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The **Hill sphere** of an astronomical body is the region in which it dominates the attraction of satellites. To be retained by a planet, a 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. One simple view of the extent of the Solar System is the Hill sphere of the Sun with respect to local stars and the galactic nucleus.^{ [1] }

- Formula and examples
- Derivation
- True region of stability
- Further examples
- Solar System
- See also
- Notes
- References
- External links

In more precise terms, the Hill sphere approximates the gravitational sphere of influence of a smaller body in the face of perturbations from a more massive body. It was defined by the American astronomer George William Hill, based on the work of the French astronomer Édouard Roche.

In the example to the right, the Earth's Hill sphere extends between the Lagrange points L_{1} and L_{2}, which lie along the line of centers of the two bodies. The region of influence of the smaller 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 small object 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.

For any given energy of the third object (considered to have a negligible mass) there is a zero-velocity surface in space which cannot be passed. This is a contour of the Jacobi integral. When the energy is low, the zero-velocity surface surrounds the second body (the smaller of the two) completely, which means the third body cannot escape. At higher energy, there will be one or more gaps or bottlenecks by which the third object may escape the second object and go into orbit around the first object. If the energy is right 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 first object at one of the nearby Lagrange points (forming a cone-like point there). At the opposite side of the planet it gets close to the other Lagrange point. This limiting zero-velocity surface around the second object is basically its Hill "sphere".

If the mass of the smaller body (e.g. the Earth) is , and it orbits a heavier body (e.g. the Sun) of mass with a semi-major axis and an eccentricity of , then the radius of the Hill sphere of the smaller body, calculated at the pericenter, is approximately^{ [2] }

When eccentricity is negligible (the most favourable case for orbital stability), this becomes

In the Earth-Sun example, the Earth (5.97×10^{24} kg) orbits the Sun (1.99×10^{30} 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. All stable satellites of the Earth (those within the Earth's Hill sphere) must have an orbital period shorter than seven months.

The previous (eccentricity-ignoring) formula can be re-stated as follows:

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.

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 ) orbiting the secondary body. Assume that the distance between masses and is , and that the test particle is orbiting at a distance from the secondary. When the test particle is on the line connecting the primary and the secondary body, the force balance requires that

where is the gravitational constant and is the (Keplerian) angular velocity of the secondary about the primary (assuming that ). The above equation can also be written as

which, through a binomial expansion to leading order in , can be written as

Hence, the relation stated above

If the orbit of the secondary about the primary is elliptical, the Hill radius is maximum at the apocenter, where 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.^{ [2] } To leading order in , the Hill radius above also represents the distance of the Lagrangian point L_{1} from the secondary.

A quick way of estimating the radius of the Hill sphere comes from replacing mass with density in the above equation:

where and are the average densities of the primary and secondary bodies, and and are their radii. The second approximation is justified by the fact that, for most cases in the Solar System, 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, because many planetary astronomers work in and remember distances in units of planetary radii.

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 orbits at a large distance from the primary is larger than the region for prograde 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.^{ [3] }

This section needs additional citations for verification .(September 2018) |

An astronaut could not have orbited the Space Shuttle (with mass of 104 tonnes), where the orbit was 300 km above the Earth, because its Hill sphere at that altitude was only 120 cm in radius, much smaller than the shuttle itself. A sphere of this size and mass would be denser than lead. In fact, in any 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 ]}

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 decreasing mass. The Hill sphere of 66391 Moshup, a Mercury-crossing asteroid that has a moon (named Squannit), measures 22 km in radius.^{ [4] }

A typical extrasolar "hot Jupiter", HD 209458 b,^{ [5] } 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,^{ [6] } 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 ]}

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.^{ [7] }

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- ↑ At average distance, as seen from the Sun. The angular size from Earth varies as Earth gets closer and further.

In celestial mechanics, the **Lagrange points** are points of equilibrium for small-mass objects under the influence of two massive orbiting bodies. Mathematically, this involves the solution of the restricted three-body problem in which two bodies are very much more massive than the third.

