The distance at which a moon stops holding together

Two things hold a moon together and pull it apart, and they scale differently with distance. That difference decides everything.

A tide is a difference of gravity across a body, and it falls as the inverse cube of the distance to the source. A body’s own gravity at its own surface does not depend on how far away anything else is; it does not fall with distance at all. So a graph of the two against distance is a steep curve and a horizontal line, and a steep curve and a horizontal line cross exactly once.

Inside the crossing, the tide is stronger than the body’s own grip and the body comes apart. Outside it, the body wins and stays whole. That crossing is the Roche limit, and the striking thing about where it falls is that it does not depend on how big either body is.

The comparison, done twice #

Take a satellite of mass $m$ and radius $r$, at distance $d$ from a primary of mass $M$. Consider a small piece of the satellite sitting on the near face.

The primary pulls that piece slightly harder than it pulls the satellite’s centre, because it is closer by $r$. Expanding the inverse square,

$$a_{\text{tide}} = \frac{GM}{(d-r)^2} - \frac{GM}{d^2} \approx \frac{2GMr}{d^3}.$$

The satellite’s own gravity pulls the same piece back toward the centre with

$$a_{\text{self}} = \frac{Gm}{r^2}.$$

Setting them equal and solving for $d$:

$$d = r\left(\frac{2M}{m}\right)^{1/3}.$$

That is already the answer, and it is worth pausing on what it contains. It has the satellite’s radius in it, which seems to say that a big moon is torn apart further out than a small one — and that is wrong, in a way the second version of the calculation fixes.

Write both masses as density times volume, $M = \tfrac{4}{3}\pi R^3 \rho_M$ and $m = \tfrac{4}{3}\pi r^3 \rho_m$. The radii cancel, all of them:

$$d = R\left(\frac{2\rho_M}{\rho_m}\right)^{1/3} \approx 1.26\,R\left(\frac{\rho_M}{\rho_m}\right)^{1/3}.$$

The Roche limit depends on the primary’s radius and on the ratio of the two densities, and on nothing else at all. A pebble and a thousand-kilometre moon of the same material come apart at the same distance. Mass has dropped out because both effects scale with it in the same way — the tide with the primary’s mass, the self-gravity with the satellite’s — and size has dropped out because both scale with the satellite’s radius identically.

Rigid and fluid: a factor of about two #

The calculation above treats the satellite as a rigid sphere: a body that keeps its shape right up to the moment it fails. Real bodies held together by gravity alone do not do that. As they approach the primary, the tide stretches them into a prolate shape along the line to it, and a stretched body is worse at holding itself together — the near end is now further from the centre of mass, so the tide on it is larger and the self-gravity smaller.

Working the equilibrium through for a fluid body gives a larger limit:

$$d_{\text{fluid}} \approx 2.44\,R\left(\frac{\rho_M}{\rho_m}\right)^{1/3},$$

a factor of 1.93 further out than the rigid one. Édouard Roche derived this in 1848, and the coefficient 2.44 is the number that carries his name.

The two limits bracket reality. A body with genuine material strength — a rock rather than a rubble pile — survives inside the fluid limit, and can survive inside the rigid one too if it is small enough that its strength beats its own weight, which for rock is true up to a few hundred metres. A loosely bound aggregate behaves as a fluid. So the region between $1.26R$ and $2.44R$ is not a boundary but a zone, and where a particular body falls in it says something about what it is made of.

What was actually measured #

Saturn’s ring system is where this stops being an argument and becomes an observation.

Saturn’s equatorial radius is 60,268 km and its mean density is 0.687 g/cm³ — the lowest of any planet, and lower than water. The rings are almost pure water ice, with a bulk density for a loosely bound body of about 0.9 g/cm³. Those three numbers give a fluid Roche limit of

$$2.44 \times 60{,}268 \times (0.687/0.9)^{1/3} = 134{,}800\ \text{km},$$

or 2.23 Saturn radii.

The outer edge of the A ring — the sharp outer boundary of the main ring system — sits at 136,775 km, which is 2.27 radii. The prediction and the observation agree to 1.5%, on a quantity computed from a density ratio and a cube root.

The agreement is close enough to invite over-reading, so two cautions are worth stating.

