A Deficit of Dark Matter from Jeans Modeling of the Ultra-diffuse Galaxy NGC 1052-DF2

Ultra-diffuse galaxies (UDGs) were recently recognized as a ubiquitous class of low-surface-brightness stellar systems with luminosities like dwarf galaxies but sizes like giants (van Dokkum et al. 2015; Yagi et al. 2016). They are found in all environments from clusters and groups to the field (e.g., Martínez-Delgado et al. 2016; van der Burg et al. 2017), and appear to originate from multiple formation channels, including an extension of normal dwarfs to lower surface brightness, as tidal debris, and perhaps as "failed" galaxies (e.g., Peng & Lim 2016; Greco et al. 2018; Pandya et al. 2018).

The failed-galaxy scenario was motivated partly by inferences of UDG halo masses based on dynamics and on number counts of globular star clusters—masses that in some cases appear significantly higher than for the overall dwarf-galaxy population (Beasley et al. 2016; van Dokkum et al. 2016, 2017; Amorisco et al. 2018; Lim et al. 2018). The implication is that the stellar-to-halo mass relation (SHMR; e.g., Moster et al. 2013; Rodríguez-Puebla et al. 2017) for luminous dwarf galaxies (L ∼ 10⁸ L_⊙) may have a much larger scatter than was presumed, requiring revisions in galaxy formation models at halo masses of ∼10¹¹ M_⊙ (see also Smercina et al. 2018).

In this context, one of the nearest known UDGs, NGC 1052-DF2 in a galaxy group at ∼ 20 Mpc (Fosbury et al. 1978; Karachentsev et al. 2000; van Dokkum et al. 2018c, hereafter vD+18a), presents a valuable opportunity for detailed dynamical study. vD+18a used deep Keck spectroscopy to measure radial velocities for 10 luminous star clusters around DF2, estimating its dynamical mass within a radius of ∼8 kpc (cf. Virgo-UDGs work by Beasley et al. 2016; Toloba et al. 2018). The result was very surprising: rather than an unusually high mass-to-light ratio (M/L) as found for previous UDGs, the M/L was unusually low, and consistent with harboring no dark matter (DM) at all.

The low/no-DM result generated spirited debate, much of which focused on how best to estimate the intrinsic velocity dispersion σ of DF2 (e.g., Laporte et al. 2018; Martin et al. 2018; van Dokkum et al. 2018a).

However, the more fundamental question is what range of halo mass profiles is permitted by the data, which we examine in detail in this Letter. We adopt a generative modeling approach where the individual velocity measurements are mapped statistically onto halo parameter space, without the intervening steps of estimating σ and applying a mass estimator. In addition to deriving constraints on the dynamical mass profile, we consider the potential impact of tidal stripping, and furthermore compare DF2 with Local Group (LG) dwarfs.

NGC 1052-DF2 has position, redshift, surface brightness fluctuation (SBF), and tip of the red giant branch measurements that are all consistent with being a satellite of the giant elliptical galaxy NGC 1052 (vD+18a; van Dokkum et al. 2018d). We adopt a distance of 19 Mpc, matching the measured SBF distance to DF2 (Cohen et al. 2018), while allowing for a ±1 Mpc uncertainty in our analysis.⁷

The UDG surface brightness follows a Sérsic profile with index n = 0.6, effective radius ${R}_{{\rm{e}}}=22\buildrel{\prime\prime}\over{.} 6$ (2.08 kpc), and total luminosity of $1.2\times {10}^{8}\ {L}_{V,\odot }$. For the stellar M/L, we adopt a Gaussian prior with mean of ${{\rm{\Upsilon }}}_{* ,V}=1.7$ in Solar units and standard deviation of 0.5 (based on stellar population modeling; vD+18a; van Dokkum et al. 2018b). We truncate this distribution to be between 0.1 and 10.

NGC 1052-DF2 has 10 star clusters with radial velocity measurements in vD+18a. We use one updated velocity from van Dokkum et al. (2018a); this has only a mild impact on the results. Although the mass uncertainties from using so few tracers are relatively large (as we will find here), there is ample precedent in the literature for drawing meaningful conclusions from small sample sizes (Aaronson 1983; Chapman et al. 2005; Kleyna et al. 2005; Brown et al. 2007; Koposov et al. 2015).

