Roche limit calculator

The Roche limit is the critical centre-to-centre distance at which a satellite held together mainly by self-gravity is torn apart by the tidal field of its primary body. In the rigid closed form the distance is primary radius R times the cube root of two times the density ratio of primary over satellite; the classical fluid approximation multiplies R by about two point four four times the cube root of the same density ratio. Students and contest authors use equal-density checks that recover roughly one point two six primary radii for the rigid model; planetary scientists estimate where rings and rubble piles become unstable; mission planners compare planned flyby altitudes against a quick tidal-breakup scale before full multipole models. This calculator accepts two strictly positive densities and a strictly positive primary radius, returns both rigid and fluid Roche distances in the same length unit as the radius, exposes the density ratio and dimensionless multiples for audit, and rejects zero density so the cube-root ratio never becomes a silent Infinity or NaN. Negative densities and non-positive radius are also rejected with clear validation errors.

How the Roche limit formula works

Start from the classical picture that a satellite orbiting inside a critical distance experiences a tidal gradient strong enough that differential gravity across its diameter exceeds its own self-gravity binding. For a rigid spherical secondary the closed-form Roche limit is d equals R times the cube root of two times rho primary over rho satellite, where R is the primary radius and the two densities are mean bulk densities of the central body and the moon or rubble pile.

The factor of two arises from balancing self-gravity against the differential gravitational acceleration in the restricted three-body tidal expansion for a strengthless rigid body. When densities are equal the rigid limit collapses to R times the cube root of two, about one point two five nine nine primary radii, which is the standard equal-density homework checkpoint.

For fluid, tidally deformable secondaries the classical approximation multiplies R by roughly two point four four times the cube root of the density ratio, placing the fluid limit farther out because a fluid body deforms and sheds mass more readily. The calculator evaluates both closed forms in deterministic floating-point arithmetic: one division for the density ratio, one cube root, multiplies by R and the rigid or fluid prefactor, and no random seeds, wall-clock timestamps, network calls, or model drift.

Primary density and satellite density must each be finite and strictly positive; zero density fails with invalid input because the formulas divide by satellite density and a zero primary density is non-physical. Negative densities are rejected. Primary radius must likewise be finite and strictly positive so the distance scale is well defined.

This page intentionally uses the textbook rigid and fluid Roche limits only; full tidal multipoles, spin-synchronous strength models, eccentric periapsis passages, and atmospheric breakup are separate search intents and separate landings.

What you send and what you get back

Primary_density is the mean density of the central body and must be strictly greater than zero; for Earth use about five thousand five hundred fourteen kilograms per cubic metre, for the Sun about one thousand four hundred eight, and for Saturn about six hundred eighty-seven.

Satellite_density is the mean density of the secondary in the same density unit and must also be strictly greater than zero; for the Moon use about three thousand three hundred forty, for a fluid icy body use lower values, and for equal-density drills set both densities equal. Primary_radius is the equatorial or mean radius of the primary in metres or any consistent length unit and must be strictly greater than zero; for Earth use about six million three hundred seventy-one thousand metres.

Classic textbook checks include equal densities with unit radius recovering cube root of two for the rigid limit, and Earth density with Moon density at Earth radius recovering a rigid Roche distance of a few Earth radii from the centre. Keep units consistent: convert grams per cubic centimetre and kilometres to matching SI or any single density unit paired with a single length unit before calling so the numeric product matches the interpretation your report claims.

Where Roche limit calculators show up in real work

Secondary-school and introductory university astronomy and planetary-science courses assign tidal-breakup drills, equal-density checks, and ring-boundary estimates that rearrange the same density-ratio formula; a single trusted oracle speeds grading and demo setup. Planetary ring science and small-body mission concept teams convert tabulated densities and radii into rigid and fluid Roche distances before full strength and multipole models. Science communicators convert equal-density and ice-body cases into multiples of primary radii to illustrate why Saturn rings sit near the classical fluid limit.

Game and simulation authors seed tidal-disruption toy demos with closed-form Roche distances when they want a transparent ground truth without floating-point surprises from ad-hoc constants. If you later need Newtonian gravitational force between two masses, orbital period from semi-major axis and central mass, surface weight on other planets, or full tidal stress tensors, treat those as separate search intents and landings rather than overloading this Roche limit calculator page.