Dark and Baryonic Matter in Bright Spiral Galaxies: II. Radial

Home , NGC 1090, NGC 157, NGC 289, NGC 3319, NGC 3953, NGC 7083

arXiv:astro-ph/0602027v1 1 Feb 2006 ad olg ak D20742 MD Park, College land, i rv,Blioe D21218 MD Baltimore, Drive, tin est fClfri,SnaCu,C 95064; CA Cruz, Santa California, [email protected] of versity versity akadBroi atri rgtSia aais I Radi II. 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F but commonly wit most all profiles is matter it alone. dark as redshifts for contraction adiabatic parameterization and NFW photometry the to surface data o on that such based For distributions, relation baryon disk. correlat the maximal tightly from are to predicted speeds close speeds rotation be observed to maximum appear that sample our in galaxies R X edcmoetertto uvso 4bih prlgalaxie spiral bright 34 of curves rotation the decompose We .W ru htti sdet h us-xoeta aueo nature quasi-exponential the to due is this that argue We ). ua .Kassin A. Susan aais udmna aaees–glxe:gnrl–galaxies: – general galaxies: – parameters fundamental galaxies: itiuin o 4Galaxies 34 for Distributions 1 , 2 olfS eJong de S. Roelof , R ABSTRACT X hnteeohrglx rpris oto the of Most properties. galaxy other these than 1 neo akmte sdmnn eg,Ver- (e.g., dominant is matter pres- dark the of where ence galaxies poorly in is even How- matter constrained, dark 2001). of distribution in Rubin the matter & ever, dark (Sofue of galaxies component spiral dominant a existence the of for evidence direct most the been Introduction 1. 3 h ans frtto uvshslong has curves rotation of flatness The ejmnJ Weiner J. Benjamin & , cldt h aisweethe where radius the to scaled n ecncet Tully-Fisher a create can ne dwt aiu rotation maximum with ed e with ies tpromd nodrto order In performed. ot o speed; ion rso aais seemingly galaxies, of arts sta ftedr matter dark the of that ls t r eeal or and poor, generally are its a rfieo h baryonic- the of profile ial R tato,broswould baryons ntraction, igcolor- ying nobroi n dark and baryonic into s n ihu including without and h t locreae very correlates also ity nly ecmaeour compare we inally, X hs aais efind we galaxies, these content sfudt correlate to found is ik n rotation and disks f R X B M/L measurements bn central -band 4 relations ao – halos al heijen 1997; de Block & McGaugh 1998). In- dynamical models of gas flow to model the terestingly, the main uncertainty in the radial two-dimensional velocity field of a barred density distribution of dark matter, as derived galaxy. They find that such models require from rotation curves, stems from the poorly 80 − 100% of the maximal disk M/LI value. known stellar mass distribution (e.g., Verhei- Palunas & Williams (2000) modeled the bary- jen 1997). onic mass distributions of 74 spirals and found In this paper, we do not rely on the typi- that a mass-follows-light model could repro- cal maximal disk assumption (van Albada et duce the overall structure of the optical rota- al. 1985; van Albada & Sancisi 1986) to de- tion curves in the majority of galaxies. These rive stellar mass profiles, but use color-mass- authors found that 75% of the galaxies in to-light ratio (M/L) relations first given by their sample have a rotation curve out to R23.5 Bell & de Jong (2001) and later updated by that is entirely accounted for by baryons. The Bell et al. (2003). These color-M/L relations mass models for 20% of their sample fail be- were determined from the analysis of spectro- cause of nonaxysymmetric structures. For a photometric spiral galaxy evolution models more inclusive discussion on maximal disks, and give an upper limit to the baryonic mass see Palunas & Williams (2000). present in spirals. They are most power- It has been known for nearly 20 years that ful when applied to data in the near-infrared the radius where dark matter begins to con- since this wavelength regime traces the older tribute to the rotation curve of a galaxy is stellar populations that contain most of the smaller for low surface brightness galaxies mass, while avoiding much of the effects of than it is for brighter galaxies (e.g., Persic dust (extinction at the near-infrared K-band & Salucci 1988, 1990; Broeils 1992; Persic, is only about 10% of that at the optical B- Salucci, & Stel 1996; de Blok & McGaugh band; Martin & Whittet 1990). The color- 1997). A number of authors have demon- M/L relations allow us to investigate radial strated this by evaluating either the mass dis- variation in stellar M/L, and provide us with crepancy of galaxy disks (ratio of total to a consistent way of scaling stellar M/L from baryonic mass) by adopting a stellar M/L, or one galaxy to the next. With the distributions a similar quantity at a chosen radius. For ex- of stellar mass in galaxies reasonably well de- ample, Persic, Salucci, & Stel (1996) find that termined, we can begin to investigate those of spiral galaxies with V = 100 km s−1 have the dark matter. > 75% of their mass in dark matter within The maximum disk assumption (Sackett Ropt ≡ 3.2h where h is the disk scale-length, 1997) can be examined with the color-M/L and that galaxies with V = 150 km/s have relations. We can obtain an upper limit to only > 40%. McGaugh & de Blok (1998) the number of galaxies that have maximal find that the total M/LB evaluated at 4h is disks, and also compare the shapes of rota- smaller for galaxies of higher surface bright- tion curves derived from baryonic mass dis- ness and brighter magnitude, and does not tributions in galaxies to the observed rota- correlate with h. These authors also find that tion curves as in Palunas & Williams (2000). the radius where dark matter begins to dom- There are many pieces of evidence in the lit- inate over the luminous mass is greater for erature for maximal disks; we mention a few higher surface brightness galaxies. Zavala et noteworthy ones here. Possibly the strongest al. (2003) find that the ratio of the velocity evidence comes from the work of Weiner, Sell- due to the baryons to the total maximum ve- wood, & Williams (2001) who used fluid- locity depends mainly on disk central surface

2 density such that denser galaxies have larger (e.g., Verheijen 1997; Bosma 1981). McGaugh ratios. These authors also find that the mass & de Blok (1998) find a regularity between discrepancy evaluated at 2.2h and 5h depends the mass discrepancy and acceleration of a on central surface brightness, but puzzlingly galaxy as a function of radius. This follows does not depend on h, baryonic mass, or B- directly from the Tully-Fisher relation, prob- band luminosity. In addition, Zavala et al. ably the best observational example of self- (2003) find that M/LB anti-correlates signifi- similarity in the dark matter component of cantly with B-band central surface brightness galaxies. The mass discrepancy-acceleration and does not correlate with B-band luminos- relation shows that knowledge of the baryonic ity, in disagreement with McGaugh & de Blok mass distribution of a galaxy allows one to (1998) and the general qualitative trend in calculate its dark matter distribution. Such a the literature. Most recently, Pizagno et al. relation steps beyond the usual Tully-Fisher (2005) evaluate the total M/Li at 2.2hi and relation, which predicts the dark matter con- find median values of 2.4 and 4.4 for galaxies tent of a galaxy at a particular radius given 10 with stellar masses greater than 10 M⊙ and its baryonic mass at that radius, to one that 9 10 4.4 between 10 –10 M⊙, respectively. These predicts radial distributions. authors are all generally in qualitative agree- Theories and simulations of galaxy forma- ment, but there are discrepancies, and we will tion should be able to reproduce the distribu- discuss their origin in the body of this paper. tions of dark matter observed in galaxies. Un- A combination of resolved Hα or CO and til now, comparisons to predictions for dark H I velocity data is necessary to determine the matter distributions have consisted of fits of dark matter distribution of a galaxy from its functional formulations for dark matter halos center to beyond ∼ 5h. Hα data are essential to rotation curve data, with stellar M/L as to measure the steep rise of rotation curves a free parameter. Now that we have a han- in the inner ∼ 2h of galaxies (e.g., Palunas dle on the stellar M/L in galaxies with the & Williams 2000), whereas H I data are key color-M/L relations, we re-examine these fits. to probe beyond where Hα can be measured In particular, we test the main incarnation of (e.g., de Blok & McGaugh 1997). Since ro- the density distribution of dark matter halos tation curves in the optical portion of galax- in N-body simulations of a ΛCDM universe: ies are usually affected by non-axisymmetric the formulation of Navarro, Frenk, & White structures that do not trace the main galac- (1996) (NFW), which is the simplest and most tic potential, such as bars and spiral arms, popular analytical description. This formula- the distributions of dark matter derived from tion predicts that dark matter is significant Hα data can be quite variable. Two examples down to the very inner radii of galaxies, in of studies probing dark matter distributions contradiction to maximal disks. Moreover, that have used both types of rotation curves dark matter halos are expected to contract in a direct manner, independent of prior pa- due to baryon collapse. As this contraction rameterizations for the dark matter, are Per- is normally implemented (Blumenthal et al. sic, Salucci, & Stel (1996) and McGaugh & 1986), even more dark matter is caused to de Blok (1998). Persic, Salucci, & Stel (1996) move toward the centers of galaxies, making coadd 1100 rotation curves and find that they the situation worse (e.g., McGaugh & de Blok could be determined by a single galactic pa- 1998). We perform fits to the NFW formula- rameter (e.g., luminosity). This formulation, tion with and without including contraction however, is refuted by a number of authors as it is normally implemented. The main as-

