THE ASTROPHYSICAL JOURNAL, 518:50–63, 1999 June 10
© 1999. The American Astronomical Society. All rights reserved. Printed in U.S.A.

Prevalence and Properties of Dark Matter in Elliptical Galaxies

MICHAEL LOEWENSTEIN 1

Laboratory for High Energy Astrophysics, NASA/GSFC, Code 662, Greenbelt, MD 20771

AND

RAYMOND E. WHITE III

Department of Physics and Astronomy, University of Alabama, Tuscaloosa, AL 35487-0324

Received 1998 June 19; accepted 1999 January 15


ABSTRACT

     Given the recently deduced relationship between X-ray temperatures and stellar velocity dispersions (the T-σ relation) in an optically complete sample of elliptical galaxies (see recent work of Davis & White), we demonstrate that L>L$\mathstrut{_{*}}$ elliptical galaxies contain substantial amounts of dark matter in general. We present constraints on the dark matter scale length and on the dark-to-luminous mass ratio within the optical half-light radius and within the entire galaxy. For example, we find that minimum values of dark matter core radii scale as r$\mathstrut{_{{\rm dm}}}$>4(L$\mathstrut{_{V}}$/3L$\mathstrut{_{*}}$)$\mathstrut{^{3{/}4}}$ h$\mathstrut{^{-1}_{80}}$ kpc and that the minimum dark matter mass fraction is ≳20% within one optical effective radius re and is ≳39%–85% within 6re, depending on the stellar density profile and observed value of βspec. We also confirm the prediction of Davis & White that the dark matter is characterized by velocity dispersions that are greater than those of the luminous stars: σ$\mathstrut{^{2}_{{\rm dm}}}$≈1.4–2σ$\mathstrut{^{2}_{*}}$. The T-σ relation implies a nearly constant mass-to-light ratio within six half-light radii: M/L$\mathstrut{_{V}}$≈25 h80 M⊙/L$\mathstrut{_{V_{{\odot}}}}$. This conflicts with the simplest extension of cold dark matter theories of large-scale structure formation to galactic scales; we consider several modifications that can better account for the observed T-σ relation.

Subject headings: dark matter—galaxies: elliptical and lenticular, cD

FOOTNOTES

     1 Also with the University of Maryland Department of Astronomy.

§1. BACKGROUND

     There is a strong consensus that spiral galaxies are embedded in massive, nonluminous halos, the structure and scaling properties of which have been intensively investigated (e.g., Persic, Salucci, & Stel 1996). Studies of galaxy kinematics and of the density and temperature distributions of hot intergalactic gas in galaxy groups (Mulchaey et al. 1996) and clusters (e.g., Mushotzky 1998) have demonstrated the dominance of dark matter on larger scales. In the context of existing theories of the formation of galaxies and large-scale structure, it is therefore natural to expect nonluminous material within elliptical galaxies as well. However, until very recently, the evidence for dark matter in elliptical galaxies has been less forthcoming and more controversial.

     Dark matter in elliptical galaxies has been elusive largely because of the lack of internal dynamical indicators at optical and radio wavelengths that are as ubiquitous, spatially extensive, and unambiguous as H I rotation curves in spirals. Although mergers have endowed some early-type galaxies (many classified as S0) with neutral (e.g., Franx, van Gorkom, & de Zeeuw 1994) or ionized (Bertola et al. 1993) gas disks, these are rare and may not be representative of the population of elliptical galaxies as a whole. Studies of stellar kinematics have, until recently, been inadequate in quality and extent to significantly constrain the large-scale mass distribution in elliptical galaxies in light of the complicating effects of projection and anisotropic velocity distributions.

     Most, if not all, giant elliptical galaxies contain extended distributions of hot X-ray–emitting gas that provide a means for probing the large-scale dark matter distribution—a means potentially as powerful as rotation curves in spiral galaxies. Since these hot gaseous halos should be close to hydrostatic equilibrium (Loewenstein & Mathews 1987), measurements of the density and temperature distributions from X-ray imaging and spectroscopy yield constraints on the mass distribution on the scale and with the accuracy and spatial resolution of X-ray temperature profiles. Observations of elliptical galaxies by the Einstein Observatory IPC were presented as strong evidence for the presence of extensive dark halos in early-type galaxies (Forman, Jones, & Tucker 1985; Fabian et al. 1986; Loewenstein & Mathews 1987); however, the temperature profiles were of sufficiently poor quality to foster a continuing skepticism (Fabbiano 1989; de Zeeuw & Franx 1991).

     Progress on a number of fronts has been rapid during recent years. In the optical, improvements in detector capabilities, data analysis, and modeling have now provided robust stellar dynamical constraints on the mass distribution out beyond the optical half-light radius re for several ellipticals (Saglia et al. 1993; Carollo et al. 1995; Rix et al. 1997; Gerhard et al. 1998), while gravitational lensing observations are providing the first evidence for dark matter in intermediate-redshift galaxies (Kochanek 1995; Brainerd, Blandford, & Smail 1996; Griffiths et al. 1996). X-ray observations using the ROSAT and ASCA satellites have improved the accuracy, spatial resolution, and extent of derived hot gas density and temperature profiles. Of particular interest is the recent study of ASCA observations of ≈30 early-type galaxies by Matsushita (1997) that shows that gas temperature profiles are relatively flat out as far as can be reliably measured (5&arcmin;–20&arcmin;, depending on the intrinsic luminosity and galaxy distance). The existence of isothermal 0.5–1 keV gaseous halos extending, in some cases, to well beyond 10re is strong evidence for the presence of extended dark matter in elliptical galaxies.

     The case for dark matter halos in a number of elliptical galaxies is now overwhelming, and the following pattern seems to be emerging from the latest results on individual galaxies: dark matter comprises a significant but not dominant fraction of the total mass within re (Saglia et al. 1993; De Paolis, Ingrosso, & Strafella 1995; Rix et al. 1997; Gerhard et al. 1998; but see Mushotzky et al. 1994 for an exceptional case: NGC 4636); dark matter becomes increasingly dominant at large radii: M/r≈5×10$\mathstrut{^{12}}$ M⊙ kpc-1 to within a factor of 2 for r>30 kpc (Mushotzky et al. 1994; Irwin & Sarazin 1996; Buote & Canizares 1997), corresponding to mass-to-light ratios more than 100 M⊙/L$\mathstrut{_{B_{{\odot}}}}$ measured at (or extrapolated to) ∼100 kpc (Bahcall, Lubin, & Dorman 1995; Griffiths et al. 1996).

§2. MOTIVATION AND OVERVIEW

     While there is now little doubt that some elliptical galaxies do contain dark matter, a number of broader issues remain to be addressed. Among the most important with respect to our understanding of galaxy formation is whether dark matter in ellipticals is ubiquitous, and how the relative scaling of dark and luminous matter varies—both within galaxies as a function of radius and between galaxies with different optical properties. In this paper we use the recently derived collection of X-ray temperatures in a complete, optically selected sample from Davis & White (1996, hereafter DW) to constrain the average properties of dark halos in the population of L>L$\mathstrut{_{*}}$ elliptical galaxies.

     DW analyzed X-ray spectra from 42 of the 43 optically brightest elliptical galaxies, using ROSAT PSPC data if available and Einstein IPC data otherwise (i.e., for four galaxies). See DW and White & Davis (1999) for details about the sample and data analysis. For the purposes of this work, we use only the mean relationship between X-ray temperatures 〈T〉 and central projected optical velocity dispersions 〈σ〉 (hereafter the T-σ relation)—&angl0;T&angr0;∝&angl0;σ&angr0;$\mathstrut{^{1.45}}$—or equivalent X-ray/optical correlations. More precisely, we consider the variation of the observable βspec with optical luminosity LV (Fig. 1), where



and μmp is the mean mass per particle. If dark matter dominates the gravitational potential on large scales, then 〈T〉 is essentially a measure of the dark matter content within the extraction radii used by DW (6re, where re is the optical effective radius). Since the observed scaling relations (i.e., the fundamental plane) for elliptical galaxies provide a link between 〈σ〉 and the global luminosity, βspec is an excellent diagnostic of the dark-to-luminous matter ratio within the optical radius.

Fig. 1

     Temperature profiles can provide additional dark matter constraints, but are measurable for only the X-ray brightest galaxies in the sample. Likewise, extended stellar velocity dispersion profiles are useful, but data of sufficient detail to be unambiguously interpreted are not generally available. On the other hand, central velocity dispersions have been accurately measured for the entire sample; central velocity dispersions are known to correlate with other optical properties and are less likely to suffer from the uncertainties caused by anisotropic stellar orbits.

     Our goal in this paper is to make general inferences about the presence of dark matter halos and their mean systematic variation with optical luminosity. Toward that end, we use a set of physically and observationally well-motivated mass models and scaling relations in order to reproduce the T-σ relation. Although we address the scatter in the T-σ relation, a detailed examination of individual galaxies is beyond the scope of this paper and is deferred to future investigation.

