Hill sphere
Encyclopedia
An astronomical body's Hill sphere 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.
In more precise terms, the Hill sphere approximates the gravitational sphere of influence of a smaller body in the face of perturbation
s from a more massive body. It was defined by the American
astronomer
George William Hill
, based upon the work of the French
astronomer Édouard Roche
. For this reason, it is also known as the Roche sphere (not to be confused with the Roche limit
). The Hill sphere extends between the Lagrangian point
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.
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 (5.97×1024 kg) orbits the Sun (1.99×1030 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. All stable satellites of the Earth (those within the Earth's Hill sphere) 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 the volume 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.
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.
(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. 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. 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 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 main belt will have a Hill sphere that can reach 220 000 km (for 1 Ceres
), 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.
A typical extrasolar
"hot Jupiter
", HD 209458 b
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 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.
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:
Natural satellite
A natural satellite or moon is a celestial body that orbits a planet or smaller body, which is called its primary. The two terms are used synonymously for non-artificial satellites of planets, of dwarf planets, and of minor planets....
. To be retained by a planet, a moon
Natural satellite
A natural satellite or moon is a celestial body that orbits a planet or smaller body, which is called its primary. The two terms are used synonymously for non-artificial satellites of planets, of dwarf planets, and of minor planets....
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.
In more precise terms, the Hill sphere approximates the gravitational sphere of influence of a smaller body in the face of perturbation
Perturbation (astronomy)
Perturbation is a term used in astronomy in connection with descriptions of the complex motion of a massive body which is subject to appreciable gravitational effects from more than one other massive body....
s from a more massive body. It was defined by the American
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
astronomer
Astronomer
An 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 Hill
George William Hill
George William Hill , was an American astronomer and mathematician.Hill was born in New York City, New York to 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 French
France
The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...
astronomer Édouard Roche
Édouard Roche
Édouard 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 (not to be confused with the Roche limit
Roche limit
The 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...
). The Hill sphere extends between the Lagrangian point
Lagrangian point
The 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.
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 axisSemi-major axis
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
Eccentricity (mathematics)
In mathematics, the eccentricity, denoted e or \varepsilon, is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.In particular,...
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 (5.97×1024 kg) orbits the Sun (1.99×1030 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. All stable satellites of the Earth (those within the Earth's Hill sphere) 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 the volume 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
Radiation 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 effect
Yarkovsky effect
The 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 ShuttleSpace Shuttle
The Space Shuttle was a manned orbital rocket and spacecraft system operated by NASA on 135 missions from 1981 to 2011. The system combined rocket launch, orbital spacecraft, and re-entry spaceplane with modular add-ons...
(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
Low Earth orbit
A 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 lead
Lead
Lead is a main-group element in the carbon group with the symbol Pb and atomic number 82. Lead is a soft, malleable poor metal. It is also counted as one of the heavy metals. Metallic lead has a bluish-white color after being freshly cut, but it soon tarnishes to a dull grayish color when exposed...
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 osmium
Osmium
Osmium is a chemical element with the symbol Os and atomic number 76. Osmium is a hard, brittle, blue-gray or blue-blacktransition metal in the platinum family, and is the densest natural element. Osmium is twice as dense as lead. The density of osmium is , slightly greater than that of iridium,...
, 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 system
Solar System
The Solar System consists of the Sun and the astronomical objects gravitationally bound in orbit around it, all of which formed from the collapse of a giant molecular cloud approximately 4.6 billion years ago. The vast majority of the system's mass is in the Sun...
, the planet with the largest Hill radius is Neptune
Neptune
Neptune is the eighth and farthest planet from the Sun in the 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 times...
, 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
Asteroid
Asteroids are a class of small Solar System bodies in orbit around the Sun. They have also been called planetoids, especially the larger ones...
from the main belt will have a Hill sphere that can reach 220 000 km (for 1 Ceres
1 Ceres
Ceres, formally 1 Ceres, is the smallest identified dwarf planet in the Solar System and the only one in the asteroid belt. With a diameter of about 950 km, Ceres is by far the largest and most-massive asteroid, comprising about a third of the mass of the asteroid belt. Discovered on 1 January 1801...
), 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.
A typical extrasolar
Extrasolar planet
An extrasolar planet, or exoplanet, is a planet outside the Solar System. A total of such planets have been identified as of . It is now known that a substantial fraction of stars have planets, including perhaps half of all Sun-like stars...
"hot Jupiter
Hot Jupiter
Hot Jupiters are a class of extrasolar planet whose mass is close to or exceeds that of Jupiter...
", HD 209458 b
HD 209458 b
HD 209458 b is an extrasolar planet that orbits the Solar analog star HD 209458 in the constellation Pegasus, some 150 light-years from Earth's solar system, with evidence of water vapor....
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 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.
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.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:
See also
- n-body problemN-body problemThe n-body problem is the problem of predicting the motion of a group of celestial objects that interact with each other gravitationally. Solving this problem has been motivated by the need to understand the motion of the Sun, planets and the visible stars...
- Sphere of influence (astrodynamics)Sphere of influence (astrodynamics)A sphere of influence in astrodynamics and astronomy is the spherical region around a celestial body where the primary gravitational influence on an orbiting object is that body...
- Interplanetary Transport NetworkInterplanetary Transport NetworkThe Interplanetary Transport Network is a collection of gravitationally determined pathways through the solar system that require very little energy for an object to follow. The ITN makes particular use of Lagrange points as locations where trajectories through space are redirected using little...