of discrete solutions for $\chi$, $a$ and $M$ labelled by the number of apocentric passages since the Big Bang. The solution corresponding to a single apocentric passage gives the smallest (and only plausible) estimate for the mass, which is about $5\times 10^{12}{\rm \,M_{\odot}}$ for current estimates of $r$, $dr/dt$ and $t$. Note that this is still a lower limit on the total mass, even within the simple point-mass binary model, since any non-radial motions in the system would increase its present kinetic energy and so increase the mass required to reverse the initial expansion and bring the pair to their observed separation by the present day [see @el82].

As @kw59 realised, this timing estimate of the total mass of the Local group exceeds by more than an order of magnitude the mass within the visible regions of the galaxies, as estimated from their internal dynamics, in particular, from their rotation curves. Thus 90% of the mass must lie outside the visible galaxies and be associated with little or no detectable light. Modern structure formation theories like $\Lambda$CDM predict this mass to be in extended dark matter halos with $M(r)$ increasing very roughly as $r$ out to the point where the 