Vol. Science · Astronomy · The critical distance
A moon that comes too close can be pulled apart, and the pieces can stay in orbit as a ring. The fluid Roche limit, the classic spherical form, is d ≈ 2.44 R · (ρprimary / ρsatellite)1/3. The rigid-body variant uses 1.26 in place of 2.44. Read them as a band, not as one magic line. Presets use NASA NSSDCA fact-sheet mean densities, volumetric mean radii, and mean orbital distances. For Saturn and Rhea the fluid distance is 116,698 km (2.004 volumetric radii) and the rigid distance is 60,262 km (1.035). Rhea’s semimajor axis is 527,040 km, outside both. Earth–Moon gives a fluid distance of 18,364 km; the Moon’s semimajor axis is 384,400 km. Mars–Phobos gives fluid 10,541 km and rigid 5,443 km; Phobos’s mean orbital distance is 9,378 km, between them. The tides desk is the near-side and far-side difference. The Lagrange desk is the five points that keep station. The sphere around a secondary, where that body dominates a test particle, is the Hill desk. Earth’s is 1,496,565 km. The Moon sits inside it. Phobos sits deep inside Mars’s, and still between the limits on this page.
Classroom schematic. One plane, top-down from the north, so the orbit runs counterclockwise. In each panel the primary disk, the two Roche arcs, and the orbit share one radial scale. When the moon is far outside, the lower panel magnifies the neighborhood of the primary, and that panel has its own scale. The satellite mark is not the moon’s radius. On Saturn, the band is the rings fact sheet’s span from 66,900 km to 136,780 km, not a photograph. Densities are means. Compressed where a moon sits far out. Not an ephemeris. Not a mission tool.
From the north · orbit counterclockwise · schematic
Pair
Density ratio · ρ primary / ρ satellite · fact sheet
The formulas
d ≈ 2.44 R · (ρprimary / ρsatellite)1/3. This is the fluid Roche limit in the classic spherical form. R is the primary’s volumetric mean radius. The densities are the fact-sheet means, in kg/m³, so the ratio has no units. A fluid moon has no strength. Inside this distance its own gravity cannot hold it together against the primary’s tide. The coefficient 2.44 already assumes a sphere. It is a classroom constant, not a fitted measurement from a spacecraft.
d ≈ 1.26 R · (ρprimary / ρsatellite)1/3. The same cube root, with 1.26 in place of 2.44. This is the rigid-body variant: a body that can bear some stress, still idealized as a sphere. It sits closer to the primary than the fluid distance. The gap between the two numbers is the band. A moon outside the fluid distance is outside both. A moon inside the rigid distance is inside both. A moon between them is inside the fluid limit and outside the rigid one.
The inputs. Saturn: volumetric mean radius 58,232 km, mean density 687 kg/m³. Rhea: mean density 1,240 kg/m³, semimajor axis 527.04 × 10³ km. Earth: volumetric mean radius 6,371.000 km, mean density 5,513 kg/m³. The Moon fact sheet’s Earth column prints 5,514; this desk uses the Earth Fact Sheet’s 5,513. Moon: mean density 3,344 kg/m³, semimajor axis 0.3844 × 106 km. Mars: volumetric mean radius 3,389.5 km, mean density 3,934 kg/m³. Phobos: mean density 1,900 kg/m³, mean orbital distance 9,378 km. The slider clamps the density ratio from 0.25 to 4 and leaves the orbit and the primary radius where the sheets put them. Equal density sets the ratio to 1.
Saturn’s rings, as a comparison. The rings fact sheet lists the D ring’s inner edge at 66,900 km and the A ring’s outer edge at 136,780 km. Those kilometers are from Saturn’s center. The same sheet’s “Saturn radii” column divides by the equatorial radius 60,268 km, so the A edge is 2.270 equatorial radii. This desk’s R is the volumetric mean, 58,232 km, so the same edge is 2.349 volumetric radii. For Rhea’s density the fluid distance is 116,698 km, and the outer A ring sits outside that number. For equal density the fluid distance is 2.44 × 58,232 km = 142,086 km, and the A edge sits inside it. The classroom point is that icy material near Saturn cannot gather into a moon. It is not that every ring particle has Rhea’s density.
Where this shows up
Saturn and Rhea. Rhea’s semimajor axis is 527,040 km, which the moon sheet also prints as 8.7449 Saturn radii using the equatorial radius. In volumetric radii that orbit is 9.051 R, far outside both limits for Rhea’s density. The axes on the sheet are 765 × 763 × 762 km. This page’s 763 km figure is the cube root of that product, labeled as derived. The ring band on the drawing is the fact-sheet span, drawn to the same scale as the arcs.
Earth and the Moon. The fluid distance is 18,364 km, 2.882 Earth radii. The rigid distance is 9,483 km, 1.488 Earth radii. The Moon’s semimajor axis is 384,400 km, 60.336 Earth radii. The Moon is not close. The sheet also says the distance from Earth ranges roughly from 357,000 km to 407,000 km. Even the near end of that range is far outside 18,364 km. The Moon’s own volumetric mean radius, 1,737.4 km, is not an input to the formula. It is printed so the moon has a size on the page.
