Group galaxy number density profiles far out: is the 'one-halo' term NFW
  out to $>10$ virial radii?
  While the density profiles (DPs) of $\Lambda$CDM haloes obey the NFW law out
to roughly one virial radius, $r_{\rm vir}$, the structure of their outer parts
is still poorly understood, since the 1-halo term describing the halo itself is
dominated by the 2-halo term representing the other haloes picked up. Using a
semi-analytical model, we measure the real-space `1-halo' number DP of groups
out to $20\,r_{\rm vir}$ by assigning each galaxy to its nearest group with
mass above $M_{\rm a}$, in units of the group $r_{\rm vir}$. If $M_{\rm a}$ is
small (large), the outer DP of groups falls rapidly (slowly). We find that
there is an optimal $M_{\rm a}$ for which the stacked DP resembles the NFW
model to $0.1$ dex accuracy out to $\simeq 13\,r_{\rm vir}$. We find similar
long-range NFW surface DPs (out to $\simeq 10\,r_{\rm vir}$) in the SDSS
observations using a galaxy assignment scheme that combines the non-linear
virialized regions of groups with their linear outer parts. The optimal $M_{\rm
a}$ scales as the minimum mass of the groups that are stacked to the power
$0.25-0.3$. Our results suggest that the NFW model does not solely originate
from violent relaxation. Moreover, populating haloes with galaxies using HOD
models must proceed out to larger radii than usually done.