The surface-density distribution of the star-cluster population is highly unconstrained. We assume an exponential distribution of tracers (i.e., a Sérsic profile with n = 1) where the half-number radius is drawn from a Gaussian prior with a mean of the observed half-number radius (32'') and standard deviation of 10''. We truncate this distribution to be between 10'' and 70''. Our adopted mean half-number radius is 40% larger than R_e of the galaxy diffuse starlight, which is consistent with studies of the star-cluster systems of other UDGs (Peng & Lim 2016; van Dokkum et al. 2017; Toloba et al. 2018; cf. Forbes 2017).

We use the Bayesian Jeans modeling formalism of Wasserman et al. (2018) to infer the mass distribution of DF2. Here, a given mass profile and a tracer density profile are linked to a predicted line-of-sight velocity dispersion profile σ_J(R). The assumptions include spherical symmetry, dynamical equilibrium, and velocity-dispersion anisotropy ($\beta =1-{\sigma }_{t}^{2}/{\sigma }_{r}^{2}$) that is constant with galactocentric radius. (There is no evidence for rotation in the system, although individual velocity uncertainties are too large for strong constraints; vD+18a). We adopt a Gaussian prior on $\tilde{\beta }=-{\mathrm{log}}_{10}(1-\beta )$ with a mean of 0 (isotropic) and standard deviation of 0.5 (truncated to the range of $\tilde{\beta }=-1$ to +1).

Because we do not directly constrain the dynamical mass beyond ∼8 kpc, we must rely on priors on the halo characteristics—on the DM profile shape, and also on expected correlations between halo mass, concentration, and stellar mass.

We model the mass distribution as the sum of the stellar mass, with spatial distribution described in Section 2, and a DM halo. For the halo density distribution we use the generalized Navarro–Frenk–White (gNFW) profile,

$$\begin{eqnarray}&&\rho (r)={\rho }_{s}{\left(\displaystyle \frac{r}{{r}_{s}}\right)}^{-\gamma }{\left(1+\displaystyle \frac{r}{{r}_{s}}\right)}^{\gamma -3}\end{eqnarray} \tag{ 1 }$$

where r_s is the scale radius, ρ_s is the scale density, and γ quantifies the inner log-slope. For γ = 1, this matches the usual Navarro–Frenk–White (NFW) halo model (Navarro et al. 1997), but letting γ vary below 1 allows for models that have a cored, shallower density profile.

We re-parameterize the halo in terms of virial mass (${M}_{200{\rm{c}}}$) and concentration (${c}_{200{\rm{c}}}$), where

$$\begin{eqnarray}&&{M}_{200{\rm{c}}}=200{\rho }_{\mathrm{crit}}\displaystyle \frac{4\pi {r}_{200{\rm{c}}}^{3}}{3}\end{eqnarray} \tag{ 2 }$$

and ${c}_{200{\rm{c}}}={r}_{200{\rm{c}}}/{r}_{s}$.

We then consider two flavors of mass models: one in which the stellar and halo masses are drawn from a SHMR, and one where the stellar and dark masses are decoupled. For the latter model, we use a uniform prior on ${\mathrm{log}}_{10}{M}_{200{\rm{c}}}/{M}_{\odot }$ between 2 and 15. This effectively allows for the case of no DM, as the stellar mass is ${\mathrm{log}}_{10}{M}_{\star }/{M}_{\odot }\sim 8.3$.

For both types of models we assume that the halo concentration is drawn from a mass–concentration relation (MCR; Diemer & Kravtsov 2015; Diemer 2017) based on the Planck 2015 cosmology. We use a log-normal distribution about this expected concentration with a scatter of 0.16 dex.

