Hill sphere

{{Short description|Region in which an astronomical body dominates the attraction of satellites}} {{for|the inner part of the Oort cloud|Hills cloud}} {{multiple issues| {{more science citations needed|date=July 2023}} {{Lead rewrite|talk=Talk:Hill_sphere#Problems_in_the_lead|date=April 2024}} }} thumb|right|300px|In sectional/side view, a two-dimensional representation of the three-dimensional concept of the Hill sphere, here showing the Earth's "gravity well" (gravitational potential of Earth, blue line), the same for the Moon (red line) and their combined potential (black thick line). Point P is the force free spot, where gravitational forces of Earth and Moon cancel. The sizes of Earth and Moon are in the proportion, but distances and energies are not to scale. {{Astrodynamics}}

The '''Hill sphere''' is a common model for the calculation of a gravitational sphere of influence. It is the most commonly used model to calculate the spatial extent of gravitational influence of an astronomical body (''m'') in which it dominates over the gravitational influence of other bodies, particularly a primary (''M'').<ref name="Souami Cresson Biernacki Pierret p. ">{{cite journal | last1=Souami | first1=D. | last2=Cresson | first2=J. | last3=Biernacki | first3=C. | last4=Pierret | first4=F. | title=On the local and global properties of gravitational spheres of influence | journal=Monthly Notices of the Royal Astronomical Society | date=2020 | volume=496 | issue=4 | pages=4287–4297 | arxiv=2005.13059 | doi=10.1093/mnras/staa1520 | doi-access=free }}</ref> It is sometimes confused with other models of gravitational influence, such as the Laplace sphere<ref name="Souami Cresson Biernacki Pierret p. "/> or the '''Roche sphere''', the latter of which causes confusion with the Roche limit.<ref name="Williams 2015 w404">{{cite web | last=Williams | first=Matt | title=How Many Moons Does Mercury Have? | website=Universe Today | date=2015-12-30 | url=https://www.universetoday.com/14335/how-many-moons-does-mercury-have/ | access-date=2023-11-08}}</ref><ref name="Hill 2022 p. ">{{cite journal | last=Hill | first=Roderick J. | title=Gravitational clearing of natural satellite orbits | journal=Publications of the Astronomical Society of Australia | publisher=Cambridge University Press | volume=39 | year=2022 | issn=1323-3580 | doi=10.1017/pasa.2021.62 | article-number=e006 | bibcode=2022PASA...39....6H | s2cid=246637375 }}</ref> It was defined by the American astronomer George William Hill, based on the work of the French astronomer Édouard Roche.{{not verified in body|date = July 2023}}

To be retained by a more massive, hence more gravitationally attracting, astrophysical object—a planet by a star, a moon by a planet—the less massive body must have an orbit that lies within the gravitational potential represented by the more massive body's Hill sphere.{{not verified in body|date = July 2023}} That moon would, in turn, have a Hill sphere of its own, and any object within that distance would tend to become a satellite of the moon, rather than of the planet itself.{{not verified in body|date = July 2023}}

[[File:Lagrange points2.svg|thumb|right|300px|A contour plot of the effective gravitational potential of a two-body system, here, the Sun and Earth, indicating the five Lagrange points.{{clarify|date = July 2023}}{{citation needed|date = July 2023}}]]

One simple view of the extent of the Solar System is that it is bounded by the Hill sphere of the Sun (engendered by the Sun's interaction with the galactic nucleus or other more massive stars).<ref name=Chebotarev1964a>{{cite journal |title=On the Dynamical Limits of the Solar System |first=G. A. |last=Chebotarev |date=March 1965 |journal=Soviet Astronomy |volume=8 |pages=787 |bibcode= 1965SvA.....8..787C}}</ref>{{verify source|date = July 2023}} A more complex example is the one at right, the Earth's Hill sphere, which extends between the Lagrange points {{L1}} and {{L2}},{{clarify|date = July 2023}} which lie along the line of centers of the Earth and the more massive Sun.{{not verified in body|date = July 2023}} The gravitational influence of the less massive body is least in that direction, and so it acts as the limiting factor for the size of the Hill sphere;{{clarify|date = July 2023}} beyond that distance, a third object in orbit around the Earth would spend at least part of its orbit outside the Hill sphere, and would be progressively perturbed by the tidal forces of the more massive body, the Sun, eventually ending up orbiting the latter.{{not verified in body|date = July 2023}}