3 sumption behind these fits, namely that the the remaining 4 galaxies can be found in Ver- NFW halos can be fit to data that subtends heijen (1997). More details on the sample only their very inner parts (. 0.1Rvirial), is selection are given in Paper I. The profiles problematic. Furthermore, the simulations were calculated in elliptical annuli of increas- that produce the NFW functions have diffi- ing distance from the centers of the galaxies culty forming disk galaxies once hydrodynam- and have been corrected for Galactic extinc- ics is included (e.g., Abadi et al. 2003), sig- tion. The galaxies NGC 1090, NGC 2841, and naling possible unknown interactions between NGC 3198, which have images from the Sloan baryons and dark matter. To address these Digital Sky Survey (SDSS) Second Data Re- concerns, we turn the problem of fitting dark lease (Abazajian et al. 2004), have images in matter halos to data around and present a which almost half of the galaxy was off of simple universal form for dark matter profiles the detector. This is also the case, but to in terms of baryonic mass profiles. a much lesser extent for NGC 3521. This ef- This paper is organized as follows: In §2 fect can be observed in the g-band images in we briefly discuss our galaxy sample. Radial Figure 1 of Paper I; these galaxies are flagged baryonic mass distributions and their rotation in the following analysis. All the analyses in curves are derived in §3. In §4 these baryonic this paper have been performed for both the rotation curves are compared with rotation approaching and receding sides of a handful curves from the literature to derive rotation of galaxies with complete imaging. No sig- curves due to dark matter. We discuss max- nificant change (to within the zero-point un- imal disks in §5. In §6 we define quantities certainties) was found between the results for to describe the baryonic and dark matter dis- each side of these galaxies. Physical param- tributions and show how they correlate with eters (i.e., Hubble type, distance, R25, inte- general galaxy properties. We investigate the grated magnitudes) and bulge-disk decompo- radial behavior of dark matter in §7, and mass sitions can also be found in Paper I. profiles are fit with the NFW profile in §8. In §9, we summarize our conclusions. Through- 3. Baryonic Matter out this paper we adopt a Hubble constant In this section, we derive the baryonic mass (H ) of 70 km s−1 Mpc−1. When distance- o surface density profiles of the galaxies, and dependent quantities have been derived from compute the component of the galaxies’ ob- the literature, we have reduced them to this served rotation that is due to the baryons. value of H and always quote the converted o These components are called “baryonic rota- values. tion curves,” to distinguish them from observed rotation curves. In §4, we will intro- 2. The Galaxy Sample duce the term “dark matter rotation curves,” Our data consist of surface brightness pro- to denote the contribution from dark matter files, physical parameters, and rotation curves to the observed rotation curves. for 34 bright spiral galaxies. These galaxies have inclinations in the range ∼ 30–65 degrees 3.1. Radial Baryonic Surface Mass- in order to reduce the effects of dust, while Density Distributions still being able to obtain accurate kinematical To determine a galaxy’s radial stellar sur- information. Surface brightness profiles for 30 face mass-density distribution, we apply a galaxies were presented in Kassin, de Jong, color-M/L relation to its surface brightness & Pogge (2005) (hereafter, Paper I); those for

4 bulge contribution (such that inclusion of a bulge component does not change the rotation curve due to baryons beyond the uncertainties of the color-M/L relations), we apply a color-M/L relation directly to the az- imuthally 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 aperture B −R color since this color-M/L relation has a shallow slope. For those galaxies with a signifi- Fig. 1.— Color-M/L relation for B −R color and cant bulge component, a bulge-disk decompo- (M/L)∗ at K from Bell et al. (2003) (solid line). sition is performed. A characteristic B − R The galaxies in the sample are plotted as filled color is adopted for each component based on circles, and a reddening vector is plotted for for the average colors of the bulge and disk, and correction to face-on for a Milky Way-type galaxy the LK profiles are multiplied by the resulting viewed at an inclination of 80◦ with the Tully et (M/L)∗. Extinction corrections are discussed al. 1998 formalism. in §3.4. We also extend the exponential disk surface brightness profiles until approximately ∞ (defined here as r = 10000′′).1 profiles. These relations were derived from spectrophotometric spiral galaxy evolution When available from the literature, we use models in Bell & de Jong (2001), and were radial H I 21 cm measurements to determine updated in Bell et al. (2003). The color-M/L the contribution of interstellar gas to the ra- relations show a relatively tight correlation dial baryonic surface mass-densities. There (∼ 0.1 dex spread for the color and metallic- are gas measurements available for NGC 1090 ity range where our galaxies lie) between the from Gentile et al. (2004), NGC 3198 from optical color of a galaxy and its stellar mass- Begeman (1987), and NGC 3949, NGC 3953, to-light ratio, (M/L)∗. Color-M/L relations and NGC 3992 from Verheijen (1997). The are most useful when applied to (M/L)∗ in the neutral gas component is included by scal- near-infrared since they are the least affected ing the H I surface mass-density by a fac- by dust obscuration, as discussed in Bell & tor of 1.32 to account for the abundance de Jong (2001). Specifically, we choose to use of helium. For those galaxies without gas the relation between B − R color and (M/L)∗ mass measurements, we assume that gas does in the K-band. This relation is composed of not contribute to its baryonic surface mass- the optical color with the largest wavelength density. This is probably not a bad assump- baseline and the reddest near-infrared band. tion for high surface brightness galaxies since It is reproduced in Figure 1. If imaging is un- a galaxy with a Hubble type between Sa and available at R, we use the relation for B − V , Scd has a gas mass fraction that is typically and if imaging is unavailable at K we use H. 1 Extended galactic stellar disks have been discovered for some galaxies via very deep imaging (see Ibata et For those galaxies without a significant al. 2005, and references within).

5 rotmod task in the GIPSY software package (van der Hulst et al. 1992), which calculates rotation curves for galaxies composed of a truncated exponential disk (Casertano 1983) and a spherical bulge when applicable. Ex- ponential disks are assumed to have a scale- height of 0.3 kpc, which is typical of bright spirals. The diﬀerence between using a diﬀer- ent scale-height for each galaxy of 0.1hIR kpc, as in Sparke & Gallagher (2000), and a global value of 0.3 kpc is negligible. The results also remain unchanged if we used the relation between the central surface brightness and the ratio of vertical scale-height to vertical scale- Fig. 2.— Rotation curve for NGC 1090 due to the length given by Bizyaev & Mitronova (2002). stellar mass component (solid line) is compared with the rotation curve due to stars and H I gas 3.3. Uncertainties in Baryonic Rota- from Gentile et al. 2004 (dashed line). The ob- tion Curves served Hα and H I rotation curves are plotted as The largest source of uncertainty for the open triangles and circles, respectively. The up- baryonic rotation curves lies in the normal- per and lower bounds for the stellar mass rotation ization of the color-M/L relation, which is curve (dotted lines) are due to the 0.1 dex uncer- mainly determined by the stellar IMF at low- tainty in the color-M/L relations. mass. Since the faint end of the IMF is relatively unconstrained, there may exist many

MHI /Mgas+stars ≈ 0.03 (Roberts & Haynes low-mass, low-luminosity stars that can con- 1994), and hence does not greatly affect its tribute significantly to the mass budget of baryonic mass. The average gas mass frac- a stellar population without creating a de- tion for the 5 galaxies in our sample that tectable increase in luminosity or change in have gas mass measurements is 0.04, consis- color. For their derivation of the color-M/L tent with this measurement. Figure 2 demon- relations, Bell & de Jong (2001) and Bell et strates the effect of including the interstel- al. (2003) adopted a truncated Salpeter IMF lar gas component in the baryonic rotation which derives from the constraint that bary- curve of NGC 1090, which is generally within onic rotation curves should not over-predict the uncertainty of the stellar mass rotation observed rotation curves for spiral galaxies curve. Therefore, by not including interstel- in Ursa Major (the Verheijen 1997 sample). lar medium contributions in the analysis, only With this constraint, they predict fewer low a minor systematic effect is introduced. mass stars than a Salpeter IMF. These relations thus give an upper limit to the stel- 3.2. Baryonic Rotation Curves lar mass present; a lower normalization of the color-M/L relation cannot be excluded. A baryonic rotation curve is calculated for We re-derive this constraint on the upper each galaxy from its baryonic surface mass- limit to the IMF by re-calculating maximum density profile and is plotted in Figure 3. The disk fits for galaxies in the Verheijen (1997) rotation curve calculation is done with the 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 Hα 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 Hα and N II rotation curves, the error bars are smaller than the points at the resolution of the plots. The eﬀects of the ±0.1 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 =2.2hIR are marked with thick and thin bars, respectively.

7 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.

8 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.

10 done in the original Verheijen (1997) fits. For these fits, we define maximal disk to be the greatest possible 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 possible without over-predicting the observed rotation curves beyond the very inner parts. In order to best match the overall shapes of the observed rotation curves, we allow over- predictions of the inner 2 points of H I rotation curves due to beam-smearing and over- predictions of the inner parts of Hα rotation curves. We also re-scale the stellar and gas mass distributions from the distance used in Verheijen (1997) to 20.7 Mpc before fitting the maximal disks, unlike the approximate re- Fig. 4.— Maximum disk stellar (M/LK)∗ versus scaling done in Bell & de Jong (2001). Gas reddening-corrected B − R colors for galaxies in distributions derived from H I measurements the Verheijen (1997) sample (Xs) and in this paper are multiplied by 1.32 to account for helium, (triangles). Plotted for reference are the color- as in Verheijen (1997). In Figure 4, the result- M/L relations of Bell & de Jong 2001 (dashed ing maximum disk (M/L)∗ are plotted against line) and Bell et al. 2003 (solid line). the B − R colors of the galaxies corrected for extinction with the Tully et al. (1998) formalism. In this figure, we also plot maximum relations, uncertainties in the zero-points of disk (M/L)∗ for those galaxies presented in these relations, uncertainties in the photomet- this paper that have BRK photometry. All ric zero-point calibrations, and uncertainties the points in Figure 4 are upper limits to the in the determination of a galaxy’s inclination (M/L)∗; galaxies cannot have values greater to the line of sight. Secondary sources of than those defined by the lower envelope in uncertainty generally have a small effect on this plot without over-predicting their rota- the baryonic rotation curves. They include tion curves beyond the very inner parts. It secondary photometric (“bootstrap”) calibra- is apparent from Figure 4 that the Bell et al. tions, the neglect of interstellar gas content, (2003) color-M/L relation is consistent with and position angle uncertainties. Dust red- all the galaxies to within ∼ 0.1 dex, which is dening also plays a role in the uncertainty of the uncertainty in the relations. the baryonic rotation curves; this will be discussed in §4.3. Note that the transformation Other than the IMF, there are five pri- of Two Micron All Sky Survey (2MASS; Jar- mary sources of uncertainty in the determi- rett et al. 2000; Cutrie et al. 2000; Jarrett et nation of the baryonic surface mass-density al. 2003) and SDSS photometry to the Kron- distributions: uncertainties in the distances 2 Cousins system introduces negligible uncer- to galaxies , 0.1 dex spread in the color-M/L tainties, as shown in Paper I. 2Of the 34 galaxies in our sample, 4 have distances As an illustration, Figure 5a–d shows the estimated from Cepheid variables for which the uncertainties are much less than other types of distance measurements, and are typically ∼ 10%.