For the SHMR we use the z = 0 relation of Rodríguez-Puebla et al. (2017), where halos with mass ${M}_{200{\rm{c}}}\sim {10}^{10.8}{M}_{\odot }$ host galaxies with M_* similar to DF2 (note that for a satellite galaxy such as DF2, the halo mass is pre-infall, before tidal stripping). We allow for variation around this mean relation through a variable scatter:

$$\begin{eqnarray}&&{\sigma }_{\mathrm{log}{M}_{* }}=0.2-0.26(\mathrm{log}{M}_{\mathrm{vir}}-\mathrm{log}{M}_{1})\end{eqnarray} \tag{ 3 }$$

below virial masses of M₁ = 10^11.5 M_⊙ (note ${M}_{\mathrm{vir}}\ne {M}_{200{\rm{c}}};$ at M₁, ${M}_{200{\rm{c}}}\sim 0.9{M}_{\mathrm{vir}}$), while at higher masses, ${\sigma }_{\mathrm{log}{M}_{* }}$ is a constant 0.2 dex scatter (Garrison-Kimmel et al. 2017; Munshi et al. 2017).

Given the wide range of possible baryonic effects on the inner slope of DM halos (Oh et al. 2011; Adams et al. 2014; Pineda et al. 2017), we adopt a uniform prior on γ between 0 and 2.

To connect the Jeans model predictions for σ_J to the velocity observations, we use a Gaussian likelihood for the probability of drawing data, v_i, given the location R_i and the various model parameters θ,

$$\begin{eqnarray}{ \mathcal L }({v}_{i}| R,\theta ) & = & { \mathcal N }({v}_{i}-{v}_{\mathrm{sys}},{\sigma }^{2}={\sigma }_{{\rm{J}}}^{2}(R| \theta )+\delta {v}_{i}^{2})\\ & = & {(2\pi {\sigma }^{2})}^{-1/2}\exp \left(-\displaystyle \frac{{({v}_{i}-{v}_{\mathrm{sys}})}^{2}}{{\sigma }^{2}}\right)\end{eqnarray} \tag{ 4 }$$

where v_sys is the systemic velocity (drawn from a Gaussian prior with a mean of the observed velocities, 1801.6 $\mathrm{km}\,{{\rm{s}}}^{-1}$, and with a 5 $\mathrm{km}\,{{\rm{s}}}^{-1}$ standard deviation), and δv_i is the measurement uncertainty.

We draw from our posterior with the emcee Markov Chain Monte Carlo (MCMC) ensemble sampler (Foreman-Mackey et al. 2013). We run our sampler with 128 walkers for 2000 iterations, rejecting the first 1500 to ensure fully mixed chains. The posterior distributions of v_sys, ϒ_*, and distance closely match those of the associated prior distributions. For the inference with the SHMR-informed prior, the posterior distribution of the star-cluster system R_e is slightly lower (with median of 26''). The weak-prior model prefers a slightly tangential orbital anisotropy, although consistent with isotropy, while the posterior anisotropy in the SHMR-prior model matches the prior.

Before discussing the best-fitting results, in Figure 1 we present a comparison between the data and a simple model with a cuspy NFW halo that follows the mean SHMR, assuming isotropic orbits. The individual star-cluster velocity measurements (absolute value relative to v_sys) versus galactocentric radius are shown along with a model line-of-sight σ profile (dashed green curve). It is clear that this is not a favorable model: ∼3 of the observed velocities should lie above the curve, which has a spatially averaged σ ∼ 36 $\mathrm{km}\,{{\rm{s}}}^{-1}$, compared to an observed σ ∼ 5–10 $\mathrm{km}\,{{\rm{s}}}^{-1}$.

