A
Hill sphere is, roughly, the volume around an astronomical body (such as a
planetA planet , is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...
) where it dominates in attraction of
satellitesA natural satellite or moon is a celestial body that orbits a planet or smaller body, which is called the primary. Technically, the term natural satellite could refer to a planet orbiting a star, or a dwarf galaxy orbiting a major galaxy, but it is normally synonymous with moon and used to identify...
to that body, rather than to a larger body (such as a
starA star is a massive, luminous ball of plasma that is held together by gravity. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth. Other stars are visible in the night sky, when they are not outshone by the Sun...
) which it
orbitIn physics, an orbit is the gravitationally curved path of one object around a point or another body, for example the gravitational orbit of a planet around a star....
s. Thus, for a planet to retain a
moonThe Moon is Earth's only natural satellite and the fifth largest satellite in the Solar System. The average centre-to-centre distance from the Earth to the Moon is , about thirty times the diameter of the Earth. The common centre of mass of the system is located at about —a quarter the Earth's...
, the moon must have an orbit that lies within the Hill sphere of the planet. That moon would, in turn, have a Hill sphere of its own. Any objects within that distance would tend to become satellites of the moon, rather than of the planet itself.
More precisely, the Hill sphere approximates the gravitational sphere of influence of a smaller body in the face of
perturbationPerturbation or perturb may refer to any of numerous concepts in several fields:* Perturbation theory, mathematical methods that give approximate solutions to problems that cannot be solved exactly...
s from a more massive body.
A
Hill sphere is, roughly, the volume around an astronomical body (such as a
planetA planet , is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...
) where it dominates in attraction of
satellitesA natural satellite or moon is a celestial body that orbits a planet or smaller body, which is called the primary. Technically, the term natural satellite could refer to a planet orbiting a star, or a dwarf galaxy orbiting a major galaxy, but it is normally synonymous with moon and used to identify...
to that body, rather than to a larger body (such as a
starA star is a massive, luminous ball of plasma that is held together by gravity. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth. Other stars are visible in the night sky, when they are not outshone by the Sun...
) which it
orbitIn physics, an orbit is the gravitationally curved path of one object around a point or another body, for example the gravitational orbit of a planet around a star....
s. Thus, for a planet to retain a
moonThe Moon is Earth's only natural satellite and the fifth largest satellite in the Solar System. The average centre-to-centre distance from the Earth to the Moon is , about thirty times the diameter of the Earth. The common centre of mass of the system is located at about —a quarter the Earth's...
, the moon must have an orbit that lies within the Hill sphere of the planet. That moon would, in turn, have a Hill sphere of its own. Any objects within that distance would tend to become satellites of the moon, rather than of the planet itself.
More precisely, the Hill sphere approximates the gravitational sphere of influence of a smaller body in the face of
perturbationPerturbation or perturb may refer to any of numerous concepts in several fields:* Perturbation theory, mathematical methods that give approximate solutions to problems that cannot be solved exactly...
s from a more massive body. It was defined by the
AmericanThe United States of America is a federal constitutional republic comprising fifty states and a federal district...
astronomerAn astronomer is a scientist who studies celestial bodies such as planets, stars, and galaxies.Historically, astronomy was more concerned with the classification and description of phenomena in the sky, while astrophysics attempted to explain these phenomena and the differences between them using...
George William HillGeorge William Hill , was a U.S. astronomer and mathematician.Hill was born in New York City, New York, son of the painter and engraver John William Hill and Catherine Smith Hill. He moved to West Nyack with his family when he was eight years old. After attending high school, Hill graduated from...
, based upon the work of the
FrenchFrance , officially the French Republic , is a country located in Western Europe, with several overseas islands and territories located on other continents. Metropolitan France extends from the Mediterranean Sea to the English Channel and the North Sea, and from the Rhine to the Atlantic Ocean...
astronomer
Édouard RocheÉdouard Albert Roche was a French astronomer and mathematician, who is best known for his work in the field of celestial mechanics...
. For this reason, it is also known as the
Roche sphere.
To illustrate, consider the specific case of the planet
JupiterJupiter is the fifth planet from the Sun and the largest planet within the Solar System. It is a gas giant with a mass slightly less than one-thousandth that of the Sun but is two and a half times the mass of all of the other planets in our Solar System combined. Jupiter is classified as a gas...
, which orbits the
SunThe Sun is the star at the center of the Solar System. The Earth and other matter orbit the Sun, which by itself accounts for about 99.86% of the Solar System's mass....
. For any point in space, one can compute the sum of the following three forces:
- gravity due to the Sun,
- gravity due to Jupiter,
- the centrifugal force
In classical mechanics, centrifugal force is an outward force associated with curved motion, that is, rotation about some center...
experienced by a particle at such a point moving with the same frequencyIn physics , angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
as Jupiter around the Sun.
The Hill sphere for Jupiter is the largest sphere, centered at Jupiter, within which the sum of the three forces is always directed towards Jupiter. In more general terms, it is the sphere around a secondary body in orbit around a primary body within which the net force is a
centripetal forceCentripetal force is a force that makes a body follow a curved path; it is always directed orthogonal to the velocity of the body, toward the instantaneous center of curvature of the path...
directed at the secondary body. Thus, the Hill sphere in our example describes the outer limit that an even smaller object such as a moon or artificial satellite can stably orbit Jupiter rather than simply entering an elliptical orbit of its own around the Sun.
