  The outer haloes of the Milky Way and Andromeda galaxies contain as much
important information on their assembly and formation history as the properties
of the discs resident in their centres. In this paper we have used the
Constrained Local UniversE Simulation project to disentangle the stellar and DM
component of three galaxies that resemble the MW, M31 and M33 using both DM
only and DM + gas-dynamical simulations. Stars that are accreted in
substructures and then stripped follow a completely different radial
distribution than the stripped DM: the stellar halo is much more centrally
concentrated than DM. In order to understand how the same physical process can
lead to different z=0 radial profiles, we examined the potential at accretion
of each stripped particle. We found that star particles sit at systematically
higher potentials than DM, making them harder to strip. We then searched for a
threshold in the potential of accreted particles \phi_[th], above which DM
particles behave as star particles. We found such a threshold at >16
\phi_{subhalo}, where \phi_{subhalo} is the potential at a subhaloes edge at
the time of accretion. Thus a rule as simple as selecting particles according
to their potential at accretion is able to reproduce the effect that the
complicated physics of star formation has on the stellar distribution. This
niversal result reproduces the stellar halo to an accuracy of within ~2%.
Studies which make use of DM particles as a proxy for stars will undoubtedly
miscalculate their proper radial distribution and structure unless particles
are selected according to their potential at accretion. Furthermore, we have
examined the time it takes to strip a given star or DM particle after
accretion. We find that, owing to their higher binding energies, stars take
longer to be stripped than DM. The stripped DM halo is thus considerably older
than the stripped stellar halo.