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This is not, however, the only plausible model, as there is scatter in the predicted SHMR and in the halo concentration. Furthermore, UDGs and luminous dwarfs in general are expected to inhabit cored DM halos (Chan et al. 2015; Di Cintio et al. 2017). Allowing for a DM core (dotted–dashed purple curve) reduces the tension with the data somewhat (σ ∼ 22  $\mathrm{km}\,{{\rm{s}}}^{-1}$). Introducing scatter in the SHMR and the MCR as discussed in Section 3, we present the best "standard" model from our MCMC fitting, including a freely varying orbital anisotropy, as a solid-blue curve with uncertainty envelope in Figure 1. This model dispersion profile is fairly constant with spatially averaged $\sigma ={17}_{-4}^{+6}$ $\mathrm{km}\,{{\rm{s}}}^{-1}$ and appears more reasonably close to the data, although it is still in tension with the many observed near-zero relative velocities. The posteriors on some key model parameters are: $\beta \,={0.0}_{-2.5}^{+0.7},$ ${\mathrm{log}}_{10}\,{M}_{200{\rm{c}}}/{M}_{\odot }={10.7}_{-0.3}^{+0.2}$, ${c}_{200{\rm{c}}}={9}_{-3}^{+4}$ (implying ${r}_{s}\,={8}_{-3}^{+4}$ kpc), and $\gamma ={0.2}_{-0.2}^{+0.3}$, although we note that the samples of γ hit the prior boundary at 0. This is a model solution with a normal halo and concentration (consistent with the priors: see Figure 2, left panel) but a large central density core—strongly disfavoring the NFW model.

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We next consider a model that allows for deviation from the standard SHMR, along with a free central DM slope, while still imposing the standard prior on halo mass versus concentration. We find that the DM halo all but disappears, with ${M}_{200{\rm{c}}}\,\lt 1.2\times {10}^{8}\ {M}_{\odot }$ (M_DM/M_* < 0.6) at the 90th percentile. The posterior velocity dispersion profile is shown in Figure 1 (right), with an average σ = 7 ± 1 $\mathrm{km}\,{{\rm{s}}}^{-1}$. This model prefers a more tangential $\beta =-{1.0}_{-2.7}^{+1.2}$.

For a measure of relative predictive accuracy of these two models, we use the Watanabe–Akaike Information Criterion (WAIC), an approximation of cross-validation (Gelman et al. 2013), defined as

$$\begin{eqnarray}\mathrm{WAIC} & = & -2\displaystyle \sum _{i}^{n}\mathrm{ln}\int { \mathcal L }({v}_{i}\ | \ \theta ){p}_{\mathrm{post}}(\theta )d\theta \\ & & +\,4\displaystyle \sum _{i}^{n}{\mathrm{var}}_{\mathrm{post}}[\mathrm{ln}{ \mathcal L }({v}_{i}\ | \ \theta )]\end{eqnarray} \tag{ 5 }$$

where p_post(θ) is the posterior distribution, ${ \mathcal L }({v}_{i}| \theta )$ is the likelihood, and var_post is the variance over the posterior. The first term measures the predictive accuracy marginalized over the posterior distribution, while the second term penalizes for model complexity by computing an approximation of the effective number of model parameters (analogous to reduced χ²). We find ΔWAIC = 1.5 (equivalent to a model odds ratio of ∼2), slightly favoring the model without the SHMR prior, although not enough to reject the SHMR model outright.

As a summary of these results, Figure 2 shows the distribution of stellar and dark mass within a three-dimensional aperture of 10 kpc, as well as the halo concentration for these two models. For the SHMR-prior model, the data prefer a lower enclosed dark mass than in the prior, with a shift in the median M_DM within 10 kpc from $1.2\times {10}^{10}\ {M}_{\odot }$ to $5.1\times {10}^{9}\ {M}_{\odot }$. For the weak-priors model, the posterior distribution of M_DM(<10 kpc) extends all the way to the lower prior boundary (∼10² M_⊙), with a 90th-percentile upper bound of $1.2\times {10}^{8}\ {M}_{\odot }$. Thus, the data prefer a relatively low amount of DM within the region probed.

The more tightly constrained quantity of interest is the total dynamical mass within 10 kpc, which is $({2.2}_{-0.6}^{+0.9})\times {10}^{8}{M}_{\odot }$, or dynamical $M/{L}_{V}={1.7}_{-0.5}^{+0.7}$. The latter value is remarkably coincident with the independent stellar population estimate for DF2 (Section 2). We conclude that data-driven dynamical modeling of DF2 allows for at most an extremely low-mass DM halo, and suggests that this UDG is comprised purely of stars.

