System
Test Particle
Orbit Presets
Visualization
Animation
Jacobi Constant
About Lagrange Points
- L1, L2, L3: Collinear, unstable
- L4, L5: Triangular, stable if μ < 0.0385
- Applications: JWST (L2), Trojans (L4/L5)
Explore the circular restricted three-body problem—find Lagrange points and halo orbits in the rotating frame
The circular restricted three-body problem (CR3BP) is a fundamental model in celestial mechanics. Two massive bodies (primaries) orbit their common center of mass in circular orbits, and we study the motion of a third body (test particle) with negligible mass that doesn't affect the primaries. By working in a rotating reference frame that co-rotates with the two primaries, we reduce the problem to an autonomous system with a single conserved quantity: the Jacobi constant.
In the rotating (synodic) frame, the two primaries remain fixed on the x-axis. The origin is at the barycenter, the larger primary (mass 1-μ) sits at x = -μ, and the smaller primary (mass μ) sits at x = 1-μ. The frame rotates with angular velocity ω = 1 (in normalized units where the orbital period is 2π and the separation is 1). This choice of coordinates eliminates the time dependence of the primaries' positions, at the cost of introducing fictitious forces: centrifugal and Coriolis.
The test particle experiences gravitational attractions from both primaries plus the centrifugal and Coriolis forces. The equations of motion are:
ẍ - 2ẏ = ∂Φₑff/∂x ÿ + 2ẋ = ∂Φₑff/∂y
where the effective potential Φₑff = -(1-μ)/r₁ - μ/r₂ - ½(x²+y²). The terms 2ẏ and -2ẋ are the Coriolis force, and the ½(x²+y²) contribution to the potential gives the centrifugal force. This effective potential determines the "landscape" in which the particle moves.
Setting ẍ = ÿ = ẋ = ẏ = 0, we find five equilibrium points where a test particle can remain stationary in the rotating frame (co-rotating with the primaries):
The James Webb Space Telescope orbits Sun-Earth L2, staying in Earth's shadow for thermal stability while maintaining constant Earth-Sun-spacecraft geometry for communications. SOHO and WMAP also used L1 and L2. The instability of these points requires active control, but the required ΔV is small—these orbits are "marginally unstable." Halo orbits are periodic three-dimensional orbits around L1 or L2, staying out of the primaries' shadow.
Despite the non-conservative Coriolis force, the CR3BP has an integral of motion:
C = 2Φₑff(x, y) - (ẋ² + ẏ²)
This is the Jacobi constant (also called Jacobi integral). It combines the effective potential energy and kinetic energy in the rotating frame. For a given initial condition, C remains constant throughout the motion. The Jacobi constant determines which regions of space are accessible to the particle.
At any point where the kinetic energy vanishes (ẋ² + ẏ² = 0), we have C = 2Φₑff. The curve defined by 2Φₑff(x,y) = C is the zero-velocity curve (ZVC). The particle cannot cross into regions where 2Φₑff < C, because that would require negative kinetic energy. As C decreases (higher energy), forbidden regions shrink and new regions become accessible. At certain critical values of C, the topology changes—necks open up allowing transitions between previously disconnected regions. This controls the possibility of escapes, captures, and transfers between the primaries.
The CR3BP possesses rich families of periodic orbits. Halo orbits around L1 and L2 are used for spacecraft missions. Lyapunov orbits are planar periodic orbits around L1, L2, and L3. Distant retrograde orbits (DROs) are highly stable and being considered for lunar Gateway station. Vertical Lyapunov orbits oscillate perpendicular to the orbital plane. Near L4 and L5, tadpole and horseshoe orbits exist—Jupiter Trojans follow tadpoles, while Saturn's moon Janus and Epimetheus execute a horseshoe co-orbital dance.
Away from special periodic orbits, trajectories in the CR3BP are often chaotic. Small changes in initial conditions lead to exponentially diverging paths. This sensitivity is quantified by positive Lyapunov exponents. Invariant manifolds—stable and unstable manifolds of periodic orbits—form a "highway network" structuring phase space. These manifolds provide low-energy pathways for transfers between moons (e.g., the Interplanetary Superhighway concept). Genesis and other missions exploited these manifolds for fuel-efficient trajectories.