     Our results can be considered in three stages. First, we demonstrate how the observed range of βspec necessarily requires the existence of dark halos in elliptical galaxies. Second, we derive conservative lower limits on the dark matter scale length and dark-to-luminous mass fraction, and discuss how the dark matter content must scale with optical luminosity so as to reproduce the observed T-σ relation. Finally, we discuss possible implications of our results for cosmology and galaxy formation by embedding our models within particular scenarios for large-scale structure formation.

§3. MODELING

     As stated above, we use the observable βspec, defined in equation (1), to constrain the dark matter distribution in elliptical galaxies. This parameter is the ratio of suitable averages (to be defined below) of the square of the stellar velocity dispersion and the gas temperature. Therefore, we must derive the expected distributions of stellar velocity dispersions σ and gas temperatures T, given specified total gravitational mass distributions.

§3.1. The Stellar Component

     Recently published Hubble Space Telescope (HST) observations of the centers of elliptical galaxies show that their surface brightness profiles can generally be characterized by the function



where R is the projected radius and rbr is a break radius such that I∝R$\mathstrut{^{-c}}$ for R&Lt;r$\mathstrut{_{{\rm br}}}$ and I∝R$\mathstrut{^{-b}}$ for R&Gt;r$\mathstrut{_{{\rm br}}}$, while a characterizes the sharpness of the break (Faber et al. 1997). In order to accurately calculate the central stellar velocity dispersion, this observed departure from distributions with "analytic cores" must be accounted for. In the Appendix we demonstrate that, for a=2 and c<1, equation (2) provides a good approximation to the projection of a space density profile ∝r-d (c<d<c+1) for r&Lt;r$\mathstrut{_{{\rm br}}}$ and ∝r-b-1 for r&Gt;r$\mathstrut{_{{\rm br}}}$. To derive the luminosity density distribution l(r), we numerically deproject the surface brightness profile



where x≡r/r$\mathstrut{_{{\rm br}}}$ and the change of variables z≡r$\mathstrut{_{{\rm br}}}$/R has been made.

     The DW sample is magnitude-limited, and we are primarily interested in the most luminous ellipticals, since they have the most accurate X-ray temperatures (see § 3.3). We adopt the typical values for M$\mathstrut{_{V}}$<-22 galaxies (adopting H$\mathstrut{_{0}}$=80 km s-1 Mpc-1) of a=2, b=1.44, and c=1/10 (Faber et al. 1997). For the nine galaxies in common with DW (excluding the highly flattened NGC 4697), the mean surface brightness parameters and variances from Faber et al. (1997) are as follows: a=1.88±0.51, b=1.38±0.16, and c=0.096±0.076. The resulting stellar density profile diverges slightly faster than r-1 for r&Lt;r$\mathstrut{_{{\rm br}}}$. The asymptotic surface brightness slope b is flatter than in conventional surface brightness parameterizations, and the integrated luminosity L(r) does not converge. Therefore, we truncate the optically luminous distribution at radius rmax, determined so that



consequently r$\mathstrut{_{{\rm max}}}$=200r$\mathstrut{_{{\rm br}}}$.

     As an alternative stellar model, we consider the density profile from Hernquist (1990),



where L is the total luminosity and r$\mathstrut{_{{\rm Hern}}}$=0.45r$\mathstrut{_{e}}$ (as adopted here) provides the best match to a de Vaucouleurs (deV) model with effective radius re (Hernquist 1990). An advantage of this model is that the density and mass profiles are analytic. We shall subsequently refer to the stellar distribution of equation (3) as the HST model, and that of equation (5) as the Hernquist model. A comparison of the luminosity density (in units of the average density, 3L/4πr$\mathstrut{^{3}_{{\rm max}}}$) as a function of normalized radius (r/rmax) in these models with the corresponding deV model is shown in Figure 2, where r$\mathstrut{_{{\rm max}}}$=200r$\mathstrut{_{{\rm br}}}$=40r$\mathstrut{_{{\rm Hern}}}$/3=6r$\mathstrut{_{e}}$. All three models produce similar slopes at very small r; however, deV and Hernquist models fail to reproduce the flattening at &ap;rbr observed by HST. On the other hand, while &ap;80% of the total luminosity is contained within &ap;0.7r$\mathstrut{_{{\rm max}}}$&ap;4r$\mathstrut{_{e}}$ in all three models, the HST model concentrates the entire remaining &ap;20% of the mass between &ap;4re and 6re as opposed to spreading it out from &ap;4re to ∞. In order to accurately characterize the observed central stellar properties as required by the definition of βspec without introducing additional parameters (i.e., an additional break radius and a distinct asymptotic form for the density profile), such a redistribution of the stellar light is necessary. If the light profile does indeed steepen outside of the HST field of view, then the only effect on our results for the HST model is that, by overestimating the stellar mass at large radii, we would underestimate the dark halo mass required to reproduce any observed value of βspec.

Fig. 2

     The stellar velocity dispersion is derived by solving the Jeans equation in the form



where ρ* is the stellar density distribution obtained from multiplying the luminosity density (see eqs. [3] and [5]) by a constant stellar mass-to-light ratio, M(<r) is the total mass within radius r (the dark matter component of which is described in the next section), A is the velocity dispersion anisotropy parameter (A≡1-σ$\mathstrut{^{2}_{t}}$/σ$\mathstrut{^{2}_{r}}$), and σr (σt) is the radial (tangential) component of the velocity dispersion. We make the reasonable assumption that the velocity dispersion becomes isotropic at small r but radial at large r (Ciotti & Pelligrini 1992) and adopt the functional form



where the transitional scale radius s is a free parameter of our models.

§3.2. The Dark Matter Component

     The dark matter mass distribution is assumed to follow the universal density profile (Navarro, Frenk, & White 1997),



that, when integrated, yields



     It is not clear how appropriate equations (8) and (9) are for elliptical galaxy dark halos. On one hand, there is evidence for cores in spiral galaxy dark matter distributions (e.g., Persic et al. 1996); on the other hand, dissipational collapse of the baryonic component in ellipticals can increase the dark matter concentration. Moreover, models with cores cannot generally be consistently coupled with stellar mass distributions as cuspy as those often observed in elliptical galaxies (Ciotti & Pelligrini 1992). We have verified the existence of self-consistent two-component mass models with dark matter following equation (8) and Hernquist stellar models (eq. [7]), in the parameter range of interest, according to the criterion of Ciotti & Pelligrini (1992).

     From the virial theorem it follows that the integrated mass can be robustly estimated from the mean temperature. Constraints on the dark matter concentration can also be placed from consideration of the T-σ relation; however, any inferences about dark matter at small or large radii are contingent on the corresponding assumed asymptotic dark matter density slopes. Our choice of equation (8) for the dark matter distribution does allow us to connect our results with predictions of dark halo structure from numerical structure formation simulations. However, the true shape of the dark matter spatial distribution can only be constrained from temperature and stellar velocity dispersion profiles of sufficient extent and resolution.

§3.3. The Hot Gas Component

     The hot, X-ray–emitting gas is assumed to be in hydrostatic equilibrium so that



The mass in hot gas is not significant compared to the stellar mass, so its contribution to the gravitational potential is neglected. Since we are interested in the average properties of the DW sample, and because spatial analysis is not available for most of the sample, we assume that ρ$\mathstrut{^{2}_{{\rm gas}}}$∝ρ$\mathstrut{_{*}}$; this has been found to be a good approximation for X-ray bright galaxies and is expected if heating is balanced by cooling locally (Loewenstein & Mathews 1987). Both assumptions, hydrostatic equilibrium and ρ$\mathstrut{^{2}_{{\rm gas}}}$∝ρ$\mathstrut{_{*}}$, are more likely to break down for galaxies with low X-ray to optical luminosity ratios, where a partial or transient outflow might obtain; however, the T-σ relation for these relatively X-ray–faint systems does not differ, within the statistics of the sample, from that of the sample as a whole.

§3.4. Scaling Relations and Model Solutions

     A significant unifying simplification for the HST stellar models follows from the observations that the ratios of core mass to total stellar mass [Mcore/M*(rmax), where Mcore is Lcore (eq. [4]) multiplied by the stellar mass-to-light ratio], and break radius rbr to re (the "true" re as observed, not the half-light radius in our models) are approximately constant: M$\mathstrut{_{{\rm core}}}$&ap;0.012M$\mathstrut{_{*}}$(r$\mathstrut{_{{\rm max}}}$) and r$\mathstrut{_{{\rm br}}}$&ap;r$\mathstrut{_{e}}$/33 (Faber et al. 1997). Our adoption of these scaling relations, along with the empirical relation M$\mathstrut{_{*}}$/L$\mathstrut{_{V}}$∝L$\mathstrut{^{1{/}4}_{V}}$, ensures that our model galaxies are consistent with the fundamental plane (FP) as long as the core is baryon dominated, since it follows that



where σc is the central velocity dispersion. 2 In order to provide us with a one-parameter family of models (the quality and quantity of the X-ray data is insufficient to attempt to derive a T-σ-re or equivalent FP-like relation), we take a Faber-Jackson cut through the FP such that L$\mathstrut{_{V}}$∝σ$\mathstrut{^{4}}$→r$\mathstrut{_{e}}$∝L$\mathstrut{^{3{/}4}_{V}}$, normalized so that r$\mathstrut{_{e}}$=6.3 kpc (and σ=250 km s-1) at M$\mathstrut{_{V}}$=-22 for H$\mathstrut{_{0}}$=80 km s-1 Mpc-1. The root mean square deviation of galaxies in the DW sample from this projection of the FP is only 34 km s-1 (13.5% of the mean). The Hernquist models (eq. [5]) likewise adhere to the FP, since we universally adopt r$\mathstrut{_{{\rm Hern}}}$=0.45r$\mathstrut{_{e}}$.

