The Hill sphere of an

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 ...

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 System Capitalization 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 ...

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. 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 stron ...

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 a ...

of a smaller body in the face of perturbations from a more massive body. It was defined by the American

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, moons, comets and galaxies – in either ...

George William Hill, based on the work of the French astronomer

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 points 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.

Formula and examples

comparison of Hill sphere and Roche limit

If the mass of the smaller body (e.g. the Earth) is

m

, and it orbits a heavier body (e.g. the Sun) of mass

M

with a semi-major axis

a

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-Centre (geometry), center, in geometry * Eccentricity (g ...

e

, then the radius

r_

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 :

r_ \approx a (1-e) \sqrt

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

r_ \approx a \sqrt

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 ...

(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

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: :

3\frac \approx \frac.

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

m

) orbiting the secondary body. Assume that the distance between masses

M

and

m

r

, and that the test particle is orbiting at a distance

r_

from the secondary. When the test particle is on the line connecting the primary and the secondary body, the force balance requires that :

\frac-\frac+\Omega^2(r-r_)=0,

where

G

is the gravitational constant and

\Omega=\sqrt

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-\frac\left(1-\frac\right)^+\frac\left(1-\frac\right)=0,

which, through a binomial expansion to leading order in

r_/r

, can be written as :

\frac-\frac\left(1+2\frac\right)+\frac\left(1-\frac\right) = \frac-\frac\left(3\frac\right)\approx 0.

Hence, the relation stated above :

\frac\approx \sqrt

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

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_/r

, 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: :

\approx \frac,

where

\rho_

and

\rho_

are the average densities of the primary and secondary bodies, and

R_

and

R_

are their radii. The second approximation is justified by the fact that, for most cases in the Solar System,

\sqrt /math> 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.

True region of stability

The Hill sphere is only an approximation, and other forces (such as

radiation pressure Radiation pressure is the mechanical pressure exerted upon any surface due to the exchange of momentum between the object and the electromagnetic field. This includes the momentum of light or electromagnetic radiation of any wavelength that is a ...

or the

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. It is usually considered in relation to meteoroids or small asteroids (about 10 cm to 10 ...

) 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 orbit Retrograde motion in astronomy is, in general, orbital or rotational motion of an object in the direction opposite the rotation of its primary, that is, the central object (right figure). It may also describe other motions such as precession or ...

s at a large distance from the primary is larger than the region for

prograde orbit Retrograde motion in astronomy is, in general, orbital or rotational motion of an object in the direction opposite the rotation of its primary, that is, the central object (right figure). It may also describe other motions such as precession or ...

s 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

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 104

ton Ton is the name of any one of several units of measure. It has a long history and has acquired several meanings and uses. Mainly it describes units of weight. Confusion can arise because ''ton'' can mean * the long ton, which is 2,240 pounds ...

Space Shuttle The Space Shuttle is a retired, partially reusable low Earth orbital spacecraft system operated from 1981 to 2011 by the U.S. National Aeronautics and Space Administration (NASA) as part of the Space Shuttle program. Its official program ...

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 than

lead Lead is a chemical element with the symbol Pb (from the Latin ) and atomic number 82. It is a heavy metal that is denser than most common materials. Lead is soft and malleable, and also has a relatively low melting point. When freshly cu ...

, and indeed, in

low Earth orbit A low Earth orbit (LEO) is an orbit around Earth with a period of 128 minutes or less (making at least 11.25 orbits per day) and an eccentricity less than 0.25. Most of the artificial objects in outer space are in LEO, with an altitude never mor ...

, 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. Within the

, 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 The asteroid belt is a torus-shaped region in the Solar System, located roughly between the orbits of the planets Jupiter and Mars. It contains a great many solid, irregularly shaped bodies, of many sizes, but much smaller than planets, c ...

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 66391 Moshup , provisional designation , is a binary asteroid, classified as a near-Earth object and potentially hazardous asteroid of the Aten group, approximately 1.3 kilometers in diameter. It was discovered on 20 May 1999, by Lincoln Near- ...

, a Mercury-crossing asteroid that has a moon (named Squannit), measures 22 km in radius. A typical extrasolar "

hot Jupiter Hot Jupiters (sometimes called hot Saturns) are a class of gas giant exoplanets that are inferred to be physically similar to Jupiter but that have very short orbital periods (). The close proximity to their stars and high surface-atmosphere tem ...

HD 209458 b HD 209458 b, which is also nicknamed Osiris after the Egyptian god, is an exoplanet that orbits the solar analog HD 209458 in the constellation Pegasus, some from the Solar System. The radius of the planet's orbit is , or one-eighth the radius ...

, 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, 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.

Solar System

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.

Explanatory notes

References

External links

Can an Astronaut Orbit the Space Shuttle?

The moon that went up a hill, but came down a planet
{{DEFAULTSORT:Hill Sphere Equations of astronomy Orbits Tides