Thread by @nfmartin1980

Remember this dwarf galaxy that was presented as lacking dark matter last week (van Dokkum et al. 2018)? #thread
arxiv.org/abs/1803.10237

Since extraordinary claims require extraordinary proof, my student @Nico_Longeard and I decided to have a look at the claim of very low velocity dispersion and M/L in NGC1052-DF2 and, how to say…

Turns out that sampling a distribution function with 10 tracers is tricky. Small number statistics are a pain and the uncertainties on the velocities of the 10 globular clusters studied in the paper are of order the intrinsic velocity dispersion. #tricky

So you can use "fancy" techniques to put together a velocity dispersion estimate (sigma < 10.5 km/s at the 90% confidence level in the paper) that you'll always have doubts about, or you can go back to the basics and write down your likelihood function. #statsareyourfriend

Let's assume all 10 tracers are members of the population we're trying to model and build a simple Gaussian generative model folding in the velocity uncertainties.

\sigma_vr is what we're interested in, the intrinsic velocity dispersion of the population. (delta_{v,sys} is small and irrelevant for the NGC1052-DF2 analysis)

So what happens if you do that? You get this marginalized PDF of \sigma_vr (assuming flat prior) and sigma_vr = 9.5 +4.8/-3.9 km/s if you want to summarise the PDF with 3 numbers.

But I hear someone in the back pointing out that some of the tracers could be outliers and that this inference is only an upper limit on the true, contamination-corrected intrinsic dispersion. That's very possible, so let's have the data tell us what they prefer. #thedataareboss

We now use a model that is the sum of the Gaussian above and a flat contamination over the considered velocity range (we actually used a very, very large Gaussian because we had that code ready) and use \eta as the fraction of the sample that is in the contamination.

And here's what the data tell us… #drumrolls Turns out that the fraction of contaminants could be as high as ~30%, but that it doesn't really affect the PDF of the intrinsic velocity dispersion! (and there's no degeneracy between the 2 parameters, we checked)

Why? Because the velocity point that we'd be the most likely to say is an outlier in the velocity histogram (here from the van Dokkum et al. paper) has a large uncertainty and could perfectly be drawn from the main population; the data are fine with that scenario that they favor.

Bottom line: sigma_vr = 9.5 +4.8/-3.9 km/s (model w/o contamination) or sigma_vr = 9.1 +4.8/-3.6 km/s (model w/ contamination). A far cry from \sigma_vr<10.5 km/s and the 90% CL.

What consequence for the mass-to-light ratio of the system? This. The black PDF is obtained using the Walker et al. (2009) mass estimator within the half-light radius and the red one is the Watkins et al. (2010) estimator used in the NGC1052-DF2 paper. Both agree #yay

Sure, M/L within the tracers could be < 3, but certainly not at the 90% CL as claimed. In fact, it could very well be 4, or 6, or maybe even higher than 8 and perfectly in sync with a dwarf galaxy with dark matter.

In fact, that measurement is perfectly compatible with the Local Group dwarf elliptical NGC 147 and NGC 185 of similar luminosity and that have M/L~4 (Geha et al. 2009).
arxiv.org/abs/0911.3654

Also, fun fact, NGC1052-DF2 sits perfectly on the "universal mass profile" proposed by Walker et al. (2009) for Local Group dwarf galaxies. NGC1052-DF2 is the green point and the red line is the expectation for an NFW profile.
arxiv.org/abs/0906.0341

So what's our take on this? Maybe NGC1052-DF2 lacks dark matter, but the current data set doesn't warrant such a strong conclusion. The current data just aren't very constraining (note that you can't reject either that the dynamical mass of the system is 0!)

And always let the data speak for themselves, especially if you have large uncertainties and small data sets. #endthread

(By the way, if you care about generative models and likelihood, (and you should) you should read this very didactic reference by @davidwhogg, @jobovy, and @dstndstn if you haven't done so already!)
arxiv.org/abs/1008.4686