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| [[File:Lagrange points2.svg|thumb|right|300px|A contour plot of the effective [[potential]] of a two-body system due to gravity and inertia at one point in time. The Hill spheres are the circular regions surrounding the two large masses. (Earth and sun radii are not drawn to scale.)]]
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| An [[astronomical body]]'s '''Hill sphere''' is the region in which it dominates the attraction of [[natural satellite|satellites]]. To be retained by a [[planet]], a [[natural satellite|moon]] must have an [[orbit]] that lies within the planet's Hill sphere. That moon would, in turn, have a Hill sphere of its own. Any object within that distance would tend to become a satellite of the moon, rather than of the planet itself. Thus, some (perhaps simplistic) definitions of "solar system" simply refer to the Hill sphere of the local star(s), namely the [[Sun]].<ref>http://adsabs.harvard.edu/full/1965SvA.....8..787C</ref>
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| In more precise terms, the Hill sphere approximates the [[gravity|gravitational]] [[sphere of influence (astrodynamics)|sphere of influence]] of a smaller body in the face of [[Perturbation (astronomy)|perturbation]]s from a more massive body. It was defined by the [[United States|American]] [[astronomer]] [[George William Hill]], based upon the work of the [[France|French]] astronomer [[Édouard Roche]]. For this reason, it is also known as the '''Roche sphere''' (not to be confused with the [[Roche limit]]).
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| In the example to the right, the Hill sphere extends between the [[Lagrangian point]]s {{L1}} and {{L2}}, which lie along the line of centers of the two bodies. The region of influence of the second body is shortest in that direction, and so it acts as the limiting factor for the size of the Hill sphere. Beyond that distance, a third object in orbit around the second (e.g. Jupiter) would spend at least part of its orbit outside the Hill sphere, and would be progressively perturbed by the tidal forces of the central body (e.g. the Sun), eventually ending up orbiting the latter.
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| == Formula and examples ==
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| If the mass of the smaller body (e.g. Earth) is ''m'', and it orbits a heavier body (e.g. Sun) of mass ''M'' with a [[semi-major axis]] ''a'' and an [[Eccentricity (mathematics)|eccentricity]] of ''e'', then the radius ''r'' of the Hill sphere for the smaller body (e.g. Earth) is, approximately<ref name="HamiltonBurns92">{{cite journal | author= D.P. Hamilton & J.A. Burns| title= Orbital stability zones about asteroids. II - The destabilizing effects of eccentric orbits and of solar radiation| journal= Icarus| year= 1992| volume= 96 | issue= 1| pages= 43| bibcode= 1992Icar...96...43H | doi= 10.1016/0019-1035(92)90005-R}}</ref>
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| :<math>r \approx a (1-e) \sqrt[3]{\frac{m}{3 M}}.</math>
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| When eccentricity is negligible (the most favourable case for orbital stability), this becomes
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| :<math>r \approx a \sqrt[3]{\frac{m}{3M}}.</math>
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| In the Earth example, the Earth (5.97×10<sup>24</sup> kg) orbits the Sun (1.99×10<sup>30</sup> kg) at a distance of 149.6 million km. 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 (astrodynamics)|sphere of influence]] of Earth and it is therefore not at risk of being pulled into an independent orbit around the Sun. All stable satellites of the Earth (those within the Earth's Hill sphere) must have an orbital period shorter than 7 months.
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| The previous (eccentricity-ignoring) formula can be re-stated as follows:
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| :<math>3\frac{r^3}{a^3} \approx \frac{m}{M}. </math>
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| This expresses the relation in terms of the volume of the Hill sphere compared with the volume of the second body's orbit around the first; specifically, the ratio of the masses is three times the ratio of the volume of these two spheres.
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| A quick way of estimating the radius of the Hill sphere comes from replacing mass with density in the above equation:
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| :<math>\frac{r}{R_{\mathrm{secondary}}} \approx \frac{a}{R_{\mathrm{primary}}} \sqrt[3]{\frac{\rho_{\mathrm{secondary}}}{3 \rho_{\mathrm{primary}}}} \approx \frac{a}{R_{\mathrm{primary}}}, </math>
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| where <math>\rho_{\mathrm{second}}</math> and <math>\rho_{\mathrm{primary}}</math> are the densities of the primary and secondary bodies, and <math>\frac{r}{R_{\mathrm{secondary}}}</math> and <math>\frac{r}{R_{\mathrm{primary}}}</math> are their radii. The second approximation is justified by the fact that, for most cases in the Solar System, <math>\sqrt[3]{\frac{\rho_{\mathrm{secondary}}}{3 \rho_{\mathrm{primary}}}}</math> happens to be close to one. (The Earth–Moon system is the largest exception, and this approximation is within 20% for most of Saturn's satellites.) This is also convenient, since many planetary astronomers work in and remember distances in units of planetary radii.
