Current Velocity Data on Dwarf Galaxy NGC 1052-DF2 do not Constrain it to Lack Dark Matter

We first assume that all 10 GCs are members of NGC 1052-DF2, with velocities v_i and velocity uncertainties ${\delta }_{v,i}$. In this case, our generative model is a simple Gaussian function with mean $\langle v\rangle$ and intrinsic dispersion, σ_int. The likelihood function can be expressed as

$$\begin{eqnarray}&&{ \mathcal L }=\displaystyle \prod _{i=1}^{i\leqslant 10}\displaystyle \frac{1}{\sqrt{2\pi }{\sigma }_{\mathrm{obs}}}\exp \left(-0.5{\left(\displaystyle \frac{{v}_{i}-\langle v\rangle }{{\sigma }_{\mathrm{obs}}}\right)}^{2}\right),\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\mathrm{with}\ {\sigma }_{\mathrm{obs}}^{2}={\sigma }_{\mathrm{int}}^{2}+{\delta }_{v,i}^{2}.\end{eqnarray} \tag{ 2 }$$

Because the uncertainties ${\delta }_{v,i}$ provided by vD18b are asymmetric, we use the positive uncertainty when ${v}_{i}\lt \langle v\rangle$ and the negative one otherwise.

We assume uniform priors on $\langle v\rangle$ and σ_int over the ranges 1750 to 1850 $\,\mathrm{km}\,{{\rm{s}}}^{-1}$ and 0 to 30 $\,\mathrm{km}\,{{\rm{s}}}^{-1}$, respectively. We then sample the posterior PDF with our own Markov Chain Monte Carlo algorithm (Martin et al. 2016; Longeard et al. 2018). The resulting joint PDF is shown in Figure 1, along with the marginalized PDFs for the two parameters. The PDF on the intrinsic velocity dispersion of NGC 1052-DF2 is well behaved and yields a significantly higher dispersion than the one reported by vD18b: ${\sigma }_{\mathrm{int}}={9.5}_{-3.9}^{+4.8}\,\mathrm{km}\,{{\rm{s}}}^{-1}$ (<18.8 $\mathrm{km}\,{{\rm{s}}}^{-1}$ at the 90% confidence level) versus ${\sigma }_{\mathrm{int}}={3.2}_{-3.2}^{+5.5}\,\mathrm{km}\,{{\rm{s}}}^{-1}$ (<10.5 $\,\mathrm{km}\,{{\rm{s}}}^{-1}$ at the 90% confidence level). Note that our measurement is by design corrected for the velocity uncertainties, as those are specifically included in the model. Our inference is compatible with the rms estimate of vD18b (σ_rms ∼ 12.2 $\,\mathrm{km}\,{{\rm{s}}}^{-1}$); this is expected, as of all three methods used by vD18b, the rms estimate most closely resembles our formalism.

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The inference described above assumes a uniform prior on σ_int, but it is known that such a prior can be biased for small values. We also test the use of Jeffreys's prior, which does not suffer from this bias, but has the uncomfortable property of being improperly defined (i.e., the PDF does not integrate to unity) if it is not bound at the lower end. Doing so and forcing σ_int > 1 $\,\mathrm{km}\,{{\rm{s}}}^{-1}$ yields ${\sigma }_{\mathrm{int}}={7.4}_{-3.3}^{+4.5}\,\mathrm{km}\,{{\rm{s}}}^{-1}$ (<15.5 $\,\mathrm{km}\,{{\rm{s}}}^{-1}$ at the 90% confidence level), which does not significantly change our inference. Alternatively, one can argue that, because the dynamical mass of NGC 1052-DF2 is the physical quantity we aim to constrain and because this quantity scales as ${\sigma }_{\mathrm{int}}^{2}$, it would be more appropriate to assume a uniform prior on ${\sigma }_{\mathrm{int}}^{2}$. Unsurprisingly, doing so yields larger value for the most likely intrinsic dispersion, with ${\sigma }_{\mathrm{int}}={13.1}_{-4.5}^{+6.6}\,\mathrm{km}\,{{\rm{s}}}^{-1}$ (<27.2 $\,\mathrm{km}\,{{\rm{s}}}^{-1}$ at the 90% confidence level).

While changing the prior on σ_int does not change the main conclusion of this paper (the velocity dispersion of the NGC 1052-DF2 velocity sample is not very well constrained), the fluctuations on the constraint stemming from the choice of prior displays the poor constraining power of the data set.