The first is that the A ring’s outer edge is also the location of a 7:6 resonance with Janus, and resonances confine ring edges sharply. So the edge is held where it is by a resonance that happens to sit almost exactly where the Roche limit falls. Which of the two is doing the work is a real question, and the honest answer is that the resonance sharpens an edge that the Roche limit put roughly there. The second is the exceptions. Pan orbits at 133,584 km — 2.216 radii, inside the fluid limit — and it is a moon, not a ring. It survives because it has some cohesion, because its density is genuinely low (about 0.42 g/cm³, which pushes its own limit inward), and because it sits in the Encke Gap, sweeping up material rather than being torn from it. Atlas at 2.28 radii and Prometheus at 2.31 are similar cases. Every one of them is unusually low in density and unusually elongated — Pan and Atlas both have equatorial ridges that make them look like walnuts, which is what happens when ring material accretes onto a body sitting at the edge of the zone where accretion is barely possible.

That is the observation doing more than confirming the formula. The bodies near the limit are visibly different from bodies elsewhere, and they are different in the direction the theory says they should be.

What the tide does long before it wins #

The Roche limit is the point at which the tide destroys a body. Everything between there and infinity is the tide doing smaller things to it, and those smaller things are why most of the solar system’s moons are the shape and temperature they are.

The tidal field is a stretch along the line to the primary and a squeeze across it, and the two are in the ratio 2:1 exactly, because that is the trace-free structure of the second derivative of an inverse-square potential. A body sitting in it deforms into a prolate ellipsoid whose long axis points at the primary — the same shape the fluid calculation above uses, just much less extreme.

If the body rotates, that bulge has to travel around it, and the flexing dissipates energy. Two consequences follow, and both are visible.

The first is tidal locking. The dissipation torques the body until its rotation matches its orbit and the bulge stops moving — which is why the Moon keeps one face toward the Earth, and why every large inner moon of every giant planet does the same. The timescale goes as $d^6$, so a moon at half the distance locks sixty-four times faster, and a moon near the Roche limit locks essentially at formation.

The second is tidal heating. If the orbit is eccentric the flexing does not stop after locking, because the tide’s strength and direction still vary through each orbit. Io is held at $e = 0.0041$ by a resonance with Europa and Ganymede, sits at 5.9 Jupiter radii, and dissipates about $10^{14}$ watts — a hundred times the Earth’s entire geothermal output, from a body a quarter of the Earth’s diameter. It is the most volcanically active object in the solar system, and the energy comes from the orbit rather than from any internal source.

Both dissipations are paid for out of the orbit, so the orbit moves: outward for a moon that goes round more slowly than its planet turns, inward for one that goes round faster.

The Roche limit is not the Hill sphere #

Two distances in this subject are both cube roots of a mass ratio, both describe a region around a body, and are constantly confused. They answer opposite questions.

The Roche limit is a distance from the primary, and it asks: how close can a satellite come before the primary pulls it apart? The Hill sphere is a radius around the satellite, and it asks: how far from the satellite can something orbit before the primary takes it away?

$$r_{\text{Hill}} = a\left(\frac{m}{3M}\right)^{1/3}.$$

The Earth’s Hill radius is 1.5 million kilometres, and the Moon at 384,000 km is comfortably inside it — which is the condition for the Moon to be the Earth’s moon rather than an independent body on a similar solar orbit. The Earth’s Roche limit for a body of lunar density is about 18,400 km, and the Moon is well outside that, which is the condition for it to be one object rather than a ring.

Both conditions have to hold, and they bound the Moon’s distance from below and above. The same pair of conditions bounds every satellite in the solar system, and the ratio between the two — roughly 80 for the Earth–Moon system — is the width of the window in which a moon can exist at all. It is a wide window, which is why moons are common. It is not infinitely wide, and the sphere of influence that trajectory design uses is a close relative of its outer edge.

Where the limit does not apply, and the comet that proved it #

Two famous cases are usually offered as evidence for the Roche limit, and only one of them is.

Shoemaker–Levy 9 passed within 1.3 Jupiter radii in July 1992 and was torn into the string of fragments that struck the planet two years later. That is well inside any version of the limit, the object was a loosely bound comet nucleus, and the disruption was observed directly. It is as clean a demonstration as the solar system provides.

Phobos, orbiting Mars at 2.76 Mars radii, is often described as being inside its Roche limit and doomed. It is not inside it — Mars’s density is 3.93 and Phobos’s about 1.87, giving a fluid limit near 2.44 × 1.28 = 3.13 radii, so Phobos at 2.76 is inside, but it is visibly not disintegrating. It survives because it has cohesion: its grooves are widely interpreted as early tidal failure, and the current estimate is that a few kilopascals of tensile strength is enough. Phobos is a body in the zone between the rigid and fluid limits, held together by being slightly more than a pile of gravel. It is also spiralling inward from tidal drag and will reach the rigid limit in some tens of millions of years, at which point the argument becomes moot.