     Equations (6) and (10) are solved in dimensionless form. There are two fundamental (luminosity dependent) dimensional quantities that we denote as r* (chosen to be rbr for the HST model or rHern for the Hernquist model) and M0 (Mcore for the HST model or total stellar mass M* for the Hernquist model). Thus, adopting the above scaling relations, the length-, mass-, density-, velocity-, and temperature-scaling relations are as follows:















and



Here h80 is the Hubble constant in units of 80 km s-1 Mpc-1, λ≡L$\mathstrut{_{V}}$/L$\mathstrut{_{0}}$ with L$\mathstrut{_{0}}$=5.2×10$\mathstrut{^{10}}$ h$\mathstrut{^{-2}_{80}}$ L$\mathstrut{_{V_{{\odot}}}}$, and μ* is defined such that the stellar mass-to-light ratio M$\mathstrut{_{*}}$/L$\mathstrut{_{V}}$=10μ$\mathstrut{_{*}}$ M&odot;/L$\mathstrut{_{V_{{\odot}}}}$ for L=L$\mathstrut{_{0}}$ and h$\mathstrut{_{80}}$=1. L0h$\mathstrut{^{2}_{80}}$ corresponds to M$\mathstrut{_{V}}$=-22, i.e., L$\mathstrut{_{0}}$&ap;3L$\mathstrut{_{*}}$.

     The dark matter mass distribution is scaled accordingly, i.e.,



where α is the ratio of dark to stellar mass at the HST stellar model cutoff radius, i.e.,



and δ≡r$\mathstrut{_{{\rm dm}}}$/r$\mathstrut{_{*}}$ is the dimensionless dark matter scale length. In the above, x≡r/r$\mathstrut{_{*}}$, x$\mathstrut{_{{\rm max}}}$≡r$\mathstrut{_{{\rm max}}}$/r$\mathstrut{_{*}}$, and the function f is defined in equation (9).

     To determine the average values of T and σ, the hydrostatic and Jeans equations are integrated inward from a zero-pressure boundary at r=∞ (see Loewenstein 1994). Although the stellar mass has a cutoff (rmax) for the HST model, the gas and dark matter halos are likely to extend beyond this and are not truncated in the models. This results in a pressure contribution to the gas virial temperature within rmax that leads to lower values of βspec for a given α and is therefore conservative in the sense of minimizing the required amount of dark matter.

     The average quantities that constitute the observable βspec are calculated as follows: we define 〈σ〉 as the average projected stellar velocity dispersion within a circular aperture of size rbr and 〈T〉 as the emission-weighted projected gas temperature within rmax (the same aperture over which the temperature is measured in DW, corresponding to the radius enclosing &ap;90% of the optical light). The value of βspec is robust to the precise choice of these radii since r$\mathstrut{_{{\rm max}}}$&Gt;r$\mathstrut{_{e}}$&Gt;r$\mathstrut{_{{\rm br}}}$. The aperture-averaged, projected velocity dispersion for all central radii more than 0.1rbr (>0.3rbr) is within 10% of its value at rbr for the HST (Hernquist) model, assuming isotropic orbits and that the central potential is dominated by the stellar component. By averaging over constant multiples of the fundamental stellar scale length (r*), we assure that our computed values of βspec depend only on the relative (to stellar) dimensionless dark matter and velocity dispersion anisotropy parameters. These parameters are the following: the ratio of dark-to-luminous matter scale length δ≡r$\mathstrut{_{{\rm dm}}}$/r$\mathstrut{_{*}}$, the mass ratio Mdm/M* within rmax α, and the dimensionless anisotropy scale length s/r*. The only additional model parameters are the stellar distribution slopes. For a given triplet [s/r*, $\mathstrut{\left(M_{{\rm dm}}{/}M_{*}\right)}$$\mathstrut{_{r_{{\rm max}}}}$, rdm/r*], a general solution for the dimensionless gas temperature and stellar velocity dispersion radial distributions follows that can be scaled to physical units for any h80 and LV using equations (12)–(16); μ* is determined by normalizing to &angl0;σ&angr0;=250 km s-1 for M$\mathstrut{_{V}}$=-22 and h$\mathstrut{_{80}}$=1; note that μ$\mathstrut{_{*}}$∝(σ$\mathstrut{_{0}}$/&angl0;σ&angr0;)$\mathstrut{^{2}}$.

FOOTNOTES

     2 We note at this point that we do not distinguish between dark (luminous) and nonbaryonic (baryonic) matter. Elliptical galaxies may very well have an additional baryonic dark component in the form of stellar remnants that may account for the variation with luminosity of the central mass-to-light ratio (Zepf & Silk 1996). Any such contribution is subsumed into our definition of the stellar (luminous) component through M*/LV.

§4. RESULTS

     In this section we consider the dependences of βspec on s/r*, rdm/r*, and $\mathstrut{\left(M_{{\rm dm}}{/}M_{*}\right)}$$\mathstrut{_{r_{{\rm max}}}}$ for both HST and Hernquist models. While the latter provide a more accurate global representation of elliptical galaxy stellar distributions, the former are more accurate around and inside the break radius rbr where 〈σ〉 is calculated. For models that are dark matter dominated beyond several re, the redistribution of the stellar mass at large radii in the HST model is irrelevant.

§4.1. Variation with Anisotropy Scale and the Necessity for Dark Matter

     In the absence of dark matter, the only free parameter in our models is the anisotropy length scale s; recall that this is defined such that orbits become isotropic for r&Lt;s and radial for r&Gt;s. Therefore, if stars dominate the gravitational potential within rmax and if the shape of the velocity ellipsoid does not vary systematically with LV, then βspec should be independent of LV and 〈σ〉 (i.e., &angl0;T&angr0;∝&angl0;σ&angr0;$\mathstrut{^{2}}$), contrary to the observed T-σ relation (DW). Moreover, as we now show, the typical observed value of β$\mathstrut{_{{\rm spec}}}$&ap;0.5 cannot be reproduced in the absence of dark matter.

     The solid and dotted curves in Figure 3 show the variation of βspec with the dimensionless anisotropy scale length s/rbr for, respectively, the HST and Hernquist stellar models without dark matter. Evidently, βspec falls with increasing s, but is essentially constant for s>10–20rbr (&ap;re/3–2re/3), since 〈σ〉 is an average over a much smaller radius. The asymptotic values of βspec are 1.2 and 0.75 for the HST and Hernquist stellar models, respectively. The lower value for the Hernquist model is the result of a temperature maximum at several rbr that substantially contributes to the emission-averaged temperature because of the high stellar (and, by assumption, gas) density there. As dark matter is introduced, the temperature profile becomes more nearly isothermal, and differences in βspec between HST and Hernquist models lessen (see next paragraph, and Fig. 3). Varying the slope parameters of the HST models (eq. [2]) does not result in any values of βspec less than 0.75.

Fig. 3

     The values of βspec derived without dark matter lie well above what is observed in a typical elliptical galaxy (in fact, no galaxy in the DW sample has a best-fit value of βspec greater than 1; see Fig. 1). One of our major conclusions is already apparent. While a typical bright elliptical galaxy has β$\mathstrut{_{{\rm spec}}}$&ap;0.5, even under the most extreme assumptions, βspec is never less than &ap;0.75 without the presence of an extended dark halo. While 10 of the 30 galaxies in the DW sample have upper limits to βspec greater than 0.75, only 3 have upper limits greater than 1.2 (see Fig. 1). Of the 16 galaxies where βspec is determined to better than 20%, only one has an upper limit greater than 0.75.

     The dash-dotted and dashed curves show the corresponding models containing dark matter halos with dark matter scale length rdm set equal to re (i.e., r$\mathstrut{_{{\rm dm}}}$=33r$\mathstrut{_{{\rm br}}}$=2.2r$\mathstrut{_{{\rm Hern}}}$) and with $\mathstrut{\left(M_{{\rm dm}}=3M_{*}\right)}$$\mathstrut{_{r_{{\rm max}}}}$ chosen so that β$\mathstrut{_{{\rm spec}}}$&ap;0.5 for s≥r$\mathstrut{_{e}}$. The nonluminous fraction of the total mass within re is &ap;0.6 and &ap;0.5 for the HST and Hernquist models, respectively.

     From here on we assume isotropic stellar orbits. As shown in Figure 3, for the practical purposes of calculating βspec, this is equivalent to assuming that the orbits are mostly isotropic inside ∼re/2. Since βspec increases with decreasing s/rbr, more dark matter is required to obtain a given value of βspec if this assumption is discarded; once again we adopt the most conservative assumption as regards the required amount of dark matter.

