Hill Sphere and Sphere of Influence

1. The question: why doesn't the Sun steal the Moon?

The Sun's gravitational pull on the Moon is actually about twice as strong as Earth's — a fact that surprises most people the first time they compute it. Yet the Moon has orbited Earth stably for over four billion years. The resolution isn't about the strength of a single force; it's about the difference between how the Sun pulls on the Moon versus how it pulls on Earth. Because the Moon and Earth are close together compared to their distance from the Sun, the Sun pulls on both of them almost equally — and it's that near-cancellation, not the raw force, that determines whether Earth can keep the Moon.

This is exactly the same restricted three-body setup that produces Lagrange points: a small body (the Moon) in the combined field of two large bodies (Earth and the Sun) whose own orbit around each other we treat as fixed for the purpose of the calculation.

2. The Hill sphere formula

The Hill sphere (named after American astronomer George William Hill, who derived it in 1878 while studying the Moon's motion) is the region around a smaller body m — orbiting a larger body M at distance a — within which m's own gravity dominates over the tidal pull of M:

Notice the cube root: because tidal forces fall off as 1/a³ (the difference of two 1/r² forces separated by a small baseline), the Hill radius scales gently with mass ratio — even a 1000× mass difference only shrinks the sphere by a factor of ten.

/** Hill sphere radius. All lengths in the same unit (e.g. AU). */
function hillRadius(a, m, M) {
  return a * Math.cbrt(m / (3 * M));
}
// Earth around the Sun: a = 1 AU, m = 3.0e-6 M_sun
hillRadius(1, 3.0e-6, 1);  // ≈ 0.0098 AU ≈ 1.47 million km

3. Why the Hill radius equals the L1/L2 distance

The Hill sphere is not a separate concept from Lagrange points — it's the same physics viewed from a different angle. In the circular restricted three-body problem, the distance from the smaller mass m to the L1 and L2 points is given, to leading order in the small mass ratio μ = m/M, by:

This is not a coincidence: L1 and L2 sit almost exactly on the boundary of the Hill sphere, along the line connecting the two large bodies. Inside that boundary, an orbiting test particle feels m's gravity more strongly than the tidal stretching from M; at L1/L2 the two effects balance. The Hill sphere is, in effect, the three-dimensional generalisation of the one-dimensional L1–L2 distance.

4. Hill radii across the Solar System

Jupiter's enormous Hill sphere is why it hosts dozens of stable moons and can capture passing asteroids as temporary "moons" for decades — its sphere of dominance is larger than most planets' entire orbits around the Sun.

5. The patched-conic sphere of influence

For mission-design purposes, engineers often use a simpler, slightly different formula: the sphere of influence (SOI), used in the classic patched-conic approximation for interplanetary trajectories (the method NASA used for early missions before full numerical integration became routine):

The SOI answers a slightly different question than the Hill sphere: instead of "where does tidal stretching overwhelm local gravity," it asks "where is the ratio of perturbing acceleration to primary acceleration equal from both bodies' point of view" — a definition tuned for patching together two-body Kepler arcs rather than describing long-term orbital stability.

6. Moon stability and the practical safety margin

A moon sitting exactly at the Hill radius is not stable in practice — perturbations from the moon's own eccentricity, solar radiation pressure, and other planets nudge it around. Numerical simulations of the restricted three-body problem show that stable circular orbits require staying within roughly 1/3 to 1/2 of the Hill radius:

This safety margin explains observed moon systems: even Jupiter's farthest confirmed irregular moons orbit well inside its Hill sphere, and no natural satellite in the Solar System is found orbiting near the theoretical edge. The same margin is why mission designers place spacecraft in "halo orbits" around L1/L2 rather than sitting exactly at the equilibrium point — any orbit that approaches the true boundary of gravitational dominance becomes progressively harder to maintain without station-keeping.