**Tidal acceleration** is an effect of the tidal forces between an orbiting natural satellite and the primary planet that it orbits. The acceleration causes a gradual recession of a satellite in a prograde orbit away from the primary, and a corresponding slowdown of the primary's rotation. The process eventually leads to tidal locking, usually of the smaller body first, and later the larger body. The Earth–Moon system is the best-studied case.

In celestial mechanics, **escape velocity** or **escape speed** is the minimum speed needed for a free, non-propelled object to escape from the gravitational influence of a primary body, thus reaching an infinite distance from it. It is typically stated as an ideal speed, ignoring atmospheric friction. Although the term "escape velocity" is common, it is more accurately described as a speed than a velocity because it is independent of direction; the escape speed increases with the mass of the primary body and decreases with the distance from the primary body. The escape speed thus depends on how far the object has already traveled, and its calculation at a given distance takes into account the fact that without new acceleration it will slow down as it travels—due to the massive body's gravity—but it will never quite slow to a stop.

In fluid mechanics, **hydrostatic equilibrium** is the condition of a fluid or plastic solid at rest, which occurs when external forces, such as gravity, are balanced by a pressure-gradient force. In the planetary physics of the Earth, the pressure-gradient force prevents gravity from collapsing the planetary atmosphere into a thin, dense shell, whereas gravity prevents the pressure-gradient force from diffusing the atmosphere into outer space.

In celestial mechanics, the **Roche limit**, also called **Roche radius**, is the distance from a celestial body within which a second celestial body, held together only by its own force of gravity, will disintegrate because the first body's tidal forces exceed the second body's gravitational self-attraction. Inside the Roche limit, orbiting material disperses and forms rings, whereas outside the limit material tends to coalesce. The Roche radius depends on the radius of the first body and on the ratio of the bodies' densities.

In Newtonian physics, **free fall** is any motion of a body where gravity is the only force acting upon it. In the context of general relativity, where gravitation is reduced to a space-time curvature, a body in free fall has no force acting on it.

The **gravitational binding energy** of a system is the minimum energy which must be added to it in order for the system to cease being in a gravitationally bound state. A gravitationally bound system has a lower gravitational potential energy than the sum of the energies of its parts when these are completely separated—this is what keeps the system aggregated in accordance with the minimum total potential energy principle.

The **orbital period** is the time a given astronomical object takes to complete one orbit around another object, and applies in astronomy usually to planets or asteroids orbiting the Sun, moons orbiting planets, exoplanets orbiting other stars, or binary stars.

In astronomy, the **barycenter** is the center of mass of two or more bodies that orbit one another and is the point about which the bodies orbit. A barycenter is a dynamical point, not a physical object. It is an important concept in fields such as astronomy and astrophysics. The distance from a body's center of mass to the barycenter can be calculated as a two-body problem.

In astronomy, **air mass** or **airmass** is a measure of the amount of air along the line of sight when observing a star or other celestial source from below Earth's atmosphere. It is formulated as the integral of air density along the light ray.

In orbital mechanics, **orbital****decay** is a gradual decrease of the distance between two orbiting bodies at their closest approach over many orbital periods. These orbiting bodies can be a planet and its satellite, a star and any object orbiting it, or components of any binary system. Orbits do not decay without some friction-like mechanism which transfers energy from the orbital motion. This can be any of a number of mechanical, gravitational, or electromagnetic effects. For bodies in low Earth orbit, the most significant effect is atmospheric drag.

A **Sun-synchronous orbit** (**SSO**), also called a **heliosynchronous orbit**, is a nearly polar orbit around a planet, in which the satellite passes over any given point of the planet's surface at the same local mean solar time. More technically, it is an orbit arranged so that it precesses through one complete revolution each year, so it always maintains the same relationship with the Sun. A Sun-synchronous orbit is useful for imaging, reconnaissance satellite, and weather satellites, because every time that the satellite is overhead, the surface illumination angle on the planet underneath it will be the same.

