A UNIVERSAL MASS PROFILE FOR DWARF SPHEROIDAL GALAXIES?^*

For the Jeans analysis in Section 3 we use empirical, projected velocity dispersion profiles for eight bright dSphs with abundant kinematic data. Walker et al. (2007b) present velocity dispersion profiles for seven of these objects, but these profiles can now be updated—and we can now provide a profile for Ursa Minor—using the data we have obtained during the past two years. For each galaxy, we compute the velocity dispersion profile after (1) discarding all stars for which the probability of dSph membership, according to the algorithm described by Walker et al. (2009c), is less than 0.95, (2) subtracting the mild velocity gradients that likely reflect the bulk transverse motion of the dSph (Walker et al. 2008), and (3) binning the data using circular annuli containing equal numbers of member stars. We estimate the velocity dispersion in each bin using the maximum-likelihood technique described by Walker et al. (2006).

Figure 1 displays the velocity dispersion profiles we measure for Carina, Draco, Fornax, Leo I, Leo II, Sculptor, Sextans, and Ursa Minor. While the profiles remain generally flat (see Figure 2 of Walker et al. 2007b), we detect a gentle decline in velocity dispersion at large projected radii (R ≳ 1 kpc) in Fornax. This behavior becomes apparent only after addition of the most recent MMFS data (Walker et al. 2009b), which contribute most of the data points at large radius. We also find hints of gently declining profiles in Sculptor and Sextans, although these data are noisier and the apparent declines hinge on the outermost one or two data points.

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Using the same data, Łokas et al. (2008) and Łokas (2009) have recently measured profiles for Carina, Fornax, Leo I, Sculptor, and Sextans that all decline more significantly at large radii than the profiles we present in Figure 1. The difference arises because Łokas et al. (2008) and Łokas (2009) adopt a rejection algorithm that attempts to remove stars that may have been tidally stripped and thus do not provide reliable tracers of the gravitational potential. The algorithm, described in detail by Klimentowski et al. (2007, see also den Hartog & Katgert 1996), iteratively estimates the mass profile from the velocities and positions of accepted members and then rejects stars with velocities that differ from the mean by more than $\sqrt{2GM(r)/r}$. The validity of this method clearly depends on the accuracy of the mass profile estimate, and this quality is difficult to assess. Łokas et al. (2008) and Łokas (2009) use a mass estimator derived from the virial theorem (Heisler et al. 1985), which holds when the tracer population is self-gravitating or, if there is a dark matter component, when the mass-to-light ratio is constant with radius ("mass follows light"). Then it is not surprising that after imposing this rejection algorithm Łokas et al. (2008) and Łokas (2009) obtain falling velocity dispersion profiles that are consistent with mass-follows-light models. While Klimentowski et al. (2007) successfully test their algorithm and mass estimator using an N-body simulation of a tidally "stirred" (Mayer et al. 2001a, 2001b) satellite in which mass actually does follow light (after 99% of the initial dark mass is stripped), they do not demonstrate that their method preserves flat velocity dispersion profiles where they may genuinely exist—e.g., in equilibrium systems with extended and intact dark matter halos.⁴

Pending evidence that the tidally stirred N-body model of Klimentowski et al. (2007) describes the evolution of the real dSphs in our sample, we opt not to apply the rejection method used by Łokas et al. (2008) and Łokas (2009). Thus, we avoid imposing any bias toward mass-follows-light models that may result from use of the virial theorem mass estimator, but at the cost of leaving our samples vulnerable to contamination from tidally unbound stars. Given the difficulty of identifying such stars, we choose to err on the side of including them in our analysis. For at least some systems—e.g., Fornax, which has orbital pericenter near its present distance of ∼140 kpc (Piatek et al. 2007; Walker et al. 2008)—we expect a negligible contribution from unbound tidal debris. Among the remaining dSphs in our sample, there exists kinematic evidence of tidal streaming motion in the outer parts of Carina (Muñoz et al. 2006) and Leo I (Sohn et al. 2007; Mateo et al. 2008). In fitting models to the velocity dispersion profiles of these two galaxies, as well as Draco, which shows a rising profile in its outer parts, we experimented with discarding the outermost one to two data points in each system. Since the results were indistinguishable from fits that included the outermost points, we decided to include the outer data points in our final analysis.

