Science lab / orbital mechanics

The restricted three-body problem: order inside gravitational complexity.

Follow a field of massless test particles through the gravity of two orbiting bodies, reveal the rotating effective potential, and compare stable Lagrange motion with unstable saddles.

Orbital field note

Three bodies, one useful restriction

Newton's two-body problem has an exact closed-form solution: two isolated masses trace conic sections around their common barycentre. Add a third massive body and that general simplicity disappears. The restricted three-body problem regains useful structure by assuming that the third body is so small that it responds to the two primaries without measurably changing their motion.

This instrument uses the planar circular restricted three-body problem (PCR3BP). The primaries follow circular orbits, every trajectory lies in one plane, and the test particles have zero dynamical mass. The model is idealised, but it captures resonant motion, transfer corridors, collision and escape, Trojan libration, and the five equilibrium locations called Lagrange points.

What “restricted” means, and what it does not mean

It is the third body's influence that is restricted, not its freedom of motion. A particle can still collide, escape, orbit either primary, or move chaotically. The two primaries do not react to it, and particles do not attract one another. The overlays show the ideal model's Lagrange points and rotating effective potential alongside the simulated paths.

From Newton's law to the rotating equations

In an inertial frame, a test particle at position \(\mathbf r\) feels the sum of the two Newtonian accelerations:

\[ \ddot{\mathbf r} = -Gm_1\frac{\mathbf r-\mathbf r_1}{\lVert\mathbf r-\mathbf r_1\rVert^3} -Gm_2\frac{\mathbf r-\mathbf r_2}{\lVert\mathbf r-\mathbf r_2\rVert^3}. \]

Choose units in which the primary separation, total mass, and angular speed are all one. Define the mass parameter \(\mu=m_2/(m_1+m_2)\). In barycentric rotating coordinates the primaries remain fixed at \((-\mu,0)\) and \((1-\mu,0)\), with distances

\[ r_1=\sqrt{(x+\mu)^2+y^2},\qquad r_2=\sqrt{(x-1+\mu)^2+y^2}. \]

Gravity and the centrifugal contribution combine into the effective potential

\[ \Omega(x,y)=\frac{x^2+y^2}{2}+\frac{1-\mu}{r_1}+\frac{\mu}{r_2}. \]

Differentiating \(\Omega\) gives the rotating-frame equations. The terms containing the velocities are the Coriolis acceleration:

\[ \ddot{x}-2\dot{y}=\frac{\partial\Omega}{\partial x},\qquad \ddot{y}+2\dot{x}=\frac{\partial\Omega}{\partial y}. \]

The Jacobi integral

The rotating problem does not conserve ordinary inertial energy, but it has one integral of motion. The Jacobi constant combines effective potential and rotating-frame speed:

\[ C=2\Omega(x,y)-\left(\dot{x}^2+\dot{y}^2\right). \]

Because speed squared cannot be negative, a chosen value of \(C\) divides the plane into allowed and forbidden regions. Their boundaries are zero-velocity curves. As \(C\) changes, narrow necks open near L1, L2, and L3; those openings are central to low-energy spacecraft transfers and escape pathways. The deep explorer's contour toggle shows levels of \(\Omega\), not one selected zero-velocity curve.

Five stationary solutions in the rotating frame

Lagrange points satisfy \(\nabla\Omega=0\). L1, L2, and L3 lie on the line through the primaries and must generally be found numerically. The triangular points have exact normalised coordinates:

\[ L_4=\left(\frac{1}{2}-\mu,\frac{\sqrt3}{2}\right),\qquad L_5=\left(\frac{1}{2}-\mu,-\frac{\sqrt3}{2}\right). \]

L1–L3 have linearly unstable saddle directions: generic small disturbances grow. L4 and L5 are linearly stable only when the mass ratio is sufficiently small,

\[ \mu < \mu_R=\frac12\left(1-\sqrt{\frac{23}{27}}\right)\approx0.03852. \]

Small motions near a stable L4 or L5 can librate around that point on tadpole paths. Larger horseshoe paths can encompass L3, L4, and L5 in the rotating frame. Stability does not imply attraction: without dissipation, objects do not spiral into the exact point. The simulator starts with a small Trojan population already near L4 and L5.

