The distance that forgets the moon

Saturn has rings out to about two and a quarter of its own radii, and its innermost round moon orbits at just over three. Between the two there is nothing large. That gap is not an accident of what happened to be there; it is a boundary, and the boundary is a distance at which a self-gravitating body stops being able to hold itself together.

The striking feature of that distance is what it does not contain. It does not know how big the satellite is.

The two accelerations #

Put a loose stone on the surface of a satellite, on the side facing the planet, and ask what holds it down.

Its own satellite pulls it inward with $Gm/r^2$, where $m$ and $r$ are the satellite’s mass and radius. Writing the mass as the density times the volume, that is $\tfrac{4}{3}\pi G\rho_{\text{s}} r$ — proportional to the satellite’s radius.

The planet pulls the stone and the satellite both, and what matters is the difference: the tide, which is what free fall cannot remove. Expanding the planet’s field over the satellite’s size gives $2GMr/d^3$ for the stretch at the near and far surfaces, where $M$ is the planet’s mass and $d$ the orbital distance — also proportional to the satellite’s radius.

What is left of a gravitational field after free fall has removed the uniform part is the tide. It stretches along the line to the source and squeezes across it, and it is the only part of gravity a falling observer can detect at all — which is why the whole of this essay is about a difference of accelerations rather than about an acceleration. The uniform part is not weak here; it is irrelevant, because everything is falling in it together.

Set the two equal and the satellite’s radius cancels from both sides. What is left is

$$d = R\left(\frac{2\rho_{\text{p}}}{\rho_{\text{s}}}\right)^{1/3}$$

with $R$ the planet’s radius. Two densities and a cube root, and nothing else.

The cancellation, checked #

A cancellation is exactly the kind of claim that is easy to assert and easy to get wrong, so it is worth doing rather than saying.

Three satellites differing by a factor of nine in radius, and therefore by a factor of seven hundred in mass, have their breaking distance computed independently by bisection on their own curves. The three answers agree to a part in a thousand billion.

That is a stronger statement than it looks. It says a boulder and a moon of the same material break up at the same place. It says a ring cannot coalesce inside the limit at any size at all — so a ring is not debris that has failed to assemble into a moon, it is material forbidden to.

The reason the cancellation happens twice is worth holding onto, because it is not a coincidence. Self-gravity at the surface grows with the size of a body because a bigger body of the same density has more mass under the same square of distance. The tide grows with the size of a body because it is a difference of field across the body, and a bigger body samples more of the gradient. Both are linear in the radius for the same underlying reason — that gravity is a field with a gradient — and so the comparison between them is scale-free.

Rigid, fluid, and the factor between them #

The derivation above treats the satellite as a rigid sphere, and a self-gravitating body is not one.

A body large enough for self-gravity to dominate its strength has already been pulled round, which is the argument that decides which objects are spheres. Such a body responds to a tide by deforming: it stretches along the line to the planet, and having stretched it presents a longer body to the tide and gives way sooner.

Doing the calculation for a fluid satellite in hydrostatic equilibrium replaces the coefficient $2^{1/3} = 1.26$ with 2.44, so the limit moves outward by nearly a factor of two. Both are drawn in the opening figure. For ice around a planet of Saturn’s density they come out at 1.15 and 2.23 planetary radii, and the rings end at 2.27 — just outside the fluid limit, which is where the physics puts the boundary between a ring and a moon.

The agreement is better than the calculation deserves. Real ring particles are held together partly by material strength and partly by contact forces, they are not spheres, and the outer edge of the A ring is shaped by resonances with Janus as well as by the Roche criterion. What the calculation gets right is the location to within its own coefficient’s uncertainty, and the mechanism exactly.

Reading the formula #

Three things about that expression are worth extracting, because each is a prediction that can be checked against a body somewhere.

The limit is a multiple of the primary’s own radius, not an absolute distance. A large planet has a proportionally large exclusion zone, so scaling a whole system up changes nothing about which bodies survive where. That is why the same criterion works for Jupiter, for Saturn, and for a hot Jupiter orbiting another star.

A denser satellite survives closer in, by the cube root of the density ratio. So an iron body can orbit at rather more than half the distance an icy one can, and the inner moons of the giant planets are systematically the rockier ones — a correlation that follows from this expression alone.

And a low-density primary has a limit that can fall inside its own surface. Saturn’s density is 687 kilograms per cubic metre, less than water’s; for a satellite denser than about 1.4 times that, the rigid limit computes to less than one Saturn radius, which means a rigid body of that density could in principle orbit at the cloud tops without disruption. The fluid limit does not have that property at the same density, and the difference between the two coefficients is therefore not a detail — it decides whether a limit exists at all.

The bodies that ignore it #

Phobos orbits Mars at 2.76 Martian radii, and Mars is about four times as dense as Phobos, so the fluid Roche limit is somewhere near 3.9 radii. Phobos is well inside it and has not come apart.

