range of $500-1,000{\rm \,kpc}$ and with negative relative radial velocities. (Note that this is the true relative velocity rather than the relative peculiar velocity, i.e. we have added the Hubble expansion to the relative peculiar velocity and have required the result to be negative.) We identify isolated pairs by keeping only those which have no “massive” companion, defined as a galaxy with $V_{max} \ge
150{{\rm \,km\,s^{-1}}}$, within a sphere of 1 Mpc radius centred on the mid-point of the binary, and no nearby cluster, defined as a halo with $M_{200}>3\times
10^{13}{\rm \,M_{\odot}}$ within 3 Mpc of the mid-point of the binary. These cuts ensure that the dynamics are dominated by mass associated with the two main systems, as appears to be the case for the Local Group. For galaxies selected in the narrower $V_{max}$ range we then find $178$ pairs when the morphology cut is applied and $1,128$ pairs when it is not. For the wider $V_{max}$ range the corresponding numbers are $2,815$ pairs and $16,479$ pairs respectively.

When calibrating the TA estimator it proves advantageous to use simulated pairs with dynamical state quite close to that of the real Local Group. As we will see below, this eliminates some 