# Hill sphere <!-- SOLSIM:BEGIN g31 — Solar System explorer state (hand-built on wt-core, specs/solar/); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *The Oort cloud in the Solar System explorer* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/solar/Solar_System.html?obj=oort&embed=1" data-title="The Oort cloud in the Solar System explorer"></div> *The Solar System explorer locked on this article's state (`?obj=oort`); every object and population of the [[PORTAL_Solar_System|Solar System portal]] has its own state in the same scene.* <!-- SOLSIM:END --> *Try: under show, choose clouds and boundaries: the faint outermost shell, near 200,000 AU, stands for the Sun's Hill sphere against the Galaxy, with the Oort cloud's sampled points inside it; set scale to true and read the caption, which reports that true scale cannot show this and keeps the logarithmic distances; then set show back to everything and scroll in to the planets at the centre, whose own Hill spheres are not drawn.* The **Hill sphere** of a body is the region around it in which its own [[Gravity|gravity]] dominates the motion of small objects over the pull of a more massive body it orbits: a moon inside a planet's Hill sphere can stay bound to the planet, but beyond it the star's tidal pull takes over.[^souami2020][^lauretta2023] It is the most widely used model of a gravitational sphere of influence, and is distinct from the Laplace sphere and from the Roche limit, the distance inside which tides tear a body apart.[^souami2020][^hill2022] The concept comes from the work of the American astronomer George William Hill, building on that of the French astronomer Édouard Roche. For a body of mass m orbiting a much larger mass M at distance a, the Hill radius is approximately r_H ≈ a (m/3M)^(1/3).[^depater2015] [[Earth]]'s Hill sphere reaches about 1.5 million km, so the [[Moon]], at 384,400 km, lies comfortably inside it (derived).[^nasa-fs] Every moon has a smaller Hill sphere of its own nested inside its planet's. At the other extreme, one simple way to bound the [[PORTAL_Solar_System|Solar System]] is the Sun's own Hill sphere in the [[Milky_Way]], estimated at up to about 230,000 [[Astronomical_unit|AU]].[^chebotarev1964][^chebotarev1965] The explorer at the top of this page is set to the [[Oort_cloud|Oort cloud]] and marks that outer limit as a faint shell at 200,000 AU, a simplified ILLUSTRATIVE boundary; the planets' own Hill spheres are too small to show at this scale. ## Definition The Hill sphere is defined in the restricted three-body problem. Two bodies attracting each other by [[Newton's_law_of_universal_gravitation|Newtonian gravity]] follow exact, closed-form orbits, but three bodies in general do not: their motion has to be computed numerically. The problem becomes tractable when the third body's mass is negligible, so that it responds to the other two without disturbing them.[^depater2015] In a frame rotating with the two massive bodies, such a test particle has a conserved quantity, the Jacobi integral, which fixes surfaces of zero velocity that it cannot cross. At low energy the zero-velocity surface closes around the smaller body and the particle is trapped; at higher energy the surface opens at the Lagrange points L1 and L2 on either side, and the particle can leak out into an orbit around the larger body.[^depater2015] The Hill radius is the distance from the smaller body to L1 and L2, to leading order in the mass ratio. With the separation taken as the orbit's semi-major axis a, it is r_H ≈ a (m/3M)^(1/3).[^depater2015][^higuchi2017] Orbital eccentricity e brings the bodies closer at pericentre, where the larger body's tide is strongest, so the practical limit is the Hill radius at pericentre, r_H ≈ a (1 − e)(m/3M)^(1/3).