For two massive bodies with gravitational potentials and any given energy of a third object of negligible mass interacting with them, one can define a zero-velocity surface in space which cannot be passed, the contour of the Jacobi integral.{{not verified in body|date = July 2023}} When the object's energy is low, the zero-velocity surface completely surrounds the less massive body (of this restricted three-body system), which means the third object cannot escape; at higher energy, there will be one or more gaps or bottlenecks by which the third object may escape the less massive body and go into orbit around the more massive one.{{not verified in body|date = July 2023}} If the energy is at the border between these two cases, then the third object cannot escape, but the zero-velocity surface confining it touches a larger zero-velocity surface around the less massive body{{verify source|date = July 2023}} at one of the nearby Lagrange points, forming a cone-like point there.{{clarify|date = July 2023}}{{not verified in body|date = July 2023}} At the opposite side of the less massive body, the zero-velocity surface gets close to the other Lagrange point.{{not verified in body|date = July 2023}}

==Definition== {{expand section | with = a comprehensive definition of the title subject, drawn from multiple secondary and tertiary sources, that can be summarised in the lead | small = no | date = July 2023}} The Hill radius or sphere (the latter defined by the former radius{{citation needed|date=July 2023}}) has been described as "the region around a planetary body where its own gravity (compared to that of the Sun or other nearby bodies) is the dominant force in attracting satellites," both natural and artificial.<ref>{{cite web | author = Lauretta, Dante and the Staff of the Osiris-Rex Asteroid Sample Return Mission | date = 2023 | title = Word of the Week: Hill Sphere | work = Osiris-Rex Asteroid Sample Return Mission (AsteroidMission.org) | location = Tempe, AZ | publisher = University of Arizona | url = https://www.asteroidmission.org/wotw-hill-sphere/ | access-date = July 22, 2023 }}</ref>{{better source needed|date = July 2023}}

As described by de Pater and Lissauer, all bodies within a system such as the Sun's Solar System "feel the gravitational force of one another", and while the motions of just two gravitationally interacting bodies—constituting a "two-body problem"—are "completely integrable ([meaning]...there exists one independent integral or constraint per degree of freedom)" and thus an exact, analytic solution, the interactions of ''three'' (or more) such bodies "cannot be deduced analytically", requiring instead solutions by numerical integration, when possible.<ref name=dePaterLissauerPlanetarySciences>{{cite book | author = de Pater, Imke & Lissauer, Jack | date = 2015 | title = Planetary Sciences | chapter = Dynamics (The Three-Body Problem, Perturbations and Resonances) | pages = 26, 28–30, 34 | edition = 2nd | location = Cambridge, England | publisher = Cambridge University Press | isbn = 9781316195697 | url = https://books.google.com/books?id=stFpBgAAQBAJ | access-date = 22 July 2023 }}</ref>{{rp|p.26}} This is the case, unless the negligible mass of one of the three bodies allows approximation of the system as a two-body problem, known formally as a "restricted three-body problem".<ref name=dePaterLissauerPlanetarySciences/>{{rp|p.26}}

For such two- or restricted three-body problems as its simplest examples—e.g., one more massive primary astrophysical body, mass of <math>m_1</math>, and a less massive secondary body, mass of <math>m_2</math>—the concept of a Hill radius or sphere is of the approximate limit to the secondary mass's "gravitational dominance",<ref name=dePaterLissauerPlanetarySciences/> a limit defined by "the extent" of its Hill sphere, which is represented mathematically as follows:<ref name=dePaterLissauerPlanetarySciences/>{{rp|p.29}}<ref name=HiguchiIdaTAJapril2017>{{cite journal | author= Higuchi1, A. & Ida, S. | title= Temporary Capture of Asteroids by an Eccentric Planet| journal= The Astronomical Journal | date= April 2017 | volume= 153 | issue= 4| pages= 155| doi= 10.3847/1538-3881/aa5daa | doi-access= free| publisher = The American Astronomical Society | location = Washington, DC| arxiv= 1702.07352| bibcode= 2017AJ....153..155H| s2cid= 119036212}}</ref> :<math>R_{\mathrm{H}} \approx a \sqrt[3]{\frac{m_2}{3(m_1+m_2)}} \approx a \sqrt[3]{\frac{1}{3(\frac{m_1}{m_2}+1)}}</math>, where, in this representation, semi-major axis "<math>a</math>" can be understood as the "instantaneous heliocentric distance" between the two masses (elsewhere abbreviated ''r<small>p</small>'').<ref name=dePaterLissauerPlanetarySciences/>{{rp|p.29}}<ref name=HiguchiIdaTAJapril2017/>