Mars and Phobos. The fluid distance is 10,541 km, 3.110 Mars radii. The rigid distance is 5,443 km, 1.606 Mars radii. Phobos’s mean orbital distance is 9,378 km, 2.767 Mars radii. That is inside the fluid limit and outside the rigid one: between. The sheet gives three axes, 13.0 × 11.4 × 9.1 km, and a mean density of 1,900 kg/m³. Phobos is irregular. The formulas assume a sphere. Treat the verdict as a schematic comparison of those published means, not as a claim that Phobos is about to come apart. That same 9,378 km is deep inside Mars’s Hill sphere. The Hill desk, from the fact-sheet masses and Mars’s 227.956 × 106 km, puts that sphere at 1,084,138 km. Phobos is 0.00865 of that radius. Hill is where Mars dominates the Sun for a test particle. Roche is where Mars can shred a moon of Phobos’s density.
Literacy · a band, not a line
Sources · public NASA pages
Classroom picture of the Roche band. Fluid distance 2.44 R times the cube root of the density ratio. Rigid distance 1.26 R times the same cube root. Saturn–Rhea: 116,698 km and 60,262 km, Rhea at 527,040 km, outside. Earth–Moon: 18,364 km and 9,483 km, the Moon at 384,400 km, outside. Mars–Phobos: 10,541 km and 5,443 km, Phobos at 9,378 km, between. The A ring’s outer edge, 136,780 km, sits inside the equal-density fluid distance for Saturn, 142,086 km. Mean densities, spherical formulas, circles about the primary. Not an ephemeris, and not a mission tool. Earthrise preview: NASA / Bill Anders (public domain).
Keep going
The difference
The same unequal pull, gentle enough to bulge an ocean. Two bulges, not a tide table.
The five points
L1 through L5. Sun–Earth L2 is 1,501,538 km from Earth. A different equilibrium.
The sphere
Where the secondary dominates. Earth’s sphere is 1,496,565 km. Phobos is deep inside Mars’s, and still between these limits.
The pull
Weight, Newton, Einstein — the force that both holds a moon and can shred it.
Neighborhood
Where Saturn, Earth, and Mars sit. Day and local time, still a schematic.
Two Roche distances for a planet and a moon, drawn as arcs around the primary. The numbers come from NASA fact-sheet densities, radii, and mean orbits. The picture is a classroom schematic, not an ephemeris and not a mission tool.
| Saturn–Rhea | The default. Saturn 58,232 km and 687 kg/m³. Rhea 1,240 kg/m³, orbit 527,040 km. Fluid 116,698 km. Rigid 60,262 km. Outside both. The ring band runs from 66,900 km to 136,780 km. |
|---|---|
| Earth–Moon | Earth 6,371.000 km and 5,513 kg/m³. Moon 3,344 kg/m³, orbit 384,400 km. Fluid 18,364 km. Rigid 9,483 km. The Moon is far outside. The lower panel is there because the arcs would otherwise hug the disk. |
| Mars–Phobos | Mars 3,389.5 km and 3,934 kg/m³. Phobos 1,900 kg/m³, mean orbit 9,378 km. Fluid 10,541 km. Rigid 5,443 km. Between: inside the fluid limit, outside the rigid one. Phobos is irregular and rubble. Schematic only. |
| Ratio slider | ρ primary / ρ satellite, clamped from 0.25 to 4. The orbit and the primary radius stay on the fact sheet. Only the two arcs move. The label says “what-if” once you leave the fact-sheet ratio. |
|---|---|
| Fact-sheet ratio | Puts the slider back on the published densities for this pair. |
| Equal density | Sets the ratio to 1. For Saturn that fluid distance is 142,086 km, which is outside the A ring’s 136,780 km edge. That is the classroom comparison for the rings. |
| Play | Walks the satellite mark counterclockwise, as seen from the north. About 12 seconds per turn. A time-lapse, not the moon’s period. Press again to pause. |
| Full view | Primary at the center, to its volumetric radius. Dashed circle: the mean orbit. Solid arc: fluid. Dashed arc: rigid. On Saturn, the band is the rings span. The satellite mark is not the moon’s radius. |
| Lower panel | Appears when the moon is far outside the arcs. Magnified, own scale, and it does not spin with Play. It is there so the two arcs are readable when the orbit is huge. |
| N | North, the view. Motion runs counterclockwise, the same sense as the other top-down desks. |
| Fluid and rigid faces | The two distances in kilometers. The line under each face is that distance in volumetric radii of the primary. Rounded to the nearest kilometer from the formulas above. |
|---|---|
| Mean orbit | The fact-sheet semimajor axis, or Phobos’s mean orbital distance. Also in volumetric radii. Rhea’s sheet also prints 8.7449 equatorial Saturn radii. This page’s 9.051 uses 58,232 km, not 60,268 km. |
| Verdict | Outside, between, or inside. Between means inside the fluid distance and outside the rigid one. The words follow the numbers. They are not a prediction. |
| Satellite size | The Moon has a published volumetric mean radius, 1,737.4 km. Rhea and Phobos have three axes. The single kilometer figure for those two is the cube root of the product, and it is labeled derived. Size is not an input to the formula. |