Movatterモバイル変換

[0]ホーム

Jump to content

Hill sphere

Edit links

From Wikipedia, the free encyclopedia

Region in which an astronomical body dominates the attraction of satellites

For the inner part of the Oort cloud, seeHills cloud.

This article has multiple issues. Please helpimprove it or discuss these issues on thetalk page.(Learn how and when to remove these messages)

This scientific articleneeds additionalcitations tosecondary or tertiary sources. Help add sources such as review articles, monographs, or textbooks. Please also establish the relevance for anyprimary research articles cited. Unsourced or poorly sourced material may be challenged and removed.(July 2023) (Learn how and when to remove this message)

This article'slead sectionmay need to be rewritten. Relevant discussion may be found onTalk:Hill sphere. Please review thelead guide and helpimprove the lead of this article if you can.(April 2024) (Learn how and when to remove this message)

(Learn how and when to remove this message)

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
Part of a series on

Orbital mechanics
Orbital elements Apsis Argument of periapsis Eccentricity Inclination Mean anomaly Orbital nodes Semi-major axis True anomaly
Types oftwo-body orbits by eccentricity Circular orbit Elliptic orbit Transfer orbit (Hohmann transfer orbit Bi-elliptic transfer orbit) Parabolic orbit Hyperbolic orbit Radial orbit Decaying orbit
Equations Dynamical friction Escape velocity Kepler's equation Kepler's laws of planetary motion Orbital period Orbital velocity Surface gravity Specific orbital energy Vis-viva equation
Celestial mechanics
Gravitational influences Barycenter Hill sphere Perturbations Sphere of influence
N-body orbits Lagrangian points (Halo orbits) Lissajous orbits Lyapunov orbits
Engineering and efficiency
Preflight engineering Mass ratio Payload fraction Propellant mass fraction Tsiolkovsky rocket equation
Efficiency measures Gravity assist Oberth effect
Propulsive maneuvers Orbital maneuver Orbit insertion
v t e

TheHill sphere is a common model for the calculation of agravitational sphere of influence. It is the most commonly used model to calculate the spatial extent of gravitational influence of anastronomical body (m) in which it dominates over the gravitational influence of other bodies, particularly aprimary (M).^[1] It is sometimes confused with other models of gravitational influence, such as theLaplace sphere^[1] or theRoche sphere, the latter of which causes confusion with theRoche limit.^[2]^[3] It was defined by theAmerican astronomer George William Hill, based on the work of theFrench astronomerÉdouard Roche.^{[not verified in body]}

To be retained by a more gravitationally attracting astrophysical object—a planet by a more massive star, amoon by a more massive planet—the less massive body must have anorbit that lies within the gravitational potential represented by the more massive body's Hill sphere.^{[not verified in body]} 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]}

A contour plot of the effective gravitational potential of a two-body system, here, the Sun and Earth, indicating the fiveLagrange points.^{[clarification needed]}^{[citation needed]}

One simple view of the extent of theSolar System is that it is bounded by the Hill sphere of theSun (engendered by the Sun's interaction with thegalactic nucleus or other more massive stars).^[4]^{[verification needed]} A more complex example is the one at right, the Earth's Hill sphere, which extends between theLagrange points L₁ andL₂,^{[clarification needed]} which lie along the line of centers of the Earth and the more massive Sun.^{[not verified in body]} 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;^{[clarification needed]} 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]}

For two massive bodies with gravitational potentials and any given energy of a third object of negligible mass interacting with them, one can define azero-velocity surface in space which cannot be passed, the contour of theJacobi integral.^{[not verified in body]} When the object's energy is low, the zero-velocity surface completely surrounds the less massive body (of thisrestricted 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]} 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^{[verification needed]} at one of the nearby Lagrange points, forming a cone-like point there.^{[clarification needed]}^{[not verified in body]} At the opposite side of the less massive body, the zero-velocity surface gets close to the other Lagrange point.^{[not verified in body]}

Definition

[edit]

This sectionneeds expansion with: a comprehensive definition of the title subject, drawn from multiple secondary and tertiary sources, that can be summarised in the lead. You can help byadding to it.(July 2023)

The Hill radius or sphere (the latter defined by the former radius^{[citation needed]}) 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.^[5]^{[better source needed]}

As described by de Pater and Lissauer, all bodies within a system such as the Sun'sSolar 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 ofthree (or more) such bodies "cannot be deduced analytically", requiring instead solutions by numerical integration, when possible.^[6]^: 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".^[6]^: p.26

For such two- or restricted three-body problems as its simplest examples—e.g., one more massive primary astrophysical body, mass of $m_{1}$ , and a less massive secondary body, mass of $m_{2}$ —the concept of a Hill radius or sphere is of the approximate limit to the secondary mass's "gravitational dominance",^[6] a limit defined by "the extent" of its Hill sphere, which is represented mathematically as follows:^[6]^: p.29^[7]