Stellar surface densities of dSphs are typically fit by Plummer (1911), King (1962) and/or Sersic (1968), profiles (e.g., Irwin & Hatzidimitriou 1995; McConnachie & Irwin 2006; Belokurov et al. 2007). The Plummer profile, I(R) = L(πr²_half)⁻¹[1 + R²/r²_half]⁻² where L is the total luminosity, is the simplest as it has only a single shape parameter, the projected half-light radius.⁵ It is also the only profile with published parameters for all dSphs, since the concentration parameters of King and Sersic profiles are not well constrained by the sparse data available for the faintest dSphs. Therefore, in what follows we adopt the Plummer profile to characterize dSph stellar densities.

Given a model I(R) for the projected stellar density, one recovers the three-dimensional density from (BT08)

$$\begin{equation} \nu (r)=-\frac{1}{\pi }\displaystyle \int _r^{\infty }\frac{dI}{dR}\frac{dR}{\sqrt{R^2-r^2}}. \end{equation} \tag{ 4 }$$

Thus, for the Plummer profile we have ν(r) = 3L(4πr³_half)⁻¹[1 + r²/r²_half]^−5/2.

We note that even though dSph surface brightness data can be fit adequately by a variety of density profiles, the choice of profile is not trivial. Evans et al. (2009) demonstrate that, even when the gravitational potential is dominated by dark matter, the adopted shape of the stellar density profile can profoundly affect the inferred shape of M(r) at small radii. In what follows, while for simplicity we present only the results obtained using the Plummer profile, we explicitly identify any results that are strongly sensitive to this choice (Section 3.5.3).

For the dark matter halo, we follow S08 (also Koch et al. 2007a, 2007b) in adopting a generalized Hernquist profile given by Hernquist (1990) and Zhao (1996)

$$\begin{equation} \rho (r)=\rho _0\biggl (\frac{r}{r_0}\biggr)^{-\gamma }\biggl [1+\biggl (\frac{r}{r_0}\biggr)^{\alpha }\biggr ]^{\frac{\gamma -3}{\alpha }}, \end{equation} \tag{ 5 }$$

where the parameter α controls the sharpness of the transition from inner slope lim_r→0dln(ρ)/dln(r) ∝ −γ to outer slope lim_r→∞dln(ρ)/dln(r) ∝ −3. In terms of these parameters, the mass profile is

$$\begin{eqnarray} &&M(r)=4\pi \displaystyle \int _{0}^{r}s^2\rho (s)ds=\frac{4\pi \rho _0r_0^3}{3-\gamma }\biggl (\frac{r}{r_0}\biggr)^{3-\gamma }\\ &&_2F_1\biggl [\frac{3-\gamma }{\alpha },\frac{3-\gamma }{\alpha };\frac{3-\gamma +\alpha }{\alpha };-\biggl (\frac{r}{r_0}\biggr)^\alpha \biggr ],\nonumber \end{eqnarray} \tag{ 6 }$$

where ₂F₁(a, b; c; z) is Gauss' hypergeometric function.

The profiles admitted by Equation (5) include the range of plausible halo shapes relevant to the ongoing controversy regarding "cores" versus "cusps" in individual dark matter halos. Profiles with γ>0 are centrally cusped, while those with γ = 0 have constant-density cores. For α = γ = 1, one recovers the cuspy Navarro–Frenk–White (hereafter "NFW"; Navarro et al. 1996, 1997) profile motivated by cosmological N-body simulations.

It is convenient to normalize the mass profile by M(r₀), the enclosed mass at the scale radius of the dark matter halo. This quantity relates to the maximum circular velocity, V_max, according to

$$\begin{equation} M(r_0)=\frac{\eta r_0V_{\rm max}^2}{G}\frac{_2F_1\bigl [\frac{3-\gamma }{\alpha },\frac{3-\gamma }{\alpha };\frac{3-\gamma +\alpha }{\alpha };-1\bigr ]}{\eta ^{3-\gamma }_2F_1\bigl [\frac{3-\gamma }{\alpha },\frac{3-\gamma }{\alpha };\frac{3-\gamma +\alpha }{\alpha };-\eta ^\alpha \bigr ]}, \end{equation} \tag{ 7 }$$

where η ≡ r_max/r₀ identifies the radius corresponding to the maximum circular velocity and is specified uniquely by α and γ. Thus, the parameter V_max sets the normalization of the mass profile.