11 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 between 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 ± 20% in the distance. The distance uncertainty causes a maximum change in |∆vb|/vb of 0.24 which corresponds to a change in the velocity of nearly 40 km −1 s . The average change is |∆vb|/vb = 0.09 −1 (|∆vb| ⋍ 13 km s ). Figure 5b shows the effect of a systematic change of ±0.1 dex in the color-M/L relations. This uncertainty causes a maximum change in |∆vb|/vb of 0.12 −1 (|∆vb| ⋍ 25 km s ), and an average change Fig. 5.— Effects of uncertainties on the baryonic −1 of 0.12 (|∆vb| ⋍ 20 km s ). In Figure 5c,d, rotation curve of NGC157. For one side of the the effects of introducing systematic errors uncertainty (e.g., +0.1 dex), the |∆vb|/vb curve in the photometric zero-point calibrations are is plotted as a dotted line, and for the other side shown. In Figure 5c, the effect of the actual (e.g., -0.1 dex) the curve is plotted as a solid line. zero-point uncertainties for NGC 157 is shown The effects of the following uncertainties are plot- (σB = 0.03,σR = 0.03,σK = 0.04). This ted: (a) ±20% uncertainty in the distance to the causes an average change in |∆vb|/vb of 0.10 galaxy, (b) ±0.1 dex scatter in the color-M/L rela- −1 (|∆vb| ⋍ 17 km s ). While most of our opti- tion, (c) actual photometric zero-point uncertain- cal photometric zero-points are good to ≤ 5%, ties for NGC 157, (d) photometric zero-point un- the worst uncertainty in an optical zero-point certainties for the galaxy with the poorest zero- calibration for any of the surface brightness point, (e) a change in the inclination of 5◦, and profiles that we use in this analysis is ±15%, (f) a change in the position angle of 5◦. The radii as given in Table 3 of Paper I. All of the uncer- 2.2hK and R25 are marked as thin and thick lines, tainties on the near-infrared zero-points are respectively, on the x-axis in panel a. ∼ 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 inclinations are both ∼±5 (Paper I). In Fig- happen to the rotation curve if photometric ure 5e, we plot the effect of a change in the position angle of ±5◦, and in Figure 5f we show zero-point errors were ±15% in the optical ◦ and ±4% in the near-infrared. Such uncer- the effect of a change in the inclination of ±5 . Uncertainties in the inclination have a larger tainties cause an average change in |∆vb|/vb −1 effect (on average |∆vb| ⋍ 0.03; |∆vb| ⋍ 5 km of 0.13 (|∆vb| ⋍ 22 km s ). s−1) than those in the position angle (on av- In Figure 5e,f, we show the effects on the erage |∆v | ⋍ 0.007; |∆v | ⋍ 1kms−1). How- baryonic rotation curve of NGC 157 of chang- b b ever, both these uncertainties are small when ing the inclination and position angle in the compared to other sources of error discussed derivation of its surface brightness profiles. in this section. Typical uncertainties in position angles and In Figure 6, to compare the color-M/L re-

12 baryonic rotation curves.

3.4. Effects Due to Dust 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 populations, as foreground dust will systematically both redden and extinguish galactic light. In this section, we discuss the possible effects of dust reddening and extinction on absolute derived stellar masses. We examine the empirical inclination- Fig. 6.— Baryonic rotation curves derived from dependent extinction correction of Tully et the color-M/L relations (plotted as dotted lines al. (1998), which should describe to first or- for the ±0.1 dex scatter in the relations) are com- der the dust content of galaxies. We correct to pared with baryonic rotation curves derived from face-on the total integrated colors and magni- constant (M/LK)∗ of 0.75 (thin solid line) and 1.0 tudes of galaxies in our sample. In doing this, (thick solid line). The thin dashed line plotted for we ignore radial dust gradients in the disk NGC 7606 is the model observed rotation curve of and the fact that dust reddening is likely to Courteau 1997. All other features of the plot are be large for the inner parts of galaxies. While the same as Figure 3. the K-band magnitudes barely change when extinction corrected (0.05 mag on average), the B − R colors do (0.09 mag on average). lations to constant (M/L )∗, baryonic rota- K This, however, should not affect the final de- tion curves created with the color-M/L rela- rived stellar masses, since the color-M/L re- tions and with (M/L )∗ of 0.75 and 1.0 are K lations that we use have a very shallow slope plotted for 4 example galaxies. The rotation in B −R color. To examine this effect further, curves created with (M/L )∗ of 0.75 and 1.0 K for 4 galaxies that span the range of inclina- are approximately consistent with the ±0.1 tions in our sample, we apply the Tully et al. dex uncertainty of the color-M/L relations. (1998) extinction correction to their radial This is similarly the case for all galaxies in K-band profiles, and use their extinction- our sample, and is due to the shallow slopes corrected B − R colors to derive (M/L )∗. of the color-M/L relations (∼ 0.14–0.18 in K From the resulting surface mass-density pro- log10M/L). files, we derive baryonic mass rotation curves. To summarize, other than the IMF, of the In Figure 7, we plot for the 4 galaxies the other sources of uncertainty in the determina- difference between the extinction-corrected tion of the baryonic rotation curves, only the baryonic rotation curve and the uncorrected distance uncertainty, the ±0.1 dex scatter in one. For NGC 157, NGC 3726, NGC 7217, the color-M/L relations, and zero-point un- and NGC 7606, the rotation curves on aver- certainties can produce non-negligible effects. age differ by 1.3, 1.1, 2.7, and 5.0 km s−1, These three sources of uncertainty vary from respectively. The greatest difference is for galaxy to galaxy and introduce scatter in the NGC 7606 which has the largest inclination

13 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 Observed rotation curves are plotted in Figure 3; Table 1 lists the tracer and literature Fig. 7.— Difference between the extinction- reference for each. Some galaxies have two ob- corrected (V ) and uncorrected (Vo) baryonic ro- served rotation curves, one from Hα or N II tation curves. The extinction-corrected baryonic observations which traces the inner parts, and rotation curves are derived from colors and magni- another from H I observations which traces tudes corrected for extinction with the formalism the outer parts. If errors for the observed ro- of Tully et al. 1998. tation curves were given in the original table or plot that they were taken from, then ◦ of the four galaxies, 63.9 . These differences these are used in this paper. If no errors are in velocity are all less than the uncertainties given, then we estimate them to be the dif- of the baryonic rotation curves themselves, as ference in rotation velocity between the ap- discussed in §3.3. proaching and receding sides of the galaxy; ro- Galaxy disks have been observed to have tation curves for which we estimate the errors a relatively constant face-on dust opacity of in this manner are noted in Table 1. This esti- ∼ 0.5 magnitude in the I-band (Holwerda mate will generally give errors larger than the et al. 2005). Correcting for this, AI would true measurement errors since it will be more increase the baryonic velocity by ∼ 20 km affected by non-axisymmetric features such as s−1. However, since the maximal disk limits spiral arms and slight warps in the gas distri- in Figure 4 would also need to be corrected, butions. Note that the 2–3 innermost points and the color-M/L relations re-scaled accord- of the H I rotation curves and the outer few ingly, the net effect would more or less can- points of the Hα and N II rotation curves have cel out. However, it is very likely that more a greater uncertainty than other points due complex dust models where the effects of dust to beam smearing and low signal-to-noise, re- depend on star/dust geometry are necessary, spectively. For many of the rotation curves we such as those by Disney (1989) and Gordon have obtained data from the authors, but for et al. (2001). We choose to leave this ap- a handful we could not. For those few galax- proach to a future analysis, since the detailed ies, rotation curves are extracted from plots radiative transfer codes are not publicly avail- in the literature with the DataThief program able and are difficult to apply to real galaxies. (Tummers 2000); these galaxies are noted in Moreover, different galaxies may have vari- Table 1. Errors inherent to the extraction of ous amounts of dust and different star/dust a rotation curve vary from plot to plot, but geometries which would cause dust to affect tend to be ≤ 5 km s−1.

14 Galaxies marked with the reference “Math- curs only in the very inner parts where the ob- ewson & Courteau” in Table 1 are rotation served rotation curve is affected by structures curves that were originally presented in Math- such as rings, bars, inner windings of spiral ewson, Ford, & Buchhorn (1992) and were arms, and/or irregular morphology. Baryonic later modeled by Courteau (1997). For these rotation curves are derived under the assump- galaxies, we plot both the actual and model tion of circular motion and need not trace such rotation curves in Figure 3. Since we are structures. For NGC 6300 and NGC 7083, primarily interested in large-scale trends, we the baryonic rotation curves for -0.1 dex do adopt the model rotation curves in the follow- not over-predict the observed rotation curves. ing analysis in order to avoid much of the fine The galaxy NGC 6300 has a bar and a ring structure inherent to the actual data. in the region where the over-prediction oc- curs, but the image of NGC 7083 shows no 4.2. Dark Matter Rotation Curves sign that its baryonic rotation curve should At those radii where the observed rotation over-predict. This is somewhat acceptable, speed of a galaxy is greater than that of its though, since the baryonic rotation curve for baryons, the additional gravitational compo- NGC 7083 for -0.1 dex does not over-predict. nent is assumed to be due to dark matter. A Two galaxies, NGC 4698 and NGC 5371, dark matter rotation curve is derived as the have baryonic rotation curves that over- square root of the difference of the squares of predict their observed rotation curves, even the observed rotation velocity and the bary- for the -0.1 dex scatter in the color-M/L re- onic rotation curve velocity at each radius lations. For both these galaxies, the over- (Binney & Tremaine 1987). In doing this, it is prediction cannot only be explained by bars assumed that the halos of galaxies are axially and/or rings in the galaxy images. To ex- symmetric, the disk and halo are aligned, and amine things further, we create baryonic ro- the observed gas is in circular orbits. Dark tation curves using the position angle and matter rotation curves for the galaxies are inclination that were used to derive their ob- plotted in Figure 3. served rotation curves; the resulting curves For 10 galaxies, the baryonic rotation also over-predict. Next, we create baryonic curves over-predict the observed rotation rotation curves for the -20% uncertainty in curves for some range in radius3. These the galaxies’ distances and both the distance galaxies are NGC 157, NGC 1559, NGC 2139, and color-M/L uncertainties; they are plot- NGC 2841, NGC 3198, NGC 4138, NGC 4698, ted in Figure 8. For NGC 4698, the baryonic NGC 5371, NGC 6300, and NGC 7083. The rotation curve created by taking into account baryonic rotation curve for the -0.1 dex scatter both these effects still over-predicts. However, in the color-M/L relations also over-predicts for this galaxy, its distance was calculated for the first six galaxies listed above. However, with a Virgocentric infall calculation and was for these six galaxies, the over-prediction oc- found to be triple-valued (Paper I). We derive a baryonic rotation curve for it with the dis- 3We do not include in this count baryonic rotation tance solution which is closer than the one curves that over-predict the inner parts of H I rota- chosen in Paper I (9.7 Mpc, as opposed to the tion curves, since they are affected by beam-smearing chosen solution of 19.1 Mpc). This baryonic in this region. Also, Figure 4 suggests that this number is less than 10, but we ignore the very inner parts rotation curve is plotted in Figure 8 and over- of the galaxies when calculating maximal (M/LK )∗s predicts only the inner portion (∼ 1.5 kpc) of in §3.1. the observed rotation curve. For NGC 5371,

