A satellite orbiting the Earth has a tangential velocity and an inward acceleration.

Two bodies of different mass orbiting a common barycenter. The relative sizes and type of orbit are similar to the Pluto–Charon system.

In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System.^[1][2] Orbits of planets are typically elliptical.

Current understanding of the mechanics of orbital motion is based on Albert Einstein's general theory of relativity, which accounts for gravity as due to curvature of space-time, with orbits following geodesics. For ease of calculation, relativity is commonly approximated by the force-based theory of universal gravitation based on Kepler's laws of planetary motion.^[3]

History[link]

Historically, the apparent motions of the planets were first understood geometrically (and without regard to gravity) in terms of epicycles, which are the sums of numerous circular motions.^[4] Theories of this kind predicted paths of the planets moderately well, until Johannes Kepler was able to show that the motions of planets were in fact (at least approximately) elliptical motions.^[5]

In the geocentric model of the solar system, the celestial spheres model was originally used to explain the apparent motion of the planets in the sky in terms of perfect spheres or rings, but after the planets' motions were more accurately measured, theoretical mechanisms such as deferent and epicycles were added. Although it was capable of accurately predicting the planets' position in the sky, more and more epicycles were required over time, and the model became more and more unwieldy.

The basis for the modern understanding of orbits was first formulated by Johannes Kepler whose results are summarised in his three laws of planetary motion. First, he found that the orbits of the planets in our solar system are elliptical, not circular (or epicyclic), as had previously been believed, and that the Sun is not located at the center of the orbits, but rather at one focus.^[6] Second, he found that the orbital speed of each planet is not constant, as had previously been thought, but rather that the speed depends on the planet's distance from the Sun. Third, Kepler found a universal relationship between the orbital properties of all the planets orbiting the Sun. For the planets, the cubes of their distances from the Sun are proportional to the squares of their orbital periods. Jupiter and Venus, for example, are respectively about 5.2 and 0.723 AU distant from the Sun, their orbital periods respectively about 11.86 and 0.615 years. The proportionality is seen by the fact that the ratio for Jupiter, 5.2³/11.86², is practically equal to that for Venus, 0.723³/0.615², in accord with the relationship.

The lines traced out by orbits dominated by the gravity of a central source are conic sections: the shapes of the curves of intersection between a plane and a cone. Parabolic (1) and hyperbolic (3) orbits are escape orbits, whereas elliptical and circular orbits (2) are captive.

This image shows the four trajectory categories with the gravitational potential well of the central mass's field of potential energy shown in black and the height of the kinetic energy of the moving body shown in red extending above that, correlating to changes in speed as distance changes according to Kepler's laws.

Isaac Newton demonstrated that Kepler's laws were derivable from his theory of gravitation and that, in general, the orbits of bodies subject to gravity were conic sections, if the force of gravity propagated instantaneously. Newton showed that, for a pair of bodies, the orbits' sizes are in inverse proportion to their masses, and that the bodies revolve about their common center of mass. Where one body is much more massive than the other, it is a convenient approximation to take the center of mass as coinciding with the center of the more massive body.

Albert Einstein was able to show that gravity was due to curvature of space-time, and thus he was able to remove Newton's assumption that changes propagate instantaneously. In relativity theory, orbits follow geodesic trajectories which approximate very well to the Newtonian predictions. However there are differences that can be used to determine which theory describes reality more accurately. Essentially all experimental evidence that can distinguish between the theories agrees with relativity theory to within experimental measuremental accuracy, but the differences from Newtonian mechanics are usually very small (except where there are very strong gravity fields and very high speeds).

However, the Newtonian solution is still used for most purposes since it is significantly easier to use.

Planetary orbits[link]

Within a planetary system, planets, dwarf planets, asteroids (a.k.a. minor planets), comets, and space debris orbit the barycenter in elliptical orbits. A comet in a parabolic or hyperbolic orbit about a barycenter is not gravitationally bound to the star and therefore is not considered part of the star's planetary system. Bodies which are gravitationally bound to one of the planets in a planetary system, either natural or artificial satellites, follow orbits about a barycenter near that planet.