R_{\mathrm {H} }\approx a{\sqrt[{3}]{\frac {m_{2}}{3(m_{1}+m_{2})}}}

where, in this representation, semi-major axis " $a {\displaystyle a}$ " can be understood as the "instantaneous heliocentric distance" between the two masses (elsewhere abbreviatedrp).^[6]^: p.29^[7]

More generally, if the less massive body, $m_{2}$ , orbits a more massive body, $m_{1}$ (e.g., as a planet orbiting around the Sun), and has asemi-major axis $a {\displaystyle a}$ , and aneccentricity of $e {\displaystyle e}$ , then the Hill radius or sphere, $R_{\mathrm {H} }$ of the less massive body, calculated at thepericenter, is approximately:^[8]^{[non-primary source needed]}^{[better source needed]}

R_{\mathrm {H} }\approx a(1-e){\sqrt[{3}]{\frac {m_{2}}{3(m_{1}+m_{2})}}}.

When eccentricity is negligible (the most favourable case for orbital stability), this expression reduces to the one presented above.^{[citation needed]}

Example and derivation

[edit]

This sectionneeds additional citations forverification. Please helpimprove this article byadding citations to reliable sources in this section. Unsourced material may be challenged and removed.
Find sources: "Hill sphere" – news ·newspapers ·books ·scholar ·JSTOR(December 2023) (Learn how and when to remove this message)

A schematic, not-to-scale representation of Hill spheres (as 2D radii) andRoche 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).^[9]

In the Earth-Sun example, the Earth (5.97×10²⁴ kg) orbits the Sun (1.99×10³⁰ kg) at a distance of 149.6 million km, or oneastronomical 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 gravitationalsphere of influence of Earth and it is therefore not at risk of being pulled into an independent orbit around the Sun.

The earlier eccentricity-ignoring formula can be re-stated as follows:

{\frac {R_{\mathrm {H} }^{3}}{a^{3}}}\approx 1/3{\frac {m_{2}}{M}}

, or

3{\frac {R_{\mathrm {H} }^{3}}{a^{3}}}\approx {\frac {m_{2}}{M}}

where M is the sum of the interacting masses.

Derivation

[edit]

This sectiondoes notcite anysources. Please helpimprove this section byadding citations to reliable sources. Unsourced material may be challenged andremoved.(July 2023) (Learn how and when to remove this message)

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 $m {\displaystyle m}$ ) orbiting the secondary body. Assume that the distance between masses $M {\displaystyle M}$ and $m {\displaystyle m}$ is $r {\displaystyle r}$ , and that the test particle is orbiting at a distance $r_{\mathrm {H} }$ from the secondary. When the test particle is on the line connecting the primary and the secondary body, the force balance requires that

{\frac {Gm}{r_{\mathrm {H} }^{2}}}-{\frac {GM}{(r-r_{\mathrm {H} })^{2}}}+\Omega ^{2}(r-r_{\mathrm {H} })=0,

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

{\frac {m}{r_{\mathrm {H} }^{2}}}-{\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,

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

{\frac {m}{r_{\mathrm {H} }^{2}}}-{\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_{\mathrm {H} }^{2}}}-{\frac {M}{r^{2}}}\left(3{\frac {r_{\mathrm {H} }}{r}}\right)\approx 0.

Hence, the relation stated above

{\frac {r_{\mathrm {H} }}{r}}\approx {\sqrt[{3}]{\frac {m}{3M}}}.

If the orbit of the secondary about the primary is elliptical, the Hill radius is maximum at theapocenter, where $r {\displaystyle r}$ 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 $r_{\mathrm {H} }/r$ , the Hill radius above also represents the distance of the Lagrangian point L₁ from the secondary.

Regions of stability

[edit]

This sectionmay be in need of reorganization to comply with Wikipedia'slayout guidelines. The reason given is:the section lacks sourcing and focus. Please help byediting the article to make improvements to the overall structure.(July 2023) (Learn how and when to remove this message)

The Hill sphere is only an approximation, and other forces (such asradiation pressure or theYarkovsky effect) can eventually perturb an object out of the sphere.^{[citation needed]} As stated, the satellite (third mass) should be small enough that its gravity contributes negligibly.^[6]^: p.26ff

The region of stability forretrograde orbits at a large distance from the primary is larger than the region forprograde orbits 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.^[10]In a two-planet system, the mutual Hill radius of the two planets must exceed $2{\sqrt {3}}$ to be stable. Multi-planet systems of three or more with semi-major-axis differences of less than ten mutual Hill radii are always unstable. This is due to the loss of angular momentum due to perturbations by a third planet.^[11]