15 4.3. Uncertainties in Dark Matter Ro- tation Curves Uncertainties in the dark matter rotation curves arise from a number of effects. The most significant are those inherent to the determination of the baryonic mass component, as discussed in §3.3. Other uncertainties in- Fig. 8.— Effects on the baryonic rotation curves clude: non-circular motions that perturb the of the -0.1 dex scatter in the color-M/L relations underlying potential (i.e., spiral arms, bars, and the -20% uncertainty in distance for galax- substructure), statistical errors from the mea- ies where the baryons over-predict the observed surement of velocities in radial bins, system- rotation curve. Original baryonic rotation curves atic errors in measuring the velocity (i.e., (thick solid lines), those derived by taking into ac- beam smearing and slit position angle er- count the distance uncertainty (dashed lines), and ror), and uncertainties in the measurement of those derived by taking into account both the dis- the dynamical centers of the galaxies. There tance uncertainty and the color-M/L scatter (dot- also may be differences between the centers ted lines) are triangles for Hα and small points for of galaxies determined from photometry and H I. For NGC4698, we also plot the baryonic rota- those determined from the observed rotation tion curve derived from the adoption of a different curves. However, center measurements are distance to the galaxy (thin solid line). Observed not expected to differ much since photometric rotation curves are plotted as points. The radius centers are always chosen to be the brightest 2.2hIR is marked as a thick solid line on the x-axis. 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 over- The radius R = 2.2h is where the rotation predicts, while that created by taking into curve of a self-gravitating exponential disk account both distance and color-M/L uncer- reaches its peak (Freeman 1970). A commonly tainties does not. The sort of over-prediction used definition of a maximal disk is given by that is observed for NGC 4698 and NGC 5371 Sackett (1997) where the galaxy disk provides can also be due to such factors as a mis- 85% ± 10% of the total rotational support of measurement of the observed rotation curves the galaxy at 2.2hR. In the following anal- and/or the effects of dust. A stellar popula- ysis, we define a galaxy to have a maximal tion affected by dust may appear redder (and disk if it has a baryonic mass (disk and bulge) hence heavier and have a faster rotation) than contribution to the observed rotation curve of it is intrinsically since the reddening effect of > 90% at 2.2h, or similarly, a dark matter dust is slightly greater than its extinction ef- contribution to the observed rotation curve fect. These galaxies could also signal a need of < 10% at 2.2h. We choose to adopt this to lower the normalization of the color-M/L definition of maximal disk over that of Sack- relations. If we did this, then many of the ett (1997) since we perform detailed bulge- other galaxies in the sample would be sub- disk decompositions, and are therefore able maximal, but the qualitative results of this to model the combined bulge plus disk bary- paper would not change. onic rotation curves. Furthermore, we use h

16 measured at K, and when K-band imaging is NGC 7606, the baryonic rotation curve that unavailable, we use H. Since near-infrared takes into account both these effects follows bands trace most of the mass of the stel- the observed rotation curve fairly well. For lar populations, a near-infrared scale-length NGC 4062, its distance was calculated us- is more analogous to the stellar mass scale- ing a Virgocentric infall calculation and was length of a disk. Disk scale-lengths measured found to be triple-valued (Paper I). We cre- at K are typically ∼ 1.2 times shorter than ate baryonic rotation curves for NGC 4062 those measured at B (de Jong 1996). with the other 2 distance solutions (17.6 and In Figure 3, there are 4 galaxies that have 24.4 Mpc), both of which are greater than submaximal disks, even if the -0.1 dex scatter the one chosen in Paper I; they are plotted in the color-M/L relations is taken into ac- in Figure 9. These baryonic rotation curves count. Since the normalization of the color- have a strange behavior since they under- M/L relations is an upper limit, the num- predict the observed rotation curve in the ber of galaxies in our sample that do not inner parts, but match it in the outer regions. have maximal disks can in principle be much This may be the effect of such factors as an greater than 4. However, for those galaxies underestimate of the stellar mass in the inner that we observe to have maximal disks, the parts of the galaxy due to blue spiral arms overall shapes4 of the inner parts of their ob- in the B-band image (see image in Figure 1 served rotation curves (within ∼ R25) are gen- of Paper I), a poor bulge/disk decomposition, erally matched by the shapes of their baryonic and/or an underestimate of the uncertainties rotation curves. This is evidence that many of the observed rotation curve. In summary, of the disks have a significant baryonic com- while NGC 4062 and NGC 7606 may not have ponent in their inner parts. submaximal disks, NGC 3319 and NGC 3992 These 4 galaxies are NGC 33195, NGC 3992, likely do. NGC 4062, and NGC 7606; they have bary- There are a few galaxies that have marginally onic mass contributions to their total masses submaximal disks. For these galaxies, the at 2.2hIR of 22%, 50%, 57%, and 68%, re- baryonic rotation curves for the +0.1 dex spectively. In Figure 9 we plot for these 4 scatter in the color-M/L relations result in galaxies the baryonic rotation curves with maximal disks. These galaxies are NGC 1241, and without uncertainties in distances and the NGC 2139, and NGC 2280, and they have color-M/L relations taken into account; these baryonic mass contributions to their total ro- baryonic rotation curves still under-predict tational support at 2.2hIR of 75%, 73%, and the observed rotation curves for NGC 3319, 72%, respectively. NGC 3992, and NGC 4062. However, for One striking example of a galaxy that is likely close to maximum disk is NGC 157. 4 This is distinguished from the “bumps and wiggles” This galaxy has Hα and H I rotation curves of the observed rotation curves that are likely due to small-scale features such as bars and spiral arm per- that have a sudden steep decline at ∼ 8kpc turbations (Palunas & Williams 2000; Kranz, Slyz, & (∼ 3hK ) and flatten afterward, which results Rix 2003; Slyz, Kranz, & Rix 2003). in a hump-like structure. While this peculiar 5For NGC 3319, we do not take into account its in- behavior cannot be absolutely confirmed by nermost 2 observed rotation curve points since they Ryder et al. (1998), there are strong lines of are measured from H I and are likely affected by beam-smearing. Also, the distance measurement for evidence presented in their paper that point NGC 3319 is from Cepheid variable stars and has an to this hump-like structure as physical. This uncertainty of only ∼ 10%. same structure is also found in the baryonic

17 mass distributions, we also tabulate the differences in their values for renormalizations NGC 3319 NGC 3992 of the color-M/L relations by ±0.1 and -0.3 dex. We choose to create the quantity Vb,max because it can be derived from imaging alone and obviates the need for much more expen- sive and time-consuming line width observations needed to obtain Vtot,max. For bright NGC 4062 NGC 7606 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 be discussed further in §6.2. We derive two quantities from the dark Fig. 9.— Same as Figure 8, but for galaxies where matter rotation curves calculated in §4.2: R10, the baryons under-predict the observed rotation the radius where dark matter contributes 10% curves, and for the +0.1 dex scatter in the color- to the velocity of the observed rotation curve, M/L relations and the +20% distance uncertainty. and RX , the radius where the dark matter For NGC 4062, baryonic rotation curves derived contribution equals that of the baryons (the for 2 different distance determinations are plotted “cross-over radius”). The quantity R10 is sim- as thin solid lines. ilar to RIBD of Salucci (2001) and Rt of Persic et al. (1996). The radius RX is analogous to rotation curve for NGC 157. Moreover, a dark R2:1 of McGaugh & de Blok (1998). The radii matter halo in the shape of a NFW model R10 and RX are listed in Table 2 along with is not consistent with this structure unless the difference in values for renormalizations baryons make a significant contribution to the of the color-M/L relations by ±0.1 and -0.3 inner ∼ 5 scale-lengths of NGC 157. In sum- dex. Note that for some galaxies we cannot mary, although a lower normalization of the measure R10 or RX ; this is in general because IMF cannot be ruled out, it seems unlikely their rotation curves do not extend far enough that at least NGC 157 is strongly submaximal. in radius.