The models considered in the previous sections were for an isolated dwarf and neglected any influence from the nearby massive elliptical galaxy, NGC 1052. In particular, infall of a satellite into a larger host initiates a process of tidal stripping, first from the outer DM halo, then from the central regions, followed by total disruption. Tidal stripping and heating has been proposed as the dominant mechanism for forming UDGs, which could be considered as exemplars for galaxies undergoing tidal disruption (Carleton et al. 2018). Some previously studied UDGs are clearly in the process of disruption (Merritt et al. 2016), while many others have undisturbed morphologies out to $\sim 4\ {R}_{{\rm{e}}}$ (Mowla et al. 2017).

vD+18a presented analysis of tidal stripping to constrain the physical separation between DF2 and NGC 1052. Here our aim is to develop a holistic model where the inferred UDG mass distribution is checked for consistency with tidal constraints, propagating uncertainties on viewing angle, satellite mass distribution, and central galaxy mass. In particular, is a no-DM scenario implausible owing to a likelihood of disruption?

We use a simple model for the tidal radius given enclosed masses of satellite and central galaxies:

$$\begin{eqnarray}&&{r}_{\mathrm{tidal}}={\left(\displaystyle \frac{{M}_{\mathrm{sat}}({r}_{\mathrm{tidal}})}{(\alpha -{\gamma }_{M}){M}_{\mathrm{cen}}(d)}\right)}^{1/3}d,\end{eqnarray} \tag{ 6 }$$

where d is the 3D distance between the two galaxies, γ_M is the local log-slope of the enclosed mass profile of the central galaxy at d, and α = 2 if we assume the orbit of the satellite is radial and α = 3 if we assume that the orbit is circular (van den Bosch et al. 2018). Without modeling any constraints on the actual orbit of DF2, we compare results assuming either radial or circular orbits for the satellite, assuming that the truth lies somewhere in between these two cases. For our sampled central mass profiles and separation distances, γ_M ∼ 1. DF2 shows no obvious evidence of tidal disturbances, with regular isophotes out to ≈2 R_e (∼4 kpc; vD+18a). This provides a tidal constraint that ${r}_{\mathrm{tidal}}\gtrsim 4\,\mathrm{kpc}$.

To estimate the central galaxy mass, we use the halo-to-stellar mass relation from Rodríguez-Puebla et al. (2017). For M_* = 10¹¹ M_⊙ (Forbes et al. 2017), we expect ${M}_{200{\rm{c}}}=4.9\,\times {10}^{12}\,{M}_{\odot }$ with a scatter of 0.25 dex (from inverting the SHMR scatter of 0.15 dex). We then adopt an NFW profile with concentration from the MCR and calculate the enclosed mass at a given radius.

To fold in all the uncertainties together (central mass, satellite mass posterior from the previous inference, and distance), we randomly sample from the underlying parameters, including a uniform distribution of projection angles. We plot the resulting distribution of tidal radii in Figure 3.

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For the data-driven model, ${r}_{\mathrm{tidal}}={4.3}_{-1.7}^{+4.7}$ (${5.7}_{-1.71}^{+5.09}$) kpc when assuming a circular (radial) orbit. Thus, there is a large fraction of model-posterior space (52% for circular, 81% for radial) where DF2 can have little/no DM yet be tidally undisturbed out to 4 kpc. We note that the low-velocity star clusters observed out to ∼ 7.5 kpc could still be bound even with ${r}_{\mathrm{tidal}}\sim 4\,\mathrm{kpc}$, if they have retrograde orbits (Read et al. 2006).

Turning to the SHMR-prior model, the dwarf would be naturally much more resistant to tides, and the tidal radius would be farther out (Figure 3). However, the predicted value of ${r}_{\mathrm{tidal}}={16}_{-7}^{+28}$ kpc (or ${24}_{-10}^{+33}$ kpc for the radial case) implies that DF2 would still likely have most of its DM stripped away by now, as ${M}_{\mathrm{DM}}({r}_{\mathrm{tidal}})/{M}_{200{\rm{c}}}\sim 0.2$ (∼0.4).

The latter point leads us to the possibility that DF2 started out with a normal DM halo, but has been tidally eroded, not only by removing the outer parts but also by stripping out much of the central DM prior to disruption of the visible galaxy. Such a solution was studied through N-body simulations by Ogiya (2018), who found that the final dark mass within 10 kpc could be ∼10⁸ M_⊙—which is consistent with our observations (see the red curves in Figure 2). We note, however, two major caveats to this interpretation: (1) there is a small range of orbital parameter space that allows for the necessary degree of stripping; (2) the dynamical time within the UDG is comparable to its orbital period, which may prevent it from relaxing into a visually undisturbed system with cold kinematics.

