since the distance to the nearest external bright galaxy is much greater than that between M31 and the Milky Way, these forces are presumably dominated by material associated with the two spirals themselves.

@kw59 set up a simple model to analyse this situation – two point masses on a radial orbit. These were at pericentre (i.e. at zero separation) at the Big Bang and must have passed through apocentre at least once in order to be approaching today. Clearly this requires an apocentric separation larger than the current separation and an orbital period less than twice the current age of the Universe. Together these requirements put a lower limit on on the total mass of the pair. A more precise estimate of the minimum possible mass is obtained from the parametric form of Kepler’s laws for a zero angular momentum orbit: $$r=a(1-\cos\chi)$$

$$t=\bigg(\frac{a^{3}}{GM}\bigg)^{1/2}(\chi-\sin\chi)$$

$$\frac{dr}{dt}=\sqrt{\frac{GM}{a}}\frac{\sin\chi}{1-\cos\chi}$$

where $r$ is the current separation, $dr/dt$ is the current relative velocity, $a$ is the semi-major axis, $\chi$ is the eccentric anomaly, $t$ is the time since the Big Bang (the age of the universe) and $M$ is total mass [@lb81]. Given observationally determined values for $r$, $dr/dt$ and $t$, these equations have an infinite set 