Circular restricted three-body problem

In the circular restricted three-body problem the two massive bodies move on circular orbits about their barycentre. The problem is autonomous in their uniformly rotating frame.

The unforced planar equation of motion in a rotating frame near a circular Kepler orbit has components $\ddot{x}-2n\dot{y}-3n^{2}x=0$ and $\ddot{y}+2n\dot{x}=0$. They describe epicyclic motion and Keplerian shear; mutual gravity adds close-encounter forcing.

The oscillatory integration constants determine the orbital eccentricity vector, the constant radial offset determines the semi-major axis, and the constant along-track offset fixes the mean longitude. The radial and tangential epicycle amplitudes have ratio $1:2$.

The photogravitational restricted three-body problem adds radial radiation pressure from one or both luminous primaries. If the pressure-to-gravity ratio of primary $i$ is ${\beta }_i$, its effective attraction on the test particle is multiplied by $1-{\beta }_i$.

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 $L_1,L_2,L_3$ 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.

The Hill sphere is the neighborhood of a smaller body in which its gravity can dominate tidal perturbations from a larger primary.

For a body of mass $m$ on a circular orbit of radius $a$ about mass $M\gg m$, the Hill radius is approximately

The Jacobi constant is the conserved energy-like integral of the circular restricted three-body problem in the rotating frame.