The resolution is that the limit is a statement about bodies held together by their own gravity, and Phobos is not one. Its self-gravity at the surface is about six millimetres per second squared; the cohesive strength of even a weakly bound rubble pile is enough to beat the tidal stress at that distance. A body small enough that its material strength dominates simply ignores the calculation, which is why spacecraft, people and pebbles are unaffected by orbiting inside a Roche limit.

The crossover is the same one that decides whether a body is round, and it depends on material rather than on orbit: a strong small body survives where a weak large one does not. Phobos is nevertheless expected to break up eventually, because it is spiralling inward and its strength is low; the grooves across its surface may be the first cracks.

What the rings are evidence of #

Every giant planet in the solar system has rings, and every one of them lies inside its planet’s Roche limit. That is four independent instances of the same statement, and the coincidence is worth taking seriously rather than filing as a curiosity.

A ring is not a lattice — its particles are not held in an arrangement by anything — but the question is the same one: what decides whether a collection of objects stays a collection. In a lattice the answer is a competition between forces with a preferred spacing; in a ring it is a competition between the tide and whatever holds a lump together, and the answer is a distance rather than a spacing.

The argument runs both ways and both directions are informative. Inside the limit, material cannot gather. Two ring particles that touch are held together only by their own mutual gravity, which the tide beats, so they separate again; accretion is forbidden at any size and the ring is stable against becoming a moon. Outside the limit, material must gather, because collisions are inelastic and gravity has nothing opposing it, so a debris disc outside the limit becomes moons on a timescale short compared with the age of the system.

So a ring is not a stage on the way to a moon. It is a region where the moon-forming process is switched off, and that is why the boundary between the two is as sharp as it is. Saturn’s A ring ends at 2.27 radii and Pan and Atlas — the tiny moons just outside — sit at 2.22 and 2.28, which is close enough to the calculation that the discrepancy is about the shape of those bodies rather than about the physics.

There is one more test available and it has been performed. The moons closest to the giant planets are unusually elongated: Pan and Atlas are flattened into shapes like ravioli, with equatorial ridges. Those shapes are what a body at the edge of its own Roche limit is expected to look like — barely holding on, with material at the sub-planetary point almost free — and they were photographed before anybody predicted them.

What a disruption actually looks like #

Crossing the limit does not produce a sudden explosion. It produces a body that can no longer hold onto material at its sub-planetary and anti-planetary points, and the material leaves from there.

The tide falls as the attracting mass rises, which is the fact that makes large black holes undramatic to fall into. Tidal stretch at a horizon goes down with mass, so a star is torn apart outside a million-solar-mass hole and swallowed whole by a billion-solar-mass one — and an observer crossing the horizon of a sufficiently large hole notices nothing locally at all. The Roche argument and the horizon are independent, and which comes first depends on the mass.

Shoemaker–Levy 9 is the observed case. It passed within about 1.3 Jovian radii in 1992, well inside Jupiter’s Roche limit for a weak body, and emerged as a string of more than twenty fragments strung out along its orbit. The string is the signature: the fragments separate along the orbital direction rather than in all directions, because that is the direction in which the tidal field is stretching, and each fragment then follows a slightly different orbit.

What comes back afterwards is not the same body. The fragments of Shoemaker–Levy each acquired their own orbits and their own rotation, and had they not struck Jupiter two years later they would have spread into a stream and then, outside the limit, slowly re-accreted into a rubble pile with none of the original’s internal structure. That is the general fate of a body that survives a close pass: it is not destroyed but rebuilt, and its interior is randomised. Several near-Earth asteroids have shapes and spin states consistent with having been through exactly that, and the resulting pile is held together by nothing but its own gravity and a little cohesion.

The same thing happens to stars near massive black holes, with one twist that runs the other way. Tidal stretch at a given mass falls as the hole grows, because the horizon grows faster than the tide, so a star is disrupted outside the horizon of a million-solar-mass hole and swallowed whole by a billion-solar-mass one. The flare a disruption produces is therefore a way of measuring a hole’s mass from the far side of the universe, and the absence of such flares from the largest holes is a prediction rather than a disappointment.

Where the same comparison decides something else #

The competition between a tide and a body’s own gravity is a general one, and the Roche limit is only its most famous instance.

The Hill sphere is the same comparison with the roles of the bodies exchanged: it asks how far from a satellite the satellite’s gravity still dominates the planet’s tide, and gives a radius $d(m/3M)^{1/3}$. A moon can only keep its own moons inside it, and the Earth’s Hill sphere at 1.5 million kilometres is why the Moon is bound to the Earth rather than to the Sun.

The Jeans length is the same comparison against a gas’s pressure rather than against another body’s tide, and it decides which disturbances collapse. And in a protoplanetary disc the two combine: material can only accrete outside the local Roche limit, which is one of several reasons planets do not form arbitrarily close to their stars.

The lobe that carries the same name #

Roche’s name is on a second surface, arrived at by the same competition in a different geometry, and it is the one that drives most of the violent events in binary stars.