The normalization can equivalently be set by specifying, rather than V_max, the enclosed mass at some particular radius. For radius x, the enclosed mass M(x) specifies M(r₀) according to

$$\begin{equation} M(r_0)=M(x)\frac{_2F_1\bigl [\frac{3-\gamma }{\alpha },\frac{3-\gamma }{\alpha };\frac{3-\gamma +\alpha }{\alpha };-1\bigr ]}{\bigl (\frac{x}{r_0}\bigr)^{3-\gamma }_2F_1\bigl [\frac{3-\gamma }{\alpha },\frac{3-\gamma }{\alpha };\frac{3-\gamma +\alpha }{\alpha };-\bigl (\frac{x}{r_0}\bigr)^{\alpha }\bigr ]}. \end{equation} \tag{ 8 }$$

S08 demonstrate that for most dSphs the Jeans analysis can tightly constrain M₃₀₀. Here, in addition to M₃₀₀, we shall consider the masses within two alternative radii as free parameters with which to normalize the mass profile. Specifically, we consider the mass within the half-light radius, M(r_half), and the mass within the outermost data point of the empirical velocity dispersion profile, M(r_last).

In order to evaluate a given halo model, we compare the projected velocity dispersion profile, σ_p(R), from Equation (3) to the empirical profile, $\sigma _{V_0}(R)$, displayed in Figure 1. For a given parameter set S ≡ {−log(1 − β), log M_X, log r₀, α, γ}, where M_X is one of {V_max, M(r_half), M₃₀₀ or M(r_last)}, we adopt uniform priors and consider the likelihood

$$\begin{equation} \zeta =\displaystyle \prod _{i=1}^N \frac{1}{\sqrt{2\pi (\mathrm{Var}[\sigma _{V_0}(R_i)])}}\exp \biggl [-\frac{1}{2}\frac{(\sigma _{V_0}(R_i)-\sigma _p(R_i))^2}{\mathrm{Var}[\sigma _{V_0}(R_i)]}\biggr ], \end{equation} \tag{ 9 }$$

where $\mathrm{Var}[\sigma _{V_0}(R_i)]$ is the square of the error associated with the empirical dispersion.

Our mass models have five free parameters (four halo parameters plus one anisotropy parameter). In order to explore the large parameter space efficiently, we employ Markov-chain Monte Carlo (MCMC) techniques. That is, we use the standard Metropolis–Hastings algorithm (Metropolis et al. 1953; Hastings 1970) to generate posterior distributions according to the following prescription: (1) from the current location in parameter space, S_n, draw a prospective new location, S', from a Gaussian probability density centered on S_n; (2) evaluate the ratio of likelihood at S_n and S'; and (3) if ζ(S')/ζ(S_n) ⩾ 1, accept such that S_n+1 = S', else accept with probability ζ(S')/ζ(S_n), such that S_n+1 = S' with probability ζ(S')/ζ(S_n) and S_n+1 = S_n with probability 1 − ζ(S')/ζ(S_n).

For this procedure, we use the adaptive MCMC engine CosmoMC⁶ (Lewis & Bridle 2002). Although it was developed specifically for analysis of cosmic microwave background data, CosmoMC provides a generic sampler that continually updates the probability density according to the parameter covariances in order to optimize the acceptance rate. For each galaxy and parameterization, we run four chains simultaneously, allowing each to proceed until the variances of parameter values across the four chains become less than 1% of the mean of the variances. Satisfaction of this convergence criterion typically requires ∼10⁴ steps for our chains. We then identify the posterior distribution in parameter space with the last 70% of all accepted points (we discard the first 30% of points, which correspond to the "burn-in" period.)

Figure 2 indicates posterior distributions of model parameters that we obtain from our MCMC analysis of Fornax, for each choice of parameter used to specify the normalization of the mass profile. We ran CosmoMC independently for each normalization parameter, so while the distributions for other parameters should be similar for a given galaxy, they need not be identical. Contours in Figure 2 identify regions of 68% and 95% confidence.

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In general, there exist degeneracies among the five parameters such that most cannot be constrained uniquely. However, we find that most allowed mass profiles, regardless of their shapes, intersect near the half-light radius. Then, as Figure 2 demonstrates, when we normalize the mass profile by either M(r_half) or M₃₀₀, the degeneracies are such that the allowed region in parameter space subtends a relatively small mass range. Therefore, while our analysis cannot place meaningful constraints on most parameters, it does constrain M(r_half) and M₃₀₀. At much smaller and much larger radii (e.g., r_last), the allowed mass profiles can diverge and the resulting constraints are weaker.

In the interest of brevity, we do not include the equivalent of Figure 2 for each of the other galaxies in our sample. Instead, for each of the eight dSphs with velocity dispersion profiles in Figure 1, Figure 3 indicates the posterior distributions of M(r_half) and anisotropy. Table 2 then lists the constraints on M(r_half), M₃₀₀, M(r_last) and V_max.