More generally, if the less massive body, <math>m_2</math>, orbits a more massive body, <math>m_1</math>(e.g., as a planet orbiting around the Sun), and has a semi-major axis <math>a</math>, and an eccentricity of <math>e</math>, then the Hill radius or sphere, <math>R_{\mathrm{H}}</math> of the less massive body, calculated at the pericenter, is approximately:<ref name="HamiltonBurns92">{{cite journal | author= Hamilton, D.P. & Burns, J.A. | title= Orbital Stability Zones About Asteroids: II. The Destabilizing Effects of Eccentric Orbits and of Solar Radiation| journal= Icarus| date= March 1992| volume= 96 | issue= 1| pages= 43–64| location = New York, NY | publisher = Academic Press| doi= 10.1016/0019-1035(92)90005-R| bibcode= 1992Icar...96...43H| doi-access= free}} See also {{cite journal | author= Hamilton, D.P. & Burns, J.A.| title= Orbital Stability Zones About Asteroids| journal= Icarus| date= March 1991| volume= 92 | issue= 1| pages= 118–131| url= https://www.astro.umd.edu/~hamilton/research/reprints/HamBurns91.pdf | access-date = 22 July 2023 | location = New York, NY | publisher = Academic Press| doi= 10.1016/0019-1035(91)90039-V| bibcode= 1991Icar...92..118H}} cited therein.</ref>{{primary source inline|date = July 2023}}{{better source needed|date = July 2023}}  :<math>R_{\mathrm{H}} = a {\frac{(1-e^2)}{(1+e \cos(\theta))}} \sqrt[3]{\frac{m_2}{3(m_1+m_2)}} = a {\frac{(1-e^2)}{(1+e \cos(\theta))}}\sqrt[3]{\frac{1}{3(\frac{m_1}{m_2}+1)}}</math> where <math>\theta</math> is the true anomaly for the less massive body in its orbit around the more massive body, an angle contained between 0° and 360°, with <math>\cos(\theta)</math> varying between +1 at the periapsis, <math>\theta = 0</math>, and -1 at the apoapsis, <math>\theta = 180</math>. :With <math>(1-e^2) = (1+e)(1-e)</math>,

:<math>R_{\mathrm{H}} = a (1-e) \sqrt[3]{\frac{m_2}{3(m_1+m_2)}} = a (1-e)\sqrt[3]{\frac{1}{3(\frac{m_1}{m_2}+1)}}</math> at the perigee and :<math>R_{\mathrm{H}} = a (1+e) \sqrt[3]{\frac{m_2}{3(m_1+m_2)}} = a (1+e)\sqrt[3]{\frac{1}{3(\frac{m_1}{m_2}+1)}}</math> at the apogee, in the case of the Earth-Moon system, for example, <math>{\frac{m_2}{m_1}} = 0.0123000371(4)</math> and <math> {\frac{m_1}{m_2}} \approx 81.300568</math>, <math>a = 384,399 \ km</math>, <math>e = 0.05490</math> and the Hill radius of the Moon at the perigee is <math>R_{\mathrm{H}} = 57,910 \ km </math>, and at the apogee is <math>64,638 \ km</math>.

