Orbiting a Moon: the Restricted Three-Body Problem

Rotating frames, the Jacobi constant, Lagrange points and zero-velocity curves, Hill spheres of moons, and the Lidov–Kozai instability of polar orbits

Moscow, 1961

Mikhail Lidov, working on the orbits of the first artificial satellites, studied what the Moon's pull does to a satellite of the Earth over long times. He found something unexpected: a nearly circular orbit, tilted steeply enough relative to the Moon's orbit, does not stay circular. Its eccentricity grows, slowly and inexorably, while its inclination falls, until the orbit's low point dips into the atmosphere. A year later, Yoshihide Kozai found the same effect independently for asteroids perturbed by Jupiter. For a spacecraft around a moon, the perturbing body is the planet — and it is enormous and close. Around Europa the effect that took Lidov's satellites years takes weeks.

Around a planet, a spacecraft can be treated as a two-body problem with small perturbations. Around a moon it cannot: the planet is close and massive. This is the circular restricted three-body problem of the unit on Lagrange points again — the model that put Aditya-L1 around Sun–Earth L1 — with the planet as the big primary and the moon as the small one.

The rotating frame and units are the same: separation, total mass and angular velocity all equal 1. The moon has mass ratio

\mu = m/(m+M)

and sits at

x = 1-\mu

; the planet at

x = -\mu

. What changes is the size of

\mu

and of everything that depends on it: Europa's

2.5 \times 10^{-5}

is eight times the Sun–Earth value, and the moon's Lagrange points crowd in close to its surface.

\ddot x - 2\dot y = \frac{\partial U}{\partial x},\quad \ddot y + 2\dot x = \frac{\partial U}{\partial y},\quad \ddot z = \frac{\partial U}{\partial z},\qquad U = \frac{x^2+y^2}{2} + \frac{1-\mu}{r_1} + \frac{\mu}{r_2}

The 2ẏ and 2ẋ terms are the Coriolis acceleration; the (x²+y²)/2 in U is the centrifugal potential.

💡The Jacobi constant

As in the unit on Lagrange points, the Jacobi constant decides where the spacecraft may go:

C = 2U(x, y, z) - v^2 = \text{const}

Jacobi's integral (1836). Because v² ≥ 0, a spacecraft with constant C can only be where 2U ≥ C: the boundary 2U = C is a zero-velocity surface.

Lagrange points and gates. The five equilibrium points of the rotating frame are where

\nabla U = 0

. Three lie on the line through the two bodies; the two that bracket the moon, L1 (towards the planet) and L2 (away from it), are the gates. For large C the zero-velocity surface encloses the moon completely: a spacecraft inside can never leave. As C decreases, a neck opens first at L1 and then at L2.

The first lab finds L1 and L2 for every major moon by solving the equilibrium condition. They sit almost exactly at the Hill radius,

r_H = a(m/3M)^{1/3}

Europa: L1 13,563 km, L2 13,749 km from the moon's centre;
Titan: about 52,000 km;
Enceladus: 948 km — less than 700 km above its surface.

Body	Hill radius	Hill radius / body radius	Height of the Hill sphere above the surface
Moon (Earth)	61,500 km	35	59,800 km
Io	10,560 km	5.8	8,740 km
Europa	13,660 km	8.7	12,100 km
Ganymede	31,720 km	12.1	29,090 km
Titan	52,410 km	20.4	49,830 km
Enceladus	949 km	3.8	697 km
Triton	14,590 km	10.8	13,240 km
Vesta (Sun)	124,000 km	473	124,000 km
Ceres (Sun)	223,000 km	476	223,000 km

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📐Worked Example: Europa's gates

For Jupiter–Europa,

\mu = 2.528\times10^{-5}

, C(L1) = 3.003643 and C(L2) = 3.003609 (second lab).

A spacecraft in a 100 km circular orbit has C = 3.0106: both gates firmly shut.
To open the L1 gate takes a prograde burn of 421 m/s; the L2 gate, 423 m/s. The two-body escape burn from the same orbit is 575 m/s. In the three-body problem the spacecraft does not need to reach parabolic speed to leave — Jupiter's tide helps it out through the neck.
Started at periapsis 100 km up with C just above C(L1), the spacecraft is still bound after 60 days. With C = C(L1) − 10⁻⁴ it escapes in 1.6 days.

Run backwards in time, the last case is an arrival: a spacecraft that comes in through a barely open gate can loiter around the moon for many revolutions before leaving — temporary ballistic capture, the idea behind low-energy routes such as the one GRAIL and many lunar missions since have used to reach the Moon.

💡The Lidov–Kozai mechanism

Averaged over the spacecraft's orbit and the moon's orbit round the planet, the planet's tide makes eccentricity and inclination trade with each other while the component of angular momentum normal to the moon's orbit,

\sqrt{1-e^2}\cos i

, is conserved. Starting from a near-circular orbit, the exchange happens only if the inclination to the moon's orbit plane exceeds a critical value:

i_{crit} = \arccos\sqrt{3/5} = 39.2^\circ,\qquad e_{\max} = \sqrt{1 - \tfrac{5}{3}\cos^2 i_0}

Between 39.2° and 140.8° the eccentricity of an initially circular orbit grows to e_max. A polar orbit can reach e → 1.

The time scale is set by the ratio of the moon's orbital rate round the planet to the spacecraft's rate round the moon:

\tau \sim n_{sc}/n_M^2

. Around the Earth, for a satellite perturbed by the Moon, that is years. Around Europa, whose 'month' round Jupiter is 3.55 days, it is weeks.

The third lab integrates the averaged equations for a 200 km orbit (impact when

e > 1 - R/a = 0.114

inclinations up to 39°: eccentricity stays below 0.02 — the orbit survives;
90°: impact after 38 days; 60° and 120°: about 50 days;
150° and 180° (retrograde, low inclination): survive.

A full integration of Hill's equations for the polar orbit gives impact after 55 days, with peak eccentricity 0.115 — same mechanism, different initial phase. Low polar orbits about Europa simply do not last without frequent correction.

💡How real missions live with it

Three strategies exist. Fly by instead of orbiting (Europa Clipper). Orbit a big, distant moon where the effect is slow, such as Ganymede, which is larger and farther from Jupiter; JUICE plans a 5,000 km circular orbit and then 500 km. Use frozen orbits — carefully chosen eccentricity and argument of periapsis at which the moon's own J₂ and the planet's tide balance, so the orbit stays put for months — and pay for regular station-keeping. Most proposed Europa orbiter designs choose slightly eccentric, inclined frozen orbits rather than exactly polar circular ones.

🛰Enceladus: orbiting inside 700 km

Enceladus's Hill sphere reaches less than 700 km above its surface. Low orbits there are possible but strongly perturbed by Saturn, and high orbits do not exist. The Enceladus Orbilander concept study (2020), prepared for the 2023–2032 decadal survey, proposed spending its orbital phase on halo-like orbits associated with Saturn–Enceladus L1 and L2 — orbits that exist only because of the three-body dynamics — over the south polar plumes, before descending to land.

↓But there is a problem

Orbiting is one thing; landing is another. From a 100 km orbit around Europa, the surface is a kilometre and a half per second away — every metre per second of it removed by an engine, with no atmosphere to help, no runway, and no way for anyone on Earth to intervene in the twelve minutes it takes. On Titan it is the opposite. The air is thicker than Earth's, gravity is weaker than the Moon's, and the hard part is not stopping but getting through the heat of entry.