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| === True region of stability ===
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| 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. This third object should also be of small enough mass that it introduces no additional complications through its own gravity. Detailed numerical calculations show that orbits at or just within the Hill sphere are not stable in the long term; it appears that stable satellite orbits exist only inside 1/2 to 1/3 of the Hill radius. The region of stability for [[retrograde orbit]]s at a large distance from the primary, is larger than the region for [[prograde orbit]]s at a large distance from the primary. 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 |last=Astakhov |first=Sergey A. |last2=Burbanks |first2=Andrew D. |last3=Wiggins |first3=Stephen |lastauthoramp=yes |last4=Farrelly |first4=David |title=Chaos-assisted capture of irregular moons |journal=[[Nature (journal)|Nature]] |volume=423 |issue=6937 |pages=264–267 |year=2003 |pmid=12748635 |doi=10.1038/nature01622 |bibcode = 2003Natur.423..264A }}</ref>
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| === Further examples ===
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| An astronaut could not orbit the [[Space Shuttle]] (with mass of 104 [[tonnes]]), where the orbit is 300 km above the Earth, since the Hill sphere of the shuttle is only 120 cm in radius, much smaller than the shuttle itself. In fact, in any [[low Earth orbit]], a spherical body must be 800 times denser than [[lead]] in order to fit inside its own Hill sphere, or else it will be incapable of supporting an orbit.<!-- Min density = 9102.6 g/cm^3, altitude 1400 km, which is given in the LEO article as the highest-altitude LEO. Lead's density 11.34 g/cm^3. Earth mass 5.9736E+24 kg --> A spherical [[geostationary satellite]] would need to be more than 5 times denser than lead to support satellites of its own; such a satellite would be 2.5 times denser than [[osmium]], the densest naturally-occurring material on Earth.<!-- Geostationary altitude 35,786 km, iridium density 22.65 g/cm^3 --> Only at twice the geostationary distance could a lead sphere possibly support its own satellite; since the Moon is more than three times further than the 3-fold geostationary distance necessary, [[lunar orbit]]s are possible.
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| 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). An [[asteroid]] from the [[asteroid belt]] will have a Hill sphere that can reach 220 000 km (for [[1 Ceres]]), diminishing rapidly with its mass. The Hill sphere of {{mpl|(66391) 1999 KW|4}}, a [[List of Mercury-crossing minor planets|Mercury-crosser asteroid]] that has a moon (S/2001 (66391) 1), measures 22 km in radius.
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| A typical [[extrasolar planet|extrasolar]] "[[hot Jupiter]]", [[HD 209458 b]]<ref>[http://exoplanet.eu/planet.php?p1=HD+209458&p2=b HD 209458 b]</ref> has a Hill sphere of radius (593,000 km) about 8 times its physical radius (approx 71,000 km). Even the smallest close-in extrasolar planet, [[CoRoT-7b]]<ref>[http://exoplanet.eu/planet.php?p1=CoRoT-7&p2=b CoRoT-7 b]</ref> still has a Hill sphere radius (61,000 km) 6 times greater than its physical radius (approx 10,000 km). Therefore these planets could have small moons close in.
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| == Derivation ==
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| A non-rigorous but conceptually accurate derivation of the Hill radius can be made by equating the orbital angular speed of the orbiter around a body (i.e. a planet) and the orbital angular speed of that planet around the host star. This is the radius at which the gravitational influence of the star roughly equals that of the planet.
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| :<math>\Omega_{\mathrm{planet}} = \Omega_\star</math>
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| :<math>\sqrt{\frac{GM_{\mathrm{planet}}}{R_H^3}} = \sqrt{\frac{GM_\star}{a^3}},</math>
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| where <math>R_H</math> is the Hill radius, a is the semi-major axis of the planet orbiting the star. With some basic algebra:
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| :<math>\frac{M_{\mathrm{planet}}}{R_H^3} = \frac{M_\star}{a^3}</math>
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| giving a Hill radius of:
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| :<math>
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| R_H = a \left(\frac{M_{\mathrm{planet}}}{M_\star}\right)^{1/3}.
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| </math>
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| == Solar System ==
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| The following plot shows the Hill radius (in km) of some bodies of the Solar System:
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| [[File:Hill sphere of the planets.png|upright=1.3|center|Radius (km) of the Hill sphere in the Solar System]]
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| == See also ==
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| * [[n-body problem|''n''-body problem]]
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| * [[Sphere of influence (astrodynamics)]]
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| * [[Interplanetary Transport Network]]
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| == References ==
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| {{reflist}}
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| == External links ==
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| * [http://www.asterism.org/tutorials/tut22-1.htm Can an Astronaut Orbit the Space Shuttle?]
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| * [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]
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| {{DEFAULTSORT:Hill Sphere}}
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| [[Category:Orbits]]
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