§4.2. Variation with Dark Matter Scale

     Figure 4a shows βspec as a function of the ratio of dark matter scale length to stellar break radius rdm/rbr for the HST (solid curve) and Hernquist (dotted curve) models with $\mathstrut{\left(M_{{\rm dm}}=3M_{*}\right)}$$\mathstrut{_{r_{{\rm max}}}}$. (For the Hernquist model, the scaling relation r$\mathstrut{_{{\rm br}}}$=r$\mathstrut{_{e}}$/33=r$\mathstrut{_{{\rm Hern}}}$/15 has been used.) As rdm decreases below ∼10rbr, dark matter plays an increasingly important role in determining 〈σ〉, so βspec rises. For fixed $\mathstrut{\left(M_{{\rm dm}}{/}M_{*}\right)}$$\mathstrut{_{r_{{\rm max}}}}$, βspec is nearly independent of rdm/rbr provided that r$\mathstrut{_{{\rm dm}}}$>r$\mathstrut{_{e}}$. It is clear that measurements of βspec can only place lower limits on rdm.

Fig. 4

     There is a fair amount of freedom in how the dark matter can be distributed and still produce β$\mathstrut{_{{\rm spec}}}$&ap;0.5. This is illustrated in Figure 4b, which displays the variation in baryon fraction at re (solid curve for the HST stellar model, dotted curve for the Hernquist model) and at rbr (dashed curve for the HST stellar model, dot-dashed curve for the Hernquist model). The observable βspec changes very slowly with rdm/rbr for r$\mathstrut{_{{\rm dm}}}$/r$\mathstrut{_{{\rm br}}}$>20, while the baryon fraction inside rbr varies from &ap;0.73 and &ap;0.66 for the HST and Hernquist stellar models, respectively, to &ap;0.98 for both models as rdm becomes very large. Meanwhile, the baryon fraction inside r$\mathstrut{_{e}}$=33r$\mathstrut{_{{\rm br}}}$ increases from &ap;0.31 (&ap;0.43) if r$\mathstrut{_{{\rm dm}}}$/r$\mathstrut{_{{\rm br}}}$=20, to &ap;0.75 (&ap;0.84) for the HST (Hernquist) model as rdm becomes large compared to the stellar cutoff radius rmax.

§4.3. Variations with Dark-to-Luminous Mass Fraction

     Figure 5 shows βspec as a function of $\mathstrut{\left(M_{{\rm dm}}{/}M_{*}\right)}$$\mathstrut{_{r_{{\rm max}}}}$ for HST and Hernquist stellar models with rdm set equal to 0.5, 1.0, and 2.0re. As discussed above, β$\mathstrut{_{{\rm spec}}}$→1.2 (0.75) as M$\mathstrut{_{{\rm dm}}}$→0 for the HST (Hernquist) model. The curves pass through β$\mathstrut{_{{\rm spec}}}$&ap;0.5 at $\mathstrut{\left(M_{{\rm dm}}{\approx}3M_{*}\right)}$$\mathstrut{_{r_{{\rm max}}}}$ if r$\mathstrut{_{{\rm dm}}}$=r$\mathstrut{_{e}}$ or 2re, while [M$\mathstrut{_{{\rm dm}}}$&ap;4(6)M$\mathstrut{_{*}}$]$\mathstrut{_{r_{{\rm max}}}}$ is required if r$\mathstrut{_{{\rm dm}}}$=r$\mathstrut{_{e}}$/2 for HST (Hernquist) models. As $\mathstrut{\left(M_{{\rm dm}}{/}M_{*}\right)}$$\mathstrut{_{r_{{\rm max}}}}$ becomes sufficiently large for fixed rdm, dark matter comes to dominate the mass even at small radii (i.e., within rbr where 〈σ〉 is calculated) and βspec becomes independent of α. That is, in order to maintain consistency with the FP and the Faber-Jackson relation, the physical mass must remain constant and the increase in dark-to-luminous mass ratio α is compensated for by decreasing the stellar mass-to-light ratio (i.e., μ* in eqs. [13]–[16]) accordingly.

Fig. 5

§4.4. Limits on Dark Matter Parameters

     We have shown (§ 4.1) that the lower limits on βspec for the HST and Hernquist stellar models without dark matter are 0.75 and 1.2, respectively. To explain observed values of βspec that are less than this requires the addition of dark matter halos with sufficient mass to raise the value of 〈T〉, but with a large enough scale length to leave 〈σ〉 (which is averaged over a much smaller scale) relatively unaltered. By calculating solutions to equations (6) and (10) for the entire [$\mathstrut{\left(M_{{\rm dm}}{/}M_{*}\right)}$$\mathstrut{_{r_{{\rm max}}}}$, rdm/r*] parameter space, we have derived limits on the amount and scale length of any dark halo able to reproduce a given observed value of βspec.

     As discussed in § 4.3 and shown in Figure 5, for a fixed value of rdm/rbr there is a minimum value of βspec that is obtained when α becomes sufficiently large that both 〈σ〉 and 〈T〉 are determined by the nonluminous mass component. Since this minimum βspec increases as rdm/rbr decreases, there is a minimum value of rdm/rbr required to explain any observed βspec. Figure 6a shows these minimum values of rdm/rbr for the HST (solid curve) and Hernquist (dotted curve) models. The minimum value of rdm for a L$\mathstrut{_{V}}$&ap;L$\mathstrut{_{0}}$&ap;3L$\mathstrut{_{*}}$ galaxy with β$\mathstrut{_{{\rm spec}}}$&ap;0.5 is &ap;2 h$\mathstrut{^{-1}_{80}}$ kpc (&ap;re/3); the dimensional value scales as L$\mathstrut{^{3{/}4}_{V}}$. These extreme models are dark matter dominated, even inside the break radius rbr, and have very low stellar mass-to-light ratios. Requiring models to have M$\mathstrut{_{*}}$/L$\mathstrut{_{V}}$≥5 M&odot;/L$\mathstrut{_{V_{{\odot}}}}$ for L$\mathstrut{_{V}}$=L$\mathstrut{_{0}}$ and h$\mathstrut{_{80}}$=1 (or, equivalently, that &gsim;60% of the mass within rbr be stellar) roughly doubles the minimum values of rdm/rbr for 0.4<β$\mathstrut{_{{\rm spec}}}$<0.6. This reasonable requirement then implies that r$\mathstrut{_{{\rm dm}}}$>4(L$\mathstrut{_{V}}$/3L$\mathstrut{_{*}}$)$\mathstrut{^{3{/}4}}$ h$\mathstrut{^{-1}_{80}}$ kpc.

Fig. 6

     The maximum baryon fractions within rbr, re, and rmax for any given βspec are shown in Figure 6b. The lower limits on the dark matter fractions in models required to explain β$\mathstrut{_{{\rm spec}}}$>0.4 are less than 5% within rbr and less than 20% within re; nonluminous halos could be quite inconspicuous unless data well beyond re are considered. The maximum baryon fractions at rmax for the HST (Hernquist) model are 0.43 (0.15), 0.53 (0.27), and 0.61 (0.45) to produce β$\mathstrut{_{{\rm spec}}}$=0.4, 0.5, and 0.6, respectively. These models have very large dark matter scale lengths (r$\mathstrut{_{{\rm dm}}}$&Gt;r$\mathstrut{_{{\rm max}}}$) and are analogous to "maximum disk" models of spiral galaxies in that they maximize the contribution of stars to the mass-to-light ratio.

§4.5. Scaling of Dark Matter Content with Optical Luminosity

     In order to investigate the scaling characteristics of dark halos with optical luminosity, we consider the equivalent of the T-σ relation in the βspec-LV plane. Correlations of βspec and log βspec with log LV are derived using the DW data set, blue magnitudes and B-V colors from the RC3 catalog (de Vaucouleurs et al. 1991), and velocity distances from Faber et al. (1989). (The βspec–log LV data and best-fit correlation are replotted in Fig. 7b.) Galaxies in the centers of cluster cooling flows and those dominated by nuclear point-source emission are excluded. The log βspec–log LV correlation, equivalent to β$\mathstrut{_{{\rm spec}}}$&ap;0.58λ$\mathstrut{^{0.13}}$ where λ is the visual luminosity in units of L$\mathstrut{_{0}}$=5.2×10$\mathstrut{^{10}}$ h$\mathstrut{^{-2}_{80}}$ L$\mathstrut{_{V_{{\odot}}}}$, is essentially identical to the convolution of the T-σ correlation and Faber-Jackson relation (see § 3.4). Nevertheless, requiring that models conform to the observations in this form makes use of the tilt of the FP for scaling of the stellar component, thus circumventing concerns about the thickness of the projection of the FP onto the LV-σ plane. Note that the galaxies considered span 0.15<λ<3, although most of the sample lies within the range 0.5<λ<2.