6. Dark and Baryonic Matter Scaling 6.1. Baryonic Scaling Relations Relations In Figure 10, basic physical parameters of galaxies are plotted versus M : total abso- In this section, we examine scaling relations b lute B and K-band magnitudes, Hubble T- for dark and baryonic matter. We use quanti- type, R25, and B and K-band central surface ties from Tables 1 and 4 of Paper I, and derive brightnesses (µ , µ ). They have the fol- others: V the maximum observed rota- o,B o,K tot,max lowing correlation coeﬃcients: 0.91 (B), 1.00 tion curve velocity, Vb,max the maximum bary- (K), 0.40 (Hubble T-type), 0.78 (R25), 0.29 onic rotation curve velocity, R(Vb,max) the ra- (µo,B), and 0.58 (µo,K). Integrated magni- dius at which V = Vb,max, and Mb the bary- tudes, sizes, µo,K, and maximum rotation ve- onic mass. These quantities are listed in Table locities correlate very well with M , such that 2, and if they are derived from the baryonic b galaxies with greater Mb are brighter, larger,

18 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 Fig. 10.— Basic physical parameters of galax- partial imaging. Error bars for T-type, LB, and ies versus baryonic mass. Open circles represent R25 are the size of the data points at the resolution galaxies from the SDSS that have partial imaging. of the figure. 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 ple have. In addition, a tight correlation indi- make the plot in panel c such that points do not cates that there is not a wide spread in the de- overlap, we add fractions with values < 1 to the gree of maximality. The quantity log10Vtot,max integer Hubble T-types. also correlates with Hubble T-type and R25 such that galaxies that rotate faster have ear- lier T-types and are larger. The quantity and have brighter µ . Morphology and µ o,K o,B R(Vb,max) correlates with log10Mb with a cor- also correlate with Mb, but to a lesser extent. relation coefficient of 0.66, but correlates relatively weakly with log10Vb,max, log10Vtot,max, In Figure 11, relations are plotted for and µo,K with correlation coefficients of 0.08, log10Vb,max and log10Vtot,max, which trace 0.35, and 0.16, respectively. the baryonic and total mass components of As discussed above, the maximum rotation the galactic potential, respectively. In Fig- speeds predicted from the baryon distribu- ure 11a, along the lines of Roberts & Haynes tions are tightly correlated with the observed (1994), we plot log10Vtot,max versus log10LB maximum rotation speeds. To examine this and find good agreement with their results. relation further, we plot the ratio of Vb,max to In Figure 11b, we plot log10V versus b,max Vtot,max versus µo,K in Figure 12, and a find log10Vtot,max; they are strongly correlated root mean square (rms) deviation from unity with a correlation coefficient of 0.82. These of only 0.18. Using this result, one can cre- velocities should be equivalent for maximal ate a Tully-Fisher relation from two passband disks, which many of the galaxies in our sam- surface photometry and a redshift alone (e.g.,

19 a b

c d

Fig. 12.— Ratio of maximum baryonic veloc- Fig. 13.— Baryonic Tully-Fisher relations for ity to maximum observed velocity versus µo,K . Vtot,max and Vb,max, and the size-baryonic mass SDSS galaxies with partial imaging are plotted as relation are plotted in panels a–c. Weighted bi- open circles. Outliers are labeled: NGC 4062 and sector fits with and without NGC 3319 are plotted NGC 3992 have submaximal disks, NGC 4138 has as black and gray lines, respectively. In panel d, evidence of kinematic disturbance, and NGC 3319 the Vtot,max residual of the Tully-Fisher relation has both a submaximal disk and a kinematic dis- is plotted versus the size residual from the size- turbance. baryonic mass relation. Galaxies from the SDSS that have partial imaging are plotted as open cir- by using the SDSS and the color-M/L rela- cles. tions for g − r and (M/Lr)∗). The 4 outliers in Figure 12 are galaxies that have submaxi- do not include the outlier in these two re- mal disks and/or are kinematically disturbed. lations, NGC 3319, we obtain the following The relation between Vb,max and Vtot,max is fits: M200 = 5.36 ± 0.03 and α = 3.3 ± 0.3 not expected to be as tight for less luminous with a rms residual of 0.19 for Vtot,max, and spirals, but could possibly be calibrated for M200 = 6.75 ± 0.03 and α = 3.1 ± 0.3 with such a population. a rms residual of 0.21 for Vb,max. The galaxy In Figure 13a,b we plot the baryonic NGC 3319 is both kinematically disturbed on Tully-Fisher relations for Vtot,max and Vb,max. one side (Moore & Gottesman 1998), and has Weighted bisector fits are given in the form: the smallest stellar mass in our sample. For α Mb = M200V where M200 is the stellar mass most of the galaxies in our sample, we do 10 in units of 10 M⊙ for a galaxy on the Tully- not have H I or molecular hydrogen measure- Fisher relation with V = 200 km s−1. They ments. If gas masses were included, the Tully- are: M200 = 5.40 ± 0.03 and α = 3.4 ± 0.3 Fisher slope of Mb on Vtot,max would flatten, with a rms residual of 0.20 for Vtot,max, and as lower mass galaxies (e.g., NGC 3319) have M200 = 6.98 ± 0.03 and α = 2.8 ± 0.3 with a larger gas content than more massive galax- a rms residual of 0.22 for Vb,max. If we ies (e.g., Verheijen 1997).

20 We refrain from comparing the slopes disks are submaximal. This argument, how- found here and in the literature (e.g., Bell & ever, is not entirely straightforward, since in de Jong 2001; McGaugh 2005; Pizagno et al. the Pizagno et al. (2005) model for disk col- 2005) for the following reasons. Such fits are lapse within a halo, variations in dark matter sensitive, among other things, to the velocity halo parameters can create enough scatter measurements used (i.e., Vtot,max,Vb,max,Vflat,V2.2, in the Tully-Fisher relation to hide the pre- W20), the range in stellar mass of the sam- dicted correlation. Furthermore, McGaugh ples, and the galaxy samples used (range & de Blok (1998) and Sellwood (1999) ar- in Hubble type, field versus cluster environ- gue that submaximal disks do not solve the ments). In particular, Gurovich et al. (2005) problem, and one is left with a fine-tuning find a break in the stellar mass and baryonic problem: either disk M/Ls correlate with Tully-Fisher relations for less massive galaxies surface brightness, halo contributions vary, 7.6 9.6 (10 . M∗ . 10 M⊙). This break is such or Newtonian dynamics falters. Figure 12 il- that the Tully-Fisher relation for less mas- lustrates this fine-tuning problem by showing sive galaxies is steeper than that for galaxies that Vb,max/Vtot,max is independent of surface 9.6 11.2 with stellar masses from ∼ 10 –10 M⊙. brightness, so that the total velocity “knows Gurovich et al. (2005) find slopes for the lower about” the baryonic contribution. Changing and upper mass ranges of their H-band sam- the normalization of the color-M/L relations ple, using W20 as a velocity measurement, re-scales the y-axis of the plot, but the ratio of 4.4 ± 0.3 and 3.3 ± 0.3, respectively. This remains independent of surface brightness. break/curvature makes a comparison between slopes of different samples risky at best, es- 6.2. Dark Matter Scaling Relations pecially when different selection criteria are In Figures 14a–d and 15, the baryon/dark used. A 2D Kolomogrov-Smirnov test would matter equality radius in units of the disk be ideal, but beyond the scope of this work scale length, RX /hIR, is plotted versus a num- considering the modest stellar mass range of ber of galactic parameters: Vtot,max, Vb,max, our sample. Hubble T-type, Mb, B, K, µo,B, and µo,K. In The size-baryonic mass relation, where size Figure 14e, we plot RX versus hIR. There are is parameterized by hIR, is plotted in Fig- fewer points in these figures than in previous ure 13c. We perform a weighted bisector fit ones since not all galaxies have a dark mat- 10 0.60±0.08 and find hIR = 0.67 ± 2.31 (Mb/10 M⊙) ter rotation curve that allows us to determine kpc with a rms of 0.16. We compare the RX . The correlation coefficients for these re- Vtot,max residuals of the Tully-Fisher relation lations and for R(Vb,max) with RX are listed in with the size residuals of the size-mass rela- Table 4. The radius RX is found to correlate tion in Figure 13d, and find a scatter plot. If most strongly with Vb,max and very strongly galaxy disks are maximal and the dark matter with Mb, Hubble T-type, and Vtot,max. There halos of galaxies of different surface brightness are no changes in the relative strengths of the are identical, then it is predicted by McGaugh relations if the color-M/L relations are renor- & de Blok (1998) and Courteau & Rix (1999) malized by ±0.1 or −0.3 dex. that there should be a correlation between The trends with RX that we find are qual- these residuals. Courteau & Rix (1999) ar- itatively in agreement with those in the lit- gue that the lack of residual correlation, es- erature (e.g., de Blok & McGaugh 1997; Mc- sentially the surface brightness independence Gaugh & de Blok 1998; Pizagno et al. 2005), of the Tully-Fisher relation, implies that all with some exceptions that are discussed here.

21 Fig. 14.— Relations for RX /hIR 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 ±0.1 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 hIR: those with hIR < 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 NGC2841, NGC3319, and NGC4138. 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′′ (Silćhenko, Vlasyuk, & Burenkov 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. 22 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 number of relations involving a quantity similar to the mass discrepancy that are in contradiction to those presented in this paper and in the literature. Here we discuss possible causes of these discrepancies. Our differences with McGaugh & de Blok (1998) may be traced to the quantities used in the analyses: they use M/LB and hB, while we use RX and hIR. Luminosity in the B-band should have more scatter in relations with galaxy properties than baryonic mass derived from a combination of optical and near-infrared data. This is because the Fig. 15.— Relations for R /h and inte- X IR B-band is more affected by star formation and grated magnitudes and central surface bright- extinction, while the K-band is a better tracer nesses. Symbols and error bars are the same as of stellar mass. In addition, a near-infrared in Figure 14. The galaxy NGC3319 has a µ o,K scale-length is more analogous to the bary- measurement, but not a K measurement. Error onic mass scale-length of a disk, and hence bars along the x-axis are approximately the point should have less scatter in its relations with size at the resolution of the plot. galaxy properties than an optical scale-length. Adding some credence to the hypothesis that McGaugh & de Blok (1998) and de Blok & different quantities are the root of the discrep- McGaugh (1997) find that µo,B plays a major ancy, McGaugh & de Blok (1998) find that role in relations with galaxy properties, in- relations between R2:1 and both µo,B and MB cluding with R2:1, which is analogous to our are such that brighter galaxies have larger val- RX parameter. We find that RX /hIR is cor- ues of R2:1, consistent with our results. That related with µo,B, but that it is even more McGaugh & de Blok (1998) find a correlation correlated with µo,K. We do not find a corre- between M/LB evaluated at 4h and µo,B is lation between the mass discrepancy and µo,K likely also due to the larger range in surface (in Figure 12 we plot the inverse mass discrep- brightness in their sample. ancy), but McGaugh & de Blok (1998) do find In Figure 14f, R10 is plotted versus RX , and a relation between total M/LB at 4h and µo,B. galaxies are plotted as different symbols ac- Also, we find that hIR correlates strongly with cording to their value of hIR. Although there RX , once outliers are removed, which is not is some scatter, as hIR increases, these two in qualitative agreement with other studies: radii move further out in the disks in tandem. McGaugh & de Blok (1998) and Zavala et al. That there is such a tight relation between (2003) find that h does not correlate with to- R10 and RX tells us that dark matter contri- tal M/LB evaluated at 4h or the mass discrep- butions to the observed rotation curves must ancy evaluated at maximum rotation velocity, increase in a characteristic way between these respectively. However, the results of Pizagno two radii for all galaxies. This is likely due et al. (2005) for relations with h are consistent to the combined effects of quasi-exponential

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 properties than RX because the observed rotation curves should not yet be ﬂat in the region where R10 is measured, which is found to be the case.