Owing to mutual gravitational perturbations, the eccentricities of the planetary orbits vary over time. Mercury, the smallest planet in the Solar System, has the most eccentric orbit. At the present epoch, Mars has the next largest eccentricity while the smallest orbital eccentricities are seen in Venus and Neptune.

As two objects orbit each other, the periapsis is that point at which the two objects are closest to each other and the apoapsis is that point at which they are the farthest from each other. (More specific terms are used for specific bodies. For example, perigee and apogee are the lowest and highest parts of an orbit around Earth, while perihelion and aphelion are the closest and farthest points of an orbit around the Sun.)

In the elliptical orbit, the center of mass of the orbiting-orbited system is at one focus of both orbits, with nothing present at the other focus. As a planet approaches periapsis, the planet will increase in speed, or velocity. As a planet approaches apoapsis, its velocity will decrease.

Understanding orbits[link]

There are a few common ways of understanding orbits:

• As the object moves sideways, it falls toward the central body. However, it moves so quickly that the central body will curve away beneath it.
• A force, such as gravity, pulls the object into a curved path as it attempts to fly off in a straight line.
• As the object moves sideways (tangentially), it falls toward the central body. However, it has enough tangential velocity to miss the orbited object, and will continue falling indefinitely. This understanding is particularly useful for mathematical analysis, because the object's motion can be described as the sum of the three one-dimensional coordinates oscillating around a gravitational center.

As an illustration of an orbit around a planet, the Newton's cannonball model may prove useful (see image below). This is a 'thought experiment', in which a cannon on top of a tall mountain is able to fire a cannonball horizontally at any chosen muzzle velocity. The effects of air friction on the cannonball are ignored (or perhaps the mountain is high enough that the cannon will be above the Earth's atmosphere, which comes to the same thing.)^[7]

Newton's cannonball, an illustration of how objects can "fall" in a curve

If the cannon fires its ball with a low initial velocity, the trajectory of the ball curves downward and hits the ground (A). As the firing velocity is increased, the cannonball hits the ground farther (B) away from the cannon, because while the ball is still falling towards the ground, the ground is increasingly curving away from it (see first point, above). All these motions are actually "orbits" in a technical sense — they are describing a portion of an elliptical path around the center of gravity — but the orbits are interrupted by striking the Earth.

If the cannonball is fired with sufficient velocity, the ground curves away from the ball at least as much as the ball falls — so the ball never strikes the ground. It is now in what could be called a non-interrupted, or circumnavigating, orbit. For any specific combination of height above the center of gravity and mass of the planet, there is one specific firing velocity (unaffected by the mass of the ball, which is assumed to be very small relative to the Earth's mass) that produces a circular orbit, as shown in (C).

As the firing velocity is increased beyond this, elliptic orbits are produced; one is shown in (D). If the initial firing is above the surface of the Earth as shown, there will also be elliptical orbits at slower velocities; these will come closest to the Earth at the point half an orbit beyond, and directly opposite, the firing point.

At a specific velocity called escape velocity, again dependent on the firing height and mass of the planet, an open orbit such as (E) results — a parabolic trajectory. At even faster velocities the object will follow a range of hyperbolic trajectories. In a practical sense, both of these trajectory types mean the object is "breaking free" of the planet's gravity, and "going off into space".