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3.5.1. Mass

3.5.2. V_max for Fornax

3.5.3. No Constraint on Cores/Cusps

We conclude that the available data tightly constrain dSph masses within r ∼ r_half. Furthermore, these constraints hold over a wide range of possible halo shapes and stellar velocity anisotropies. S08 reach the same conclusion regarding the robustness of M₃₀₀, while Peñarrubia et al. (2008a) show that M(r_half) is well constrained for an NFW halo regardless of the particular values of V_max and r₀. Our analysis thus confirms S08's result and generalizes the conclusion of Peñarrubia et al. (2008a) to include non-NFW halos. In short: we know M(r_half) for the dwarf spheroidals.

Because the MCMC method requires significant computational effort, it is reasonable to wonder whether the measurement of such a bulk quantity as M(r_half) might adequately be made using more conventional techniques. Therefore, let us define a simple analytic model that relates M(r_half) to observed quantities. Specifically, suppose that the stellar component is distributed as a Plummer sphere with a velocity distribution that is isotropic (β = 0) and has constant dispersion $\sigma _{V_0}^2(R)=\bar{v_r^2}=\sigma ^2$. Since these conditions are broadly consistent with all available dSph data,⁹ this exercise amounts to consideration of a subset of the models not ruled out by the MCMC analysis discussed above. Then the Jeans equation gives

$$\begin{equation} M(r)=-\frac{r^2\bar{v_{r}^2}}{G\nu }\frac{d\nu }{dr}=\frac{5r_{\rm half}\sigma ^2\bigl (\frac{r}{r_{\rm half}}\bigr)^3}{G\bigl [1+r^2/r_{\rm half}^2\bigr ]}, \end{equation} \tag{ 10 }$$

where the last expression is specific to the assumption of a Plummer profile for the stellar density. Equation (10) immediately provides the convenient estimate

$$\begin{equation} M(r_{\rm half})=\mu r_{\rm half}\sigma ^2, \end{equation} \tag{ 11 }$$

where μ ≡ 580 M_☉ pc⁻¹ km⁻² s², which differs only by a factor of 1.5 from the core-fitting formula commonly used to estimate dSph dynamical masses (e.g., Illingworth 1976; Richstone & Tremaine 1986; Mateo 1998).

Table 2 lists estimates of M(r_half) and M₃₀₀ obtained from Equations (10) and (11) for the eight bright dSphs with velocity dispersion profiles, and Figure 3 indicates the estimates obtained from Equation (11) for easy comparison to estimates obtained from the full Jeans/MCMC analysis. We find that the simple estimates from Equation (11) generally stand in excellent agreement with constraints we obtain from the full Jeans/MCMC analysis. The only (marginal) exception is Draco, for which Equation (11) overestimates the MCMC mass by a factor of ∼3; however, even in that case the estimates (barely) overlap within the 1σ errors (Figure 3 and Table 2). Furthermore, as shown in Figure 4, we reproduce the flat M₃₀₀–luminosity relation of S08 (compare their Figure 1) accurately merely by applying Equation (10) to the data listed in Table 1. We conclude that the mass estimates given by Equations (10) and (11) are robust against a wide range of halo models when evaluated near the half-light radius. At radii much different from the half-light radius, the estimates become model dependent. In what follows, the values of M(r_half) that we consider are those obtained from Equation (11).

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The observable properties of the Local Group dSphs must be influenced to some degree by the tidal forces exerted by the Milky Way and M31. Early work on this problem (e.g., Oh et al. 1995; Piatek & Pryor 1995) suggested that tides do not significantly alter the central velocity dispersion of a satellite as it passes through pericenter, lending credibility to estimates of dSph dynamical masses. More recent N-body simulations (e.g., Read et al. 2006; Muñoz et al. 2008; Peñarrubia et al. 2008b; Klimentowski et al. 2009) monitor the evolution of both stellar and dark matter components over several pericentric passages and a wide range of orbits. Peñarrubia et al. (2008b, "P08" hereafter) provide a set of analytic formulae to describe the tidal evolution of the parameters relevant to the present work. In particular, they find that for a dSph consisting of a stellar component described by a King profile embedded within an NFW halo, all parameters evolve according to the fraction of mass lost within the core radius (here taken to be approximately the half-light radius). For tidal stripping that results in the loss of as much as 99% of the original luminosity, the evolution of parameters is described by L/L₀ ≈ 2⁷(σ/σ₀)^6.6[1 + σ²/σ²₀]⁻⁷ and $r_{\rm half}/r_{half_0}\approx 2^{3/2}(\sigma /\sigma _0)^{1.3}[1\,{+}\, \sigma ^2/\sigma _0^2]^{-3/2}$, where subscripts of naught indicate initial values (P08).