When eccentricity is negligible (the most favourable case for orbital stability), this expression reduces to the one presented above.{{citation needed|date = July 2023}}

== Example and derivation == {{more citations needed|section|dated=December 2023|date=December 2023}} [[File:comparison of Hill sphere and Roche limit.svg|thumb|A schematic, not-to-scale representation of Hill spheres (as 2D radii) and Roche limits of each body of the Sun-Earth-Moon system. The actual Hill radius for the Moon is on the order of 60,000 km (i.e., extending less than one-sixth the distance of the 378,000 km between the Moon and the Earth).<ref>{{cite web | author = Follows, Mike | date = 4 October 2017 | title = Ever Decreasing Circles | work = NewScientist.com | url = https://www.newscientist.com/lastword/mg23631461-200-ever-decreasing-circles/ | access-date = 23 July 2023 | quote = The moon's Hill sphere has a radius of 60,000 kilometres, about one-sixth of the distance between it and Earth.}}</ref>]] In the Earth-Sun example, the Earth ({{val|5.97e24|u=kg}}) orbits the Sun ({{val|1.99e30|u=kg}}) at a distance of 149.6 million km, or one astronomical unit (AU). The Hill sphere for Earth thus extends out to about 1.5 million km (0.01 AU). The Moon's orbit, at a distance of 0.384 million km from Earth, is comfortably within the gravitational sphere of influence of Earth and it is therefore not at risk of being pulled into an independent orbit around the Sun. The Sun's Hill sphere has an unstable maximum boundary of {{convert|230000|AU|pc ly|abbr=on}}.<ref name=Chebotarev1964b>{{cite journal |bibcode=1964SvA.....7..618C |title=Gravitational Spheres of the Major Planets, Moon and Sun |last1=Chebotarev |first1=G. A. |volume=7 |date=1964 |pages=618 |journal=Soviet Astronomy}}</ref>

The earlier eccentricity-ignoring formula can be re-stated as follows: :<math>\frac{R^3_{\mathrm{H}}}{a^3} \approx 1/3 \frac{m_2}{M}</math>, or <math>3\frac{R^3_{\mathrm{H}}}{a^3} \approx \frac{m_2}{M}</math>, where M is the sum of the interacting masses.

=== Derivation === {{unreferenced section|date = July 2023}} The expression for the Hill radius can be found by equating gravitational and centrifugal forces acting on a test particle (of mass much smaller than <math>m</math>) orbiting the secondary body. Assume that the distance between masses <math>M</math> and <math>m</math> is <math>r</math>, and that the test particle is orbiting at a distance <math>r_{\mathrm{H}}</math> from the secondary. When the test particle is on the line connecting the primary and the secondary body, the force balance requires that

:<math>\frac{Gm}{r^2_{\mathrm{H}}}-\frac{GM}{(r-r_{\mathrm{H}})^2}+\Omega^2(r-r_{\mathrm{H}})=0,</math>

where <math>G</math> is the gravitational constant and <math>\Omega=\sqrt{\frac{GM}{r^3}}</math> is the (Keplerian) angular velocity of the secondary about the primary (assuming that <math>m\ll M</math>). The above equation can also be written as

:<math>\frac{m}{r^2_{\mathrm{H}}}-\frac{M}{r^2}\left(1-\frac{r_{\mathrm{H}}}{r}\right)^{-2}+\frac{M}{r^2}\left(1-\frac{r_{\mathrm{H}}}{r}\right)=0,</math>

which, through a binomial expansion to leading order in <math>r_{\mathrm{H}}/r</math>, can be written as

:<math>\frac{m}{r^2_{\mathrm{H}}}-\frac{M}{r^2}\left(1+2\frac{r_{\mathrm{H}}}{r}\right)+\frac{M}{r^2}\left(1-\frac{r_{\mathrm{H}}}{r}\right) = \frac{m}{r^2_{\mathrm{H}}}-\frac{M}{r^2}\left(3\frac{r_{\mathrm{H}}}{r}\right)\approx 0.</math> Hence, the relation stated above

:<math>\frac{r_{\mathrm{H}}}{r}\approx \sqrt[3]{\frac{m}{3 M}}.</math>

If the orbit of the secondary about the primary is elliptical, the Hill radius is maximum at the apocenter, where <math>r</math> is largest, and minimum at the pericenter of the orbit. Therefore, for purposes of stability of test particles (for example, of small satellites), the Hill radius at the pericenter distance needs to be considered.

To leading order in <math>r_{\mathrm{H}}/r</math>, the Hill radius above also represents the distance of the Lagrangian point L<sub>1</sub> from the secondary.