The velocity relationship of two moving objects with mass can thus be considered in four practical classes, with subtypes:

1. No orbit
2. Suborbital trajectories
   • Range of interrupted elliptical paths
3. Orbital trajectories (or simply "orbits")
   • Range of elliptical paths with closest point opposite firing point
   • Circular path
   • Range of elliptical paths with closest point at firing point
4. Open (or escape) trajectories
   • Parabolic paths
   • Hyperbolic paths

Newton's laws of motion[link]

In many situations relativistic effects can be neglected, and Newton's laws give a highly accurate description of the motion. The acceleration of each body is equal to the sum of the gravitational forces on it, divided by its mass, and the gravitational force between each pair of bodies is proportional to the product of their masses and decreases inversely with the square of the distance between them. To this Newtonian approximation, for a system of two point masses or spherical bodies, only influenced by their mutual gravitation (the two-body problem), the orbits can be exactly calculated. If the heavier body is much more massive than the smaller, as for a satellite or small moon orbiting a planet or for the Earth orbiting the Sun, it is accurate and convenient to describe the motion in a coordinate system that is centered on the heavier body, and we say that the lighter body is in orbit around the heavier. For the case where the masses of two bodies are comparable, an exact Newtonian solution is still available, and qualitatively similar to the case of dissimilar masses, by centering the coordinate system on the center of mass of the two.

Energy is associated with gravitational fields. A stationary body far from another can do external work if it is pulled towards it, and therefore has gravitational potential energy. Since work is required to separate two bodies against the pull of gravity, their gravitational potential energy increases as they are separated, and decreases as they approach one another. For point masses the gravitational energy decreases without limit as they approach zero separation, and it is convenient and conventional to take the potential energy as zero when they are an infinite distance apart, and then negative (since it decreases from zero) for smaller finite distances.

With two bodies, an orbit is a conic section. The orbit can be open (so the object never returns) or closed (returning), depending on the total energy (kinetic + potential energy) of the system. In the case of an open orbit, the speed at any position of the orbit is at least the escape velocity for that position, in the case of a closed orbit, always less. Since the kinetic energy is never negative, if the common convention is adopted of taking the potential energy as zero at infinite separation, the bound orbits have negative total energy, parabolic trajectories have zero total energy, and hyperbolic orbits have positive total energy.

An open orbit has the shape of a hyperbola (when the velocity is greater than the escape velocity), or a parabola (when the velocity is exactly the escape velocity). The bodies approach each other for a while, curve around each other around the time of their closest approach, and then separate again forever. This may be the case with some comets if they come from outside the solar system.

A closed orbit has the shape of an ellipse. In the special case that the orbiting body is always the same distance from the center, it is also the shape of a circle. Otherwise, the point where the orbiting body is closest to Earth is the perigee, called periapsis (less properly, "perifocus" or "pericentron") when the orbit is around a body other than Earth. The point where the satellite is farthest from Earth is called apogee, apoapsis, or sometimes apifocus or apocentron. A line drawn from periapsis to apoapsis is the line-of-apsides. This is the major axis of the ellipse, the line through its longest part.

Orbiting bodies in closed orbits repeat their paths after a constant period of time. This motion is described by the empirical laws of Kepler, which can be mathematically derived from Newton's laws. These can be formulated as follows:

1. The orbit of a planet around the Sun is an ellipse, with the Sun in one of the focal points of the ellipse. [This focal point is actually the barycenter of the Sun-planet system; for simplicity this explanation assumes the Sun's mass is infinitely larger than that planet's.] The orbit lies in a plane, called the orbital plane. The point on the orbit closest to the attracting body is the periapsis. The point farthest from the attracting body is called the apoapsis. There are also specific terms for orbits around particular bodies; things orbiting the Sun have a perihelion and aphelion, things orbiting the Earth have a perigee and apogee, and things orbiting the Moon have a perilune and apolune (or periselene and aposelene respectively). An orbit around any star, not just the Sun, has a periastron and an apastron.
2. As the planet moves around its orbit during a fixed amount of time, the line from the Sun to planet sweeps a constant area of the orbital plane, regardless of which part of its orbit the planet traces during that period of time. This means that the planet moves faster near its perihelion than near its aphelion, because at the smaller distance it needs to trace a greater arc to cover the same area. This law is usually stated as "equal areas in equal time."
3. For a given orbit, the ratio of the cube of its semi-major axis to the square of its period is constant.