The general point is that the Roche limit is a statement about bodies held together by gravity alone. Anything with strength — a rock, a spacecraft, an astronaut — has a limit computed from its strength instead, and for small objects that limit is far inside the planet.

The generalisation: why there are rings at all #

The Roche limit explains a fact about the solar system that would otherwise be strange: every giant planet has rings and no giant planet has a large moon close in.

Inside the limit, material cannot accrete. Two boulders that touch are pulled apart again before they can stick, so a disc of debris inside the limit stays a disc indefinitely. Outside it, accretion proceeds and the debris collects into moons on a timescale of years. So a planet that formed with a disc of material around it ends up with rings inside the limit and moons outside — which is precisely the arrangement observed at all four giant planets.

It also explains why the rings are thin. A ring particle’s orbit is confined by collisions to the plane, and there is nothing to gather the particles vertically because there is nothing to gather them at all. Saturn’s main rings are around ten metres thick and 280,000 kilometres across — an aspect ratio of 30 million to one, far flatter than a sheet of paper — and they are that flat because accretion, which is what makes a disc lumpy, is switched off.

The same argument runs at every scale. Protoplanetary discs have an inner region where tidal shear from the star prevents accretion. A star passing too close to a black hole is disrupted at the black hole’s Roche limit and produces a tidal-disruption flare; for a Sun-like star and a million-solar-mass black hole that limit is about $10^{13}$ cm, comfortably outside the event horizon, which is why such flares are visible at all. The arithmetic is identical, the densities are the input, and the answer is a cube root.

It is worth separating the two ways a body can fail the comparison, because they have different thresholds and are often quoted as one. A body held together only by its own gravity — a rubble pile, or a fluid — comes apart at the limit this essay has computed. A body with material strength survives further in, because the tide has to exceed the strength rather than the self-gravity, and for a small enough body the strength wins at any distance: a metre-sized rock in a low orbit is in no danger at all. The limit is therefore a statement about large bodies, and the size at which it starts to apply is set by where self-gravity overtakes strength, which for rock is a few hundred kilometres.

Where the model stops #

Strength is ignored. The whole derivation assumes gravity is the only thing holding the satellite together, and for anything smaller than a few hundred metres that is false.

The satellite is assumed spherical, and it is not. The rigid calculation uses a sphere that stays a sphere; the fluid one uses a Roche ellipsoid. A real body is somewhere between and is also rotating, which changes the balance again — a synchronously rotating satellite has centrifugal support that makes it easier to disrupt.

Nothing here is a dynamical calculation. The Roche limit says where a body cannot hold itself together, not what happens next. Whether the debris spreads into a ring, re-accretes further out, or falls onto the planet is a question about angular momentum and collisions that the density ratio does not answer.

The figures are not to scale. The survey plots put ring edges and moons on one axis in planetary radii, which is honest about position and silent about size — the moons drawn as tick marks span four orders of magnitude in radius, and the rings are a sheet ten metres thick. Nothing about the vertical dimension of this system can be shown on a page.

That distinction is also why the limit is quoted with a density ratio attached: a satellite denser than its primary survives closer in, and the coefficient in front is different for a rigid body than for a fluid one by about a factor of two.

The limit is a ratio of densities, so it is worth reading at a ratio well outside the solar system’s range and beside the spin barrier that is the same arithmetic turned inward.

The ladder from here #

Later rungs on this anchor: the Roche ellipsoid derived properly, and where the 2.44 comes from. Tidal disruption of a rotating body, and how spin changes the limit. Shoemaker–Levy 9’s fragment train, and what its spacing measures. Ring confinement by shepherd moons and resonances, which is what actually sets a sharp edge. Tidal heating, and why Io is the most volcanically active body in the solar system. Tidal-disruption events at black holes, and the light curve their debris produces.

Roche published the calculation in 1848 in the Mémoires de la section des sciences of Montpellier, in a paper about the figures of planets, and it attracted almost no attention for forty years. It became famous only when it turned out to say something about Saturn’s rings — which is a common enough pattern, and worth noticing: the result was not derived to explain the rings, and the rings were not the reason anyone looked.