== Regions of stability == The Hill sphere is only an approximation, and other forces such as radiation pressure or the Yarkovsky effect can eventually perturb an object out of the sphere.<ref>{{Cite web |date=2025-10-31 |title=Yarkovsky Effect - Definition & Detailed Explanation - Astronomical Units & Measurements Glossary - Sentinel Mission |url=https://sentinelmission.org/astronomical-units-measurements-glossary/yarkovsky-effect/ |access-date=2025-12-04 |language=en-US}}</ref> As stated, the satellite (third mass) should be small enough that its gravity contributes negligibly.<ref name=dePaterLissauerPlanetarySciences/>{{rp|p.26ff}}

At large distances from a primary body, retrograde orbits remain stable over a wider region than prograde orbits. This was thought to explain the preponderance of retrograde moons around Jupiter; however, Saturn has a more even mix of retrograde/prograde moons so the reasons are more complicated.<ref>{{Cite journal |last1=Astakhov |first1=Sergey A. |last2=Burbanks |first2=Andrew D. |last3=Wiggins |first3=Stephen |last4=Farrelly |first4=David |name-list-style=amp |date=2003 |title=Chaos-assisted capture of irregular moons |journal=Nature |volume=423 |issue=6937 |pages=264–267 |bibcode=2003Natur.423..264A |doi=10.1038/nature01622 |pmid=12748635 |s2cid=16382419}}</ref> In a two-planet system, the mutual Hill radius of the two planets must exceed <math>2\sqrt{3}</math> to be stable. For systems with three or more planets, configurations in which neighbouring planets are separated by fewer than ten mutual Hill radii are inherently unstable, mainly because a third planet introduces perturbations that drain angular momentum.<ref>Chambers, J. E., Wetherill, G. W., & Boss, A. P. (1996). [https://d1wqtxts1xzle7.cloudfront.net/47958397/icar.1996.001920160810-4925-1isamnf-libre.pdf?1470874395=&response-content-disposition=inline%3B+filename%3DThe_Stability_of_Multi_Planet_Systems.pdf&Expires=1734045827&Signature=bWdVw3AQ2siVDrvhkojr4gWwcmVoxl84SRLdN0TJlLWpVuIZgNU5rI1lmMyQ1vIL3RBrmdyP8LNXaUAs6jpTYIoGkuyh36KOAnvZENl9DORqtFIkUCkHGq~5FMo1-MOHwdFUdjLCa12FMHOvZx55fqef0ouvVWbV4S-lYOLmk-Vw-hQnanixbFrWdMF8dhFBb1K-msiyAe-xm8ILNY25B8Kn6BcYaVWf4v0wvij-bD3llu2DrSuPvyfqkYY8ksYVjVEEiphqDJ44RhrB-OkPMKmYZHR~8htFTDb9NY8CXOD410ptQMl6bNztSntL35x31mxx2IK8lEvjKgHfpYmhWg__&Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA The stability of multi-planet systems]. Icarus, 119(2), 261-268.</ref>

== Further examples == {{More citations needed section|date=September 2018}} It is possible for a Hill sphere to be so small that it is impossible to maintain an orbit around a body. For example, an astronaut could not have orbited the 104 ton Space Shuttle at an orbit 300 km above the Earth, because a 104-ton object at that altitude has a Hill sphere of only 120 cm in radius, much smaller than a Space Shuttle. A sphere of this size and mass would be denser than lead, and indeed, in low Earth orbit, a spherical body must be more dense than lead in order to fit inside its own Hill sphere, or else it will be incapable of supporting an orbit. Satellites further out in geostationary orbit, however, would only need to be more than 6% of the density of water to fit inside their own Hill sphere.{{citation needed|date=November 2016}}

Within the Solar System, the planet with the largest Hill radius is Neptune, with 116 million km, or 0.775 au; its great distance from the Sun amply compensates for its small mass relative to Jupiter (whose own Hill radius measures 53 million km). If Planet Nine exists, however, then assuming a mass of ~10 Earths, radius of ~15,000 km, distance of ~500 AU and eccentricity of ~0.25, it would have a Hill radius of 1.2 billion km, over 10 times Neptune's Hill radius. An asteroid from the asteroid belt will have a Hill sphere that can reach 220,000 km (for 1 Ceres), diminishing rapidly with decreasing mass. The Hill sphere of 66391 Moshup, a Mercury-crossing asteroid that has a moon (named Squannit), measures 22 km in radius.<ref name="Moshup">{{cite web |first1 = Robert |last1 = Johnston |title = (66391) Moshup ''and'' Squannit |website = Johnston's Archive |date = 20 October 2019 |url = http://www.johnstonsarchive.net/astro/astmoons/am-66391.html |access-date = 30 March 2017}}</ref>