[^hamilton1992] For the Moon around Earth, e ≈ 0.055, which makes the Moon's Hill radius about 10% smaller at perigee than at apogee, since (1 − e)/(1 + e) ≈ 0.90 (derived).[^nasa-fs] The Hill sphere is not a hard wall. It marks where the balance of forces tips, not where orbits become safe: only orbits well inside it are stable over long times, as the section on regions of stability describes. ## Example and derivation For Earth, with a mass of 5.97 × 10²⁴ kg orbiting the Sun, of mass 1.99 × 10³⁰ kg, at 149.6 million km (1 AU), the circular formula gives r_H ≈ 1.5 million km, about 0.01 AU. Including Earth's eccentricity of 0.017 at perihelion gives 1.47 million km (derived).[^nasa-fs][^nasa-sun] The Moon's orbit lies at about 26% of that distance, so it is in no danger of being pulled into an independent orbit around the Sun. The Sun's own sphere of influence in the Galaxy has an unstable outer boundary of about 230,000 AU.[^chebotarev1964] Rearranged as 3 (r_H/a)³ ≈ m/M, the formula says that the Hill sphere's size relative to the orbit depends only on the cube root of the mass ratio. Jupiter, with about a thousandth of the Sun's mass, has r_H ≈ a (1/3,000)^(1/3) ≈ 0.07a, while Earth, with about three millionths, has r_H ≈ 0.01a (derived).[^nasa-fs][^nasa-sun] ### Derivation Consider a test particle on the line joining the primary (mass M) and secondary (mass m), at distance r from the secondary on the side facing the primary, so that it is a − r from the primary. If it co-rotates with the secondary's orbit at angular speed Ω, where Ω² = G(M + m)/a³ ≈ GM/a³ by [[Kepler's_laws_of_planetary_motion|Kepler's third law]], balance of the forces along the line requires GM/(a − r)² − Gm/r² = Ω²(a − r).[^depater2015] Substituting Ω² and multiplying through by a²/(GM), for r much smaller than a the left-hand side expands as 1 + 2r/a − (m/M)(a/r)² and the right-hand side as 1 − r/a. The leading terms cancel, leaving 3r/a ≈ (m/M)(a/r)², that is r³ ≈ m a³/(3M), or r ≈ a (m/3M)^(1/3) (derived). The point found this way is L1; L2 lies at the same distance on the far side.[^depater2015] On an elliptical orbit the Hill radius is largest at apocentre and smallest at pericentre, so for the stability of small satellites the pericentre value is the one that matters.[^hamilton1992] ## Regions of stability The Hill sphere is only an approximation. Other forces, such as the pressure of sunlight on small particles, can push an object out over time, and the formula assumes that the satellite's own mass is negligible.[^hamilton1992][^depater2015] Hamilton and Burns showed, for satellites of asteroids, that stable orbits extend only part of the way to the Hill radius, and that the stable zone shrinks further when the asteroid's orbit is eccentric and when radiation pressure acts on small particles.[^hamilton1991][^hamilton1992] The direction of the orbit matters. Far from the primary body, retrograde orbits remain stable over a wider region than prograde orbits, which was long thought to explain why so many of [[Jupiter]]'s distant irregular moons are retrograde; [[Saturn]], however, has a more even mix, and the capture of irregular moons appears to involve chaotic dynamics near the Hill sphere's edge.[^astakhov2003] The Hill radius also sets the spacing of planets. For two planets, the mutual Hill radius, r_H,m = ((a₁ + a₂)/2) ((m₁ + m₂)/3M)^(1/3), gives the natural unit of separation: two planets on circular orbits are stable against close encounters if they are separated by more than about 2√3 ≈ 3.5 mutual Hill radii. Systems of three or more planets closer than about ten mutual Hill radii become unstable over time, mainly because each additional planet adds perturbations.