Note that that while bound orbits around a point mass or around a spherical body with an Newtonian gravitational field are closed ellipses, which repeat the same path exactly and indefinitely, any non-spherical or non-Newtonian effects (as caused, for example, by the slight oblateness of the Earth, or by relativistic effects, changing the gravitational field's behavior with distance) will cause the orbit's shape to depart from the closed ellipses characteristic of Newtonian two-body motion. The two-body solutions were published by Newton in Principia in 1687. In 1912, Karl Fritiof Sundman developed a converging infinite series that solves the three-body problem; however, it converges too slowly to be of much use. Except for special cases like the Lagrangian points, no method is known to solve the equations of motion for a system with four or more bodies.

Instead, orbits with many bodies can be approximated with arbitrarily high accuracy. These approximations take two forms:

One form takes the pure elliptic motion as a basis, and adds perturbation terms to account for the gravitational influence of multiple bodies. This is convenient for calculating the positions of astronomical bodies. The equations of motion of the moons, planets and other bodies are known with great accuracy, and are used to generate tables for celestial navigation. Still, there are secular phenomena that have to be dealt with by post-Newtonian methods.

The differential equation form is used for scientific or mission-planning purposes. According to Newton's laws, the sum of all the forces will equal the mass times its acceleration (F = ma). Therefore accelerations can be expressed in terms of positions. The perturbation terms are much easier to describe in this form. Predicting subsequent positions and velocities from initial values corresponds to solving an initial value problem. Numerical methods calculate the positions and velocities of the objects a short time in the future, then repeat the calculation. However, tiny arithmetic errors from the limited accuracy of a computer's math are cumulative, which limits the accuracy of this approach.

Differential simulations with large numbers of objects perform the calculations in a hierarchical pairwise fashion between centers of mass. Using this scheme, galaxies, star clusters and other large objects have been simulated.

Analysis of orbital motion[link]

It has been suggested that this article or section be merged into Kepler's_laws_of_planetary_motion#Derivation from Newton's laws of motion and Newton's law of gravitation. (Discuss) Proposed since November 2009.

(See also Kepler orbit, orbit equation and Kepler's first law.)

Note that the following is a classical (Newtonian) analysis of orbital mechanics, which assumes that the more subtle effects of general relativity, such as frame dragging and gravitational time dilation are negligible. Relativistic effects cease to be negligible when near very massive bodies (as with the precession of Mercury's orbit about the Sun), or when extreme precision is needed (as with calculations of the orbital elements and time signal references for GPS satellites.^[8])

To analyze the motion of a body moving under the influence of a force which is always directed towards a fixed point, it is convenient to use polar coordinates with the origin coinciding with the center of force. In such coordinates the radial and transverse components of the acceleration are, respectively:

Failed to parse (Missing texvc executable; please see math/README to configure.): a_r = \ddot{r}-r\dot{\theta }^2 \,

and

Failed to parse (Missing texvc executable; please see math/README to configure.): a_\theta =\frac{1}{r}\frac{d}{dt}\left( r^2 \dot \theta \right)

Since the force is entirely radial, and since acceleration is proportional to force, it follows that the transverse acceleration is zero. As a result,

Failed to parse (Missing texvc executable; please see math/README to configure.): a_\theta = 0 \,

After integrating, we have

Failed to parse (Missing texvc executable; please see math/README to configure.): r^2 \dot \theta = \text{constant} \,

which is actually the theoretical proof of Kepler's second law (A line joining a planet and the Sun sweeps out equal areas during equal intervals of time). The constant of integration, h, is the angular momentum per unit mass. It then follows that

Failed to parse (Missing texvc executable; please see math/README to configure.): \dot\theta =\frac{h}{r^2} = hu^2 \,

where we have introduced the auxiliary variable

Failed to parse (Missing texvc executable; please see math/README to configure.): u = { 1 \over r }.

The radial force ƒ(r) per unit mass is the radial acceleration a_r defined above. Solving the above differential equation with respect to time^[9](See also Binet equation) yields:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{d^2u}{d\theta^2} + u = -\frac{f(1 / u)}{h^2u^2}.