Fig. 7

     Given our homologous elliptical galaxy models, which are based on the scaling properties of the FP, the observed variation in βspec implies that there is a systematic variation of relative dark matter properties that breaks the self-similarity for galaxies of different luminosity. The sense of the observed T-σ relation could also be explained by a fine-tuned systematic variation of the velocity dispersion anisotropy (having more radial orbits for more luminous galaxies), or if more luminous galaxies have cuspier stellar distributions (larger values of c in eq. [2]). However, the latter is contrary to observations, and as we have shown, the observed values of βspec are too low to be explained without the introduction of dark matter. It is possible to explain the βspec-LV trend, assuming the dark-to-luminous mass ratio within rmax is constant, if the ratio of dark matter to stellar break (or, equivalently, effective) radius rdm/rbr decreases with increasing galaxy luminosity LV. However, since βspec is flat for r$\mathstrut{_{{\rm dm}}}$/r$\mathstrut{_{{\rm br}}}$>10 and steeply rises for r$\mathstrut{_{{\rm dm}}}$/r$\mathstrut{_{{\rm br}}}$<10 (Fig. 4a) this requires fine tuning. Moreover, as we discuss shortly, r$\mathstrut{_{{\rm dm}}}$/r$\mathstrut{_{{\rm br}}}$<10 is not expected.

     More naturally, the increase in βspec with LV could be explained if $\mathstrut{\left(M_{{\rm dm}}{/}M_{*}\right)}$$\mathstrut{_{r_{{\rm max}}}}$ decreases with increasing LV (that is, if more luminous galaxies are less dark matter dominated) and rdm/rbr is relatively large (see Fig. 5). We now discuss the plausibility of such a correlation within the context of the hierarchical clustering formalism of structure formation, simulations of which produced the dark matter distribution expressed by equations (8) and (9). This formalism provides specific predictions for rdm as a function of virial mass (defined here as the mass within a radius where the mean dark matter density is 200 times the average density in the universe) for a given cosmology and power spectrum of initial fluctuations.

     In order to extend our models to the scale of the virial radius and mass, we transform the dark matter parameters rdm/r* and $\mathstrut{\left(M_{{\rm dm}}{/}M_{*}\right)}$$\mathstrut{_{r_{{\rm max}}}}$ that are directly constrained by observations into the following pair of global parameters: the dark matter concentration parameter



and the global dark-to-luminous mass ratio



where r200 refers to the radius within which the overdensity of the dark matter (only) is 200. [Mdm(r200) and r200 refer to the original mass of the dark halo, but this may have been subsequently reduced in rich environments through tidal stripping.] Note that for C=10 and α$\mathstrut{_{{\rm virial}}}$=10, r$\mathstrut{_{200}}$=320 h$\mathstrut{^{-1}_{80}}$ kpc, r$\mathstrut{_{{\rm dm}}}$/r$\mathstrut{_{{\rm br}}}$=170, and $\mathstrut{\left(M_{{\rm dm}}{/}M_{*}\right)}$$\mathstrut{_{r_{{\rm max}}}}$=1.6 for λ=1 (L&ap;3L$\mathstrut{_{*}}$).

     In bottom-up hierarchical clustering theories of structure formation, lower mass halos are more concentrated—a reflection of the higher density in the universe at the earlier epoch of their formation (Navarro et al. 1997). We parameterize this dependence with the function



for the concentration, where χ≡M$\mathstrut{_{{\rm dm}}}$(r$\mathstrut{_{200}}$)/M$\mathstrut{_{0}}$ and M$\mathstrut{_{0}}$≡10$\mathstrut{^{13}}$ h$\mathstrut{^{-1}_{80}}$ M&odot; is on the order of the present-day characteristic mass. The second term in the denominator of equation (21) is introduced to prevent the concentration from exceeding a maximum value Cmax, which we set equal to 100, and driving the stellar mass-to-light ratio to very small values. Note that B=[C$\mathstrut{_{{\rm max}}}$(1+γ)/C$\mathstrut{_{0}}$]$\mathstrut{^{-{(}1+{\gamma}{)}{/}{\gamma}}}$ becomes insignificant for γ&Lt;1 (e.g., in standard cold dark matter models).

     Using the "characteristic density" program kindly provided by J. Navarro, a standard cold dark matter (CDM) fluctuation spectrum results in C$\mathstrut{_{0}}$&ap;10 and γ&ap;0.1. For these parameters, r$\mathstrut{_{{\rm dm}}}$/r$\mathstrut{_{{\rm br}}}$&ap;150(α$\mathstrut{_{{\rm virial}}}$/10)$\mathstrut{^{13{/}30}}$λ$\mathstrut{^{-5{/}24}}$ (r$\mathstrut{_{{\rm dm}}}$/r$\mathstrut{_{e}}$&ap;4.5 for α$\mathstrut{_{{\rm virial}}}$&ap;10 and λ&ap;1) and rdm is indeed much greater than 10rbr in the luminosity range of interest. The dot-dashed curve in Figure 7a shows $\mathstrut{\left(M_{{\rm dm}}{/}M_{*}\right)}$$\mathstrut{_{r_{{\rm max}}}}$ as a function of λ for α$\mathstrut{_{{\rm virial}}}$=16 (i.e., baryon fraction 0.06) CDM models (and for two alternative models to be discussed in the next section). Clearly $\mathstrut{\left(M_{{\rm dm}}{/}M_{*}\right)}$$\mathstrut{_{r_{{\rm max}}}}$ increases with optical luminosity, implying a decrease of βspec with increasing LV as shown by the dot-dashed curve in Figure 7b. A comparison with the observed βspec-λ correlation (Fig. 7b, solid curve), demonstrates that a constant baryon fraction CDM model for elliptical galaxy dark matter halos clearly fails to match the observed correlation, even over the limited range of λ where the correlation is well established. A successful alternative model must produce dark halos where $\mathstrut{\left(M_{{\rm dm}}{/}M_{*}\right)}$$\mathstrut{_{r_{{\rm max}}}}$ decreases with increasing optical luminosity in this range. We will discuss the form such possible alternatives must take in the next section.

§5. DISCUSSION

     A natural explanation for why βspec, the ratio of the central stellar velocity dispersion to the globally averaged gas temperature, increases with 〈σ〉 and therefore LV is that the dark-to-luminous mass ratio within the luminous parts of elliptical galaxies (r<r$\mathstrut{_{{\rm max}}}$=6r$\mathstrut{_{e}}$) decreases with LV (see Figs. 5, 7a, and 7b). In fact, the observed trend requires that the total mass-to-light ratio within rmax be nearly constant. This can be understood from a simple argument based on the virial theorem and FP, as follows. The virial temperature within rmax is T$\mathstrut{_{{\rm virial}}}$∝(M$\mathstrut{_{{\rm dm}}}$+M$\mathstrut{_{*}}$)/r$\mathstrut{_{{\rm max}}}$∝(1+M$\mathstrut{_{{\rm dm}}}$/M$\mathstrut{_{*}}$)(M$\mathstrut{_{*}}$/r$\mathstrut{_{e}}$) since r$\mathstrut{_{{\rm max}}}$∝r$\mathstrut{_{e}}$ (all masses are evaluated at rmax). Since from the FP &angl0;σ&angr0;$\mathstrut{^{2}}$∝M$\mathstrut{_{*}}$/r$\mathstrut{_{e}}$ it follows that &angl0;σ&angr0;$\mathstrut{^{2}}$/T$\mathstrut{_{{\rm virial}}}$∝(1+M$\mathstrut{_{{\rm dm}}}$/M$\mathstrut{_{*}}$)$\mathstrut{^{-1}}$ or β$\mathstrut{_{{\rm spec}}}$∝(1+M$\mathstrut{_{{\rm dm}}}$/M$\mathstrut{_{*}}$)$\mathstrut{^{-1}}$ if we associate Tvirial with the integrated X-ray temperature, which should be a fair approximation.

     The implication from the observed βspec-LV trend that the total mass-to-light ratio within rmax be nearly constant (see below) is contrary to expectations from CDM models of large-scale structure if all elliptical galaxies have the same global baryon fraction. The dark-to-luminous mass ratio is predicted to increase with LV, in such models, since the ratio of the dark matter scale length rdm to the optical half-light radius is large and weakly varying, thus producing a predicted correlation between βspec and LV in the opposite sense to what is observed (Figs. 7a and 7b). We consider two simple variations on the above model to try and recover the observed βspec-LV trend. First we assume that the dark matter concentration depends only weakly on halo mass, as CDM predicts, but relax the constant baryon fraction assumption. Then we assume that the concentration varies much more steeply with virial mass than the CDM prediction.