7. Radial Behavior of Dark Matter

In Figure 16, the dimensionless parameter β(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 potential as a function of radius in a galaxy, and is akin to the β parameter defined by Salucci (2001) and similar parameters used in many other papers, except here it is evaluated at all radii. Where β(r) = 1, the baryonic mass of a galaxy accounts for its observed Fig. 16.— β(r) ≡ Mb(r)/Mtot(r) and its inverse rotation curve, β(r) = 0.5 at r = R , and X for galaxies with an appreciable dark matter con- β(r) = 0 where the dark matter accounts tribution. β(r) is plotted versus r/hIR and r/RX for its observed rotation curve. If the bary- in parts a and b, respectively; its inverse is plotted onic model over-predicts the observed rota- versus r/RX in panel c. Different line types corre- tion curve, then β(r) > 1. In Figure 16a, β(r) spond to the Vb,max of the galaxies: Vb,max > 250 is plotted versus radius in terms of hIR. For −1 km s (solid line), 201 < Vb,max ≤ 250 (dot- many galaxies, the observed rotation curves ted line), 120 < Vb,max ≤ 201 (dashed line), and are entirely accounted for by baryons in the Vb,max ≤ 120 (dot-dash line line). inner parts. Beyond this region, baryonic mass falls off as dark matter begins to dom- inate. Other galaxies are dark matter domi- nate causes the curves to overlap at r = RX nated throughout. Curves in Figure 16a have (where β(r) = 0.5), and thus allows for a bet- different line types that correspond to ranges ter comparison of their radial behavior. In of Vb,max. For most of the galaxies, as Vb,max Figure 16c, we plot the somewhat less intu- −1 increases, so does the proportion of baryonic itive function β (r) since it increases linearly to dark matter at all radii such that the fastest with r/RX . There is little variation in the be- rotators are observed to be dominated by havior of the curves in Figures 16b,c. Such baryons until quite far out into their disks. regularity can be explained in terms of quasi- However, there is clearly much scatter about exponential disks and flat rotation curves as this trend; this conclusion can also be inferred follows: The rotation curve of a galaxy is from Figure 14f. nearly flat beyond r ∼ 2hIR, or it is at least In Figure 16b, β(r) is plotted versus radius a slow function of r and varies less than Vb(r) in this region. In addition, much of the bary- in units of RX . This choice of radial coordi-

24 onic mass of a galaxy is enclosed at r ∼ 2hIR, used. To test the most common implementa- which causes the baryonic rotation curves to tion of dark matter contraction (Blumenthal be roughly Keplerian (Vb(r) ∝ 1/r) beyond et al. 1986), we adiabatically contract the grid this radius. Therefore, for all galaxies, beyond of halo models according to the radial density ∼ 2hIR, it should be the case that β(r) ≡ distribution of baryons in the galaxies follow- 2 2 Vb /Vtot ∝ 1/r, which is what we observe. All ing the formalism of Dutton et al. (2005), and the β(r) curves overlap nicely when plotted perform the fits again. The NFW fitting for- versus radius normalized to RX since this ra- mula has 2 free parameters that we fit for dius is located beyond 2hIR, and is generally using a grid covering: 0.5 ≤ c200 ≤ 20 and in the falling part of the baryonic rotation 0.5 Vtot,max ≤ V200 ≤ 2.5 Vtot,max. −1 curves. We parametrize the trend of β with In Figure 17, we plot the best-fit halo mod- r/RX in a simple universal manner that holds els for 8 example galaxies, and in Table 3 −1 for all galaxies: β = 1.71(r/RX ) + 0.021. we list the best-fit parameters along with the This relation is consistent with the predic- reduced χ2 of all the fits. Most of the fits tion of Palunas & Williams (2000) that ei- are very poor; the average reduced χ2 values ther the contribution of dark matter within with and without adiabatic contraction are: the optical radius of galaxies is small or that 7.1 and 4.2 for the original color-M/L rela- the distribution of dark matter is coupled to tions, 14.0 and 7.3 for the +0.1 dex renormal- that of the luminous matter. This relation ization of the color-M/L relations, 4.8 and 3.4 is also similar in spirit to the parameteriza- for −0.1 dex, and 3.9 and 4.3 for −0.3 dex. We tion of the mass discrepancy-acceleration re- do not perform fits for galaxies when most of lation of McGaugh (2004). Since β−1 is the the observed rotation curve is accounted for mass discrepancy, and a = V 2/r ∝ 1/r for flat by the baryonic rotation curve. rotation curves, a natural mass discrepancy- For baryonic rotation curves derived from acceleration relation arises. the original color-M/L relations, nearly all fits to the NFW models without contraction have 8. Comparison With Theories of Halo a smaller reduced χ2 than those where con- Density Distributions traction was performed. The exceptions are NGC 4062 and NGC 3992, which likely have We compare the derived dark matter pro- submaximal disks. For galaxies that have a files with an analytical function designed to large baryonic contribution to the inner parts parametrize the density profile of dark matter of their rotation curves, adiabatic contrac- halos in N-body simulations. In particular, tion over-contracts the inner parts of their we compare our data to the NFW formulation dark matter halos such that the resulting to- for dark matter halo density profiles (Navarro, tal mass rotation curves over-predict the mea- Frenk, & White 1996) with and without tak- sured rotation curves, as in Weiner, Sellwood, ing into account adiabatic contraction of the & Williams (2001), for example. For fits to halos. To do this, for each galaxy, we fit its the NFW models where the baryonic rotation observed rotation curve with a total mass ro- curves were derived from color-M/L relations tation curve created from the addition of its that were renormalized by +0.1 dex, the situa- baryonic rotation curve to a grid of halo mod- tion is exactly the same. For -0.1 dex, 7 galax- els, with a reduced χ2 statistic. Uncertain- ies have better fits when contraction is taken ties in the observed rotation curves are taken into account, and for -0.3 dex, 15 galaxies have to be 10 km s−1; results do not differ signifi- a better fit. This implementation of dark mat- cantly if the error bars plotted in Figure 3 are

25 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 better 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 ΛCDM universe. In addition, Alam, Bullock, & Weinberg (2002) examined low surface brightness galaxies and found their dark matter halos to be under- concentrated compared to what is expected from a standard ΛCDM universe, even if they assumed the galaxies to be dark matter- dominated.

9. Summary

We have decomposed the rotation curves of 34 nearby bright spiral galaxies into baryonic and dark matter components by applying Fig. 18.— Best-fit parameters for NFW fits with color-M/L relations to near-infrared and op- and without adiabatic contraction (filled and open tical photometry, and find the following: symbols, respectively). The median and the 68 per cent of the c200 values measured in numerical N- • The dark-to-luminous matter distribu- body simulations of dark matter halos are plotted tions are self-similar once scaled by as solid and dashed lines, respectively. When the RX , the radius where the baryonic and best-fit halo is at the edge of the parameter space dark matter contributions to the rota- searched, it is plotted as a triangle, otherwise it tion curve are equivalent. This behav- is plotted as a circle. In part a, best-fit parame- ior is parameterized by a simple func- ters for the original and ±0.1 dex renormalizations tion whose form is due to the quasi- of the color-M/L relations are plotted; in part b, exponential nature of galaxy disks and best-fit parameters for the -0.3 dex renormaliza- rotation curves that are nearly flat after tion are plotted. an initial rise. This result is independent of the normalization of the color- M/L relations. ter contraction works best when baryons do not account for all of the inner parts of the • The radii R10 and RX , where dark observed rotation curve. matter contributes 10% and 50% to In Figure 18, we plot the best-fit values the rotation of galaxies, respectively, correlate with galaxy properties. The of V200 and c200 and the range of these parameters found in N-body simulations (Bul- strongest correlation with RX is with lock et al. 2001; Eke, Navarro, & Steinmetz the maximum baryonic rotation speed 2001). Even under the assumption of no adia- such that galaxies with RX measurements that lie further out in their disks

27 rotate faster. The next strongest corre- be calibrated for lower luminosity galax- lations are equivalently baryonic mass, ies. observed rotation speed, and Hubble T- type. Contrary to what is expected from • We find generally poor fits for the NFW previous studies (de Block & McGaugh parameterization for dark matter ha- 1998; Zavala et al. 2003), B-band cen- los due to the significant baryonic con- tral surface brightness is not found to tributions found in the inner parts of be the main driving force in relations most galaxies. The concentrations of the best-fit NFW halos are therefore with RX for this sample of bright galax- much lower than what is expected for ies. The radii RX and R10 move out in galaxy disks in tandem, consistent with galaxies in a standard ΛCDM universe. a self-similar dark-to-luminous matter This is even the case when the color- distribution among galaxies. M/L relations are renormalized by -0.3 dex. In order to have better fits, a • We confirm the normalization of the normalization even lower than -0.3 dex, color-M/L relations given in Bell et al. where baryons contribute very little to (2003), which is analogous to an upper the total mass in the inner parts of ro- limit on the IMF based on the dynamics tation curves, would have to be imple- of disk galaxies. A more careful analy- mented. Adiabatic contraction, as it is sis of the data used in both Bell & de normally implemented, makes these fits Jong (2001) and Bell et al. (2003) is per- worse by adding more dark matter to formed, and data from this paper are the inner parts of galaxies. included in the sample.

