5 6.0 0.5

5.8 7.3 0.8 0.7 22.7 6.1 8.7 6.5 1.3 1.5 4.7 6.6 3.4 2.3 0.1 0.7 5.1 7.0 4.0 3.7 1.2 0.3 0.8 2.3 0.2 2.6 7.0 11.6 0.6 1.5 7.1 2.6 0.2 0.4 2.2 2.1 0.6 0.4

37


Table 3--Continued
origonal color-M /L c V200 2 (km/s) 4.0 0.5 2.0 0.5 0.5 0.5 20.0 8.0 2.0 0.5 18.5 1.5 1.0 0.5 3.0 0.5 ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 17.5 0.5 159 74 390 85a 563a 113a 149 164 382 137 97 372 375a 75a 517 112a ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 142a 447 2.3 3.7 2.2 4.5 2.0 5.3 0.3 0.3 1.9 3.8 1.7 0.6 15.7 23.6 2.3 6.1 ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 1.7 3.0 c +0.1 dex V200 (km/s) 334 74a 365 85a 113a 113a 154 234 322 77a 382 352a 295 75a 557a 112a ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 142a 142a
2

c

Galaxy NGC 3893 NGC 3949 NGC 3953 NGC 3992 NGC 4051 NGC 4062 NGC 4138 NGC 4651 NGC 4698 NGC 5371 NGC 5806 NGC 6300 NGC 7083 NGC 7217 NGC 7606

-0.1 dex V200 (km/s) 94 204 405 280 558 183 159 164 352 322 102 357 115 75a 142 307 ÇÇÇ ÇÇÇ 217 107a 96a 101 174 209a ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 157 357



2

c

-0.3 dex V200 (km/s) 119 119 245 385 183 493 164 153 152 352 117 377 105 375 152 382 525 105a 137 197 96a 96a 109 119a 126 361 176 106a 192 187



2

0.5 0.5 0.5 0.5 0.5 0.5 14.0 2.0 1.0 0.5 3.5 0.5 0.5 0.5 1.0 0.5 ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 9.5 0.5

4.4 8.9 5.5 10.8 6.5 22.9 0.3 0.3 2.9 4.9 1.5 1.1 30.1 43.0 4.9 11.1 ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 2.5 3.6

18.0 0.5 3.0 0.5 1.5 1.5 19.5 11.0 3.0 0.5 20.0 2.5 8.5 0.5 15.0 1.0 ÇÇÇ ÇÇÇ 0.5 0.5 14.0 1.0 5.0 0.5 ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 20.0 2.5

2.2 2.3 0.7 2.9 0.3 2.2 0.4 0.3 1.3 2.5 1.7 0.5 9.9 15.4 2.0 5.8 ÇÇÇ ÇÇÇ 6.2 15.1 1.6 0.7 9.5 7.0 ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 1.4 2.7

20.0 8.5 7.0 2.0 9.5 1.0 20.0 20.0 10.0 2.0 20.0 4.0 20.0 1.0 20.0 3.5 3.5 0.5 9.0 0.5 20.0 6.0 20.0 8.0 13.0 0.5 20.0 19.0 20.0 12.0

4.0 2.1 0.03 0.8 0.02 0.2 1.2 0.4 1.0 1.7 2.0 0.7 7.3 10.8 3.0 5.7 19.2 29.2 0.9 0.8 3.7 1.2 9.3 4.6 1.1 0.9 15.6 4.0 2.0 2.3

a

Limit of parameters searched.

Note.--The first and second rows for each galaxy list information for the uncontracted and contracted halo models.

38


Table 4
Correlation Coefficients All Data Without Outliersa RX /hI R RX RX /hI R RX 0.41 0.85 ÇÇÇ 0.89 0.65 ÇÇÇ ÇÇÇ 0.36 0.41 0.51 0.64 ÇÇÇ ÇÇÇ 0.22 ÇÇÇ ÇÇÇ 0.21 0.94 ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ 0.90 0.96 ÇÇÇ 0.90 0.90 ÇÇÇ ÇÇÇ 0.77 0.80 0.68 0.87 ÇÇÇ ÇÇÇ 0.58 ÇÇÇ ÇÇÇ 0.80 0.98 ÇÇÇ ÇÇÇ ÇÇÇ ÇÇÇ

Galaxy Property Vtot,max Vb,max R(Vb,max ) T-type log10 Mb hI R R10 B Kd Åo,B Åo,K
a b c

b

c

Outliers are NGC 2841 and NGC 4138. Outliers for this relation are NGC 3319 and NGC 4062.

NGC 3319 is not included because it is an outlier in this relation.

NGC 3319 is not included because it does not have K -band surface brightness profiles with high enough signal-to-noise to derive an integrated magnitude.

d

39