[^chambers1996] ## Further examples A Hill sphere can be too small to hold any orbit at all. A 104-tonne spacecraft in low Earth orbit, 300 km up (a = 6,671 km from Earth's centre), has a Hill radius of about 1.2 m, smaller than the spacecraft itself, so an astronaut could not orbit it (derived).[^nasa-fs] Turning the argument round, a sphere of density ρ fits inside its own Hill sphere only if ρ > 9M/(4πa³): about 14,400 kg/m³ in low Earth orbit, denser than lead, but only about 57 kg/m³, under 6% of the density of [[Water|water]], at geostationary distance, 42,164 km from Earth's centre (derived).[^nasa-fs] Among the planets, [[Neptune]] has the largest Hill sphere, about 115 million km: its great distance from the Sun more than makes up for its mass being only about a nineteenth of [[Jupiter]]'s, whose Hill radius is about 51 million km (derived).[^nasa-fs][^nasa-sun] Small bodies have tiny Hill spheres: that of the Mercury-crossing asteroid 66391 Moshup, which has a moon named Squannit, is about 22 km in radius.[^johnston2019] The same reasoning applies around other stars. The hot Jupiter HD 209458 b has a Hill radius of about 593,000 km, some eight times its own radius of about 71,000 km, and even the small close-in planet CoRoT-7b has a Hill radius of about 61,000 km, six times its radius. Both could in principle hold small moons close in, though not inside their Roche limits.[^exoplanet-hd209458][^exoplanet-corot7] ## Hill spheres for the Solar System The table gives Hill radii for the planets and [[Pluto]], computed with the pericentre formula r_H ≈ a(1 − e)(m/3M)^(1/3) from the masses, mean distances and eccentricities in the NASA planetary fact sheet and a solar mass of 1.9884 × 10³⁰ kg (all derived).[^nasa-fs][^nasa-sun] | Body | Hill radius (million km) | Hill radius (AU) | In body radii | |---|---|---|---| | [[Mercury_(planet)|Mercury]] | 0.175 | 0.0012 | 72 | | [[Venus]] | 1.00 | 0.0067 | 166 | | Earth | 1.47 | 0.0098 | 231 | | [[Mars]] | 0.98 | 0.0066 | 289 | | Jupiter | 50.5 | 0.338 | 707 | | Saturn | 62.0 | 0.414 | 1,029 | | [[Uranus]] | 66.7 | 0.446 | 2,610 | | Neptune | 115 | 0.770 | 4,650 | | Pluto | 5.8 | 0.039 | 4,870 | The numbers show the two effects at work. The Hill radius grows in proportion to distance from the Sun but only as the cube root of mass, so Uranus and Neptune, far out, have larger Hill spheres than Jupiter despite having about a twentieth of its mass. Measured in their own radii, the outer planets and Pluto have enormous Hill spheres, room for the distant irregular moons that the giant planets hold.[^astakhov2003] For the Moon, Earth's Hill sphere is roomy: the Moon orbits at about a quarter of the Hill radius (derived), well inside the region where prograde orbits are stable.[^hamilton1992] The explorer at the top of this page shows none of these planetary spheres; its only Hill-sphere marker is the Sun's, the outermost shell at 200,000 AU, which lies beyond the Oort cloud's sampled points and marks the Solar System's gravitational edge.[^chebotarev1965] ## See also - [[Oort_cloud]] · [[Hills_cloud]] - [[Orbit]] · [[Kepler's_laws_of_planetary_motion]] - [[Newton's_law_of_universal_gravitation]] - [[Tide]] - Roche limit · Lagrange point · Sphere of influence (plain text: not yet on Wikitube) ## Explanatory notes Derived values are marked "(derived)". They use r_H ≈ a(1 − e)(m/3M)^(1/3) with masses, mean distances, eccentricities and diameters from the NASA planetary fact sheet and the solar mass from the NASA Sun fact sheet. The spacecraft and density examples use Earth's mass, 5.9722 × 10²⁴ kg, and orbital radii of 6,671 km and 42,164 km. The Moon's orbit uses a semi-major axis of 384,400 km and