First Clear Signature of an Extended Dark Matter Halo in the Draco Dwarf Spheroidal

We present the first clear evidence for an extended dark matter halo in the Draco dwarf spheroidal galaxy based on a sample of new radial velocities for 159 giant stars out to large projected radii. Using a two parameter family of halo models spanning a range of density profiles and velocity anisotropies, we are able to rule out (at about the 2.5 sigma confidence level) haloes in which mass follows light. The data strongly favor models in which the dark matter is significantly more extended than the visible dwarf. However, haloes with harmonic cores larger than the light distribution are also excluded. When combined with existing measurements of the proper motion of Draco, our data strongly suggest that Draco has not been tidally truncated within ~1 kpc. We also show that the rising velocity dispersion at large radii represents a serious problem for modified gravity (MOND).


introduction
The central velocity dispersions of many Local Group dwarf spheroidal (dSph) galaxies are significantly larger than expected for self-gravitating systems (e.g. Mateo 1998). Assuming virial equilibrium, the implied M/L ratios reach as high as ∼ 250, making the dSphs among the most dark matter dominated systems in the universe. Given the apparent absence of dark matter in globular clusters (e.g. Dirsch & Richtler 1995), dSphs are also the smallest dark matter dominated stellar systems in the universe. As such, they have emerged as crucial testing grounds for competing theories of dark matter.
Despite their importance, dynamical models of dSphs to date have been very simple. Most analyses have relied on the use of single mass, isotropic King models, with their associated assumptions that mass-follows-light and that the stellar velocity distribution is isotropic (though see Kormendy 1991 andLokas 2001 for more general models). Hitherto, the validity of such assumptions has remained unchallenged because of the small size of the data sets. When only small numbers of radial velocities are available, there is a well-known degeneracy between mass and velocity anisotropy (e.g., Binney & Tremaine 1987). An increase in the line of sight velocity dispersion at large radii may by due to either (1) the presence of large amounts of mass at large radii, or (2) tangential anisotropy in the velocity distribution. The primary motivation of this Letter is to break this degeneracy for Draco by means of improved modelling and a larger data set with many more stars in the outer parts.

data
Observations were conducted from June 23 to 26 2000 at the 4m William Herschel Telescope using the AF2/WYFFOS multifibre positioner and spectrograph. A total of 284 stars were observed, spanning the magnitude range V ≈ 17.0 to V ≈ 19.8. Of these, 159 were Draco members (extending to 25 ′ ) with spectra of sufficient quality to be included in our dynamical analyses. The median velocity uncertainty for these 159 stars was 1.9 kms −1 . Of our stars, 62 were previously observed by Olszewski, Pryor & Armandroff (1996): the agreement between our data and the previous data is consistent with a binary fraction of ∼ 40%. Full details of the data are presented in Kleyna et al. (2001).
Previous studies (e.g., Hargreaves et al. 1996) have reported weak evidence of net rotational motion in Draco. N-body simulations show that tidal disruption of dSphs leads to an apparent rotation about an axis perpendicular to the orbit of the dSph (Oh, Lin & Aarseth 1995;Klessen & Zhao 2001). In our data set, we find that Draco appears to rotate at 6 kms −1 at a radius of 30 ′ , with a rotation axis position angle (PA) of 62 • . However, a Monte-Carlo analysis shows that this rotation signal is not statistically significant . Further, the solar rest-frame proper motion of Draco (Scholz & Irwin 1994) implies that the axis for tidally induced apparent rotation should have a PA of 138 •+23 −23 . The 3σ disagreement with the observed rotation axis argues against Draco having been significantly influenced by tides, a conclusion supported by the lack of alignment between the PA of Draco's major axis (88 ± 3 • , Odenkirchen et al. 2001, henceforth OD01) and its orbit.
From our data set, it is possible to measure the line of sight velocity dispersion profile for a strongly dark matter dominated dSph for the first time. We divide the data into radial bins and use Bayes' theorem to obtain the probability distribution and uncertainties of the velocity dispersion v 2 1/2 in each radial bin (see Kleyna et al. 2001 for details). The top panel of Figure 1 shows a plot of the radial variation of v 2 1/2 . The velocity dispersion is clearly flat or gently rising with increasing radius. This plot already rules out the best-fit mass-follows-light King model (OD01), for which the dispersion falls to zero at r tidal = 49.4 ′ , and should have fallen to ∼ 0.59 times the central dispersion (or ∼ 5 kms −1 ) by 22 ′ . 1 3. modelling