Where the model appears in nature and engineering

  • Sun–Jupiter: a large population of known Trojan asteroids shares Jupiter's orbit in leading L4 and trailing L5 swarms. NASA's Lucy mission is designed to visit objects in both swarms.
  • Sun–Earth L1: solar-monitoring spacecraft use orbits around the inner equilibrium region, maintaining an uninterrupted view toward the Sun.
  • Sun–Earth L2: the James Webb Space Telescope follows a halo orbit around L2 rather than sitting at the mathematical point.
  • Saturn's moons: small moons such as Telesto and Calypso share Tethys's orbit near its triangular Lagrange regions.
  • Mission design: invariant manifolds associated with periodic orbits near unstable Lagrange points provide low-energy transport routes through multi-body systems.

Reading this simulator

This is a qualitative numerical demonstration, not a precision ephemeris. The trajectory integrator uses a small gravitational-softening length to avoid singular accelerations during very close approaches. The equations, Lagrange-point locations, effective-potential contours, and linear-stability guides describe the ideal unsoftened PCR3BP, so they agree closely away from the primaries but are not exact representations of the softened force near a collision.

The primary is held visually at the centre while the calculation includes the indirect acceleration of that moving origin. Thin lines show the rotating effective potential. Faint green and red clouds provide a qualitative, position-only guide derived from local linearised growth and curvature. They are not boundaries of stability, which also depends on velocity and the Jacobi constant. As the mass ratio approaches the Routh threshold, the green L4/L5 guide contracts and weakens. Beyond it, the same points appear as unstable red structures. Closed markers at L4/L5 denote linear stability for the selected mass ratio; hyperbolic markers at L1–L3 denote unstable saddle structure. Fast-forward watches for an unchanged survivor count; this does not establish that the remaining orbits are stable. Particles are removed at the model's collision radii or outer cutoff, with an additional approximate escape check for outward-moving particles.

The co-rotating view holds both primaries still without changing the integration. A placed probe begins at rest in that rotating frame, making its local trajectory a direct test of nearby motion. Multiple probes are compared by superimposing their placement points, so each trace shows displacement from its own origin. The trajectory graph can instead use the primary body as one shared origin, showing every probe's co-rotating position across the full simulation extent.

References and further reading

Updated · factual and copy review

Deep instrument

Restricted orbital workbench

Place a probe and compare its path in non-rotating and co-rotating views.

System running

Placed probes start at rest in the co-rotating frame. Matching labels and colours identify them on both maps. The minimap superimposes every placement origin for direct displacement comparison.

Under the hood

The Mathematics Used in This Simulation

Updated

The workbench integrates massless test particles under the gravity of two circularly orbiting primaries. The article above gives the derivation; this disclosure summarises exactly what the numerical instrument assumes.

Acceleration model

a = -m1(r-r1)/(|r-r1|2 + ε2)3/2 - m2(r-r2)/(|r-r2|2 + ε2)3/2

This equation gives the softened gravitational acceleration in an inertial frame, with G = 1. The code uses a non-rotating origin attached to the primary, so it also subtracts that primary's acceleration towards the secondary. The primary separation is one and the angular speed is √(m1 + m2).

Rotating-frame quantities

Ω(x,y) = (x2 + y2)/2 + (1 - μ)/r1 + μ/r2

C = 2Ω - (vx2 + vy2)

Lagrange points and effective-potential overlays are computed from the ideal unsoftened equations, so the overlay is explanatory rather than a precision ephemeris.

Assumptions and limits

Motion is planar, the primaries follow circular orbits, test particles have zero mass, particles do not attract one another, and no relativistic, atmospheric, radiation, or non-spherical gravity effects are included.

Further mathematics

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