Roche Limit Calculator

Background.

The Roche limit is the distance at which the tide raised by a planet on its satellite overwhelms the satellite's own gravity and pulls it apart. Inside that distance a moon held together by nothing but self-gravity cannot exist as one object; what forms instead is a ring. The limit is set by a strikingly small amount of information — the primary's radius, and the ratio of the two bodies' bulk densities — and this calculator returns it under four different assumptions about what the satellite is made of, together with the Roche critical density and a verdict on whether the orbit you supplied is inside or outside.

The four models exist because the answer depends heavily on whether the satellite can deform. A rigid sphere held rigid by assumption resists stretching and survives closest in; a body that flows into an elongated equilibrium shape presents a longer lever to the tide and breaks up much farther out. The calculator writes all four in one common form, d = C·R_p·(ρ_p/ρ_s)^(1/3), where the coefficient C follows from a shape factor γ as C = (4π/γ)^(1/3). Using the γ values published by Tiscareno, Hedman, Burns and Castillo-Rogez in 2013 gives C = 2.454 for an incompressible fluid, 1.988 for an elongated rubble pile, 1.442 for a rigid sphere in synchronous rotation, and 1.260 for a rigid sphere when the centrifugal term is omitted — the last of these being the familiar textbook 1.26.

That spread of coefficients is not sloppiness; it is a real physical range, and quoting one number without saying which model produced it is the single most common error in writing about the Roche limit. For Earth and a Moon-density satellite the four give 18 472 km, 14 961 km, 10 855 km and 9 483 km — a factor of nearly two from end to end. The page therefore shows all four at once and labels the one you selected.

The critical-density output turns the question around, and it is how ring scientists actually use the relation. Instead of asking 'how close can a body of this density get', it asks 'given an orbit at this distance, how dense must a body be to survive there'. The answer is ρ_Roche = ρ_p(C·R_p/d)³, and comparing a ring's location against it is what constrains what the ring is made of. At the Moon's distance from Earth the critical density is 0.00037 g/cm³, which is why the Moon is in no danger whatever: it would have to be four orders of magnitude less dense than it is.

One limitation belongs beside the answer, not after it. The Roche limit assumes the satellite is held together by self-gravity alone, and completely ignores material strength. That assumption is excellent for a body hundreds of kilometres across, where self-gravity dominates, and useless for a small one: a boulder on the surface of Mars sits deep inside Mars's Roche limit and is held together by chemical bonds, which the calculation knows nothing about. Small solid objects routinely survive far inside the limit for exactly this reason. The model also treats the primary as a point mass, the satellite as uniform and the orbit as circular, and N-body work by Leinhardt and colleagues in 2012 showed that a differentiated satellite with a dense core disrupts closer in than a homogeneous body of the same mean density.

What is roche limit calculator?

Édouard Roche worked out in the 1840s that a fluid satellite orbiting a planet cannot remain a single body inside a certain distance. The reason is that gravity is not uniform across an extended object: the near side is pulled harder than the centre, and the centre harder than the far side. That difference — the tide — tries to stretch the satellite along the planet–satellite line, while the satellite's own gravity tries to hold it together. Where the two balance is the Roche limit.

Because both the tidal stretching and the self-gravity scale with the satellite's size in the same way, the satellite's radius cancels out. What survives is the primary's radius and the ratio of bulk densities: d = C·R_p·(ρ_p/ρ_s)^(1/3). A denser satellite can come closer; a denser primary pushes the limit farther out. Nothing about the satellite's absolute size enters at all.

The practical consequence is visible from Earth. Saturn's rings lie inside Saturn's Roche limit for icy material, which is why they are a ring rather than a moon, and the sharp outer edges of ring systems often sit close to where a plausible particle density crosses the critical-density curve. Comet Shoemaker–Levy 9 was torn into a chain of fragments when it passed inside Jupiter's Roche limit in 1992, then struck the planet two years later — an unusually direct demonstration.

How to use this calculator.

1. 01Choose the satellite model. Fluid is the most conservative (largest limit); rigid tide-only is the classic textbook 1.26 form.
2. 02Enter the primary's radius in kilometres. Use a volumetric mean radius so it is consistent with the bulk density.
3. 03Enter both bulk densities in g/cm³. Only their ratio matters, so any consistent unit gives the same answer.
4. 04Enter the satellite's orbital distance from the centre of the primary, for the inside-or-outside verdict.
5. 05Compare the four limits shown. If they straddle your orbit, the model assumption is doing the work and the result should be quoted as a range.
6. 06Read the critical density: it tells you how dense a body must be to survive where it is, which is usually the more useful framing.
7. 07Remember what the model ignores — material strength above all. A small solid body can sit far inside the limit unharmed.

The formula.