Further examples

[edit]

This sectionneeds additional citations forverification. Please helpimprove this article byadding citations to reliable sources in this section. Unsourced material may be challenged and removed.(September 2018) (Learn how and when to remove this message)

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 104ton 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 thanlead, and indeed, inlow 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 ingeostationary orbit, however, would only need to be more than 6% of the density of water to fit inside their own Hill sphere.^{[citation needed]}

Within theSolar System, the planet with the largest Hill radius isNeptune, 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). Anasteroid from theasteroid belt will have a Hill sphere that can reach 220,000 km (for1 Ceres), diminishing rapidly with decreasing mass. The Hill sphere of66391 Moshup, aMercury-crossing asteroid that has a moon (named Squannit), measures 22 km in radius.^[12]

A typicalextrasolar "hot Jupiter",HD 209458 b,^[13] 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,^[14] 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 respectiveRoche limits.^{[citation needed]}

Hill spheres for the solar system

[edit]

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 theJPL DE405 ephemeris and from the NASA Solar System Exploration website.^[15]

Radius of the Hill spheres of some bodies of the Solar System
Body	Million km	au	Body radii	Arcminutes^{[note 1]}	Farthest moon (au)
Mercury	0.1753	0.0012	71.9	10.7	—
Venus	1.0042	0.0067	165.9	31.8	—
Earth	1.4714	0.0098	230.7	33.7	0.00257
Mars	0.9827	0.0066	289.3	14.9	0.00016
Jupiter	50.5736	0.3381	707.4	223.2	0.1662
Saturn	61.6340	0.4120	1022.7	147.8	0.1785
Uranus	66.7831	0.4464	2613.1	80.0	0.1366
Neptune	115.0307	0.7689	4644.6	87.9	0.3360
Ceres	0.2048	0.0014	433.0	1.7	—
Pluto	5.9921	0.0401	5048.1	3.5	0.00043
Eris	8.1176	0.0543	6979.9	2.7	0.00025

Logarithmic plot of the Hill radii (in km) for the bodies of the solar system

Explanatory notes

[edit]

^At average distance, as seen from the Sun. Theangular size as seen from Earth varies depending on Earth's proximity to the object.

References

[edit]

^^a ^bSouami, 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 (4):4287–4297.arXiv:2005.13059.doi:10.1093/mnras/staa1520.
^Williams, Matt (2015-12-30)."How Many Moons Does Mercury Have?".Universe Today. Retrieved2023-11-08.
^Hill, Roderick J. (2022). "Gravitational clearing of natural satellite orbits".Publications of the Astronomical Society of Australia.39 e006. Cambridge University Press.Bibcode:2022PASA...39....6H.doi:10.1017/pasa.2021.62.ISSN 1323-3580.S2CID 246637375.
^Chebotarev, G. A. (March 1965). "On the Dynamical Limits of the Solar System".Soviet Astronomy.8: 787.Bibcode:1965SvA.....8..787C.
^Lauretta, Dante and the Staff of the Osiris-Rex Asteroid Sample Return Mission (2023)."Word of the Week: Hill Sphere".Osiris-Rex Asteroid Sample Return Mission (AsteroidMission.org). Tempe, AZ: University of Arizona. RetrievedJuly 22, 2023.
^^a ^b ^c ^d ^e ^fde Pater, Imke & Lissauer, Jack (2015). "Dynamics (The Three-Body Problem, Perturbations and Resonances)".Planetary Sciences (2nd ed.). Cambridge, England: Cambridge University Press. pp. 26,28–30, 34.ISBN 9781316195697. Retrieved22 July 2023.{{cite book}}: CS1 maint: multiple names: authors list (link)
^^a ^bHiguchi1, A. & Ida, S. (April 2017)."Temporary Capture of Asteroids by an Eccentric Planet".The Astronomical Journal.153 (4). Washington, DC: The American Astronomical Society: 155.arXiv:1702.07352.Bibcode:2017AJ....153..155H.doi:10.3847/1538-3881/aa5daa.S2CID 119036212.{{cite journal}}: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link)
^Hamilton, D.P. & Burns, J.A. (March 1992)."Orbital Stability Zones About Asteroids: II. The Destabilizing Effects of Eccentric Orbits and of Solar Radiation".Icarus.96 (1). New York, NY: Academic Press:43–64.Bibcode:1992Icar...96...43H.doi:10.1016/0019-1035(92)90005-R.{{cite journal}}: CS1 maint: multiple names: authors list (link) See alsoHamilton, D.P. & Burns, J.A. (March 1991)."Orbital Stability Zones About Asteroids"(PDF).Icarus.92 (1). New York, NY: Academic Press:118–131.Bibcode:1991Icar...92..118H.doi:10.1016/0019-1035(91)90039-V. Retrieved22 July 2023.{{cite journal}}: CS1 maint: multiple names: authors list (link) cited therein.
^Follows, Mike (4 October 2017)."Ever Decreasing Circles".NewScientist.com. Retrieved23 July 2023.The moon's Hill sphere has a radius of 60,000 kilometres, about one-sixth of the distance between it and Earth.
^Astakhov, Sergey A.; Burbanks, Andrew D.; Wiggins, Stephen & Farrelly, David (2003). "Chaos-assisted capture of irregular moons".Nature.423 (6937):264–267.Bibcode:2003Natur.423..264A.doi:10.1038/nature01622.PMID 12748635.S2CID 16382419.
^Chambers, J. E., Wetherill, G. W., & Boss, A. P. (1996).The stability of multi-planet systems. Icarus, 119(2), 261-268.
^Johnston, Robert (20 October 2019)."(66391) Moshupand Squannit".Johnston's Archive. Retrieved30 March 2017.
^"HD 209458 b".Extrasolar Planets Encyclopaedia.Archived from the original on 2010-01-16. Retrieved2010-02-16.
^"Planet CoRoT-7 b".Extrasolar Planets Encyclopaedia. 2024.
^"NASA Solar System Exploration". NASA. Retrieved2020-12-22.