A Kolmogorov–Smirnov test yields a high probability of 0.4 to 0.8 that the sparse data set is drawn from the range of models constrained in Section 2.1. It is therefore not possible to reject the simple Gaussian model as a bad model for this data set. Nevertheless, it is a priori possible that the sample of 10 GCs includes some contamination by field GCs (e.g., from the neighboring NGC 1052) and we now test a model that allows for contamination. We assume a uniform contamination model, ${ \mathcal U }$ over the range $1750\lt {v}_{r}\lt 1850\,\mathrm{km}\,{{\rm{s}}}^{-1}$. With η the fraction of the data that is in the contamination, the likelihood function becomes

$$\begin{eqnarray}&&{ \mathcal L }=\displaystyle \prod _{i=1}^{i\leqslant 10}\left[\eta \,{ \mathcal U }+\displaystyle \frac{1-\eta }{\sqrt{2\pi }{\sigma }_{\mathrm{obs}}}\exp \left(-0.5{\left(\displaystyle \frac{{v}_{i}-\langle v\rangle }{{\sigma }_{\mathrm{obs}}}\right)}^{2}\right)\right],\end{eqnarray} \tag{ 3 }$$

with σ_obs as defined in Equation (2). The resulting PDFs are shown in Figure 2 for uniform priors. Interestingly, the inference on the intrinsic velocity dispersion of the GC sample remains unchanged, despite η reaching an upper limit of ∼0.3. While this may seem surprising at first, it can easily be explained by the datum with the most discrepant velocity (GC98; $v={1764}_{-14}^{+11}\,\mathrm{km}\,{{\rm{s}}}^{-1}$) having one of the largest velocity uncertainties. The model does not feel the need to separate this datum and fold it in the contamination model (indeed, that GC has the high probability of ∼0.9 to belong to the dwarf galaxy part of the model). After marginalization, we infer ${\sigma }_{\mathrm{int}}\,={9.2}_{-3.6}^{+4.8}\,\mathrm{km}\,{{\rm{s}}}^{-1}$ (<17.3 $\,\mathrm{km}\,{{\rm{s}}}^{-1}$ at the 90% level) for our baseline model with contamination. If we use a less-constraining contaminant model using a second Gaussian with only loose priors on the contamination (uniform from 1700 to 1900 $\mathrm{km}\,{{\rm{s}}}^{-1}$ for the center and uniform between 100 and 200 $\mathrm{km}\,{{\rm{s}}}^{-1}$ for the dispersion of this Gaussian representing the contamination), we get ${\sigma }_{\mathrm{int}}={11.4}_{-4.5}^{+5.8}\,\mathrm{km}\,{{\rm{s}}}^{-1}$. Finally, even if we nevertheless decide to forego the outcome of the modeling with contamination and abruptly remove GC98 from the data set (which is not advisable) to fit a single Gaussian model to the velocities of the remaining 9 GCs, we still only infer $\sigma ={7.1}_{-3.0}^{+3.6}\,\mathrm{km}\,{{\rm{s}}}^{-1}$ (<14.3 $\,\mathrm{km}\,{{\rm{s}}}^{-1}$ at the 90% confidence level). These variable results for different contamination assumptions highlight the challenges in interpreting such small-number data sets, while also demonstrating that these cases yield dispersions significantly higher than the vD18b limit.

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In the following, we will use the model with the uniform contamination as our baseline model as it is among the most agnostic models discussed above.

To infer the mass-to-light ratio of NGC 1052-DF2, we rely on the velocity dispersion from the model with contamination and use the mass estimator of Walker et al. (2009) that provides the mass within the half-light radius of the dwarf galaxy (∼2.2 kpc) under the usual assumption of dynamical equilibrium and sphericity. This estimator yields $M({r}_{\mathrm{half}})\lt 3.7\times {10}^{8}\,{M}_{\odot }$ at the 90% confidence level. Because this radius naturally includes half of the light of the system (∼0.55 × 10⁸$\,{L}_{\odot }$), we can infer the mass-to-light ratio ${\text{}}M/L({r}_{\mathrm{half}},V)$ within the half-light radius. The corresponding PDF is shown in black in Figure 3 and yields an upper limit of 6.7 at the 90% confidence level. Wolf et al. (2010) describe an alternate mass estimator to the one of Walker et al. (2009) that, beyond highlighting the difficulty of modeling the dynamical mass from a population of tracers, yields larger masses than the ones we give here (${\text{}}M/L({r}_{\mathrm{half}},V)\lt 10.7$). The difference is driven by different choices for the profiles of the tracers, their (axi)symmetry and/or anisotropy assumption (see the discussion in Appendix C of Wolf et al. 2010). We focus on the Walker et al. (2009) estimator to allow for an easier comparison with vD18b, but recognize that the mass-to-light ratio limit of NGC 1052-DF2 would be even higher than the one we infer if we had used the Wolf et al. (2010) estimator.

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The mass estimator favored by vD18b and based on Watkins et al. (2010) gives the mass within the last datum, i.e., within 7.6 kpc for the sample of GCs.⁵ The resulting mass-to-light ratio inference is similar but slightly less constrained (the red curve in Figure 3; $M/{L}_{\mathrm{TME}}\lt 8.1$ at the 90% confidence level).

