astronomical body
An astronomical object, celestial object, stellar object or heavenly body is a naturally occurring physical entity, association, or structure that exists in the observable universe. In astronomy, the terms ''object'' and ''body'' are often us ...
is the region in which it dominates the attraction of satellites. To be retained by a
planet
A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...
, a
moon
The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...
must have an
orbit
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a p ...
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
The Solar SystemCapitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Solar S ...
is the Hill sphere of the
Sun
The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
with respect to local stars and the
galactic nucleus
An active galactic nucleus (AGN) is a compact region at the center of a galaxy that has a much-higher-than-normal luminosity over at least some portion of the electromagnetic spectrum with characteristics indicating that the luminosity is not pr ...
.
In more precise terms, the Hill sphere approximates the
gravitational
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the strong ...
sphere of influence
In the field of international relations, a sphere of influence (SOI) is a spatial region or concept division over which a state or organization has a level of cultural, economic, military or political exclusivity.
While there may be a formal al ...
of a smaller body in the face of
perturbation
Perturbation or perturb may refer to:
* Perturbation theory, mathematical methods that give approximate solutions to problems that cannot be solved exactly
* Perturbation (geology), changes in the nature of alluvial deposits over time
* Perturbatio ...
s from a more massive body. It was defined by the
American
American(s) may refer to:
* American, something of, from, or related to the United States of America, commonly known as the "United States" or "America"
** Americans, citizens and nationals of the United States of America
** American ancestry, pe ...
astronomer
An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth. They observe astronomical objects such as stars, planets, natural satellite, moons, comets and galaxy, g ...
George William Hill
George William Hill (March 3, 1838 – April 16, 1914) was an American astronomer and mathematician. Working independently and largely in isolation from the wider scientific community, he made major contributions to celestial mechanics and t ...
Édouard Roche
Édouard Albert Roche (; 17 October 1820 – 27 April 1883) was a French astronomer and mathematician, who is best known for his work in the field of celestial mechanics. His name was given to the concepts of the Roche sphere, Roche limit, and ...
.
In the example to the right, the Earth's Hill sphere extends between the
Lagrange point
In celestial mechanics, the Lagrange points (; also Lagrangian points or libration points) are points of equilibrium for small-mass objects under the influence of two massive orbiting bodies. Mathematically, this involves the solution of th ...
s and , 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
The zero-velocity surface is a concept that relates to the N-body problem of gravity. It represents a surface a body of given energy cannot cross, since it would have zero velocity on the surface. It was first introduced by George William Hill. The ...
in space which cannot be passed. This is a contour of the
Jacobi integral
In celestial mechanics, Jacobi's integral (also known as the Jacobi integral or Jacobi constant) is the only known conserved quantity for the circular restricted three-body problem.
semi-major axis
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 (major semiaxis) is the long ...
and an
eccentricity
Eccentricity or eccentric may refer to:
* Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal"
Mathematics, science and technology Mathematics
* Off-center, in geometry
* Eccentricity (graph theory) of a v ...
of , then the radius of the Hill sphere of the smaller body, calculated at the
pericenter
An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, the apsides of the Earth are called the aphelion and perihelion.
General description
There are two apsides in any el ...
, is approximately
:
When eccentricity is negligible (the most favourable case for orbital stability), this becomes
:
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 one
astronomical unit
The astronomical unit (symbol: au, or or AU) is a unit of length, roughly the distance from Earth to the Sun and approximately equal to or 8.3 light-minutes. The actual distance from Earth to the Sun varies by about 3% as Earth orbits t ...
(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
In the field of international relations, a sphere of influence (SOI) is a spatial region or concept division over which a state or organization has a level of cultural, economic, military or political exclusivity.
While there may be a formal al ...
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.
Derivation
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
Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, German mathematician, mathematician, astrologer, Natural philosophy, natural philosopher and writer on music. He is a key figure in the 17th-century Scienti ...
) 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
An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, the apsides of the Earth are called the aphelion and perihelion.
General description
There are two apsides in any el ...
, 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.
To leading order in , the Hill radius above also represents the distance of the Lagrangian point L1 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,