Using Velocity Dispersion to Estimate Halo Mass: Is the Local Group in
  Tension with $\Lambda$CDM?
  Satellite galaxies are commonly used as tracers to measure the line-of-sight
velocity dispersion ($\sigma_{\rm LOS}$) of the dark matter halo associated
with their central galaxy, and thereby to estimate the halo's mass. Recent
observational dispersion estimates of the Local Group, including the Milky Way
and M31, suggest $\sigma\sim$50 km/s, which is surprisingly low when compared
to the theoretical expectation of $\sigma\sim$100s km/s for systems of their
mass. Does this pose a problem for $\Lambda$CDM? We explore this tension using
the {\small{SURFS}} suite of $N$-body simulations, containing over 10000
(sub)haloes with well tracked orbits. We test how well a central galaxy's host
halo velocity dispersion can be recovered by sampling $\sigma_{\rm LOS}$ of
subhaloes and surrounding haloes. Our results demonstrate that $\sigma_{\rm
LOS}$ is biased mass proxy. We define an optimal window in $v_{\rm LOS}$ and
projected distance ($D_p$) -- $0.5\lesssim D_p/R_{\rm vir}\lesssim1.0$ and
$v_{\rm LOS} \lesssim0.5V_{\rm esc}$, where $R_{\rm vir}$ is the virial radius
and $V_{\rm esc}$ is the escape velocity -- such that the scatter in LOS to
halo dispersion is minimised - $\sigma_{\rm LOS}=(0.5\pm0.1)\sigma_{v,{\rm
H}}$. We argue that this window should be used to measure line-of-sight
dispersions as a proxy for mass, as it minimises scatter in the $\sigma_{\rm
LOS}-M_{\rm vir}$ relation. This bias also naturally explains the results from
\cite{mcconnachie2012a}, who used similar cuts when estimating $\sigma_{\rm
LOS,LG}$, producing a bias of $\sigma_{\rm LG}=(0.44\pm0.14)\sigma_{v,{\rm
H}}$. We conclude that the Local Group's velocity dispersion does not pose a
problem for $\Lambda$CDM and has a mass of $\log M_{\rm LG,
vir}/M_\odot=12.0^{+0.8}_{-2.0}$.