• All but 4 of the 34 galaxies in our sample We would like to thank Richard Pogge, are close to maximal disk. Two of these James Pizagno, James Bullock, Jay Frogel, 4 galaxies have maximal disks within Chris Kochanek, and David Weinberg for uncertainties. A prime example of a valuable discussions. The referee, Stacy Mc- galaxy that cannot have a submaximal Gaugh, is thanked for constructive comments disk given the current formulations of that greatly improved the quality of this pa- dark matter halos is NGC 157 which has per. We are grateful to the following authors a pronounced hump-like structure in its who provided rotation curves in tabular form observed and baryonic rotation curves. via email: Kor Begeman, Gianfranco Gentile, • Maximum rotation velocities predicted Thilo Kranz, Povilas Palunas, Stuart Ryder, from the baryon distributions of galax- Michele Thornley, and Wilfred Walsh. We are ies (Vb,max) are tightly correlated with grateful to Marc Verheijen who made avail- the observed maximum rotation speeds able his high-quality imaging and rotation (Vtot,max). Using this, a baryonic Tully- curves of galaxies in the Ursa Major cluster. Fisher relation can be created based on SAK would like to acknowledge ﬁnancial sup- two-passband surface photometry and port from The Space Telescope Science Insti- a redshift alone (e.g., with the SDSS tute Director’s Discretionary Research Fund and color-M/L relations for g − r and (DDRF). We thank the CTIO TAC for gen- (M/Lr)∗, or for more distant redshift erous allocation of time for The Ohio State surveys with Hubble Space Telescope University Galaxy Survey and the many peo- imaging). Such a relation could possibly ple over the years who helped collect these

28 observations. Funding for The Ohio State This research also made use of the HyperLeda University Bright Spiral Galaxy Survey was database and the VizieR catalog access tool, provided by grants from The National Sci- CDS, Strasbourg, France. ence Foundation (grants AST-9217716 and AST-9617006), with additional funding by REFERENCES The Ohio State University. Abadi, M. G., Navarro, J. F., Steinmetz, M., This paper makes use of data from both the & Eke, V. R. 2003, ApJ, 591, 499 Sloan Digital Sky Survey and the Two Micron All Sky Survey. The Two Micron All Sky Sur- Abazajian, K. et al. 2004, AJ, 128, 502 vey is a joint project of the University of Mas- Alam, S. M. K., Bullock, J. S. & Weinberg, sachusetts and the Infrared Processing and D. H. 2002, ApJ, 572, 34 Analysis Center/California Institute of Tech- nology, funded by the National Aeronautics Begeman, K. G. 1987, Ph.D. thesis, Univ. and Space Administration and the National Groningen Science Foundation. Funding for the creation Bell, E. F., McIntosh, D. H., Katz, N., & and distribution of the SDSS Archive has been Weinberg, M. D. 2003, ApJS, 149, 289 provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Bell, E. F. & de Jong, R. S. 2001, ApJ, 550, Aeronautics and Space Administration, the 212 National Science Foundation, the U.S. De- partment of Energy, the Japanese Monbuk- Binney, J., & Tremaine, S. 1987, Galactic Dy- agakusho, and the Max Planck Society. The namics (Princeton : Princeton Univ. Press) SDSS Web site is http://www.sdss.org/. The Bizyaev, D. & Mitronova, S. 2002, A&A, SDSS is managed by the Astrophysical Re- 389,795 search Consortium (ARC) for the Participat- ing Institutions. The Participating Institu- Blais-Ouellette, S., Amram, P., Carignan, C., tions are The University of Chicago, Fermi- & Swaters, R. 2004, A&A, 420, 147 lab, the Institute for Advanced Study, the Blumenthal, G. R., Faber, S. M., Flores, R., Japan Participation Group, The Johns Hop- & Primack, J. R. 1986, ApJ, 301, 27 kins University, Los Alamos National Labora- tory, the Max-Planck-Institute for Astronomy Bosma, A. 1981, AJ, 86, 1791 (MPIA), the Max-Planck-Institute for Astro- physics (MPA), New Mexico State University, Broeils, A. H. 1992, Ph.D. thesis, Rijksuniver- University of Pittsburgh, Princeton Univer- siteit Groningen sity, the United States Naval Observatory, and Bullock, J. S. et al. 2001, MNRAS, 321, 559 the University of Washington. This research has made use of NASA’s As- Buta, R. et al. 2001, AJ, 121, 225 trophysics Data System, the NASA/IPAC Ex- Cardelli, J. A., Clayton, G. C., & Mathis, J. tragalactic Database (NED), and the Hyper- S. 1989, ApJ, 345, 245 Leda database and the VizieR catalog access tool. NED that is operated by the Jet Casertano, S. 1983, MNRAS, 203, 735 Propulsion Laboratory, California Institute Corradi, R. L. M., Boulesteix, J., Bosma, A., of Technology, under contract with the Na- Amram, P., & Capaccioli, M. 1991, A&A, tional Aeronautics and Space Administration. 244, 27

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31 Table 1

Galaxy Tracera Reference(s)

NGC 157 FP Hα, HI Fridman et al. 2001, Ryder et al. 1998b NGC 289 H I Walsh, Staveley-Smith, & Oosterloo 1997b NGC 488 Hα Peterson 1980b,c NGC 908 Hα Mathewson, Ford, & Buchhorn 1992 NGC 1087 Hα Rubin et al. 1985 NGC 1090 Hα, HI Courteau1997,Gentile et al.2004 NGC 1241 Hα Mathewson, Ford, & Buchhorn 1992 NGC 1385 Hα Mathewson, Ford, & Buchhorn 1992 NGC 1559 Hα Mathewson, Ford, & Buchhorn 1992 NGC 1832 Hα Mathewson, Ford, & Buchhorn 1992 NGC 2090 Hα Mathewson, Ford, & Buchhorn 1992 NGC 2139 Hα Mathewson, Ford, & Buchhorn 1992 NGC 2280 Hα Mathewson, Ford, & Buchhorn 1992 NGC 2841 FP Hα, H I Blais-Ouellette et al. 2004, Giraud 1998c NGC 3198 FP Hα, HI Corradi et al. 1991b , van Albada et al. 1985 NGC 3223 Hα Mathewson, Ford, & Buchhorn 1992 NGC3319 HI Moore&Gottesman1998b,c NGC3521 HI Sanders1996c NGC3726 HI Verheijen1997b NGC 3893 Hα, HI Kranz2002,Verheijen 1997b NGC3949 HI Verheijen1997b NGC3953 HI Verheijen1997b NGC3992 HI Verheijen1997b NGC4051 HI Verheijen1997b NGC 4062 Hα Rubin et al. 1985 NGC 4138 NII,HI Jore,Broeils,&Haynes1996c ,Verheijen 1997b NGC 4651 Hα Rubin, Waterman, & Kenney 1999 NGC 4698 Hα Rubin, Waterman, & Kenney 1999 NGC5371 HI Begeman1987 NGC 5806 Hα Courteau 1997 NGC 6300 FP Hα, HI Buta et al. 2001, Ryder et al. 1996 NGC 7083 Hα Mathewson, Ford, & Buchhorn 1992 NGC 7217 Hα Rubin et al. 1985 NGC 7606 Hα Mathewson, Ford, & Buchhorn 1992 aThe notation “FP Hα” is used for rotation curves derived from Fabry-Perot measurements of Hα. bErrors are taken as the diﬀerence in velocity between the approaching and receding sides. cThe rotation curve has been extracted electronically from a plot in the referenced paper.

32 Table 2

Observed Baryonic Matter Dark Matter Vtot,max Vb,max R(Vb,max) Mb R10 RX ∆+0.1dex ∆+0.1dex ∆+0.1dex ∆+0.1dex ∆−0.1dex ∆−0.1dex ∆−0.1dex ∆−0.1dex ∆−0.3dex ∆−0.3dex ∆−0.3dex ∆−0.3dex 10 Galaxy (km/s) (km/s) (kpc) (10 M⊙) (kpc) (kpc)

NGC 157 205 205 6.2 7.1 ······ +25 +1.8 ······ -22 -1.4 ······ -60 -3.5 ······ NGC 289 182 189 3.6 4.1 9.1 11.3 +23 +1.0 +1.7 +4.2 -20 -0.8 -1.8 -2.7 -55 -2.0 -4.0 -4.7 NGC 488 350 296 10.1 31.1 ······ +36 +7.4 ······ -32 -5.9 ······ -86 -14.3 ······ NGC 908 200 201 7.3 7.2 ······ +24 +1.8 ······ -22 -1.5 ······ -59 -3.5 ······ NGC 1087 136 141 5.9 2.8 ······ +17 +0.7 ······ -15 -0.6 ······ -41 -1.4 ······ NGC 1090 170 165 7.3 5.4 16.4 23.2 +20 +1.4 +4.4 +7.8c -18 -1.1 -5.3 -4.9 -49 -2.7 -12.5 -18.2 NGC 1241 300 250 7.9 19.2 12.4 ··· +30 +4.8 +4.4d ··· -27 -3.8 -5.6 ··· -73 -9.2 -12.4d ··· NGC 1385 140 133 4.0 2.6 ······ +16 +0.6 ······ -14 -0.5 ······ -39 -1.2 ······ NGC 1559 150 136 4.5 2.3 ······ +17 +0.6 ······ -15 -0.5 ······ -40 -1.1 ······

33 Table 2—Continued

NGC 1832 200 209 3.8 5.4 ······ +25 +1.4 ······ -23 -1.1 ······ -61 -2.7 ······ NGC 2090a 160 153 1.6 1.6 ······ +19 +0.4 ······ -16 -0.3 ······ -45 -0.8 ······ NGC 2139 140 121 4.5 1.9 ······ +15 +0.5 ······ -13 -0.4 ······ -35 -0.9 ······ NGC 2280a 210 177 6.9 6.8 8.7 ··· +5 +1.8 +1.8 ··· -19 -1.4 -3.5 ··· -32 -3.4 -5.4 ··· NGC 2841 325 284 4.7 13.0 9.0 12.6 +49 +3.3 +3.4 +5.8 -20 -2.6 -0.6 -2.9 -35 -6.4 -6.2 -3.7 NGC 3198 152 120 5.6 2.3 6.8 10.5 +14 +0.6 +3.1 +2.6 -13 -0.5 -2.1 -2.5 -35 -1.1 -4.6 -7.3 NGC3223 320 314 10.4 30.8 ······ +38 +7.6 ······ -34 -6.1 ······ -92 -14.7 ······ NGC 3319b 132 50 9 0.7 2.6 3.5 +7 +0.1 +0.7 +1.1 -5 -0.1 -0.6 -0.8 -14 -0.3 -2.6d -3.5d NGC 3521 221 263 3.1 8.0 10.3 13.3 +32 +1.9 +2.3 +4.2 -29 -1.5 -1.8 -2.4 -34 -3.6 -5.8d -6.3