Visualize Lagrange Points: Start with the Earth-Moon preset (μ = 0.0121). The five Lagrange points are marked on the canvas. L1 lies between Earth and Moon, L2 beyond the Moon, L3 on the opposite side of Earth. L4 and L5 form equilateral triangles. Try placing a particle near each point and observe stability.
L1 Halo Orbit: Click "L1 Halo" preset. You'll see a periodic orbit looping around the L1 point. This orbit stays in the rotating frame—in an inertial frame it traces a 3D path around the Earth-Moon line. The James Webb Space Telescope uses an L2 halo orbit in the Sun-Earth system. Notice the Jacobi constant remains fixed.
L4 Stability: Click "L4 Stable". The particle orbits near the L4 point in a bounded region. This is the tadpole orbit characteristic of Trojan asteroids. The orbit is stable because Earth-Moon has μ ≈ 0.0121 < 0.0385. Now increase μ to 0.1 (sliding the mass ratio slider) and reset—L4/L5 become unstable and the particle escapes.
Zero-Velocity Curves: Enable "Show Zero-Velocity Curves". The dark contours show forbidden regions for the current Jacobi constant. Place a particle near L1 with low velocity—observe how it's confined by the ZVC. Increase velocity (increase vy0) and reset: the ZVC opens up, allowing the particle to explore wider regions or escape.
Mass Ratio Effects: Change μ from 0.0121 (Earth-Moon) to 0.001 (Sun-Earth) to 0.0009537 (Sun-Jupiter). As μ decreases, the smaller primary becomes negligible and L1/L2 move closer to it. With μ → 0, the CR3BP reduces to a Kepler orbit around the primary. With μ = 0.5 (equal masses), symmetry changes—L1 sits exactly at the origin.
Effective Potential: Enable "Show Effective Potential". The background coloring represents Φₑff—bright regions are high potential (hilltops in the energy landscape), dark regions are low (valleys). The primaries are deep potential wells. L1, L2, L3 are saddle points—peaks along one axis, valleys along another. L4 and L5 are hilltops.
Chaotic Trajectory: Click "Figure-8" or manually place a particle near but not on a periodic orbit. Watch the trajectory wander chaotically through accessible regions, repeatedly approaching and receding from the primaries. The path never repeats exactly—evidence of deterministic chaos. Even though the Jacobi constant is conserved, the motion is unpredictable.
Escapes and Captures: Start a particle near one primary with moderate velocity. If the Jacobi constant is low enough (high energy), the ZVC opens a "neck" through which the particle can escape to the other primary or to infinity. Adjust initial conditions to find capture orbits (inertially unbound but temporarily trapped) and escape trajectories.
Coriolis Effect: Notice the particle's trajectory curves even in "straight" motion. This is the Coriolis force, -2ω × v, deflecting the particle perpendicular to its velocity. In the rotating frame, it appears as a force; in the inertial frame, it's just the centripetal acceleration. Try setting ẋ = 0, ẏ ≠ 0 and watch the particle curve.
Sun-Jupiter Trojans: Switch to Sun-Jupiter preset (μ ≈ 0.000954). The Trojan asteroids orbit near L4 and L5. Click "L4 Stable" to see a tadpole orbit. Thousands of real asteroids occupy these regions, librating (oscillating) around the Lagrange points with amplitudes up to ~30°. They're named after heroes of the Trojan War.
The CR3BP is the simplest model exhibiting the rich dynamics of multi-body gravitational systems. It bridges the exactly-solvable two-body problem and the chaotic full three-body problem. Key applications:
The CR3BP was studied by Euler, Lagrange, Jacobi, Poincaré, and Hill—it's a cornerstone of dynamical systems theory. Poincaré's work on this problem laid foundations for chaos theory and topology. Today, it remains essential for spacecraft navigation, asteroid dynamics, and understanding the architecture of multi-planet systems. Despite being "restricted" and "circular," this idealized model captures the essence of three-body gravitational choreography.