§5.1. Alternatives to Constant Baryon Fraction CDM

     If the weak dependence of concentration on halo mass predicted in CDM simulations (eq. [21] with C$\mathstrut{_{0}}$=10 and γ=0.1) is maintained, then the assumption of a constant baryon fraction within r200 must be relaxed: the required increase in dark matter fraction within rmax for less luminous galaxies must be a reflection of a larger total (i.e., within r200) dark-to-luminous mass ratio. If all galaxies start off with identical baryon fractions, this then implies that smaller galaxies lose an increasingly large fraction of their original baryonic mass. (Similar conclusions for spiral galaxies have been reached by Persic et al. 1996.) We parameterize this variation in the dark/stellar mass ratio as



where, as previously defined, α$\mathstrut{_{{\rm virial}}}$≡$\mathstrut{\left(M_{{\rm dm}}{/}M_{*}\right)}$$\mathstrut{_{r_{200}}}$, λ is the visual luminosity in units of L$\mathstrut{_{0}}$=5.2×10$\mathstrut{^{10}}$ h$\mathstrut{^{-2}_{80}}$ L$\mathstrut{_{V_{{\odot}}}}$, and λ0 is a fitting parameter. Equation (22) has the properties α$\mathstrut{_{{\rm virial}}}$→∞ as λ→0 (complete baryonic mass loss), and α$\mathstrut{_{{\rm virial}}}$→α$\mathstrut{_{{\rm min}}}$ as λ→∞ [presumably, (1+α$\mathstrut{_{{\rm min}}}$)$\mathstrut{^{-1}}$ is the average baryon fraction in the universe]. The observed βspec-LV correlation is satisfactorily reproduced in models with ν=5/4, λ$\mathstrut{_{0}}$=3, and α$\mathstrut{_{{\rm min}}}$=0.5–2.5. The dotted curves in Figures 7a and 7b represent HST stellar models with α$\mathstrut{_{{\rm min}}}$=1.5; the observed trend is very well matched (Fig. 7b). (Hernquist stellar models require even larger values of ν; however, this is an artifact of the underestimate of βspec for low Mdm; see § 4.1.)

     Alternatively, if the dark matter halo concentration increases much more steeply with virial mass than predicted for a CDM fluctuation spectrum, the observed βspec-LV correlation can be reproduced assuming a constant total baryon fraction (i.e., αvirial). In this case, the necessary decrease in $\mathstrut{\left(M_{{\rm dm}}{/}M_{*}\right)}$$\mathstrut{_{r_{{\rm max}}}}$ for more luminous galaxies results not from a smaller relative global amount of dark matter, but from a larger fraction of the dark matter lying outside of the luminous portion of the galaxy (i.e., rmax). We have found that equation (21) with γ=1, C$\mathstrut{_{0}}$=4, and α$\mathstrut{_{{\rm virial}}}$>5 provides models that adequately reproduce the data (smaller values of γ require larger values of αvirial). $\mathstrut{\left(M_{{\rm dm}}{/}M_{*}\right)}$$\mathstrut{_{r_{{\rm max}}}}$ and βspec are shown as functions of LV for α$\mathstrut{_{{\rm virial}}}$=16 by the dashed curves in Figures 7a and 7b. Such a steep dependence of the concentration on total mass would seem to indicate a flatter-than-CDM primordial fluctuation spectrum (Navarro et al. 1997).

     These alternative scenarios were designed to reproduce the βspec-LV correlation without consideration of the scatter or that of galaxies about the mean FP. We address these concerns by computing βspec for individual pairs (LV, σ) corresponding to each of the 25 non–cluster-cooling flow, non–point-source dominated galaxies in the DW sample, for the three scenarios of Figures 7a and 7b. A comparison with the observed values of βspec, as shown in Figures 8a–8c, confirms that the constant baryon fraction CDM scenario provides the poorest match to the observations, on average, and further illustrates the magnitude of variations about the mean correlation. Further investigation into these variations will be pursued in a future work that examines the individual galaxies in detail.

Fig. 8

     Both of these alternative scenarios have drawbacks. Mass loss induced by supernovae-driven galactic winds during the star-forming epoch of elliptical galaxies has been inferred from both the color-magnitude diagram and the enrichment of the intracluster medium (e.g., Loewenstein & Mushotzky 1996; Gibson 1997). However the magnitude of this effect must be extreme (i.e., ν in eq. [22] on the order of 1 or greater) if it is to explain the increasing dominance of dark matter in less luminous galaxies required to explain the observed T-σ relation. The mass loss implied by equation (22) with the required set of parameters, when integrated over the elliptical galaxy luminosity function (Marzke, Huchra, & Geller 1994; Lin et al. 1996), corresponds to ∼80% of the original baryonic mass of L>L$\mathstrut{_{*}}$ galaxies. Moreover, in such a scenario, all elliptical galaxies with L$\mathstrut{_{V}}$<3×10$\mathstrut{^{11}}$ h80 L$\mathstrut{_{V_{{\odot}}}}$ would span a range in total mass of less than a factor of 3; the large range in mass loss could not, therefore, be due to less luminous galaxies lying within shallower potential wells. Instead, one must presume that variations in the relative timing of star formation and the merging of pregalactic fragments is responsible for the large range in mass loss, with galaxies with smaller present-day luminosities having lost the largest fraction of their original baryonic mass. Light would be a very poor tracer of mass for elliptical galaxies, and the luminosity function would be a reflection of the rarity of elliptical galaxies sufficiently relaxed at the star formation epoch to prevent copious mass loss from galactic winds.

     It might therefore seem more appealing to resort to a concentration-mass relationship for dark halos (eq. [21]) that is steeper than predicted in CDM simulations in such a way that optically less luminous galaxies are more dark matter dominated within rmax, despite having the same total (within r200) baryon fraction. The basic requirement is that the dark matter scale length rdm increase from ∼re at the low-luminosity end of the observed range, to r$\mathstrut{_{{\rm dm}}}$∼r$\mathstrut{_{{\rm max}}}$ at L$\mathstrut{_{V}}$∼L$\mathstrut{_{0}}$=5.2×10$\mathstrut{^{10}}$ h$\mathstrut{^{-2}_{80}}$ L$\mathstrut{_{V_{{\odot}}}}$, to r$\mathstrut{_{{\rm dm}}}$∼r$\mathstrut{_{200}}$ (C∼1) at the high luminosity range. These criteria are met for only a rather narrow volume in (αvirial, C0, γ) parameter space, and thus this scenario seems to suffer from a fine-tuning problem; nonetheless, the range that successfully reproduces the observed T-σ relation is not out of line for cosmogenic models with flat primordial fluctuation spectra (Navarro et al. 1997).

§5.2. Mass-to-Light Ratios, Baryon Fractions, and Velocity Dispersions

     Figures 9a, 9b, and 9c show the luminosity dependences of mass, mass-to-light ratio, and baryon fraction at various radii for the two models discussed above that reproduce the observed T-σ (βspec-LV) relation: dotted lines show the model with CDM-predicted halo concentration as a function of mass and increasing mass loss with decreasing luminosity (C$\mathstrut{_{0}}$=10 and γ=0.1 in eq. [21]; α$\mathstrut{_{{\rm min}}}$=1.5, ν=5/4, and λ$\mathstrut{_{0}}$=3 in eq. [22]); dashed curves show the model with steeper-than-CDM concentration as a function of mass and constant baryon fraction (C$\mathstrut{_{0}}$=4 and γ=1 in eq. [21]; α$\mathstrut{_{{\rm min}}}$=16 and λ$\mathstrut{_{0}}$=0 in eq. [22]). We refer to these as "mass-loss" and "non-CDM" scenarios, respectively. The stellar model is assumed to follow equation (3), and the luminosity range in the plots roughly corresponds to that of the DW sample (&ap;0.6–2L$\mathstrut{_{*}}$). The radii of interest are r$\mathstrut{_{e}}$=6.3λ$\mathstrut{^{3{/}4}}$ h$\mathstrut{^{-1}_{80}}$ kpc, r$\mathstrut{_{{\rm br}}}$=0.03r$\mathstrut{_{e}}$, and r$\mathstrut{_{{\rm max}}}$=6r$\mathstrut{_{e}}$; r200h80 increases from 270 to 400 kpc for the mass-loss scenario and r$\mathstrut{_{200}}$&ap;370λ$\mathstrut{^{5{/}12}}$ h$\mathstrut{^{-1}_{80}}$ kpc for the α$\mathstrut{_{{\rm virial}}}$=16, non-CDM scenario, where λ≡L$\mathstrut{_{V}}$/L$\mathstrut{_{0}}$ with L$\mathstrut{_{0}}$=5.2×10$\mathstrut{^{10}}$ h$\mathstrut{^{-2}_{80}}$ L$\mathstrut{_{V_{{\odot}}}}$&ap;3L$\mathstrut{_{*}}$.

Fig. 9

     There are a number of robust properties that hold for any model (including Hernquist models) to successfully reproduce the T-σ relation. Since the models are constrained by the observed properties of the stars and by the average temperature of hot gas in hydrostatic equilibrium within r$\mathstrut{_{{\rm max}}}$=6r$\mathstrut{_{e}}$, the inferred quantities at rmax are nearly model-independent: M(r$\mathstrut{_{{\rm max}}}$)/L$\mathstrut{_{V}}$&ap;25 h80 M&odot;/L$\mathstrut{_{V_{{\odot}}}}$, f$\mathstrut{_{{\rm baryon}}}$(r$\mathstrut{_{{\rm max}}}$)&ap;0.35λ$\mathstrut{^{1{/}4}}$. Note also that M$\mathstrut{_{*}}$&ap;4.8×10$\mathstrut{^{11}}$λ$\mathstrut{^{5{/}4}}$ h$\mathstrut{^{-1}_{80}}$ M&odot; for any model where the central potential is dominated by the stellar component.

