Simple approximations to the positions of the Lagrangian points

The Roche potential is the sum of the gravitational and rotational potentials experienced by a massless body rotating alongside two massive bodies in a circular orbit. The Lagrangian points are five stationary points in the Roche potential. The positions of two of the Lagrangian points (L4 and L5) are fixed. The other three (L1, L2 and L3) are along the line joining the two masses: their positions depend on the mass ratio, $q$, and can be calculated numerically by finding the roots of a quintic polynomial. Analytical approximations to their positions are useful in several situations, but existing ones are designed for small mass ratios. We present new approximations valid for all mass ratios from zero to unity: \begin{eqnarray*} x_{\rm L1} & = & 1 - \frac{q^{0.33071}}{0.51233\,q^{0.49128} + 1.487864} \\ x_{\rm L2} & = & 1 + \frac{q^{0.8383} + 2.891\,q^{0.3358}}{1.525\,q^{0.848} + 4.046596} \\ x_{\rm L3} & = & -1 + \frac{q^{1.007}}{1.653\,q^{0.9375} + 1.66308} \end{eqnarray*} in a rotating frame of reference where the more massive body is at $x=0$ and the less massive body at $x=1$. The three approximations are precise to $6 \times 10^{-5}$ for all mass ratios.