Take a small mass sitting on the surface of a satellite of radius r, at distance d from a primary of mass M_p. The differential (tidal) pull trying to lift it off is approximately 2GM_p·r/d³, and the satellite's own gravity holding it down is GM_s/r². Setting those equal and expressing both masses through densities makes r cancel, leaving d = R_p(2ρ_p/ρ_s)^(1/3) — the coefficient 2^(1/3) = 1.260 of the tide-only model.

If the satellite is tidally locked, as any body this close will be, the co-rotating frame adds a centrifugal term of GM_p·r/d³, bringing the total to three units instead of two and the coefficient to 3^(1/3) = 1.442. Tiscareno, Hedman, Burns and Castillo-Rogez (2013) write all of these in the single form d = R_p(4πρ_p/(γρ_s))^(1/3) and give γ = 4π/3 for a sphere, which is exactly this 1.442 case. Their γ ≈ 1.6 covers an elongated self-gravitating aggregate, after Porco et al. (2007), and their γ ≈ 0.85 covers an incompressible fluid deforming into an equilibrium ellipsoid, after Chandrasekhar (1969).

SOURCE CONFLICT, STATED NOT HIDDEN. Much of the literature quotes the fluid coefficient as 2.456, which corresponds to γ = 0.848252. Tiscareno et al.'s rounded γ = 0.85 gives 2.454315. This page ships 2.454315, because that is the figure in the source that was actually retrieved and checked, and it states the alternative in the results note. The disagreement is 0.069 %, or about 13 km on the 18 472 km Earth–Moon fluid limit — smaller than the uncertainty in any real satellite's bulk density, but recorded rather than quietly resolved.

ROUNDING STAGE. Nothing is rounded part-way through. All arithmetic runs at 40 significant digits in a dedicated high-precision decimal context; rounding happens once, at the return boundary, to 12 significant digits for distances and densities and 10 decimal places for the dimensionless ratios.

SIGNIFICANT FIGURES. Two or three figures is the honest ceiling, and the limiting factor is almost never the arithmetic. Bulk densities of small bodies are often uncertain by 10 % or more, and the choice of model moves the answer by a factor approaching two — both of which swamp everything else.

APPROXIMATION REGIME AND INVALID DOMAIN. Self-gravity only, no material strength; point-mass primary; uniform satellite; circular orbit; synchronous locking assumed for the synchronous and fluid cases. All four inputs must be strictly positive: a zero satellite density makes the density ratio infinite, and a zero orbital distance makes the critical density infinite. Each raises a labelled error against the offending field rather than returning infinity.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Tiscareno, M.S., Hedman, M.M., Burns, J.A. and Castillo-Rogez, J. (2013). 'Compositions and origins of outer planet systems: insights from the Roche critical density'. The Astrophysical Journal Letters 765, L28. Equation 1, a_Roche = R_p(4πρ_p/γρ)^(1/3); Equation 3, ρ_Roche = 3M_p/γa³; γ = 4π/3 ≈ 4.2 for a sphere, γ ≈ 1.6 for an elongated body after Porco et al. (2007), γ ≈ 0.85 for an incompressible fluid after Chandrasekhar (1969). Primary source for all four coefficients on this page. Peer-reviewed, open access. Retrieved 2026-07-29.
2. [2]Leinhardt, Z.M., Ogilvie, G.I., Latter, H.N. and Kokubo, E. (2012). 'Tidal disruption of satellites and formation of narrow rings'. Monthly Notices of the Royal Astronomical Society; doi:10.1111/j.1365-2966.2012.21328.x. N-body result quoted on this page: 'the Roche limit for a rubble pile is closer to the planet than for a fluid body of the same mean density. The Roche limit for a differentiated body is also closer to the planet than for a homogeneous satellite of the same mean density.' Peer-reviewed; abstract open access, journal version paywalled. Retrieved 2026-07-29.
3. [3]NASA JPL Solar System Dynamics, 'Planetary Physical Parameters': Earth volumetric mean radius 6371.0084 ± 0.0001 km and bulk density 5.5134 ± 0.0003 g/cm³; Saturn mean radius 58 232 km and bulk density 0.6871 g/cm³. Independent, open access. Retrieved 2026-07-29.
4. [4]NASA JPL Solar System Dynamics, 'Planetary Satellite Physical Parameters': Earth's Moon, mean radius 1737.4 ± 0.1 km and mean density 3.344 ± 0.001 g/cm³, from ephemeris DE440. Independent, open access. Retrieved 2026-07-29.
5. [5]NASA Science, 'Moon Facts' (page updated 12 February 2026): 'The Moon is an average of 238,855 miles (384,400 kilometers) away.' Source of the default orbital distance. Independent, open access. Retrieved 2026-07-29.
6. [6]Chandrasekhar, S. (1969). Ellipsoidal Figures of Equilibrium. New Haven: Yale University Press — the equilibrium-ellipsoid analysis behind the fluid coefficient. Print/bibliographic reference, cited here through Tiscareno et al. (2013) rather than consulted directly.