ner_tags: O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O B-Citation O O O O O O O O O B-Citation O O O O O O O O O O B-CelestialObject I-CelestialObject O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O B-Citation O B-Citation O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
tokens: is σ = 0 km s −1 (so any apparent velocity dispersion would be due entirely to binaries and measurement error); in scenario 2, we assume an intrinsic dispersion of σ = 2.7 km s −1 , which is close to our best-fitting dispersion of Ret II for the mean period assumed here. For each realization, we first calculate the uncorrected dispersion using 3σ-clipping: we find the maximum likelihood values of μ, σ, then remove stars that lie beyond 3σ, and repeat as needed until no such outliers are present. In order to find cases that roughly resemble Ret II, we repeat many random realizations and keep three cases for which the uncorrected dispersion σ clip lies between 2.5–3.5 km s −1 . This required several realizations before three such cases were found for either scenario. In scenario 2 (σ = 2.7 km s −1 ), we did choose one realization with σ clip = 2.3 km s −1 to see if a non-zero intrinsic dispersion can be recovered even for a lower σ clip value. After applying our analysis to these mock data, the resulting posteriors in the intrinsic dispersion are plotted in Fig. 5(a) for the scenario where σ = 0 km s −1 . In all three cases we infer that σ 1.5 km s −1 at greater than 90 per cent confidence; the most probable intrinsic dispersion is well below 1 km s −1 in all cases. Thus, our method appears unlikely to find a peak at σ > 2 km s −1 if the apparent velocity dispersion is dominated by binaries. In Fig. 5(b) we plot the resulting posteriors for the cases where the true σ = 2.7 km s −1 . Here we see that σ = 0 km s −1 is ruled out in all cases; in none of these cases do we find a most probable intrinsic dispersion lower than 2 km s −1 . This builds confidence that, despite the relatively small sample size, our binary-corrected velocity dispersions can be trusted. Figure 5. Posteriors in velocity dispersion for simulated data sets, where the velocity measurements are randomly resampled for each star in Ret II. In (a) we simulate a ‘galaxy’ with intrinsic dispersion σ = 0 km s −1 , while in (b) the simulated galaxy has intrinsic dispersion σ = 2.7 km s −1 . Although each case has a high uncorrected dispersion σ clip determined from sigma-clipping, our method can clearly differentiate between these two scenarios despite the relatively small sample sizes. As we mentioned above, despite Ret II having a most probable intrinsic dispersion of 3 km s −1 in our analysis, there is still a non-zero probability of σ = 0 km s −1 . What are we to make of this? First keep in mind that if the K15 measurements are not systematically offset from the higher resolution J15 measurements by ≈1–3 km s −1 , the probability of Ret II having σ = 0 km s −1 essentially vanishes. If we retain the possibility of a large offset between the K15 and J15 data sets, then there are three possible explanations: either (i) Ret II’s true intrinsic dispersion is in fact somewhere between 0–3 km s −1 , and the peak appearing at 3 km s −1 is merely due to the small sample size; (ii) Ret