Jeans Equations
The Jeans equations for a spherical stellar system lead directly to a model-independent mass estimator (see Binney & Tremaine 1987, Eq. 4-55 & 4-56) requiring only knowledge of the velocity anisotropy and the true radial velocity dispersion v 2 r . Assuming spherical symmetry, it is straightforward to obtain v 2 r from the line of sight velocity dispersion v 2 using Abel integrals. We consider two dispersion profiles which bracket the true situation: (1) a flat profile with amplitude 9.3 kms −1 (2) a profile rising linearly from 8.5 kms −1 in the centre to 11.25 kms −1 at 22 ′ . The light is assumed to follow a Plummer law with a core radius r 0 = 9.7 ′ , as this is an excellent fit to the star count data . The solid and dotted curves in the lower panel of Figure 1 show the mass profiles obtained based on these two dispersion profiles, assuming velocity isotropy. The mass enclosed within three core radii ranges from 6.3 − 18.0 × 10 7 M ⊙ , implying M/L ratios of 350−1000, where the V-band luminosity of Draco is 1.8 × 10 5 L ⊙,V (Irwin & Hatzidimitriou 1995, henceforth IH95).   Wilkinson et al. (2001). The contours are at enclosed two-dimensional χ 2 probabilities of 0.68, 0.90, 0.95, and 0.997. The most likely value is indicated by a plus sign. The top and right panels represent the probability distributions of α and ν, respectively; the median of each distribution is represented by a square, and the triangles show the 1σ, 2σ and 3σ limits. The 3σ limits are omitted from the projection of α as the upper limit is strongly determined by the cut-off at α = 1. A 40% binary fraction is assumed.

Halo Models and Distribution Functions
We now use a two parameter family of models which spans a range of possibilities for the halo of Draco. The halo potential ψ(r) is given by The parameter α determines the mass distribution of Draco's halo. If α = 1, the dark matter has the same density distribution as the light. In models with α = 0, the halo has an asymptotically flat rotation curve. Finally, when α = −2, the visible dwarf lies in the approximately harmonic core of a much larger halo. The distribution functions for dSph density profiles in potentials (1) are derived in Wilkinson et al. (2001). They contain a second parameter ν which measures the velocity anisotropy. Isotropic models have ν = 0 and models with radial (tangential) anisotropy have ν > 0 (ν < 0). We generate line profiles from our models by numerically integrating the distribution functions over all transverse velocities. The line profile is then convolved with a binary distribution and Gaussian velocity measurement uncertainty profile of width σ to giveL(v, R, σ; α, ν). The probability of observing an ensemble of points R i with velocity v i and velocity uncertainty σ i is then given by the product of thẽ Figure 2 shows the likelihood contours for the model parameters α, ν assuming a 40% binary fraction. The most likely values are α = −0.34 and ν = −0.18, indicating a model with significant dark matter at large radii and a tangentially anisotropic velocity distribution. We rule out both a mass-follows-light (α = 1) and a harmonic core (α = −2) model at the ∼ 2.5σ confidence level. Our best fit model implies a mass inside 3 core radii of 8 +3 −2 ×10 7 M ⊙ . The implied V-band M/L ratio of Draco (within 3 core radii, or 29 ′ ) is ∼ 440 ± 240. This mass profile lies between the two curves in Figure 1.
All our halo models (1) have cores. Cosmological simulations favor cusped halos, such as the Navarro-Frenk-White (NFW) profile. The present data are insufficient to discriminate strongly between cored and cusped profiles for Draco's dark halo -unlike the case of the Milky Way galaxy (Binney & Evans 2001). Cosmological implications of data sets of dSph radial velocities are discussed elsewhere (Wilkinson et al., in prep.).

