L2 Lagrange point ($L_2$)

The collinear Lagrange point beyond the smaller primary, away from the larger one, is $L_2$. For a small mass ratio it lies approximately one Hill radius from the secondary and has a growing linear mode.

In unit binary separation and mean motion, displacements obey $\ddot{X}-2\dot{Y}=(1+2B)X$ and $\ddot{Y}+2\dot{X}=(1-B)Y$, where $B={\mu }_1/r_1^3+{\mu }_2/r_2^3$ at $L_2$. The characteristic polynomial is ${\lambda }^4+(2-B){\lambda }^2+(1+2B)(1-B)$. The small-secondary limit has $B\to 4$.

For a small secondary-to-primary mass ratio, the unstable exponent at the outer collinear equilibrium is $n_{\mathrm{bin}}\sqrt{1+2\sqrt{7}}$. Its inverse is approximately $0.0634$ binary periods. Reaching a finite target distance takes this time multiplied by the logarithm of the ratio of final to initial unstable amplitudes.