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# Abundance matching with the mean star formation rate: there is no missing satellites problem in the Milky Way

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### Abstract

Classical abundance matching has been shown to produce mass estimates, $$M^{\rm abund}_{200}$$, that agree well with independent dynamical estimates, $$M^{\rm dyn}_{200}$$, for isolated dwarfs. However, for satellite galaxies, it is expected to fail. This is because tidal stripping lowers $$M_*$$ and $$M^{\rm dyn}_{200}$$, causing satellites to scatter above the $$M_*-M_{200}$$ relation for isolated dwarfs, while ram-pressure stripping quenches star formation on infall, causing satellites to scatter below the relation. In this paper, we introduce a novel abundance matching technique that produces a more accurate estimate of $$M_{200}$$ for satellite galaxies. To achieve this, we abundance match with the mean star formation rate, averaged over the time when a galaxy was forming stars, $$\langle {\rm SFR}\rangle$$, instead of $$M_*$$. Using data from the Sloan Digital Sky Survey and the Bolshoi simulation, we obtain a statistical $$\langle {\rm SFR}\rangle-{\rm M}_{200}$$ relation in $$\Lambda{\rm CDM}$$. We then compare $$M^{\rm abund}_{200}$$ derived from this relation with $$M^{\rm dyn}_{200}$$ for 21 nearby dSph and dIrr galaxies, finding a good agreement between the two. As a first application, we use our new $$\langle {\rm SFR}\rangle-{\rm M}_{200}$$ relation to empirically measure the cumulative mass function of a volume-complete sample of bright Milky Way satellites within 280 kpc of the Galactic centre. We compare this with a suite of cosmological 'zoom' simulations of Milky Way-mass halos that account for subhalo depletion by the Milky Way disc. Including a conservative lower bound on the number of 'ultra-faint' dwarfs in this same volume, we find no missing satellites problem above $$M_{200} \sim 10^9$$M$$_\odot$$ in the Milky Way. We discuss how this empirical method can be applied to a larger sample of spiral galaxies in the Local Volume.

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### Halo occupation numbers and galaxy bias

(2000)
We propose a heuristic model that displays the main features of realistic theories for galaxy bias. We show that the low-order clustering statistics of the dark-matter distribution depend almost entirely on the locations and density profiles of dark-matter haloes. A hypothetical galaxy catalogue depends on (i) the efficiency of galaxy formation, as manifested by the halo occupation number -- the number of galaxies brighter than some sample limit contained in a halo of a given mass; (ii) the location of these galaxies within their halo. The first factor is constrained by the empirical luminosity function of groups. For the second factor, we assume that one galaxy marks the halo centre, with any remaining galaxies acting as satellites that trace the halo mass. These simple assumptions amount to a recipe for non-local bias, in which the probability of finding a galaxy is not a simple function of its local mass density. We have applied this prescription to some CDM models of current interest, and find that the predictions are close to the observed galaxy correlations for a flat $$\Omega=0.3$$ model ($$\Lambda$$CDM), but not for an $$\Omega=1$$ model with the same power spectrum ($$\tau$$CDM). This is an inevitable consequence of cluster normalization for the power spectra: cluster-scale haloes of given mass have smaller core radii for high $$\Omega$$, and hence display enhanced small-scale clustering. Finally, the pairwise velocity dispersion of galaxies in the $$\Lambda$$CDM model is lower than that of the mass, allowing cluster-normalized models to yield a realistic Mach number for the peculiar velocity field. This is largely due to the strong variation of galaxy-formation efficiency with halo mass that is required in this model.
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### Author and article information

###### Journal
18 July 2018
###### Article
1807.07093