A **sphere of influence** (**SOI**) in astrodynamics and astronomy is the oblate-spheroid-shaped region around a celestial body where the primary gravitational influence on an orbiting object is that body. This is usually used to describe the areas in the Solar System where planets dominate the orbits of surrounding objects such as moons, despite the presence of the much more massive but distant Sun. In the patched conic approximation, used in estimating the trajectories of bodies moving between the neighbourhoods of different masses using a two body approximation, ellipses and hyperbolae, the SOI is taken as the boundary where the trajectory switches which mass field it is influenced by.

In orbital mechanics, **mean motion** is the angular speed required for a body to complete one orbit, assuming constant speed in a circular orbit which completes in the same time as the variable speed, elliptical orbit of the actual body. The concept applies equally well to a small body revolving about a large, massive primary body or to two relatively same-sized bodies revolving about a common center of mass. While nominally a mean, and theoretically so in the case of two-body motion, in practice the mean motion is not typically an average over time for the orbits of real bodies, which only approximate the two-body assumption. It is rather the instantaneous value which satisfies the above conditions as calculated from the current gravitational and geometric circumstances of the body's constantly-changing, perturbed orbit.

The **free-fall time** is the characteristic time that would take a body to collapse under its own gravitational attraction, if no other forces existed to oppose the collapse. As such, it plays a fundamental role in setting the timescale for a wide variety of astrophysical processes—from star formation to helioseismology to supernovae—in which gravity plays a dominant role.

A **gravity train** is a theoretical means of transportation for purposes of commuting between two points on the surface of a sphere, by following a straight tunnel connecting the two points through the interior of the sphere.

For most numbered asteroids, almost nothing is known apart from a few physical parameters and orbital elements and some physical characteristics are often only estimated. The physical data is determined by making certain standard assumptions.

The **gravity of Earth**, denoted by **g**, is the net acceleration that is imparted to objects due to the combined effect of gravitation and the centrifugal force . It is a vector (physics) quantity, whose direction coincides with a plumb bob and strength or magnitude is given by the norm .

In geometry, the **major axis** of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The **semi-major axis** is the longest semidiameter or one half of the major axis, and thus runs from the centre, through a focus, and to the perimeter. The **semi-minor axis** of an ellipse or hyperbola is a line segment that is at right angles with the semi-major axis and has one end at the center of the conic section. For the special case of a circle, the lengths of the semi-axes are both equal to the radius of the circle.

**Orbit modeling** is the process of creating mathematical models to simulate motion of a massive body as it moves in orbit around another massive body due to gravity. Other forces such as gravitational attraction from tertiary bodies, air resistance, solar pressure, or thrust from a propulsion system are typically modeled as secondary effects. Directly modeling an orbit can push the limits of machine precision due to the need to model small perturbations to very large orbits. Because of this, perturbation methods are often used to model the orbit in order to achieve better accuracy.

- ↑ Chebotarev, G. A. (March 1965). "On the Dynamical Limits of the Solar System".
*Soviet Astronomy*.**8**: 787. Bibcode:1965SvA.....8..787C. - 1 2 D.P. Hamilton & J.A. Burns (1992). "Orbital stability zones about asteroids. II - The destabilizing effects of eccentric orbits and of solar radiation".
*Icarus*.**96**(1): 43–64. Bibcode:1992Icar...96...43H. doi:10.1016/0019-1035(92)90005-R. - ↑ 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. - ↑ Johnston, Robert (20 October 2019). "(66391) Moshup
*and*Squannit".*Johnston's Archive*. Retrieved 30 March 2017. - ↑ HD 209458 b
^{[ dead link ]} - ↑ "The Extrasolar Planet Encyclopaedia — CoRoT-7 b".
*exoplanet.eu*. - ↑ "NASA Solar System Exploration". NASA. Retrieved 2020-12-22.

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