A typical extrasolar "hot Jupiter", HD 209458 b,<ref>{{Cite encyclopedia |url=https://exoplanet.eu/catalog/hd_209458_b--10/ |title=HD 209458 b |access-date=2010-02-16 |archive-date=2010-01-16 |archive-url=https://web.archive.org/web/20100116063057/http://www.exoplanet.eu/planet.php?p1=HD+209458&p2=b |encyclopedia=Extrasolar Planets Encyclopaedia |url-status=live }}</ref> has a Hill sphere radius of 593,000 km, about eight times its physical radius of approx 71,000 km. Even the smallest close-in extrasolar planet, CoRoT-7b,<ref>{{Cite encyclopedia|url=https://exoplanet.eu/catalog/corot_7_b--526/|encyclopedia=Extrasolar Planets Encyclopaedia|title=Planet CoRoT-7 b|date=2024 }}</ref> still has a Hill sphere radius (61,000 km), six times its physical radius (approx 10,000 km). Therefore, these planets could have small moons close in, although not within their respective Roche limits.{{citation needed|date=November 2016}}

== Hill spheres for the Solar System == The following table and logarithmic plot show the radius of the Hill spheres of some bodies of the Solar System calculated with the first formula stated above (including orbital eccentricity), using values obtained from the JPL DE405 ephemeris and from the NASA Solar System Exploration website.<ref name="mnras_style">{{cite web |url=https://solarsystem.nasa.gov/ |title=NASA Solar System Exploration |author= |publisher=NASA |access-date=2020-12-22}}</ref>

{| class="wikitable" |+ Radius of the Hill spheres of some bodies of the Solar System |- ! Body !! Million km !! au !! Body radii !! Arcminutes<ref group="note">At average distance, as seen from the Sun. The angular size as seen from Earth varies depending on Earth's proximity to the object.</ref> !! Farthest moon (au) !Ratio of farthest moon to Hill sphere radius |- | Mercury || 0.1753 || 0.0012 || 71.9 || 10.7 || {{NA}} |{{NA}} |- | Venus || 1.0042 || 0.0067 || 165.9 || 31.8 || {{NA}} |{{NA}} |- | Earth || 1.4714 || 0.0098 || 230.7 || 33.7 || 0.00257 |0.262 |- | Mars || 0.9827 || 0.0066 || 289.3 || 14.9 || 0.00016 |0.0242 |- | Jupiter || 50.5736 || 0.3381 || 707.4 || 223.2 || 0.1662 |0.491 |- | Saturn || 61.6340 || 0.4120 || 1022.7 || 147.8 || 0.1785 |0.433 |- | Uranus || 66.7831 || 0.4464 || 2613.1 || 80.0 || 0.1366 |0.306 |- | Neptune || 115.0307 || 0.7689 || 4644.6 || 87.9 || 0.3360 |0.437 |- | Ceres || 0.2048 || 0.0014 || 433.0 || 1.7 || {{NA}} |{{NA}} |- | Pluto || 5.9921 || 0.0401 || 5048.1 || 3.5 || 0.00043 |0.0107 |- |Haumea |9.9186 |0.0663 |12155.1 |5.3 |0.00033 |0.0049 |- |Makemake |4.8217 |0.0322 |6743.7 |2.4 |0.00014 |0.0046 |- | Eris || 8.1176 || 0.0543 || 6979.9 || 2.7 || 0.00025 |0.0046 |- |Sedna |14.3491 |0.0959 |~15943,8 |0.7 |{{?}} |{{?}} |- |Planet Nine (hypothetical) |1209.0621 |8.082 |~80604.1 |55.6 |{{?}} |{{?}} |} left|frame|Logarithmic plot of the Hill radii (in km) for the bodies of the Solar System {{Clear}}

== See also == * {{annotated link|Laplace sphere}} * {{annotated link|Interplanetary Transport Network}} * {{annotated link|n-body problem|''n''-body problem}} * {{annotated link|Roche lobe}} * {{annotated link|Sphere of influence (astrodynamics)}} * {{annotated link|Sphere of influence (black hole)}}