In the case of gravity, Newton's law of universal gravitation states that the force is proportional to the inverse square of the distance:

Failed to parse (Missing texvc executable; please see math/README to configure.): f(1/u) = a_r = { -GM \over r^2 } = -GM u^2

where G is the constant of universal gravitation, m is the mass of the orbiting body (planet) - note that m is absent from the equation since it cancels out, and M is the mass of the central body (the Sun). Substituting into the prior equation, we have

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{d^2u}{d\theta^2} + u = \frac{ GM }{h^2}.

So for the gravitational force — or, more generally, for any inverse square force law — the right hand side of the equation becomes a constant and the equation is seen to be the harmonic equation (up to a shift of origin of the dependent variable). The solution is:

Failed to parse (Missing texvc executable; please see math/README to configure.): u(\theta) = \frac{ GM }{h^2} + A \cos(\theta-\theta_0)

where A and θ₀ are arbitrary constants.

The equation of the orbit described by the particle is thus:

Failed to parse (Missing texvc executable; please see math/README to configure.): r = \frac{1}{u} = \frac{ h^2 / GM }{1 + e \cos (\theta - \theta_0)},

where e is:

Failed to parse (Missing texvc executable; please see math/README to configure.): e \equiv \frac{h^2A}{G M}.

In general, this can be recognized as the equation of a conic section in polar coordinates (r, θ). We can make a further connection with the classic description of conic section with:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{h^2}{GM} = a(1-e^2)

If parameter e is smaller than one, e is the eccentricity and a the semi-major axis of an ellipse.

Orbital planes[link]

The analysis so far has been two dimensional; it turns out that an unperturbed orbit is two-dimensional in a plane fixed in space, and thus the extension to three dimensions requires simply rotating the two-dimensional plane into the required angle relative to the poles of the planetary body involved.

The rotation to do this in three dimensions requires three numbers to uniquely determine; traditionally these are expressed as three angles.

Orbital period[link]

The orbital period is simply how long an orbiting body takes to complete one orbit.

Specifying orbits[link]

Six parameters are required to specify an orbit about a body. For example, the 3 numbers which describe the body's initial position, and the 3 values which describe its velocity will describe a unique orbit that can be calculated forwards (or backwards). However, traditionally the parameters used are slightly different.

The traditionally used set of orbital elements is called the set of Keplerian elements, after Johannes Kepler and his laws. The Keplerian elements are six:

• Inclination (i)
• Longitude of the ascending node (Ω)
• Argument of periapsis (ω)
• Eccentricity (e)
• Semimajor axis (a)
• Mean anomaly at epoch (M₀)

In principle once the orbital elements are known for a body, its position can be calculated forward and backwards indefinitely in time. However, in practice, orbits are affected or perturbed, by other forces than simple gravity from an assumed point source (see the next section), and thus the orbital elements change over time.

Orbital perturbations[link]

An orbital perturbation is when a force or impulse which is much smaller than the overall force or average impulse of the main gravitating body and which is external to the two orbiting bodies causes an acceleration, which changes the parameters of the orbit over time.

Radial, prograde and transverse perturbations[link]

A small radial impulse given to a body in orbit changes the eccentricity, but not the orbital period (to first order). A prograde or retrograde impulse (i.e. an impulse applied along the orbital motion) changes both the eccentricity and the orbital period. Notably, a prograde impulse given at periapsis raises the altitude at apoapsis, and vice versa, and a retrograde impulse does the opposite. A transverse impulse (out of the orbital plane) causes rotation of the orbital plane without changing the period or eccentricity. In all instances, a closed orbit will still intersect the perturbation point.

Orbital decay[link]

If an orbit is about a planetary body with significant atmosphere, its orbit can decay because of drag. Particularly at each periapsis, the object experiences atmospheric drag, losing energy. Each time, the orbit grows less eccentric (more circular) because the object loses kinetic energy precisely when that energy is at its maximum. This is similar to the effect of slowing a pendulum at its lowest point; the highest point of the pendulum's swing becomes lower. With each successive slowing more of the orbit's path is affected by the atmosphere and the effect becomes more pronounced. Eventually, the effect becomes so great that the maximum kinetic energy is not enough to return the orbit above the limits of the atmospheric drag effect. When this happens the body will rapidly spiral down and intersect the central body.