eccentricity 0.055. ## References [^souami2020]: Souami, D.; Cresson, J.; Biernacki, C.; Pierret, F. (2020). "On the local and global properties of gravitational spheres of influence". *Monthly Notices of the Royal Astronomical Society* 496: 4287–4297. https://doi.org/10.1093/mnras/staa1520 [^lauretta2023]: Lauretta, D.; OSIRIS-REx team (2023). "Word of the week: Hill sphere". OSIRIS-REx Asteroid Sample Return Mission. https://www.asteroidmission.org/wotw-hill-sphere/ [^hill2022]: Hill, R. J. (2022). "Gravitational clearing of natural satellite orbits". *Publications of the Astronomical Society of Australia* 39: e006. https://doi.org/10.1017/pasa.2021.62 [^depater2015]: de Pater, I.; Lissauer, J. J. (2015). *Planetary Sciences* (updated 2nd ed.). Cambridge University Press, pp. 26–34. ISBN 978-1-316-19569-7. [^nasa-fs]: NASA NSSDCA. "Planetary fact sheet – metric" and "Moon fact sheet" (fetched 2026-09-18). https://nssdc.gsfc.nasa.gov/planetary/factsheet/ [^chebotarev1964]: Chebotarev, G. A. (1964). "Gravitational spheres of the major planets, Moon and Sun". *Soviet Astronomy* 7: 618. Bibcode 1964SvA.....7..618C. [^chebotarev1965]: Chebotarev, G. A. (1965). "On the dynamical limits of the Solar System". *Soviet Astronomy* 8: 787. Bibcode 1965SvA.....8..787C. [^higuchi2017]: Higuchi, A.; Ida, S. (2017). "Temporary capture of asteroids by an eccentric planet". *The Astronomical Journal* 153: 155. https://doi.org/10.3847/1538-3881/aa5daa [^hamilton1992]: Hamilton, D. P.; Burns, J. A. (1992). "Orbital stability zones about asteroids. II. The destabilizing effects of eccentric orbits and of solar radiation". *Icarus* 96: 43–64. https://doi.org/10.1016/0019-1035(92)90005-R [^nasa-sun]: NASA NSSDCA. "Sun fact sheet" (fetched 2026-09-18). https://nssdc.gsfc.nasa.gov/planetary/factsheet/sunfact.html [^hamilton1991]: Hamilton, D. P.; Burns, J. A. (1991). "Orbital stability zones about asteroids". *Icarus* 92: 118–131. https://doi.org/10.1016/0019-1035(91)90039-V [^astakhov2003]: Astakhov, S. A.; Burbanks, A. D.; Wiggins, S.; Farrelly, D. (2003). "Chaos-assisted capture of irregular moons". *Nature* 423: 264–267. https://doi.org/10.1038/nature01622 [^chambers1996]: Chambers, J. E.; Wetherill, G. W.; Boss, A. P. (1996). "The stability of multi-planet systems". *Icarus* 119: 261–268. https://doi.org/10.1006/icar.1996.0019 [^johnston2019]: Johnston, W. R. (20 October 2019). "(66391) Moshup and Squannit". *Johnston's Archive*. http://www.johnstonsarchive.net/astro/astmoons/am-66391.html [^exoplanet-hd209458]: *The Extrasolar Planets Encyclopaedia*. "HD 209458 b". https://exoplanet.eu/catalog/hd_209458_b--10/ [^exoplanet-corot7]: *The Extrasolar Planets Encyclopaedia*. "CoRoT-7 b". https://exoplanet.eu/catalog/corot_7_b--526/ ## Further reading - de Pater, I.; Lissauer, J. J. (2015). *Planetary Sciences* (updated 2nd ed.). Cambridge University Press. ISBN 978-1-316-19569-7. - OpenStax (2016). *University Physics Volume 1*, ch. 13 "Gravitation". https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-1 ## External links - OSIRIS-REx mission. "Word of the week: Hill sphere". https://www.asteroidmission.org/wotw-hill-sphere/ - JPL Solar System Dynamics. "Approximate positions of the planets". https://ssd.jpl.nasa.gov/planets/approx_pos.html ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Hill_sphere) : [Wikitube](https://en.wikitube.io/wiki/Hill_sphere) · pinned revision [1372526144](https://en.wikipedia.org/w/index.php?oldid=1372526144) · 2026-09-18 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Solar_System]]. --- *Solar System portal child articles, wave 1 · 2026-09-18 · drafted · row SOL-080 · explorer state `?obj=oort`.* <!-- hub_tags: Life_Physics · PORTAL_Solar_System -->