Both mass estimators are therefore consistent with each other, and the data set is not strongly constraining, contrary to the finding of vD18b who found $M/{L}_{\mathrm{TME},V}\lt 3.3$ at the 90% confidence level. It is also worth noting that folding in the uncertainties on r_half and L_V would make the constraint weaker, but vD18b unfortunately do not provide those for their measurement of the size and luminosity of NGC 1052-DF2. As such, the confidence limits provided here should only be seen as lower limits.

2.5.1. Measuring the Velocity Dispersion by Resampling the Observed Data

Even though it amounts to making the data more noisy than they truly are and we do not recommend it, a common technique for measuring the dispersion from a small number of data points with significant uncertainties (i.e., similar to the size of the dispersion this is being measured), is to run a Monte Carlo resampling of the data. Here, we take the observed velocities of the vD18b sample and perturb them based on their uncertainties by randomly sampling from a Gaussian centered on the velocity measurement, with a dispersion equal to the uncertainties quoted by vD18b. We then follow their method for measuring the observed dispersion by recomputing the bi-weighted midvariance for this perturbed sample. We repeat this process 10,000 times, resulting in a distribution of values for ${\sigma }_{\mathrm{obs},\mathrm{bi}}$ (see Figure 4). From this process, we can use the mean and standard deviation of the distribution as a value for the observed dispersion, giving ${\sigma }_{\mathrm{obs},\mathrm{bi}}=14.3\pm 3.5\,\mathrm{km}\,{{\rm{s}}}^{-1}$ (very comparable with the observed r.m.s dispersion from vD18b). Following this, we must also correct for the effects of the observational uncertainties, which will inflate this measurement. We follow the process of Pryor & Meylan (1993), using the average uncertainty from our 10,000 realizations, resulting in ${\sigma }_{\mathrm{int},\mathrm{bi}}=12.0\pm 2.5\,\mathrm{km}\,{{\rm{s}}}^{-1}$. This value is considerably higher than the ${\sigma }_{\mathrm{int}}={3.2}_{-3.2}^{+5.5}\,\mathrm{km}\,{{\rm{s}}}^{-1}$ from vD18b, and lies above their proposed upper limit of ${\sigma }_{\mathrm{int}}\lt 10.5\,\mathrm{km}\,{{\rm{s}}}^{-1}$, but is consistent with the value we compute with our generative model (with or without contamination).

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2.5.2. On the Reliability of Using Small Samples of Globular Clusters to Compute the "True" Dispersion

Two key issues with interpreting any measured velocity dispersion in this instance are (1) the small number of tracers available and (2) knowing whether these are truly relaxed tracers of the underlying dark-matter halo. For the latter, we know from observations of the outskirts of both the Milky Way and Andromeda that GCs are often associated with substructure at large radii (e.g., Mackey et al. 2010; Veljanoski et al. 2014). In Andromeda in particular, between 50% and 80% of all GCs at distances beyond 30 kpc show both spatial and kinematic correlations with stellar streams (Mackey et al. 2010), meaning that they are not fully relaxed mass tracers.

Given point (2), the effects of point (1) could be severe. Measuring a single dispersion from 10 tracers that may not be relaxed could lead to either a significant over- or under-estimate of the halo velocity dispersion. This can be straightforwardly demonstrated using the GC system of M31. We take the kinematics for 72 clusters from Veljanoski et al. (2014). As the globular cluster system of M31 is known to rotate, we use their rotation-corrected velocities to ensure we are not artificially inflating our measured mass. We then randomly draw 10 clusters and measure the biweight-midvariance of their velocity distribution, following the technique used by vD18b. This sample has a much larger intrinsic dispersion than DF2, but the data are of similar quality (mean velocity uncertainties of ∼10 $\,\mathrm{km}\,{{\rm{s}}}^{-1}$). Repeating this process 10,000 times gives us a distribution of observed velocity dispersions (see Figure 5) that we can compare to both the biweight from the full sample (${\sigma }_{\mathrm{bi},\mathrm{all}}=105.0\,\mathrm{km}\,{{\rm{s}}}^{-1}$; the dashed line in Figure 5), and the average velocity dispersion of the M31 halo from its stars (${\sigma }_{{\rm{M}}{31}_{\mathrm{stars}}}\sim 90\,\mathrm{km}\,{{\rm{s}}}^{-1}$, the dashed–dotted line; Gilbert et al. 2018).

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The result here is clear: a single biweight dispersion measure from 10 GCs can give a huge range of dispersion measures. The mean value from this redraw gives ${\sigma }_{\mathrm{bi},10}=96\pm 24\,\mathrm{km}\,{{\rm{s}}}^{-1}$, but the tails extend to far higher and smaller values. Such a large statistical uncertainty would mean that, from a sample of 10 GCs in M31, halo masses ranging from $0.2\lt M\lt 1.2\times {10}^{12}\,{M}_{\odot }$ could be measured within the 90% confidence limit. Given that M31 has a stellar mass of ∼10¹¹ M_⊙ (Sick et al. 2015; Williams et al. 2017), the mass-to-light ratio could also be compatible with no dark matter based on this analysis.