     These values agree strikingly well with the constraints from statistical studies of gravitational lensing (Kochanek 1995; Brainerd et al. 1996; Griffiths et al. 1996). The solid lines in Figure 9a shows the mass as a function of luminosity, evaluated at re and rmax for the softened isothermal sphere (SIS) model that provides a best-fit to the observed weak gravitational shear from Griffiths et al. (1996). It is remarkable that the mass-to-light scaling derived from the shear that is dominated by ∼L* galaxies at intermediate redshifts is in such excellent agreement with our results using a completely independent method on local galaxies primarily with L>L$\mathstrut{_{*}}$. For the constant baryon fraction CDM model, the total mass within rmax is a significantly steeper function of luminosity than is demonstrated by the curves in Figure 9a.

     On scales both larger and smaller than rmax, the two scenarios depicted in Figures 9a–9c diverge. In the mass-loss scenario the baryon fraction within re is generally greater than 0.7 and M(r$\mathstrut{_{e}}$)/L$\mathstrut{_{V}}$(r$\mathstrut{_{e}}$)&ap;12λ$\mathstrut{^{1{/}4}}$ h$\mathstrut{_{80}}$ M&odot;/L$\mathstrut{_{V_{{\odot}}}}$, while in the non-CDM scenario the mass-to-light ratio is nearly constant, M(r$\mathstrut{_{e}}$)/L$\mathstrut{_{V}}$(r$\mathstrut{_{e}}$)&ap;11 h80 M&odot;/L$\mathstrut{_{V_{{\odot}}}}$, so that the baryon fraction within re becomes less than 0.5 below L&ap;0.6L$\mathstrut{_{*}}$.

     Extrapolated out to the dark matter virial radius r200, the total mass scalings differ (by assumption). For the mass-loss scenario



where acceptable models have A&ap;1.5–5 and B&ap;4–8 (A=2.5, B&ap;6 for the model shown by dotted curves in Figures 7a, 7b, and 9a–9c). For the non-CDM (or any other constant baryon fraction) scenario,



the (constant) total baryon fraction f$\mathstrut{_{{\rm baryon}}}$&ap;0.06 for the dashed curves shown in Figures 7a and 7b and 9a–9c. We found that f$\mathstrut{_{{\rm baryon}}}$<0.15 in non-CDM models is required to reproduce the observed βspec-LV correlation. Non-CDM models with relatively small global fbaryon have larger baryon fractions within re for λ<1 (because of larger dark matter scale lengths) and agree somewhat better with the lensing constraints at this radius.

     Integrating over the early-type galaxy luminosity function (Marzke et al. 1994; Lin et al. 1996) yields an average total (i.e., within the virial radius) mass-to-light ratio more than 65 h80 M&odot;/L$\mathstrut{_{V_{{\odot}}}}$ for the non-CDM (constant baryon fraction) scenario and Ω$\mathstrut{_{{\rm ellipticals}}}$>0.02, which can be compared to an average total mass-to-light ratio of &ap;150 h80 M&odot;/L$\mathstrut{_{V_{{\odot}}}}$ for spirals (Persic et al. 1996), and Ω$\mathstrut{_{{\rm spirals}}}$=0.04–0.06. In the mass-loss scenario, the average total mass-to-light ratio is sensitive to the lower luminosity cutoff Lmin for the luminosity function and can be as high as ∼1000 h80 M&odot;/L$\mathstrut{_{V_{{\odot}}}}$ (Ω$\mathstrut{_{{\rm ellipticals}}}$∼0.4) for L$\mathstrut{_{{\rm min}}}$=10$\mathstrut{^{8}}$ h$\mathstrut{^{-2}_{80}}$ L$\mathstrut{_{V_{{\odot}}}}$ since even the lowest luminosity elliptical galaxy has a total mass of ∼3×10$\mathstrut{^{12}}$ h$\mathstrut{^{-1}_{80}}$ M&odot;. The average density in the universe of virialized mass on galactic scales is more than 0.07 of the critical density.

     We have calculated the velocity dispersion distributions for both the dark matter and stellar density distributions, assuming isotropic orbits. These are compared in Figure 10 for an L$\mathstrut{_{V}}$=L$\mathstrut{_{0}}$ galaxy for both the mass-loss and non-CDM scenarios. Both velocity dispersion profiles have maxima since the total gravitational potential is not isothermal. The ratio (dark matter to stars) of the squares of these maxima is greater than 1.4 over the luminosity range in Figures 9a–9c for the two models under consideration and is &ap;2 over the range L$\mathstrut{_{*}}$<L$\mathstrut{_{V}}$<5L$\mathstrut{_{*}}$. In fact, the minimum value of this ratio for any HST or Hernquist model that produces β$\mathstrut{_{{\rm spec}}}$<0.7 is greater than one. In this sense, the dark matter is hotter than the stars, as predicted by Davis & White (1996).

Fig. 10

§6. SUMMARY AND CONCLUSIONS

     In this work, we have addressed two essential features of the X-ray temperatures derived by Davis & White (1996) for an optically complete sample of elliptical galaxies: (1) the X-ray–emitting gas is always hotter than the stars and, typically, twice as hot (β$\mathstrut{_{{\rm spec}}}$≡μm$\mathstrut{_{p}}$&angl0;σ&angr0;$\mathstrut{^{2}}$/k&angl0;T&angr0;&ap;0.5); (2) the gas/stellar temperature ratio tends to be higher for galaxies with lower velocity dispersions (the T-σ relation). We have constructed physically plausible models of the mass distribution in bright elliptical galaxies in an effort to constrain their average dark matter properties by matching these observations. The stellar models (described in § 3) are fully consistent with the fundamental plane scaling relations and are designed to either conform to the latest published HST results on the structure of the centers of elliptical galaxies or to follow the Hernquist (1990) approximation to a de Vaucouleurs profile (see Fig. 2). We have made other plausible, but conservative, assumptions, such as (1) maximizing the nongravitational (i.e., pressure) contribution to the gas temperature by allowing the gas and dark matter distributions to extend to infinity and (2) assuming that the stellar orbits are isotropic for r&Lt;r$\mathstrut{_{e}}$.

     A basic and general result of our calculations is that, in the absence of dark matter, β$\mathstrut{_{{\rm spec}}}$≥0.75 (see Fig. 3). Since β$\mathstrut{_{{\rm spec}}}$&ap;0.5 is observed, a convincing case is made that dark matter is an extremely common, if not ubiquitous, constituent of elliptical galaxies. Although X-ray (and other) observations have been used to infer the presence of dark matter in individual cases, we have shown that dark halos are generic to luminous, nearby elliptical galaxies. Furthermore, the dark matter/stellar temperature ratios (derived at the peak of their velocity dispersion distributions, assuming isotropic orbits) are greater than one for models with β$\mathstrut{_{{\rm spec}}}$<0.7. Thus, the observation that the temperature of the extended hot gas exceeds the central stellar temperature is a reflection of the fact that the dark matter is dynamically "hotter" than the stars, as suggested by Davis & White (1996).

     In § 4 we described how βspec varies as functions of the two relative parameters of the universal dark matter distribution (eq. [8])—the ratio of dark matter to stellar scale lengths, δ≡r$\mathstrut{_{{\rm dm}}}$/r$\mathstrut{_{*}}$, and the ratio of dark-to-luminous matter within r$\mathstrut{_{{\rm max}}}$=6r$\mathstrut{_{e}}$, α≡$\mathstrut{\left(M_{{\rm dm}}{/}M_{*}\right)}$$\mathstrut{_{r_{{\rm max}}}}$ (see Figs. 4a, 4b, and 5). Because we attempt to match only the single global observable βspec, there is an allowed range in the details of the dark matter spatial distribution; however, we have derived absolute limits on the dark matter parameters required to obtain any particular value of βspec (see Figs. 6a and 6b).

     The observations do not require that dark matter dominate the inner luminous regions of elliptical galaxies: more than half of the mass within re is baryonic for models with β$\mathstrut{_{{\rm spec}}}$=0.5 if r$\mathstrut{_{{\rm dm}}}$>r$\mathstrut{_{e}}$.

     The most natural explanation of the tendency for galaxies with lower stellar temperatures to have larger gas-to-stellar temperature ratios is that $\mathstrut{\left(M_{{\rm dm}}{/}M_{*}\right)}$$\mathstrut{_{r_{{\rm max}}}}$ decreases with LV in such a way that, on average, the total mass-to-light ratio inside rmax is nearly independent of optical luminosity. This ratio, &ap;25 h80 M&odot;/L$\mathstrut{_{V_{{\odot}}}}$, is exactly what is predicted for mass models of elliptical galaxies designed to explain the gravitational shear of background field galaxies.

     If one specifies a scaling relation for the dark halo concentration, one can extend the dark matter distribution out to the virial radius and calculate the total baryon fraction and mass-to-light ratio. When we attempt to embed our models within the CDM theory of hierarchical halo formation, the implied dark matter scaling badly fails to reproduce the observed T-σ relation unless smaller galaxies lose an increasingly larger fraction of their initial baryonic content (see Fig. 7b), such that the average L>L$\mathstrut{_{*}}$ galaxy has lost most of its initial baryonic mass. Alternatively, the global dark-to-luminous mass ratio could be constant if the dark halo concentration declines much more steeply with virial mass than CDM models predict, so that the decrease in $\mathstrut{\left(M_{{\rm dm}}{/}M_{*}\right)}$$\mathstrut{_{r_{{\rm max}}}}$ for larger systems is a result of a more diffuse dark matter halo rather than a less massive (relative to the stars) halo. In this latter scenario, dark matter may become increasingly important inside re as LV decreases, becoming dominant for L&Lt;L$\mathstrut{_{*}}$. This deviation from CDM predictions of dark halo scaling could conceivably be due to a relatively flat primordial fluctuation spectrum on mass scales less than 1014 M&odot; or to the effects on the dark matter density profile of the evolution of the baryonic component; to date, large-scale structure numerical simulations that can resolve halos on galactic scales and include a dissipational component have not been attempted.