Perigalacticon and Tidal Cut-off
A dSph in orbit about the Milky Way is truncated by the Galactic tidal field during perigalacticon passages (Oh & Lin 1992). Based on SDSS images of Draco, OD01 find no evidence for a tidal tail of Draco beyond 50 ′ and no break in its luminosity profile. Thus it seems likely that the actual tidal truncation radius of Draco is ∼ > 1 • . We use the radial velocity and proper motions to generate possible Draco orbits. The proper motions are given by Scholz & Irwin (1994) as: µ α cos δ = 0.09 ± 0.05 ′′ cent −1 , µ δ = 0.1 ± 0.05 ′′ cent −1 . The line of sight velocity (corrected for the solar motion) is −98 ± 1 kms −1 , while the distance of Draco is 82±7 kpc (Mateo 1998). We represent the Galactic potential by ψ(r) = v 2 0 log[( √ a 2 + r 2 + a)/r] (e.g., Wilkinson & Evans 1999) where the scale length a ∼ 170kpc and the normalization v 0 is obtained by taking the circular speed to be 220 kms −1 at the solar radius. We assume log-normal errors on a with 3σ range 44kpc to 489kpc; these limits correspond to masses within 100kpc of 4.6 × 10 11 M ⊙ and 1.1 × 10 12 M ⊙ , respectively. Assuming Gaussian errors on all other observed quantities, we generate sets of initial conditions consisting of a position and velocity for Draco and a scale length for the Milky Way. Each set is integrated in the Galactic potential for a few orbital periods and the perigalacticon D p and eccentricity e of the orbit is determined. Dynamical friction is unimportant on such short time scales. The contours in Figure 3 show the 1, 2 and 3σ limits on the perigalacticon and eccentricity of Draco's orbit. While the distribution of eccentricities is roughly flat in the range [0.3, 0.95], the distribution of perigalacticons is strongly peaked at ∼ 75 kpc. This suggests that Draco is presently at or close to perigalacticon.
The tidal radius for a dSph moving in the above Galactic potential is given by (c.f. King 1962) (2) where M D (r t ) is the mass of Draco within radius r t , M g is the total mass of the Galaxy, and ψ p and ψ a are the values of the potential at perigalacticon and apogalacticon. The solid curves in Figure 3 show the variation of D p with e assuming a range of values for r t . Only models with r t > 3 • fall within the 1σ contour of Draco's perigalacticon and eccentricity derived from its space velocity.
If Draco had recently been subject to strong tidal forces, we would expect significant elongation of the stellar distribution, in conflict with the roundish (ǫ = 0.3) appearance of Draco on the sky and the absence of tidal tails (OD01). Chance alignment effects (e.g. Kroupa 1997) are unlikely to be responsible for Draco's regular appearance. Since any tidal extension of Draco should be aligned with its orbit, elongation approximately along the line of sight would require Draco's orbit to be similarly aligned. However, Monte Carlo sampling of the orbital velocities using the observational errorbars rules out alignment within 45 • with 98% confidence, assuming that the prior (no proper motion information) probability of the line of sight angle is uniform. The absence of statistically significant apparent rotation argues further against line of sight elongation.