Arrows in Figures 5 and 6 indicate the directions and magnitudes of the tidal tracks followed by a satellite as it loses 99% of its stellar mass under the P08 relations. In the σ–r_half plane of observables (Figure 6), at all points along the tidal track the displacement vector points primarily downward such that, as the satellite loses mass, its velocity dispersion decreases more quickly than its size. As a result, tides tend to carry individual objects off the empirical relation rather than moving them along it. While the tidal track in the M(r_half)–r_half plane (Figure 5, left) is more parallel to the empirical relation, the tidal track in the 〈ρ〉–r_half plane (the right panel of Figure 5) provides strong evidence against a tidal origin for the empirical relation. There we find that tides tend to decrease both the half-light radius and the mean density within the half-light radius. Thus, tides do not generate the circumstance in which the smallest objects also have the largest mean densities interior to r_half. These considerations lead us to conclude that tides did not introduce the correlations exhibited in Figures 5 and 6. This is not to say that tides are irrelevant—we now discuss two ways in which tides may have altered the character of the correlations we now observe.

First, we can expect at least some of the scatter in the correlations to be due to the varying degrees to which dSphs on different orbits have been tidally sculpted. In this regard, the dSphs Cetus and Tucana merit specific attention because they are the only objects in our sample that have no clear association with either the Milky Way or M31. Cetus lies at a distance of D ∼ 755 kpc (McConnachie & Irwin 2006) and Tucana is even more remote, at D ∼ 880 kpc (Saviane et al. 1996). It is therefore reasonable to assume that neither object has lost appreciable amounts of mass to tidal stripping. Then it is perhaps not a coincidence that Cetus and Tucana are the two most extreme outliers with respect to the empirical relations in Figures 5 and 6. Both objects have large velocity dispersions (and hence large M(r_half)) compared to other objects of similar size. We speculate that tides may have influenced all other dSphs to a significantly higher degree than Cetus and Tucana, moving the population systematically along P08's tidal tracks while leaving Cetus and Tucana in relatively pristine condition. Note that even strong tidal stripping, if it acts to a similar extent on most dSphs, can preserve the slope of an initial correlation despite shifting the locus of the affected population.

Second, tides might alter the slopes of the correlations in Figures 5 and 6 if systems of a particular size happen to be the most prone to tidal stripping. According to the tidal tracks, tides act to move objects primarily downward, toward lower velocity dispersion in the σ–r_half plane of Figure 6. But we must consider that since the objects with smallest half-light radii tend also to be found closest to the Milky Way, tides may have displaced them systematically farther along the tidal track than larger, more distant systems. This may have increased the slope of the relation in Figure 6. Under this scenario, the dSph population may have formed according to a flatter relation better described by an isothermal sphere (M ∝ r), but then evolved tidally such that the relation we observe today is the steeper power law M ∝ r^1.4. However, we re-emphasize that while tides may have increased both the scatter and slope of the empirical correlations, movement of individual systems along tidal tracks would not have introduced correlations where none existed initially.

Sagittarius (Sgr) is the only one of the Local Group dSphs that is unambiguously in the throes of tidal disruption (Ibata et al. 1995; Mateo et al. 1996; Majewski et al. 2003). The largest of the objects in our sample, the main body of Sgr has a half-light radius that is slightly larger than the scale radius of the best-fitting universal NFW halo. This is compatible with the fact that Sgr continues to lose stars, as its streams of debris wrap more than once around the sky (Majewski et al. 2003). Perhaps strangely, Sgr is otherwise inconspicuous in our scaling relations, falling neatly onto the best-fitting power-law profile. According to P08's tidal tracks, the progenitor of Sgr had larger velocity dispersion than Sgr exhibits today, such that Sgr may originally have been a more extreme outlier than Cetus and Tucana are now.

Since they extend the range of dSph sizes downward by an order of magnitude, the ultrafaint dSphs provide much of the leverage in discerning the scaling relations underlying Figures 5 and 6. Therefore, we must note some caveats due to the fact that the least luminous objects are necessarily the least well studied. First, their velocity dispersions are the most uncertain, a result of small sample sizes (e.g., just five stars in the case of Leo V (Walker et al. 2009a)) and the fact that their small dispersions are near the resolution limits of the spectrographs used to measure them (Simon & Geha 2007). In some cases, the measured dispersions (and corresponding masses and densities) represent just upper limits, so the data for objects like Leo V and Bootes II remain consistent with these objects having tiny dispersions that would render them more similar to star clusters than to bona fide dSph galaxies.

