rotation at $R\sim 10-20$ kpc can be approximated.

In either case, the Galaxy’s mass is by an order of magnitude smaller than the total mass required to gravitationally bind the Local Group. If we assume that the two galaxies are gravitationally bound to the Local Group, the total mass is estimated to be on the order of $M_{\rm tot} \sim V^2 R/G \sim 4.8 \times 10^{12}\Msun$. Here, the mutual velocity of the two galaxies is taken as $V \sim \sqrt{3} V_{\rm GC}$ with $V_{\rm GC}=-134$ for M31, and the radius of the orbit is $R=385$ kpc, where the factor $\sqrt{3}$ was adopted as a correction for the freedom of motion. It is required that the Local Group contains a dark matter core of mass comparable to this binding mass, $\sim 4.8 \times 10^{12}\Msun$. The mean mass density required to stabilize the Local Group is, thus, estimated to be $\rho_{\rm LG} \sim 2 \times 10^{-5}\Msun{\rm pc}^{-3}$.

We here define the boundary of the Galaxy as the radius, at which the density of the galactic dark halo becomes equal to the density of dark matter of the Local Group. The thus defined radius is $R_{\rm G}\sim 100$ kpc for the NFW model. 