tokens: at low redshifts. Despite this some uncertainty remains as to whether the smaller scatter of Aquarius haloes is mainly due to differences between the particle tagging approach and hydrodynamical simulations. The way to verify this would be to construct mock stellar halo catalogues for apostle haloes and make direct comparisons. Such mock catalogues are not yet available for apostle haloes. Nevertheless, we note that as there are only five Aquarius haloes used for our analysis, it is quite possible that these haloes happen to be good cases simply due to statistical fluctuations. To quantify the statistical significance of the difference, we perform a one-dimensional KolmogorovSmirnov (KS) test to estimate the probability that the fitted parameters of the two samples are drawn from the same distribution. This is done separately for the two parameters. The resulting p -values are 0.3656 for M 200 and 0.6196 for c 200 , This means that we have 36.56percent (61.96percent) probability of observing an equal or even larger difference in the two mass (concentration) samples, which implies that the two samples are not statistically distinguishable for either parameter. Repeating the one-dimensional KS test for the parameter combination corresponding to anticorrelation direction we find a p -value of 0.6506, again compatible with the smaller scatter of Aquarius haloes appearing by chance due to the small sample size. With the larger sample of apostle haloes, we found a significantly larger scatter in the best-fitting parameters compared to those based on Aquarius haloes. The nearly a factor of 2 scatter in M 200 and a factor of 3 scatter in c 200 is worrying for dynamical modelling of MW halo mass. As discussed above, this amount of systematic uncertainty is irreducible and applies to any steady-state models. The typical size of the statistical errors is smaller than the symbol size in Fig. 2 , and is about a factor of 0.03 of the scatter of the systematic uncertainty (blue ellipse). With an average sample size of 4.5 10 4 star particles in each apostle halo, we find the effective number of independent star particles (see Section 4 ) to be N eff 0.03 2 4.5 10 4 40. Recall that in Section 4 , we estimated the effective number of dark matter particles in MW-like haloes to be about 1000. This means, star particles in our hydrodynamical simulations are far more correlated than dark matter particles. This is not surprising, because we know stars have much higher binding energy than dark matter and are more centrally concentrated. They are less relaxed and less phase mixed than dark matter. In addition, we expect about 3040percent of dark matter particles are smoothly accreted instead of being accreted as part of a bound substructure (see details in Wang etal. 2011 , where a detailed investigation on possible dependencies on the resolution has been made), whereas we have checked that more than 80percent of star particles beyond 20kpc of the halo centre
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