The
gravitational two-body problemIn classical mechanics, the two-body problem is to determine the motion of two point particles that interact only with each other. Common examples include a satellite orbiting a planet, a planet orbiting a star, two stars orbiting each other , and a classical electron orbiting an atomic...
concerns the motion of two point particles that interact only with each other, due to gravity. This means that influences from any third body are neglected. For approximate results that is often suitable. It also means that the two bodies stay clear of each other, that is, the two do not
collideA collision is an isolated event in which two or more moving bodies exert relatively strong forces on each other for a relatively short time.-Dynamics:Collisions involve forces...
, and one body does not pass through the other's
atmosphereAn atmosphere is a layer of gases that may surround a material body of sufficient mass, by the gravity of the body, and are retained for a longer duration if gravity is high and the atmosphere's temperature is low...
. Even if they do, the theory still holds for the part of the orbit where they don't.
The
gravitational two-body problemIn classical mechanics, the two-body problem is to determine the motion of two point particles that interact only with each other. Common examples include a satellite orbiting a planet, a planet orbiting a star, two stars orbiting each other , and a classical electron orbiting an atomic...
concerns the motion of two point particles that interact only with each other, due to gravity. This means that influences from any third body are neglected. For approximate results that is often suitable. It also means that the two bodies stay clear of each other, that is, the two do not
collideA collision is an isolated event in which two or more moving bodies exert relatively strong forces on each other for a relatively short time.-Dynamics:Collisions involve forces...
, and one body does not pass through the other's
atmosphereAn atmosphere is a layer of gases that may surround a material body of sufficient mass, by the gravity of the body, and are retained for a longer duration if gravity is high and the atmosphere's temperature is low...
. Even if they do, the theory still holds for the part of the orbit where they don't. Apart from these considerations a spherically symmetric body can be approximated by a point mass.
Common examples include the parts of a
spaceflightSpaceflight is the use of space technology to achieve the flight of spacecraft into and through outer space.Spaceflight is used in space exploration, and also in commercial activities like space tourism and satellite telecommunications...
where the rockets are off and there is no atmosphere (except on a flight from one celestial body to another), the orbit of 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...
around 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,...
, and of 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,...
around 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...
, and two
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...
s orbiting each other (a
binary starA binary star is a star system consisting of two stars orbiting around their common center of mass. The brighter star is called the primary and the other is its companion star, comes, or secondary...
).
The
reduced massReduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. This is a quantity with the unit of mass, which allows the two-body problem to be solved as if it were a one-body problem. Note however that the mass determining the gravitational force is not...
multiplied by the relative acceleration between the two bodies is equal to the gravitational force. The latter is proportional to the product of the two masses, which is equal to the reduced mass multiplied by the sum of the masses. Thus in the differential equation the two occurrences of the reduced mass cancel each other, and we get the same differential equation as for the position of a very small body orbiting a body with a mass equal to the sum of the two masses.
- Assume:
-
- the vector r is the position of one body relative to the other
- r, v, the semi-major axis
In 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 the specific relative angular momentumIn astrodynamics, the specific relative angular momentum of an orbiting body with respect to a central body is the relative angular momentum of the first body per unit mass...
h are defined accordingly (hence r is the distance)
- , the standard gravitational parameter
In astrodynamics, the standard gravitational parameter of a celestial body is the product of the gravitational constant and the mass :The units of the standard gravitational parameter are km3s-2...
(the sum of those for each mass)
- where:
-
- and are the masses of the two bodies.
- Then:
-
- the orbit equation
In astrodynamics an orbit equation defines the path of orbiting body around central body relative to , without specifying position as a function of time. Under standard assumptions, a body moving under the influence of a force, directed to a central body, with a magnitude inversely proportional...
applies; the positions of the bodies with respect to the barycenter are and times r, respectively, so the two bodies' orbits are similarTwo geometrical objects are called similar if they both have the same shape. More precisely, one is congruent to the result of a uniform scaling of the other. Corresponding sides of similar polygons are in proportion, and corresponding angles of similar polygons have the same measure...
conic sectionIn mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...
s; the same ratios apply for the velocities, and, without the minus, for the angular momentumAngular momentum is a quantity that is useful in describing the rotational state of a physical system. For a rigid body rotating around an axis of symmetry , the angular momentum can be expressed as the product of the body's moment of inertia and its angular velocity...
and for the kinetic energies, all with respect to the barycenter
- for circular orbit
thumb|200px|Two bodies with a slight difference in mass orbiting around a common [[barycenter]] with circular orbits.In astrodynamics or celestial mechanics a circular orbit is an elliptic orbit with the eccentricity equal to 0...
s
- for elliptic orbit
In astrodynamics or celestial mechanics an elliptic orbit is a Kepler orbit with the eccentricity greater than 0 and less than 1. In a gravitational two-body problem with the eccentricity in this range both bodies follow similar elliptic orbits with the same orbital period around their common...
s: (with a expressed in AU and T in years, and with M the total mass relative to that of the Sun, we get )
- for parabolic trajectories
In astrodynamics or celestial mechanics a parabolic trajectory is a Kepler orbit with the eccentricity equal to 1. When moving away from the source it is called an escape orbit, otherwise a capture orbit....
is constant and equal to
- h is the total angular momentum
Angular momentum is a quantity that is useful in describing the rotational state of a physical system. For a rigid body rotating around an axis of symmetry , the angular momentum can be expressed as the product of the body's moment of inertia and its angular velocity...
divided by the reduced mass
- the specific orbital energy
In astrodynamics the specific orbital energy of an orbiting body traveling through space under standard assumptions is the sum of its potential energy and kinetic energy per unit mass...