Second, many (e.g., BooII, Coma, Segue 1, Segue 2, UMaII, Willman 1) of the ultrafaint satellites are found within ⩽ 50 kpc of the Sun, where they are most vulnerable to Milky Way tides (see also Section 4.1). While extreme central densities (≲5 M_☉ pc⁻³) should be sufficient to protect the smallest systems against tidal disruption, many of the ultrafaint satellites appear elongated (e.g., Hercules has axis ratio 3:1; Coleman et al. 2007) or otherwise exhibit irregular morphologies (e.g., UMaII has an apparent double core; Zucker et al. 2006a). On one hand, one does not expect unbound tidal debris to inflate the velocity dispersions of the smallest dSphs—Peñarrubia et al. (2008c) use simulations to show that unbound debris escapes from a disrupting dSph on a timescale similar to the crossing time, which for the smallest satellites is just ∼r_half/σ ∼ 10 Myr. On the other hand, if distorted morphologies result from strong tidal processes, then it is difficult to understand how these systems could be embedded in intact dark matter halos similar to those inferred for more regular dSphs. It may turn out that some of the recently discovered satellites are unbound star clusters in the process of dissolving. In that case, it would be puzzling that these systems manage to fall on or near the empirical relations defined by bona fide dSphs.

We now investigate whether the correlations in Figures 5 and 6 can be described in terms of universal versions of the halo models considered in Section 3. For simplicity, we restrict our consideration to two specific halos of interest: the cuspy NFW halo with α = γ = 1 and a cored halo with α = 1, γ = 0. In order to compare these halo models directly with observable quantities, we equate M(r_half) from Equation (7) with the model-independent estimate given by Equation (11). For the NFW halo, this gives the scaling relation

$$\begin{equation} r_{\rm half}\sigma ^2=\frac{2\eta r_0V_{\rm max}^2}{5}\frac{\ln [1+r_{\rm half}/r_0]-\frac{r_{\rm half}/r_0}{1+r_{\rm half}/r_0}}{\ln [1+\eta ]-\frac{\eta }{1+\eta }}, \end{equation} \tag{ 12 }$$

where η ∼ 2.16. For the cored halo, the scaling relation becomes

$$\begin{eqnarray} r_{\rm half}\sigma ^2&=&\frac{2\eta r_0V_{\rm max}^2}{5(\ln [1+\eta ]+\frac{2}{1+\eta }-\frac{1}{2(1+\eta)^2}-\frac{3}{2})}\bigg [ \ln [1+r_{\rm half}/r_0]\nonumber \\ &&+\frac{2}{1+r_{\rm half}/r_0}-\frac{1}{2(1+r_{\rm half}/r_0)^2}-\frac{3}{2}\bigg ],\quad \end{eqnarray} \tag{ 13 }$$

now with η ∼ 4.42 for α = 1, γ = 0.

Let us consider whether the correlations in Figures 5 and 6 can be fit with a single halo of either the cusped or cored variety. Contours in the V_max–r₀ plane in Figure 7 indicate constraints we obtain by fitting, via Equations (12) and (13), NFW and cored profiles to the r_half–$\sigma _{V_0}$ data. Accounting for measurement errors in both dimensions, the best-fitting NFW halo has V_max ∼ 15 km s⁻¹, r₀ ∼ 795 pc, while the best-fitting cored halo has V_max ∼ 13 km s⁻¹, r₀ ∼ 150 pc. The best-fitting NFW and cored profiles are plotted over the empirical relations in Figures 5 and 6.

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For both NFW and cored halos, the scale radius is constrained by the slope in the empirical σ–r_half relation (Figure 6). Cored halos with scale radii larger than r₀ ≳ 200 pc generate correlations with slopes in the σ–r_half plane that are systematically steeper than that of the empirical relation. We have tried other forms of the cored halo (e.g., α = 0.5, α = 2) and find that in all cases, the scale radius must be ≲200 pc. Another way of understanding this constraint is that it is set by the fact that the empirical relation in Figure 5 (right panel) does not show signs of turning over as r → 0. Thus a universal dSph halo, if it is cored, must have a small-scale radius.