mond and draco
As an alternative to dark matter, Milgrom (1983) proposed a modification to Newton's law of gravity (MOND) at low accelerations. A number of attempts have been made to apply MOND to the dSphs of the Milky Way (Gerhard 1994, Milgrom 1995. These used only the global velocity dispersion to obtain a MOND M/L ratio and were unable to construct an unambiguous case for or against MOND. Lokas (2001), using published velocities, obtained MOND fits to the dispersion profiles of several dSphs, treating the global MOND acceleration scale as a free parameter.
We model the luminosity density I(r) of Draco using a Plummer sphere with core radius r 0 . The velocity anisotropy parameter β (Binney 1981) is assumed to vary with radius as β(r) = (γ/2)r 2 /(r 2 0 + r 2 ) yielding a velocity distribution which approaches isotropy in the centre and becomes increasingly radial (tangential) if γ is positive (negative). With these assumptions, the Jeans equations can be integrated to obtain the radial velocity dispersion σ 2 r (r) as where g M (r) is the MOND acceleration due to the mass interior to radius r. The central projected V-band luminosity density I 0 is 2.2 × 10 6 L ⊙ kpc −2 (IH95). Since all the mass in the MOND picture is provided by the stars, the Newtonian acceleration g N (r) is given simply by g N (r) = −GM * (r)/r 2 , where the stellar mass M * (r) is obtained from I(r) and depends on the M/L ratio of the stars. The MOND acceleration is then obtained via g M = g N /µ(g M /a 0 ), where a 0 = 2 × 10 −8 cm s −2 is the MOND acceleration scale, µ(x) → 1 for x ≫ 1 and µ(x) → x for x → 0. Our assumed a 0 is an upper limit to the likely value (Sanders & Verheijen 1998), and favors models with lower stellar M/L ratios.
Our analysis assumes that Draco is an isolated MOND stellar system. This is justified by noting that for stars between ∼ 1.7 ′ and ∼ 20 ′ the acceleration due to the baryonic mass of the Milky Way disk (∼ 6×10 10 M ⊙ ) at the distance of Draco is smaller than the internal accelerations. Since only ∼ 10% of our sample lies outside 20 ′ the assumption of isolation is reasonable. Representing a slightly flattened light distribution by a spherical model has greater impact when applying MOND since all the mass is in the stars. However, in the case of Draco this effect is not significant as our spherical model underestimates the mass interior to radius r by 20%, resulting in M/L ratios which are at most 20% overestimated.
From the projected dispersions, we obtain the MOND line profiles (assumed Gaussian) which we convolve with the binary velocity distribution and the error distribution to obtain the observable line profiles. Using these profiles, we analyse the velocity data set as in Section 3.2, except that the model parameters are now the stellar M/L ratio and the velocity anisotropy. Figure 4 shows the contours in the space of M/L and ν. Again, we find that some tangential anisotropy is required to reproduce the flat projected velocity dispersion profile. More interestingly, the most likely M/L ratio is 19 with a 3σ lower limit of 8.6. The assumed value of I 0 implies a total luminosity of 3.2 × 10 5 L ⊙ out to 3r 0 , approximately the 2σ upper limit of the V-band luminosity (IH95). Our lower bound is therefore a true lower limit on the MOND M/L ratio of Draco.
Estimates of the M/L ratio of Draco's stellar population based on comparison with globular clusters have a 3σ range of 1 − 4.3 (Parmentier & Gilmore 2001, Feltzing, Gilmore & Wyse 1999). Even our 3σ lower limit on the MOND M/L ratio of Draco is at variance with these values. Thus, even by modifying the law of gravity as prescribed by MOND, we are unable to explain the observed motions of Draco's stars without the presence of significant quantities of dark matter.

discussion and conclusions
This Letter has presented a new set of radial velocities for 159 stars in the Draco dSph. Our data extend to ∼ 25 ′ and are the first observations to probe the outermost regions of a strongly dark matter dominated dSph. The velocity dispersion profile is flat or slowly rising at large radii, which provides the first clear signature of an extended dark matter halo in any dSph. We analyse these data using both the Jeans equation and halo models with distribution functions. Our best fit model has a total mass within 3r 0 ≈ 29 ′ of 8 +3 −2 × 10 7 M ⊙ and a slightly tangentially anisotropic velocity distribution. We are able to rule out the traditional mass-follows-light models and extended harmonic core models at about the 2.5σ significance level.
From orbit integrations in a Galactic potential, we argue that Draco is currently close to perigalacticon and its actual tidal radius is > 1 • , several times the characteristic length scale of the luminous matter. Draco's lack of significant apparent rotation and the significant misalignment of its major axis with its orbit argue against tidal forces having played a major role in its evolution. Our results also have consequences for the Modified Newton Dynamics (MOND): a Bayesian analysis of the observational data, incorporating radially varying velocity anisotropy, yields a 3σ lower limit of 8.6 on the MOND M/L ratio still requiring the presence of a significant amount of dark matter in Draco.
NWE is supported by the Royal Society, while MIW and JK acknowledge help from PPARC. The authors gratefully thank Colin Frayn for help during the observations, and Mike Irwin, HongSheng Zhao, and the referee Slawomir Piatek for valuable discussions.