A Lagrange point is an equilibrium of the circular restricted three-body problem in the frame rotating with the two massive bodies.
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 lie on the line through the two massive bodies.
In barycentric coordinates with unit primary separation, a small secondary mass fraction gives distances 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 .
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 . 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 and , where at . The characteristic polynomial is . The small-secondary limit has .
For a small secondary-to-primary mass ratio, the unstable exponent at the outer collinear equilibrium is . Its inverse is approximately binary periods. Reaching a finite target distance takes this time multiplied by the logarithm of the ratio of final to initial unstable amplitudes.
The Lagrange point lies beyond the more massive primary, on the side opposite the secondary.
Articles by others on the same topic
A Lagrange point is a position in space where the gravitational forces of two large bodies, such as a planet and a moon or a planet and the sun, balance out the centripetal force experienced by a smaller body. This results in a stable or semi-stable location where the smaller body can maintain a position relative to the two larger bodies, effectively "parking" in that location.