The difference in predicted tidal radii between the DM and no-DM models motivates looking beyond 4 kpc for tidal features around DF2 to help distinguish between these two scenarios.

We have found through Jeans modeling that the observations of cold kinematics in DF2 imply a very low DM content. However, Martin et al. (2018) disputed the unusual nature of this galaxy by noting its similar σ and dynamical M/L to previously studied dwarfs. Here we emphasize that such comparisons neglect the different measurement radii used, and we clarify the position of DF2 in a wider context by constructing a plot relating galaxy stellar masses, halo masses, and sizes (Figure 4).

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We take the compilation of LG dwarf galaxies from Fattahi et al. (2018), selecting only galaxies with M_* > 10⁵ M_⊙ and updating with sizes from Martin et al. (2016) where available. Taking the dynamical mass within ${r}_{1/2}\approx 1.3{R}_{{\rm{e}}}$ and subtracting the associated stellar mass, we compute the DM contribution to the circular velocity,

$$\begin{eqnarray}&&{v}_{\mathrm{circ},\mathrm{DM}}=\sqrt{\displaystyle \frac{{{GM}}_{\mathrm{DM}}(\lt {r}_{1/2})}{{r}_{1/2}}},\end{eqnarray} \tag{ 7 }$$

propagating uncertainties in the distance, size, luminosity, stellar M/L, and velocity dispersion. We color these points in Figure 4 by their stellar mass, with different symbols for field dwarfs versus satellites. We compare these measurements with halo circular velocity profiles for several halo masses, adopting MCR concentrations and γ = 0.3 cores, while color-coding these profiles by the SHMR-predicted stellar mass. The halo-concentration scatter is illustrated by the red band for the 10¹¹ M_⊙ halo.

This figure shows that some dwarfs track cored-halo profiles appropriate to their stellar masses. Others have higher velocities and perhaps cuspy halos (e.g., Spekkens et al. 2005; Oñorbe et al. 2015; Genina et al. 2018). A few have low velocities; as most of these are satellites, they may be examples of ongoing tidal stripping that has depleted their central DM content (Collins et al. 2013; Buck et al. 2018; Fattahi et al. 2018). DF2, however, stands out from all of these galaxies by having the lowest DM-velocity estimate, despite the much larger measurement radius. Andromeda XIX is closest in σ–r_1/2 space, but has ∼300× lower stellar mass and thus does not appear as DM-depleted as DF2. IC 1613 has a high stellar mass, but the smaller measurement radius allows for a larger range of halo masses.

We therefore strengthen the conclusion of vD+18a that DF2 is an extreme outlier in the usual dwarf–DM scaling relations. There are then two main possible explanations. One is that the galaxy formed with little or no DM, and the other is that it has been severely stripped of DM. We cannot definitively discriminate between the two scenarios, but in Section 5 we pointed out potentially major flaws in the tidal argument. Furthermore, there is an additional clue that has so far been generally overlooked: the very star-cluster system used to probe the dynamics of DF2 is itself so far unique in the known universe. The clusters are on average far more luminous than in other galaxies including the Milky Way, and they are also unusually extended and elongated (van Dokkum et al. 2018b). The presence of either a normal or a stripped DM halo provides no explanation for this novel observation. On the other hand, if DF2 formed through a rare pathway without DM (e.g., scenarios discussed in vD+18a), then it is more plausible that its star-cluster system would show unusual properties as well.

The peculiar case of DF2 demonstrates the rich yield of information that can be obtained through detailed observations of dwarfs beyond the LG, which will help challenge and refine our understanding of galaxy formation and of the nature of DM.

We thank the anonymous referee for useful suggestions. A.W. thanks Viraj Pandya for helpful discussions. This work was supported by NSF grants AST-1616598 and AST-1616710. A.J.R. is a Research Corporation for Science Advancement Cottrell Scholar.