External links

[edit]

v t e Moon
Outline
Physical properties	Internal structure Topography Atmosphere Gravity field Hill sphere Magnetic field Sodium tail Moonlight Earthlight
Orbit	Orbital elements Distance Perigee and apogee Libration Nodes Nodal period Precession Syzygy New moon Full moon Eclipses Lunar eclipse Total penumbral lunar eclipse Tetrad Solar eclipse Solar eclipses on the Moon Eclipse cycle Supermoon Tide Tidal force Tidal locking Tidal acceleration Tidal range Kordylewski clouds Lunar station
Surface and features	Selenography Terminator Limb Hemispheres Near side Far side Poles North pole South pole Face Maria List Mountains Peak of eternal light Valleys Volcanic features Domes Calderas Lava tubes Craters List Ray systems Permanently shadowed craters South Pole–Aitken basin Regolith swirls Rilles Wrinkle ridges Rocks Lunar basalt 70017 Changesite-(Y) Water Space weathering Micrometeorite Sputtering Quakes Transient lunar phenomenon Selenographic coordinate system
Science	Observation Libration Lunar theory Origin Giant-impact hypothesis Theia Lunar magma ocean Geology Timescale Late Heavy Bombardment Lunar meteorites KREEP Volcanism Experiments Lunar laser ranging ALSEP Lunar sample displays Apollo 11 Apollo 17 Lunar seismology
Exploration	Missions Apollo program Explorers Probes Landing Colonization Moonbase Tourism Lunar habitation Lunar resources
Time-telling andnavigation	Lunar calendar Lunisolar calendar Month Lunar month Nodal period Fortnight Sennight Lunar station Lunar distance
Phases and names	New Full Names Crescent Super and micro Blood Blue Black Dark Wet Tetrad
Daily phenomena	Moonrise Meridian passage Moonset
Related	Lunar deities Lunar effect Earth phase Moon illusion Pareidolia Man in the Moon Moon rabbit Craters named after people Artificial objects on the Moon Memorials on the Moon Moon in science fiction list Apollo era futuristic exploration Hollow Moon Moon landing conspiracy theories Moon Treaty "Moon is made of green cheese" Natural satellite Double planet Lilith (hypothetical second moon) Splitting of the Moon
Category

Gravitationalorbits

Types

General	Box Circular Elliptical /Highly elliptical Horseshoe Hyperbolic trajectory Inclined /Non-inclined Kepler Lagrange point Osculating Parabolic / Escape / Capture Parking Prograde / Retrograde Radial Synchronous semi sub Transfer orbit
Geocentric	Geosynchronous Geostationary Geostationary transfer Graveyard High Earth Low Earth Medium Earth Molniya Near-equatorial Orbit of the Moon Polar Sun-synchronous Transatmospheric Tundra Very low Earth
About other points	Mars Areocentric Areosynchronous Areostationary Lagrange points Distant retrograde Halo Lissajous Libration Lunar Sun Heliocentric Earth's orbit Mars cycler Heliosynchronous Other Lunar cycler

Parameters

Shape Size	e Eccentricity a Semi-major axis b Semi-minor axis Q, q Apsides
Orientation	i Inclination Ω Longitude of the ascending node ω Argument of periapsis ϖ Longitude of the periapsis
Position	M Mean anomaly ν,θ,f True anomaly E Eccentric anomaly L Mean longitude l True longitude
Variation	T Orbital period n Mean motion v Orbital speed t₀ Epoch