formulas apply, with specific potential and kinetic energy and their sum taken as the totals for the system, divided by the reduced mass; the kinetic energy of the smaller body is larger; the potential energy of the whole system is equal to the potential energy of one body with respect to the other, i.e. minus the energy needed to escape the other if the other is kept in a fixed position; this should not be confused with the smaller amount of energy one body needs to escape, if the other body moves away also, in the opposite direction: in that case the total energy the two need to escape each other is the same as the aforementioned amount; the conservation of energy for each mass means that an increase of kinetic energy is accompanied by a decrease of potential energy, which is for each mass the inner product of the force and the change in position relative to the barycenter, not relative to the other mass
- for elliptic and hyperbolic orbits is twice the semi-major axis times the absolute value of the specific orbital energy
- For example, consider two bodies like the Sun orbiting each other:
-
- the reduced mass is one half of the mass of one Sun (one quarter of the total mass)
- at a distance of 1 AU: the orbital period
The orbital period is the time taken for a given object to make one complete orbit about another object.When mentioned without further qualification in astronomy this refers to the sidereal period of an astronomical object, which is calculated with respect to the stars.There are several kinds of...
is year, the same as the orbital period of the Earth would be if the Sun would have twice its actual mass; the total energy per kg reduced mass (90 MJ/kg) is twice that of the Earth-Sun system (45 MJ/kg); the total energy per kg total mass (22.5 MJ/kg) is one half of the total energy per kg Earth mass in the Earth-Sun system (45 MJ/kg)
- at a distance of 2 AU (each following an orbit like that of the Earth around the Sun): the orbital period is 2 years, the same as the orbital period of the Earth would be if the Sun would have one quarter of its actual mass
- at a distance of AU: the orbital period is 1 year, the same as the orbital period of the Earth around the Sun
-
- Similarly, a second Earth at a distance from the Earth equal to times the usual distance of geosynchronous orbit
A geosynchronous orbit is an orbit around the Earth with an orbital period matching the Earth's sidereal rotation period. This synchronization means that for an observer at a fixed location on Earth, a satellite in a geosynchronous orbit returns to exactly the same place in the sky at exactly the...
s would be geosynchronous.
Examples
Any classical system of two particles is, by definition, a two-body problem. In many cases, however, one particle is significantly heavier than the other, e.g., the
EarthEarth is the third planet from the Sun. It is the fifth largest of the eight planets in the solar system, and the largest of the terrestrial planets in the Solar System in terms of diameter, mass and density...
and 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....
. In such cases, the heavier particle is approximately the center of mass, and the reduced mass is approximately the lighter mass. Hence, the heavier mass may be treated roughly as a fixed center of force, and the motion of the lighter mass may be solved for directly by one-body methods.
In other cases, however, the masses of the two bodies are roughly equal, so that neither of them can be approximated as being at rest. Astronomical examples include:
- a binary star
A binary star is a star system consisting of two stars orbiting around their common center of mass. The brighter star is called the primary and the other is its companion star, comes, or secondary...
, e.g. Alpha CentauriAlpha Centauri ; is the brightest star in the southern constellation of Centaurus and an established binary star system, Alpha Centauri AB...
(approx. the same mass)
- a double planet
"Double planet" is an informal term used to describe a planet with a moon that may be large enough to be considered a planet in its own right; a common definition is that the objects orbit a centre of gravity that is above their surfaces. The formal term is "binary system"...
, e.g. PlutoPluto, formal designation 134340 Pluto, is the second-largest known dwarf planet in the Solar System and the tenth-largest body observed directly orbiting the Sun...
with its moon CharonCharon, discovered in 1978 at the United States Naval Observatory Flagstaff Station, is the largest satellite of the dwarf planet Pluto. Following the 2005 discovery of two other natural satellites of Pluto , Charon may also be referred to as Pluto I...
(mass ratio 0.147)
- a binary asteroid
A binary asteroid is a system of two asteroids orbiting their common center of mass, in analogy with binary stars. 243 Ida was the first binary asteroid to be identified when the Galileo spacecraft did a flyby in 1993...
, e.g. 90 Antiope90 Antiope is an asteroid discovered on October 1, 1866 by Robert Luther. The 90th asteroid to be discovered, it is named after Antiope from Greek mythology, though it is disputed as to whether this is Antiope the Amazon or Antiope the mother of Amphion and Zethus.Antiope orbits in the outer third...
(approx. the same mass)
See also
- Kepler orbit
In celestial mechanics, a Kepler orbit describes the motion of an orbiting body as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space...
- Kepler's laws of planetary motion
In astronomy, Kepler's three laws of planetary motion are:#The orbit of every planet is an ellipse with the sun at a focus.#A line joining a planet and the sun sweeps out equal areas during equal intervals of time....
- Virial theorem
In mechanics, the virial theorem provides a general equation relating the average over time of the total kinetic energy, , of a stable system, bound by potential forces, with that of the total potential energy, , where angle brackets represent the average over time of the enclosed quantity...
- n-body problem
To understand the motion of celestial bodies, the sun, planets and the visible stars has been the main motivation for the -body problem. The first complete mathematical formulation of this problem appeared in Newton's Principia...
- Bertrand's theorem
In classical mechanics, Bertrand's theorem states that only two types of potentials produce stable, closed orbits: an inverse-square central force such as the gravitational or electrostatic potentialand the radial harmonic oscillator potential...
- Kepler problem
In classical mechanics, Kepler’s problem is a special case of the two-body problem, in which the two bodies interact by a central force F that varies in strength as the inverse square of the distance r between them. The force may be either attractive or repulsive...