Take two stars in a circular orbit and work in the frame that turns with them. The effective potential has three pieces: the gravity of each star and the centrifugal term. Close to either star its own gravity dominates and the equipotentials are nearly spherical; further out they distort toward the companion, and at one particular value the surface around one star touches the surface around the other, at the point between them where the two pulls and the rotation balance.

The largest closed surface around each star is its Roche lobe, and the point where the two touch is the inner Lagrange point. Material at that point is not bound to either star: the self-gravity holding it and the companion’s tide plus the rotation pulling it away cancel exactly. That is the same equality this essay computes for a satellite, evaluated on a different surface.

The consequence is that a star which fills its lobe cannot keep the material at the top of it. A star can come to fill its lobe two ways — by swelling, which every star does as it leaves the main sequence, or by the orbit shrinking, which happens as the pair radiates gravitational waves or loses angular momentum in a magnetised wind. Either way, material begins to flow through the inner Lagrange point onto the companion.

Once it starts, a great deal follows. The stream has angular momentum and cannot fall straight in, so it forms a disc. If the companion is a white dwarf, the accumulating hydrogen eventually ignites in a runaway on its surface and the system is seen as a nova, repeatedly. If enough mass accumulates for the dwarf to approach the mass beyond which cold matter cannot hold itself up, the result is a type Ia supernova. If the companion is a neutron star or a black hole, the material heats to millions of kelvin on the way in and the system is one of the bright X-ray binaries.

There is a neat piece of evidence that this really happens, and it was a paradox before it was an explanation. In the eclipsing binary Algol the less massive star is the more evolved — a subgiant beside a main-sequence companion — which stellar evolution flatly forbids, since a more massive star evolves faster. The resolution is that the subgiant was originally the heavier of the two, filled its lobe as it expanded, and transferred most of its mass to its companion. The mass ratio was reversed by the transfer, and what is observed is the aftermath.

So one criterion — self-gravity against a tide — bounds where a moon can orbit, forbids a ring from becoming a moon, and decides which stars hand their material to their neighbours.

The planets found at their own limit #

The Roche limit is not confined to the solar system, and the exoplanet population has its inner edge where the calculation puts it.

For a gas giant of Jupiter’s density orbiting a star of the Sun’s mass and radius, the fluid limit sits at around a hundredth of an astronomical unit. The observed hot Jupiters pile up at three to five hundredths — roughly twice that — and the pile-up has a straightforward reading. A planet that arrived on a highly elongated orbit and was then circularised by tides ends up at about twice its original closest approach, and the closest approach that survives is the Roche limit. So an inner edge at twice the limit is the signature of that arrival route, and it is what is seen.

Some planets sit closer, and at least one is visibly being destroyed. WASP-12b orbits its star once every twenty-six hours, at a distance where the tide is large enough to distort it measurably: its light curve shows the periodic brightening and dimming of an ellipsoid rather than a sphere, so the planet’s shape is an observable. It is overflowing its Roche lobe and losing material to the star, and its orbital period is shortening by a few tens of milliseconds a year — an orbital decay measured directly, over a decade of transit timings, in a system four hundred parsecs away.

That is an unusually direct confirmation. The essay’s argument says a body inside its limit cannot hold its outermost material; here is a planet whose outermost material is observed leaving, whose shape is observed to be the elongated one the tide requires, and whose orbit is observed shrinking toward the point where the rest of it goes.

What the pictures cannot show #

The satellite is treated as a sphere throughout. A real body near its Roche limit is elongated, and its elongation feeds back on the tide it experiences. The fluid coefficient of 2.44 comes from solving that self-consistently for an equilibrium ellipsoid; the figures here draw the rigid comparison and quote the fluid number.

Rotation is left out. A satellite rotating synchronously — which is what tides quickly enforce — has a centrifugal contribution at its sub-planetary point that adds to the tide, and including it changes the coefficient again.

The primary is a point mass. A planet is not, and the correction from its own oblateness matters at the distances rings occupy: Saturn’s quadrupole moment is large enough to shift resonance locations by hundreds of kilometres, which is why the sharp edges of real rings are set by resonances rather than by the smooth criterion drawn here.

And material strength has no place in the calculation at all. Every number here is for a body with none, and the strength of a real rubble pile is small but not zero. What it does is turn the sharp limit into a size-dependent one: below a certain size, strength wins; above it, gravity does; and the transition depends on the material rather than on where the body is.

The ladder from here #

Later rungs on this anchor: the equilibrium ellipsoid calculation that produces the 2.44, and the Jacobi and Roche sequences it belongs to; the tidal heating that a body inside but not through the limit experiences, which is what keeps Io molten; the shepherding resonances that sharpen a ring’s outer edge beyond what the Roche criterion alone would give; and the tidal disruption of stars, where the same comparison is made against a black hole and the observable is a flare.

The neighbouring ladders are the term free fall cannot remove, which is the tide as a statement about geometry rather than about breaking things, the size at which a body becomes round, which is the same self-gravity compared against strength instead, and the orbit that cannot be made smaller, which is a different kind of innermost limit set by relativity rather than by tides.