The bounds of an atmosphere vary wildly. During a solar maximum, the Earth's atmosphere causes drag up to a hundred kilometres higher than during a solar minimum.

Some satellites with long conductive tethers can also experience orbital decay because of electromagnetic drag from the Earth's magnetic field. As the wire cuts the magnetic field it acts as a generator, moving electrons from one end to the other. The orbital energy is converted to heat in the wire.

Orbits can be artificially influenced through the use of rocket engines which change the kinetic energy of the body at some point in its path. This is the conversion of chemical or electrical energy to kinetic energy. In this way changes in the orbit shape or orientation can be facilitated.

Another method of artificially influencing an orbit is through the use of solar sails or magnetic sails. These forms of propulsion require no propellant or energy input other than that of the Sun, and so can be used indefinitely. See statite for one such proposed use.

Orbital decay can occur due to tidal forces for objects below the synchronous orbit for the body they're orbiting. The gravity of the orbiting object raises tidal bulges in the primary, and since below the synchronous orbit the orbiting object is moving faster than the body's surface the bulges lag a short angle behind it. The gravity of the bulges is slightly off of the primary-satellite axis and thus has a component along the satellite's motion. The near bulge slows the object more than the far bulge speeds it up, and as a result the orbit decays. Conversely, the gravity of the satellite on the bulges applies torque on the primary and speeds up its rotation. Artificial satellites are too small to have an appreciable tidal effect on the planets they orbit, but several moons in the solar system are undergoing orbital decay by this mechanism. Mars' innermost moon Phobos is a prime example, and is expected to either impact Mars' surface or break up into a ring within 50 million years.

Orbits can decay via the emission of gravitational waves. This mechanism is extremely weak for most stellar objects, only becoming significant in cases where there is a combination of extreme mass and extreme acceleration, such as with black holes or neutron stars that are orbiting each other closely.

Oblateness[link]

The standard analysis of orbiting bodies assumes that all bodies consist of uniform spheres, or more generally, concentric shells each of uniform density. It can be shown that such bodies are gravitationally equivalent to point sources.

However, in the real world, many bodies rotate, and this introduces oblateness and distorts the gravity field, and gives a quadrupole moment to the gravitational field which is significant at distances comparable to the radius of the body.

Multiple gravitating bodies[link]

The effects of other gravitating bodies can be significant. For example, the orbit of the Moon cannot be accurately described without allowing for the action of the Sun's gravity as well as the Earth's.

When there are more than two gravitating bodies it is referred to as an n-body problem. Most n-body problems have no closed form solution, although some special cases have been formulated.

Light radiation and stellar wind[link]

For smaller bodies particularly, light and stellar wind can cause significant perturbations to the attitude and direction of motion of the body, and over time can be significant. Of the planetary bodies, the motion of asteroids is particularly affected over large periods when the asteroids are rotating relative to the Sun.

Astrodynamics[link]

Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and Newton's law of universal gravitation. It is a core discipline within space mission design and control. Celestial mechanics treats more broadly the orbital dynamics of systems under the influence of gravity, including spacecraft and natural astronomical bodies such as star systems, planets, moons, and comets. Orbital mechanics focuses on spacecraft trajectories, including orbital maneuvers, orbit plane changes, and interplanetary transfers, and is used by mission planners to predict the results of propulsive maneuvers. General relativity is a more exact theory than Newton's laws for calculating orbits, and is sometimes necessary for greater accuracy or in high-gravity situations (such as orbits close to the Sun).

Earth orbits[link]

Scaling in gravity[link]

The gravitational constant G has been calculated as:

• (6.6742 ± 0.001) × 10⁻¹¹ (kg/m³)⁻¹s⁻².