The Hill sphere extends between the
Lagrangian pointThe Lagrangian points , are the five positions in an orbital configuration where a small object affected only by gravity can theoretically be stationary relative to two larger objects...
s and , 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.
The
Roche sphere is not to be confused with the
Roche lobeThe Roche lobe is the region of space around a star within which orbiting material is gravitationally bound to that star. If the star expands past its Roche lobe, then the material can escape the gravitational pull of the star. If the star is in a binary system then the material will fall in...
and
Roche limitThe Roche Limit , sometimes referred to as the Roche Radius, is the distance within which a celestial body, held together only by its own gravity, will disintegrate due to a second celestial body's tidal forces exceeding the first body's gravitational self-attraction...
which were also described by Roche. The Roche limit is the distance at which an object held together only by gravity begins to break up due to tidal forces. The Roche lobe describes the limits at which an object which is in orbit around two other objects will be captured by one or the other.
Formula and examples
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 axisIn geometry, the semi-major axis is used to describe the dimensions of ellipses and hyperbolae.- Ellipse :The major axis of an ellipse is its longest diameter, a line that runs through the centre and both foci, its ends being at the widest points of the shape...
a and an eccentricity of
e, then the radius
r of the Hill sphere for the smaller body (e.g. Earth) is, approximately
When eccentricity is negligible (the most favourable case for orbital stability), this becomes
In the Earth example, the Earth (5.97×10
24 kg) orbits the Sun (1.99×10
30 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 of Earth and it is therefore not at risk of being pulled into an independent orbit around the Sun. In terms of orbital period: all stable satellites of the Earth must have an orbital period shorter than 7 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 these two spheres.
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 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, since many planetary astronomers work in and remember distances in units of planetary radii.
True region of stability
The Hill sphere is but an approximation, and other forces (such as
radiation pressureRadiation pressure is the pressure exerted upon any surface exposed to electromagnetic radiation. If absorbed, the pressure is the power flux density divided by the speed of light...
or the
Yarkovsky effectThe Yarkovsky effect is a force acting on a rotating body in space caused by the anisotropic emission of thermal photons, which carry momentum...
) 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.
Further examples
An astronaut could not orbit the
Space ShuttleThe Space Shuttle, part of the Space Transportation System , is a spacecraft operated by NASA for orbital human spaceflight missions. It began operations in the 1980s and is scheduled to be retired from service in 2010 after 134 launches...
(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 orbitA low Earth orbit is generally defined as an orbit within the locus extending from the Earth’s surface up to an altitude of 2,000 km...
, a spherical body must be 800 times denser than
leadLead is a main-group element with symbol Pb and atomic number 82. Lead is a soft, malleable poor metal, also considered to be one of the heavy metals. Lead has a bluish-white color when freshly cut, but tarnishes to a dull grayish color when exposed to air...
in order to fit inside its own Hill sphere, or else it will be incapable of supporting an orbit. 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
osmiumOsmium is a chemical element that has the symbol Os and atomic number 76. Osmium is a hard, brittle, blue-gray or blue-black transition metal in the platinum family, and is the densest natural element. The density of osmium is , slightly greater than that of iridium, the second densest element....
, the densest naturally-occurring material on Earth. 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 orbits are possible.
Within the
solar systemThe Solar System consists of the Sun and those celestial objects bound to it by gravity, all of which formed from the collapse of a giant molecular cloud approximately 4.6 billion years ago...
, the planet with the largest Hill radius is
NeptuneNeptune is the eighth planet from the Sun in our Solar System. Named for the Roman god of the sea, it is the fourth-largest planet by diameter and the third-largest by mass. Neptune is 17 times the mass of Earth and is slightly more massive than its near-twin Uranus, which is 15 Earth masses and...
, 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
asteroidthumb|260px|right|[[253 Mathilde]], a [[C-type asteroid]] measuring about across. Photograph taken in 1997 by the [[NEAR Shoemaker]] probe.Asteroids, sometimes called minor planets or planetoids, are small Solar System bodies in orbit around the Sun, especially in the inner Solar System; they are...
from the main belt will have a Hill sphere that can reach 220 000 km (for
1 CeresCeres, formal designation 1 Ceres, is the smallest identified dwarf planet in the Solar System and the only one in the asteroid belt. It was discovered on 1 January 1801, by Giuseppe Piazzi, and is named after the Roman goddess Ceres — the goddess of growing plants, the harvest, and motherly...
), diminishing rapidly with its mass. In the case of (66391) 1999 KW₄, a Mercury-crosser asteroid which has a moon (S/2001 (66391) 1), its Hill sphere measures 22 km in radius.
Derivation
A non-rigorous, but conceptually accurate derivation of the Hill radius can be made by equating the orbital velocity of the orbiter around a body (i.e., a planet) and the orbital velocity 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. This is accurate to within factors of order unity.
Where is the Hill radius, a is the semi-major axis of the planet orbiting the star. With some basic algebra:
Giving a Hill radius of:
External links