A massless test particle moves under the gravity of two massive primaries (masses 1−μ and μ) that orbit their barycentre in circular orbits. In the corotating frame, five equilibrium Lagrange points emerge: three collinear saddle points (L1–L3) and two stable triangular points (L4, L5).
Effective potential:
U* = -(x²+y²)/2 - (1-μ)/r₁ - μ/r₂
Equations of motion (corotating frame):
ẍ - 2ẏ = ∂U*/∂x
ÿ + 2ẋ = ∂U*/∂y
Jacobi integral (conserved):
C_J = 2·U*(x,y) - (ẋ²+ẏ²)
r₁ = √((x+μ)²+y²) [dist to m₁]
r₂ = √((x-1+μ)²+y²) [dist to m₂]
The JWST telescope orbits the Sun-Earth L2 Lagrange point, 1.5 million km from Earth, where it stays in a stable halo orbit. The Trojan asteroids cluster at Jupiter's L4 and L5, and are estimated to number over one million objects larger than 1 km. Mass transfer through the L1 point in binary stars is responsible for nova and Type Ia supernova explosions.
What is the restricted three-body problem?
The restricted three-body problem considers a massless test particle moving under the gravitational influence of two massive primary bodies that orbit their common centre of mass (barycentre) in circular orbits. Because the test particle has negligible mass, it does not affect the primaries' motion, greatly simplifying the equations while retaining rich dynamical complexity including chaos.
What are Lagrange points L1 through L5?
Lagrange points are the five equilibrium positions in the corotating reference frame where the net effective force on a test particle is zero. L1, L2, and L3 lie on the line joining the two primaries and are unstable saddle points. L4 and L5 form equilateral triangles with the primaries and are stable when the mass ratio is below the Routh critical value (~0.0385), explaining the Trojan asteroids at Jupiter's L4 and L5.
What is the Jacobi integral and why is it conserved?
The Jacobi integral C_J = 2U*(x,y) − v² is a conserved quantity in the restricted three-body problem, where U* is the effective potential in the rotating frame and v is the particle speed. It is conserved because the rotating-frame problem has a time-independent Hamiltonian. The Jacobi constant determines which regions of space are accessible to the particle; zero-velocity curves bound the allowed regions.
Zero-velocity curves are contours of the effective potential U* at which a particle with given Jacobi constant C_J would have zero kinetic energy. They define the boundaries of regions forbidden to the particle. As C_J decreases, the forbidden regions shrink; when C_J equals the potential at L1, the inner Roche lobe opens and mass transfer between the two primaries becomes possible.
A Roche lobe is the region around each star within which material is gravitationally bound to that star in the corotating frame. The two Roche lobes meet at the L1 Lagrange point. When an evolving star expands to fill its Roche lobe, mass spills through L1 onto its companion — a process called Roche lobe overflow, driving mass transfer in cataclysmic variables and X-ray binaries.
The three-body problem exhibits sensitive dependence on initial conditions because the equations of motion are nonlinear and have no general closed-form solution. Even in the restricted case, trajectories near the saddle points L1, L2, L3 are exponentially unstable — small differences in initial position lead to exponentially diverging trajectories, characteristic of positive Lyapunov exponents.
This simulation uses the classical fourth-order Runge-Kutta (RK4) method, which achieves O(h⁵) local truncation error per step. The Jacobi integral serves as a built-in accuracy check: a well-integrated trajectory should conserve C_J to within a small tolerance. For long-term orbital studies, symplectic integrators would be preferred because they conserve a modified Hamiltonian exactly.
The mass ratio parameter μ = m₂/(m₁+m₂) is the only free parameter in the restricted three-body problem (in units where total mass and orbital separation equal 1). It ranges from 0 (negligible secondary) to 0.5 (equal masses). Examples: Earth–Moon μ ≈ 0.0121, Sun–Jupiter μ ≈ 0.000953. Equal-mass binaries (μ = 0.5) have symmetric Roche lobes.
The corotating frame rotates at the same angular velocity as the two primaries, so the primaries appear stationary. In exchange, two fictitious forces appear: the centrifugal force and the Coriolis force. These give rise to the effective potential U* and the Coriolis coupling terms (ẍ−2ẏ and ÿ+2ẋ) in the equations of motion.
The restricted three-body problem describes: asteroid motion in the Sun–Jupiter system (Kirkwood gaps, Trojan asteroids); spacecraft trajectories at libration points (JWST at Sun–Earth L2, SOHO at L1); mass transfer in binary stars (cataclysmic variables, X-ray binaries); stability of moons in binary asteroid systems; and tidal disruption of stellar clusters near galactic centres.