Lagrange point

A Lagrange point is a stationary position in the rotating frame of the circular restricted three-body problem. The two primaries' gravitational forces and the centrifugal force balance there. Their gravitational pulls alone need not cancel.

Three Collinear Lagrange points lie on the line through the primaries; two Triangular Lagrange points complete equilateral triangles with them. Stability requires the full rotating-frame equations, including the Coriolis force. The collinear equilibria are unstable, while the triangular equilibria are linearly stable for a sufficiently small secondary mass fraction. Spacecraft can use nearby trajectories with stationkeeping; an equilibrium calculation alone does not make a spacecraft remain there.^[ref]

The rotating-frame effective potential of the circular restricted three-body problem has negative singular wells at both primaries. Its three Collinear Lagrange points are planar saddles, and its two Triangular Lagrange points are planar local maxima for nonzero primary masses. These geometric labels do not replace a dynamical stability analysis including the Coriolis force.

The three collinear Lagrange points $L_1,L_2,L_3$ lie on the line through the two massive bodies.

In barycentric coordinates with unit primary separation, a small secondary mass fraction $\alpha$ gives distances $(\alpha /3{)}^{1/3}$ from that secondary to the two neighboring Lagrange points at leading order. This is the dimensionless Hill radius. The opposite exterior L3 Lagrange point has coordinate $-1-5\alpha /12+O({\alpha }^2)$.

The Lagrange point lying between the components of a circular restricted three-body problem. The critical effective potential through this saddle defines their Roche lobes; gas passing through it can initiate Roche-lobe overflow.

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.

The $L_3$ Lagrange point lies beyond the more massive primary, on the side opposite the secondary.

The triangular Lagrange points $L_4,L_5$ complete equilateral triangles with the two massive bodies.