     In this paper we have shown that the observed relationship between optical velocity dispersions and X-ray temperatures in giant elliptical galaxies implies that they have dark matter halos with M/L$\mathstrut{_{V}}$&ap;25 h80 M&odot;/L$\mathstrut{_{V_{{\odot}}}}$ within 6re. In the future, we plan to fully use the available X-ray and optical imaging and spectroscopic data for our sample on a case-by-case basis in order derive the mass distributions in individual galaxies in the highest possible detail and to investigate the scatter in the T-σ relationship. For galaxies with X-ray temperature profiles, we will be able to constrain the detailed form of the dark matter distribution, as well as its integrated properties.

ACKNOWLEDGMENTS

     We thank Lars Hernquist for reminding us of the consistency considerations of Ciotti & Pelligrini (1992) and Richard Mushotzky for feedback on the original manuscript. Comments from an anonymous referee led to significant improvements in the quality of this paper. R. E. W. acknowledges partial support from NASA grants NAG 5-1718 and NAG 5-1973.

APPENDIX

PROJECTION OF A BROKEN POWER LAW

     Suppose the stellar density profile is given by a broken power law of the form



The projected surface mass,



can be recast, using the change of variable y=(r$\mathstrut{^{2}}$-R$\mathstrut{^{2}}$)/R$\mathstrut{^{2}}$, into the form



where



b≡R$\mathstrut{^{2}}$/r$\mathstrut{^{2}_{c}}$, and c≡b(1+b)$\mathstrut{^{-1}}$. Note that the bracketed term in equation (A3) has the asymptotic slopes at small and large R that one would expect from dimensional analysis (i.e., Σ$\mathstrut{_{*}}$∝ρ$\mathstrut{_{*}}$r). The definite integral of equation (A4) can be evaluated as follows (Gradshteyn & Ryzhik 1980):



where γ=σ+0.5κ is one-half the asymptotic density slope as r→∞, B is the beta function, and 2F1 is the hypergeometric function (Gradshteyn & Ryzhik 1980). We have empirically found that we can introduce a parameter τ, recast equation (A3) as



where



and determine a value of τ (for any κ≤1) that results in a very nearly constant J&arcmin; as a function of R (with no significant structure near R=r$\mathstrut{_{c}}$). Equation (A6) has the expected slope as R→∞, and we find τ→0 as κ→0 (the "β model") and τ→0.5 as κ→1. We conclude that the projection of a density profile of the form expressed by equation (A1) is well described by equation (2) (at least for a=2), thus perhaps lending some physical justification to its use.

REFERENCES

FIGURES


Full image (60kb) | Discussion in text
     FIG. 1.—βspec vs. log LV for the DW sample (points, with vertical lines representing 90% confidence limits), and best-fit βspec–log LV (dashed curve) and log βspec–log LV (dot-dashed curve) linear regressions. See § 4.5 for details.

Full image (59kb) | Discussion in text
     FIG. 2.—Comparison of luminosity density profiles l(r) for deV (solid curve), HST (dotted curve), and Hernquist (dashed curve) stellar models (see text). The radii are normalized to r$\mathstrut{_{{\rm max}}}$=6r$\mathstrut{_{e}}$, l(r) to the average luminosity density within rmax, l$\mathstrut{_{{\rm avg}}}$=3L/4πr$\mathstrut{^{3}_{{\rm max}}}$, where L is the total stellar luminosity.

Full image (72kb) | Discussion in text
     FIG. 3.—βspec vs. velocity dispersion anisotropy scale length s in units of the break radius rbr for HST (solid curve, α=0; dashed curve, α=3) and Hernquist (dotted curve, α=0; dot-dashed curve, α=3) stellar models. [α≡$\mathstrut{\left(M_{{\rm dm}}{/}M_{*}\right)}$$\mathstrut{_{r_{{\rm max}}}}$].

Full image (55kb) FIG. 4a Full image (86kb) FIG. 4b | Discussion in text
     FIG. 4.—(a) βspec vs. rdm/rbr for $\mathstrut{\left(M_{{\rm dm}}=3M_{*}\right)}$$\mathstrut{_{r_{{\rm max}}}}$ (solid curve, HST stellar model; dotted curve, Hernquist model). (b) Baryon fractions at rbr (dashed curve, HST; dot-dashed curve, Hernquist model) and re (solid curve, HST; dotted curve, Hernquist model) vs. rdm/rbr for $\mathstrut{\left(M_{{\rm dm}}=3M_{*}\right)}$$\mathstrut{_{r_{{\rm max}}}}$.

Full image (72kb) | Discussion in text
     FIG. 5.—βspec vs. $\mathstrut{\left(M_{{\rm dm}}{/}M_{*}\right)}$$\mathstrut{_{r_{{\rm max}}}}$ for HST (solid curves) and Hernquist (dotted curves) stellar models and r$\mathstrut{_{{\rm dm}}}$=0.5, 1.0, and 2.0re (upper, middle, and lower curves, respectively).

Full image (61kb) FIG. 6a Full image (69kb) FIG. 6b | Discussion in text
     FIG. 6.—(a) Minimum values of rdm/rbr for HST and Hernquist models. (b) Maximum values of baryon fractions within (from top to bottom) rbr, re, and rmax for HST and Hernquist models. The HST (Hernquist) model is represented by solid (dotted) curves.

Full image (71kb) FIG. 7a Full image (70kb) FIG. 7b | Discussion in text
     FIG. 7.—(a) Ratio of dark-to-luminous matter within rmax vs. dimensionless luminosity, λ≡L$\mathstrut{_{V}}$/L$\mathstrut{_{0}}$ (L$\mathstrut{_{0}}$=5.2×10$\mathstrut{^{10}}$ h$\mathstrut{^{-2}_{80}}$ L$\mathstrut{_{V_{{\odot}}}}$), for HST stellar models. The dot-dashed curve denotes constant (within the virial radius) baryon fraction (f$\mathstrut{_{{\rm baryon}}}$=0.06) model and CDM scaling of dark matter concentration; the dotted curve has fbaryon increasing with optical luminosity; the dashed curve has a steeper-than-CDM scaling of concentration with dark halo mass. (b) Predicted variation of βspec with dimensionless luminosity. Line types are as in (a). Data points and best-fit correlation (solid curve) from Fig. 1 are replotted. (For clarity, three galaxies where βspec is uncertain to more than 50% are omitted from this and the following figure.)

Full image (66kb) | Discussion in text
     FIG. 8.—(a) βspec calculated using pairs (LV, σ) as measured for the DW sample and CDM scaling of dark matter concentration with constant fbaryon vs. observed βspec. (b) Same as (a) for model where fbaryon increases with optical luminosity. (c) Same as (a) for model with steeper-than-CDM scaling of concentration with dark halo mass. The solid line represents β$\mathstrut{_{{\rm spec}}}$=β$\mathstrut{_{{\rm obs}}}$.

Full image (89kb) FIG. 9a Full image (72kb) FIG. 9b Full image (67kb) FIG. 9c | Discussion in text
     FIG. 9.—(a) Mass vs. LV (in solar units) within, from bottom to top, re, r$\mathstrut{_{{\rm max}}}$=6r$\mathstrut{_{e}}$ (stars only), rmax (total), and r200. The solid curves show the mass at re and rmax from weak gravitational lensing studies. (b) Same as (a) for mass-to-light ratio (in solar units) at (from lowest to uppermost curves): r=0, re, rmax, and r200. (c) Same as (a) for baryon fraction vs. LV at (from uppermost to lowest curves) r=r$\mathstrut{_{{\rm br}}}$, re, rmax, and r200. Dotted and dashed line types have the same connotations as in Fig. 7.

Full image (82kb) | Discussion in text
     FIG. 10.—One-dimensional velocity dispersion distributions, assuming isotropic orbits, for an L$\mathstrut{_{V}}$=L$\mathstrut{_{0}}$=5.2×10$\mathstrut{^{10}}$ h$\mathstrut{^{-2}_{80}}$ L$\mathstrut{_{V_{{\odot}}}}$ galaxy. Mass-loss and non-CDM models are denoted by dotted and solid curves, respectively, for the stellar profiles and dot-dashed and dashed curves, respectively, for the dark matter profiles. $\mathstrut{\left(M_{{\rm dm}}{/}M_{*}\right)}$$\mathstrut{_{r_{{\rm max}}}}$=1.5(1.3) and r$\mathstrut{_{{\rm dm}}}$/r$\mathstrut{_{e}}$=4.1(12) for the mass-loss (non-CDM) models at this luminosity, and the (observational and model) value of βspec is 0.62.