34 Table 2—Continued

NGC 3726 169 144 10.8 6.0 20.4 25.3 +22 +1.5 +3.2 +3.5 -12 -1.2 -10.6 -2.2 -42 -2.9 -20.4d -16.3 NGC 3893 210 186 5.3 6.8 13.3 19.4 +23 +1.6 +4.5 +2.6c -20 -1.3 -5.5 -4.7 -54 -3.2 -10.3 -13.4 NGC 3949 169 158 3.9 2.7 7.4 ··· +18 +0.6 +0.7d ··· -17 -0.5 -1.6 ··· -46 -1.2 -7.4d ··· NGC 3953 225 227 7.7 12.0 17.8 ··· +28 +3.1 +0.3c ··· -25 -2.4 -4.2 ··· -66 -5.9 -14.0 ··· NGC3992 272 188 13.1 11.2 ······ +21 +2.8 ······ -21 -2.3 ······ -56 -5.5 ······ NGC 4051 170 167 0.3 2.4 5.8 ··· +20 +0.6 +2.1 ··· -18 -0.5 -1.3 ··· -49 -1.1 -5.8d ··· NGC 4062 162 110 3.4 1.1 0d 3.9 +14 +0.3 +1 +2.4 -11 -0.2 0d -2.0 -32 -0.6 0d -3.9d NGC 4138 195 272 1.0 4.2 7.5 17.8 +33 +1.0 +8.9 +3.1 -30 -0.8 -1.4 -5.8 -79 -2.0 -7.5d -12.6 NGC 4651 210 173 2.6 3.7 ······ +32 +0.9 ······ -10 -0.8 ······ -45 -1.8 ······

35 Table 2—Continued

NGC 4698 220 223 3.8 6.5 ······ +14 +1.6 ······ -35 -1.3 ······ -68 -3.2 ······ NGC5371 242 289 13.9 33.5 ······ +21 +8.5 ······ -43 -6.7 ······ -87 -16.4 ······ NGC 5806 200 190 1.1 5.3 ······ +27 +1.3 ······ -17 -1.1 ······ -27 -2.6 ······ NGC 6300 208 220 4.3 6.7 13.7 20.4 +12 +1.7 +5.8 +2.6 -35 -1.4 -1.3 -5.5 -67 -3.3 -8.8 -13.5 NGC 7083 210 245 6.5 13.1 ······ +30 +3.3 ······ -27 -2.7 ······ -72 -6.4 ······ NGC 7217 284 283 2.0 7.9 ······ +35 +2.0 ······ -31 -1.6 ······ -84 -3.9 ······ NGC 7606 280 228 9.8 12.9 2.2 ··· +28 +3.2 +0.8c ··· -25 -2.6 -0.7 ··· -67 -6.2 -2.2d ··· aQuantities are calculated from B − V instead of B − R b Quantities are calculated from (M/L)∗,H instead of (M/L)∗,K cUpper limit d Lower limit

36 Table 3

origonal color-M/L +0.1 dex -0.1 dex -0.3 dex 2 2 2 2 c V200 χ c V200 χ c V200 χ c V200 χ Galaxy (km/s) (km/s) (km/s) (km/s)

NGC 157 0.5 111 9.5 0.5 61a 19.9 4.0 96a 6.7 16.0 96 5.8 0.5 61a 13.8 0.5 61a 28.2 0.5 61a 8.2 2.0 131 7.3 NGC 289 8.0 143 0.7 6.5 148 0.8 10.5 138 0.8 11.5 138 0.8 3.0 168 1.2 1.0 238 1.7 5.0 133 0.8 7.5 143 0.7 NGC488 16.5 199a 9.1 8.0 204 6.5 20.0 214a 12.6 20.0 279 22.7 3.0 204 5.3 0.5 199a 6.8 8.0 199a 5.2 18.5 199a 6.1 NGC 908 ··························· 20.0 116 8.7 ··························· 14.0 96a 6.5 NGC 1087 ··························· 16.5 68a 1.3 ··························· 4.0 68a 1.5 NGC 1090 1.0 400 6.4 0.5 400 9.7 5.0 160 5.3 12.0 125 4.7 0.5 105 11.4 0.5 80a 15.6 0.5 215 8.7 1.0 360 6.6 NGC 1241 1.5 745 4.7 0.5 150a 9.3 3.0 745 3.4 20.0 180 3.4 0.5 150a 7.3 0.5 150a 22.2 0.5 354 4.0 1.5 585 2.3 NGC 1385 ··························· 4.0 328 0.1 ··························· 0.5 143 0.7 NGC 1559 ··························· 3.5 350a 5.1 ··························· 0.5 350a 7.0 NGC1832 0.5 89a 8.8 ········· 0.5 394 4.2 20.0 94 4.0 0.5 89a 19.1 ········· 0.5 394 6.9 0.5 389a 3.7 NGC 2090 ·················· 20.0 77a 0.8 20.0 122 1.2 ·················· 2.5 77a 0.4 13.5 77a 0.3 NGC 2139 2.5 301 1.5 0.5 71a 2.2 4.0 336 1.1 7.5 261 0.8 0.5 71a 2.7 0.5 71a 5.6 0.5 156 2.5 2.0 336 2.3 NGC 2280 2.0 470 1.2 1.5 475 0.5 3.0 495 0.2 5.0 375 0.2 0.5 105a 3.9 0.5 105a 2.4 0.5 265 2.7 1.0 525 2.6 NGC 2841 8.5 242 13.4 3.0 402 27.6 12.5 217 11.0 15.0 207 7.0 0.5 582 23.3 0.5 452a 53.9 1.0 647 18.6 3.5 337 11.6 NGC 3198 6.0 140 1.7 4.0 160 2.8 8.0 130 1.1 11.5 120 0.6 0.5 345 4.0 0.5 285 6.5 1.5 235 2.8 5.0 145 1.5 NGC 3223 0.5 733 5.3 0.5 158a 14.8 9.5 178 5.0 20.0 178 7.1 0.5 158a 10.1 0.5 158a 39.9 0.5 233 3.7 11.5 158 2.6 NGC 3319 2.0 269 0.3 1.5 314 0.3 2.5 233 0.2 3.5 189 0.2 1.0 304 0.9 1.5 314 1.3 1.5 259 0.7 1.5 294 0.4 NGC 3521 12.5 121 3.1 7.0 136 4.3 18.5 116 2.6 20.0 121 2.2 0.5 281 4.1 0.5 171 6.2 4.5 146 3.5 13.5 116 2.1 NGC 3726 1.0 334 1.0 0.5 379 3.6 1.0 394 0.3 6.0 154 0.6 0.5 159 5.9 0.5 94 10.2 0.5 239 2.9 0.5 414 0.4

37 Table 3—Continued

origonal color-M/L +0.1 dex -0.1 dex -0.3 dex 2 2 2 2 c V200 χ c V200 χ c V200 χ c V200 χ Galaxy (km/s) (km/s) (km/s) (km/s)

NGC 3893 4.0 159 2.3 0.5 334 4.4 18.0 94 2.2 20.0 119 4.0 0.5 74 3.7 0.5 74a 8.9 0.5 204 2.3 8.5 119 2.1 NGC 3949 2.0 390 2.2 0.5 365 5.5 3.0 405 0.7 7.0 245 0.03 0.5 85a 4.5 0.5 85a 10.8 0.5 280 2.9 2.0 385 0.8 NGC3953 0.5 563a 2.0 0.5 113a 6.5 1.5 558 0.3 9.5 183 0.02 0.5 113a 5.3 0.5 113a 22.9 1.5 183 2.2 1.0 493 0.2 NGC 3992 20.0 149 0.3 14.0 154 0.3 19.5 159 0.4 20.0 164 1.2 8.0 164 0.3 2.0 234 0.3 11.0 164 0.3 20.0 153 0.4 NGC 4051 2.0 382 1.9 1.0 322 2.9 3.0 352 1.3 10.0 152 1.0 0.5 137 3.8 0.5 77a 4.9 0.5 322 2.5 2.0 352 1.7 NGC 4062 18.5 97 1.7 3.5 382 1.5 20.0 102 1.7 20.0 117 2.0 1.5 372 0.6 0.5 352a 1.1 2.5 357 0.5 4.0 377 0.7 NGC4138 1.0 375a 15.7 0.5 295 30.1 8.5 115 9.9 20.0 105 7.3 0.5 75a 23.6 0.5 75a 43.0 0.5 75a 15.4 1.0 375 10.8 NGC 4651 3.0 517 2.3 1.0 557a 4.9 15.0 142 2.0 20.0 152 3.0 0.5 112a 6.1 0.5 112a 11.1 1.0 307 5.8 3.5 382 5.7 NGC 4698 ··························· 3.5 525 19.2 ··························· 0.5 105a 29.2 NGC 5371 ·················· 0.5 217 6.2 9.0 137 0.9 ·················· 0.5 107a 15.1 0.5 197 0.8 NGC 5806 ·················· 14.0 96a 1.6 20.0 96a 3.7 ·················· 1.0 101 0.7 6.0 96a 1.2 NGC 6300 ·················· 5.0 174 9.5 20.0 109 9.3 ·················· 0.5 209a 7.0 8.0 119a 4.6 NGC 7083 ··························· 13.0 126 1.1 ··························· 0.5 361 0.9 NGC 7217 ··························· 20.0 176 15.6 ··························· 19.0 106a 4.0 NGC7606 17.5 142a 1.7 9.5 142a 2.5 20.0 157 1.4 20.0 192 2.0 0.5 447 3.0 0.5 142a 3.6 2.5 357 2.7 12.0 187 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 Galaxy Property All Data Without Outliersa RX /hIR RX RX /hIR RX

Vtot,max 0.41 ··· 0.90 ··· Vb,max 0.85 ··· 0.96 ··· b R(Vb,max) ··· 0.22 ··· 0.58 T-type 0.89 ··· 0.90 ··· log10Mb 0.65 ··· 0.90 ··· c hIR ··· 0.21 ··· 0.80 R10 ··· 0.94 ··· 0.98 B 0.36 ··· 0.77 ··· Kd 0.41 ··· 0.80 ··· µo,B 0.51 ··· 0.68 ··· µo,K 0.64 ··· 0.87 ···

aOutliers are NGC 2841 and NGC 4138. bOutliers for this relation are NGC 3319 and NGC 4062. cNGC 3319 is not included because it is an outlier in this relation. dNGC 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.