Roche Limit Calculator

Compare fluid and rigid Roche-limit models from primary radius and density contrast.

Worked example — Roche Limit Calculator

Density ratio η	1.182
Roche limit distance	1.837×10⁴km
Altitude above surface	1.20×10⁴km
Fluid limit (km)	1.837×10⁴km
Rigid limit (km)	9.485×10³km

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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Saturn is less dense than its own rings 🖖

Saturn is less dense than the ice in its own rings — 0.687 against 1.0. That inverts the usual intuition about the Roche limit. The textbook coefficient of 2.44 gets multiplied by the cube root of the density ratio, and here that factor is 0.88 rather than something above 1. It pulls the limit inward, to 2.15 Saturn radii, about 125,000 km, and Saturn's dense main rings lie inside it. Material there cannot accrete into a moon at all: tides undo gravity's work faster than gravity does it. The coefficient itself depends on what the body is made of — 2.44 with no internal strength, 1.26 when material strength adds to self-gravity.

What the Roche limit really measures 🖖

Picture a moon held together only by its own gravity. As it nears a planet, the planet tugs harder on the moon's near side than its far side, and that difference stretches the moon. Cross the Roche limit and the stretching wins, tearing the moon apart. The moon's mass cancels out of the equation entirely. Only the two densities and the planet's radius set the distance, so a denser moon can safely orbit closer than a fluffy one.

Mars is slowly growing its own rings 🖖

Phobos, the larger Martian moon, orbits below the synchronous distance, so tidal drag is spiraling it inward by roughly 2 meters per century. In about 30-50 million years it will cross Mars's Roche limit and shatter, likely spreading into a temporary ring around the planet. The strange parallel grooves scarring Phobos may already be stress fractures from tidal stretching — a moon showing its first cracks long before the final breakup.

Problem solved in full

Fluid limit before tides pull a moon apart 5 steps
How close can a moon get to Earth before tides pull it apart? Work out the fluid limit, and then explain why the answer for a rigid body is only half as far.
1. The competition is between the moon's own gravity holding it together and the difference in Earth's pull across its diameter. Both scale with the moon's size in a way that makes the moon's radius cancel — which is why the limit depends on densities, not on how big the moon is.
  
  $d_{fluid} = 2.44 R (\frac{ρ _{M}}{ρ _{m}})^{1/3}$
2. So only the density ratio enters, and it enters as a cube root, which flattens it hard. Earth is 1.65 times denser than the Moon; the cube root of that is 1.182.
  
  $(\frac{5.51}{3.34})^{1/3} = (1.6497)^{1/3} = 1.182$
3. Multiply by the primary's radius and the constant for a fluid body. The result is 18 368 km from Earth's centre.
  
  $d = 2.44 \times 6371 \times 1.182 = 1.837 \times 1 0^{4} km$
4. Subtracting Earth's radius gives an altitude of about 12 000 km, which is where the panel puts it.
  
  $d - R = 18368 - 6371 = 1.20 \times 1 0^{4} km altitude$
5. The rigid constant is 1.26 against 2.44 for a fluid, because a solid body resists deformation until it fails, whereas a fluid one deforms first, which stretches it, which increases the tidal difference across it — a runaway the rigid case does not have.
  
  $\frac{2.44}{1.26} = 1.94, \frac{384 400}{18 368} = 20.9 (the Moon)$
Answer

The tool prints a density ratio of 1.182, a fluid limit of 1.837 × 10⁴ km and a rigid limit of 9.485 × 10³ km — a factor of 1.94 between two answers to the same question. The Moon sits at 384 400 km, about 21 times the fluid limit, so it is in no danger. Saturn's rings are the other case: they lie inside Saturn's Roche limit, which is the standard explanation for why they are rings and not a moon. Click the Saturn preset and the fluid limit for an icy body reads 1.254 × 10⁵ km, while the A ring's outer edge sits at 136,800 km. The bright rings are mostly, but not entirely, inside it.

$d \propto R (\frac{ρ _{M}}{ρ _{m}})^{1/3}; rigid = 9485 km, fluid = 18368 km$

References (1)

Insight block 3 — the grooves on Phobos read as tidal stress fractures: T. A. Hurford et al., "Tidal disruption of Phobos as the cause of surface fractures." Journal of Geophysical Research: Planets 121(6), 1054–1065, 2016.

By the 42Math team · Updated 2026-07-13

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

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Example problems

Earth-Moon densities - The fluid limit sits at 1.837 × 10⁴ km from Earth's centre, about 12,000 km up. The Moon orbits at 384,400 km, twenty-one times further out.
Saturn + icy body - Saturn at 0.687 against ice at 1.0 gives a density factor below 1, so the limit comes in to 1.254 × 10⁵ km: 2.15 Saturn radii, not the textbook 2.44.
Jupiter + rocky moon - The rigid limit for rock at Jupiter is 6.710 × 10⁴ km, inside Jupiter itself, so a solid rock could orbit as low as it liked. Comet Shoemaker-Levy 9 broke up in 1992 regardless: a rubble pile has no rigid strength, and the fluid limit is 1.299 × 10⁵ km.
White dwarf extreme - A primary a million times denser than water throws the limit out to 1.203 × 10⁶ km, 138 times the white dwarf's own radius.