== Explanatory notes == {{reflist|group="note"}}

== References == {{reflist}}

==Further reading== * {{cite book | author = de Pater, Imke & Lissauer, Jack | date = 2015 | title = Planetary Sciences | chapter = Dynamics (The Three-Body Problem, Perturbations and Resonances) | pages = 28–30, 34 | edition = 2nd | location = Cambridge, England | publisher = Cambridge University Press | isbn = 9781316195697 | url = https://books.google.com/books?id=stFpBgAAQBAJ | access-date = 22 July 2023 }} * {{cite book | author = de Pater, Imke & Lissauer, Jack | date = 2015 | title = Planetary Sciences | chapter = Planet Formation (Formation of the Giant Planets, Satellites of Planets and Minor Planets) | pages = 539, 544 | edition = 2nd | location = Cambridge, England | publisher = Cambridge University Press | isbn = 9781316195697 | url = https://books.google.com/books?id=stFpBgAAQBAJ | access-date = 22 July 2023 }} * {{cite book | author = Gurzadyan, Grigor A. | date = 2020 | title = Theory of Interplanetary Flights | chapter = The Sphere of Attraction, the Sphere of Action and Hill's Sphere | pages = 258–263 | location = Boca Raton, FL | publisher = CRC Press | isbn = 9781000116717 | url = https://books.google.com/books?id=LasLEAAAQBAJ | access-date = 22 July 2023 }} * {{cite book | author = Gurzadyan, Grigor A. | date = 2020 | title = Theory of Interplanetary Flights | chapter = The Roche Limit | pages = 263f | location = Boca Raton, FL | publisher = CRC Press | isbn = 9781000116717 | url = https://books.google.com/books?id=LasLEAAAQBAJ | access-date = 22 July 2023 }} * {{cite book | author = Ida, S.; Kokubo, E. & Takeda, T. | date = 2012 | title = Collisional Processes in the Solar System | chapter = N-Body Simulations of Moon Accretion | pages = 206, 209f | editor = Marov, Mikhail Ya. & Rickman, Hans | series = Astrophysics and Space Science Library | volume = 261 | location = Berlin, Germany | publisher = Springer Science & Business Media | isbn = 9789401007122 | url = https://books.google.com/books?id=3uHzCAAAQBAJ | access-date = 22 July 2023 }} * {{cite book | author = Ip, W.-H. & Fernandez, J.A. | date = 2012 | title = Collisional Processes in the Solar System | chapter = Accretional Origin of the Giant Planers and its Consequences | pages = 173f | editor = Marov, Mikhail Ya. & Rickman, Hans | series = Astrophysics and Space Science Library | volume = 261 | location = Berlin, Germany | publisher = Springer Science & Business Media | isbn = 9789401007122 | url = https://books.google.com/books?id=3uHzCAAAQBAJ | access-date = 22 July 2023 }} * {{cite book | author = Asher, D.J.; Bailey, M.E. & Steel | date = 2012 | title = Collisional Processes in the Solar System | chapter = The Role of Non-Gravitational Forces in Decoupling Orbits from Jupiter | pages = 122 | editor = Marov, Mikhail Ya. & Rickman, Hans | series = Astrophysics and Space Science Library | volume = 261 | location = Berlin, Germany | publisher = Springer Science & Business Media | isbn = 9789401007122 | url = https://books.google.com/books?id=3uHzCAAAQBAJ | access-date = 22 July 2023 }}

== External links == * [https://web.archive.org/web/20060621062915/http://www.asterism.org/tutorials/tut22-1.htm Can an Astronaut Orbit the Space Shuttle?] * [http://blogs.discovermagazine.com/badastronomy/2008/09/29/the-moon-that-went-up-a-hill-but-came-down-a-planet The moon that went up a hill, but came down a planet] {{Webarchive|url=https://web.archive.org/web/20080930094142/http://blogs.discovermagazine.com/badastronomy/2008/09/29/the-moon-that-went-up-a-hill-but-came-down-a-planet/ |date=2008-09-30 }}

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{{DEFAULTSORT:Hill Sphere}} Category:Equations of astronomy Category:Orbits Category:Tides