Figure 7 also demonstrates that for a universal NFW halo, the allowed region of parameter space intersects the mass–concentration relation that is derived independently from cosmological N-body simulations (Navarro et al. 1996; Eke et al. 2001; Bullock et al. 2001; Peñarrubia et al. 2008a). Thus, the single NFW halo that best fits all the data is broadly consistent with the ΛCDM framework.

From both the cusped and cored halo fits there is a clear prediction that deviates from that of the power-law fit: collectively, dSph velocity dispersions increase with half-light radius until reaching a maximum of σ ∼ 10 km s⁻¹ for r_half ∼ 1 kpc (Figure 6). Larger systems falling on the same relation will have smaller velocity dispersions, σ ≲ 10 km s⁻¹, as the half-light radius becomes similar to the scale radius of the dark matter halo. We will soon be able to test this prediction as kinematic data become available for more dSph satellites of M31, many of which have r_half ≳ 1 kpc (McConnachie & Irwin 2006). The universality of the best-fitting cusped or cored halos would imply that dSphs represent a class of spheroid that is at once distinct from globular clusters (r_half ∼ 1–10 pc and σ ∼ 5–10 km s⁻¹) and larger dwarf elliptical galaxies (r_half ∼ 1 kpc, $\sigma _{V_0}\sim 50$ km s⁻¹)—both types of object have velocity dispersions too large for the dSph relation for cored or cusped halos. In contrast, a power-law profile would imply that velocity dispersion increases with half-light radius indefinitely (dashed line in Figure 6), admitting the possibility that dSph mass profiles are similar to those of larger ellipticals and perhaps spiral galaxies (P. Salucci 2009, private communication; S. McGaugh 2009, private communication).

Irrespective of whether the halo is cored or cusped, the hypothesis of universality leads to another clear prediction. If the right-hand side of Equation (1) is universal, then the left-hand side must also be universal such that there is a universal functional relationship between the dSph stellar density, velocity dispersion, and anisotropy. This is simple to work out in the case of isotropy and constant velocity dispersion. Two dSphs with velocity dispersions σ₁ and σ₂ must have luminosity profiles ν₁ and ν₂ related by

$$\begin{equation} \nu _1^{\sigma _1^2} = \nu _2^{\sigma _2^2}. \end{equation} \tag{ 14 }$$

In other words, the two profiles are related via a power-law transformation. Even in the more complicated case when the anisotropy β is nonzero, the luminosity profiles of different dSphs are still related to each other, though the transformation is now more complicated than a power law.

A universal dSph dark matter halo must account not only for the bulk kinematic properties of the dSph population, but also for the velocity dispersion profiles, where available, of individual dSphs. Therefore, let us test whether the four candidate profiles considered in this work—a singular isothermal sphere with M ∝ r, a power-law profile with M ∝ r^1.4, an NFW cusp with scale radius r₀ = 795 pc, and a core (α = 1; γ = 0) with r₀ = 150 pc—are consistent with the velocity dispersion profiles in Figure 1. For each profile, we fit to the empirical velocity dispersion profiles using the Jeans equation (Equation (3)), allowing now for just two free parameters—anisotropy and the normalization, V_max. For the power-law profile, the circular velocity curve is unbounded as r → ∞, so we normalize the power-law profile with the value of M₃₀₀. In order to consider only the most realistic cases, we restrict the anisotropy to values between −0.6 ⩽ β ⩽ +0.3.

For each of the four universal profiles considered, Table 3 lists the values of β and V_max that produce the best fits to the empirical velocity dispersion profiles of the bright dSphs. We find that for each halo profile, the data for all eight dSphs are well fit if we allow the normalization to vary over a factor of ≲2. For the power-law profile, we can fit all the empirical velocity dispersion profiles if we let M₃₀₀ take values between [1–2] × 10⁷ M_☉—note that this is the same range for M₃₀₀ found by S08, but unlike in that study, here M₃₀₀ serves as the normalization for mass profiles of the same shape. The isothermal and cored profiles require V_max in the range 10–20 km s⁻¹, and the NFW profile requires V_max ∼ 15–25 km s⁻¹. The best fits for each profile are plotted against the empirical velocity dispersion profiles in Figure 1, and parameters of those fits are listed in Table 3.

Table 3. Anisotropy and Normalization of Universal Profile, for dSphs in Figure 1

	$\underline{M\propto r}$	$\underline{M\propto r^{1.4}}$	NFW (r0 = 795pc)	Core (r0 = 150pc)
Galaxy	β; Vmax (M☉)	β; M300 (M☉)	β; Vmax (km s−1)	β; Vmax (km s−1)
Carina	−0.6+0.7; 11+5−5	−0.5+0.9−0.1; 0.8+0.2−0.2 × 107	+0.1+0.3−0.6; 15+1−1	−0.2+0.5−0.4; 12+1−1
Draco	−0.5+0.7−0.1; 13+12−4	−0.6+0.9−0.2; 1.9+0.7−0.4 × 107	−0.4+0.8−0.2; 22+3−2	−0.5+0.9−0.8; 18+3−1
Fornax	−0.6+0.4; 18+7−7	+0.0+0.3−0.5; 1.6+0.2−0.2 × 107	+0.0+0.3−0.4; 20+1−1	−0.6+0.4−0.4; 19+1−1
Leo I	−0.5+0.9−0.1; 17+12−9	+0.0+0.7−0.6; 2.1+0.7−0.6 × 107	+0.3+0.0−0.9; 24+2−4	−0.5+0.4−0.5; 20+3−3
Leo II	−0.4+0.8−0.2; 13+10−7	+0.4+0.3−0.9; 1.4+0.4−0.5 × 107	+0.4+0.0−0.9; 19+3−3	+0.4+0.0−0.9; 16+2−3
Sculptor	−0.5+0.3−0.1; 15+8−4	−0.3+0.6−0.3; 1.9+0.4−0.3 × 107	+0.4+0.0−0.5; 24+1−2	+0.1+0.2−0.6; 20+2−2
Sextans	−0.5+0.8−0.3; 11+7−7	−0.5+0.7−0.1; 0.7+0.2−0.2 × 107	−0.3+0.6−0.3; 13+1−2	−0.6+0.7−0.1; 11+1−1
UMi	−0.6+0.6; 15+10−8	−0.6+0.8; 1.6+0.4−0.4 × 107	−0.3+0.7−0.3; 21+2−2	−0.6+0.8−0.1; 18+2−2

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We now evaluate the feasibility of a universal dSph mass profile in terms of the amount of scatter with respect to the best candidates for that profile. Figure 8 plots residuals Δ(log M) ≡ log [M(r_half)/M_☉] − log [M(r_half)_universal/M_☉] (where the first term represents the data and the second term represents the universal profile) with respect to the best-fitting isothermal, power-law, NFW and cored profiles considered in this work. For comparison, the bottom panel plots the residuals in the common–M₃₀₀ relation, log [M₃₀₀/M_☉] = 7, where M₃₀₀ is estimated from Equation (10). Text indicates for each profile the statistic χ² = N⁻¹∑^N_i=1(Δ(log M))²/δ², where $\delta \equiv \sigma _{M(r_{\rm half})}[M(r_{\rm half})\ln (10)]^{-1}$ represents the measurement error in log space. The power-law (M(r) ∝ r^1.4) provides the best fit (χ² = 3.2), followed by the NFW profile (χ² = 5.1), the cored profile (χ² = 5.8), and the isothermal profile (χ² = 11.9). While these values of χ² indicate scatter larger than what is expected from the quoted errors if the profile is truly universal, we note that the scatter of M₃₀₀ values about a common mass of 10⁷ M_☉ corresponds to χ² = 8.8. In addition to χ², Figure 8 also indicates the rms of the residuals. There we also find that the scatter about a universal mass profile (rms = 0.35 for the power-law profile) is similar to the scatter about a common value of M₃₀₀ (rms = 0.42).

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Thus in terms of empirical scatter, the hypothesis that dSphs follow a universal mass profile receives as much support from the data as the claim that all dSphs have a common M₃₀₀. Note that the former is the stronger claim, since it implies the latter (not vice versa), and both statements involve an extrapolation of sorts. S08's argument for a common M₃₀₀ rests on extrapolation, since for the smallest objects (r_half ∼ 30 pc) it requires evaluation of mass profiles at radii that are larger by an order of magnitude than the region containing kinematic data. In contrast, the claim for a universal mass profile emerges from the pattern of masses robustly measured at the half-light radius of each dSph. Yet it is important to acknowledge that we know the enclosed mass only at this one particular radius in each dSph, so we cannot definitively rule out the situation in which any given dSph has a mass profile that diverges wildly from the hypothesized universal profile at radii far from its own half-light radius. However, if this circumstance were true for more than just a few outliers, then it would require a remarkable coincidence for the masses at dSph half-light radii to generate the appearance of a universal profile over two orders of magnitude in r_half. We conclude that if one is impressed by the commonality of M₃₀₀ among dSphs, one is obliged to be impressed by the apparent commonality of dSph masses at all radii.