Thus the constant has dimension density⁻¹ time⁻². This corresponds to the following properties.

Scaling of distances (including sizes of bodies, while keeping the densities the same) gives similar orbits without scaling the time: if for example distances are halved, masses are divided by 8, gravitational forces by 16 and gravitational accelerations by 2. Hence velocities are halved and orbital periods remain the same. Similarly, when an object is dropped from a tower, the time it takes to fall to the ground remains the same with a scale model of the tower on a scale model of the Earth.

Scaling of distances while keeping the masses the same (in the case of point masses, or by reducing the densities) gives similar orbits; if distances are multiplied by 4, gravitational forces and accelerations are divided by 16, velocities are halved and orbital periods are multiplied by 8.

When all densities are multiplied by 4, orbits are the same; gravitational forces are multiplied by 16 and accelerations by 4, velocities are doubled and orbital periods are halved.

When all densities are multiplied by 4, and all sizes are halved, orbits are similar; masses are divided by 2, gravitational forces are the same, gravitational accelerations are doubled. Hence velocities are the same and orbital periods are halved.

In all these cases of scaling. if densities are multiplied by 4, times are halved; if velocities are doubled, forces are multiplied by 16.

These properties are illustrated in the formula (derived from the formula for the orbital period)

Failed to parse (Missing texvc executable; please see math/README to configure.): GT^2 \sigma = 3\pi \left( \frac{a}{r} \right)^3,

for an elliptical orbit with semi-major axis a, of a small body around a spherical body with radius r and average density σ, where T is the orbital period. See also Kepler's Third Law.

Further reading[link]

• Andrea Milani and Giovanni F. Gronchi. Theory of Orbit Determination (Cambridge University Press; 378 pages; 2010). Discusses new algorithms for determining the orbits of both natural and artificial celestial bodies.

See also[link]

	Astronomy portal
	Spaceflight portal

• List of orbits
• Escape velocity
• Gravity
• Kepler orbit
• Kepler's laws of planetary motion
• Molniya orbit
• Orbit (dynamics)
• Orbital spaceflight/Sub-orbital spaceflight
• Perifocal coordinate system
• Lagrangian point
• Rosetta (orbit)
• Klemperer rosette
• Trajectory, Hyperbolic trajectory, Parabolic trajectory and Radial trajectory
• Polar Orbits
• VSOP (planets)

References[link]

Further reading[link]

• Abell, Morrison, and Wolff (1987). Exploration of the Universe (fifth ed.). Saunders College Publishing. 

External links[link]

Look up orbit in Wiktionary, the free dictionary.

Wikimedia Commons has media related to: Orbits

• CalcTool: Orbital period of a planet calculator. Has wide choice of units. Requires JavaScript.
• Browser Based Three Dimension Simulation of Orbital Motion. Objects and distance are drawn to scale. Run on JavaScript-enabled browser such as Internet Explorer, Mozilla Firefox and Opera.
• Java simulation on orbital motion. Requires Java.
• NOAA page on Climate Forcing Data includes (calculated) data on Earth orbit variations over the last 50 million years and for the coming 20 million years
• On-line orbit plotter. Requires JavaScript.
• Orbital Mechanics (Rocket and Space Technology)
• Orbital simulations by Varadi, Ghil and Runnegar (2003) provide another, slightly different series for Earth orbit eccentricity, and also a series for orbital inclination. Orbits for the other planets were also calculated, by F. Varadi, B. Runnegar, M. Ghil (2003). "Successive Refinements in Long-Term Integrations of Planetary Orbits". The Astrophysical Journal 592: 620–630. Bibcode 2003ApJ...592..620V. DOI:10.1086/375560. , but only the eccentricity data for Earth and Mercury are available online.
• Understand orbits using direct manipulation. Requires JavaScript and Macromedia
• Linton, Christopher (2004). From Eudoxus to Einstein. Cambridge: University Press. ISBN 0-521-82750-7
• Swetz, Frank; et al. (1997). Learn from the Masters!. Mathematical Association of America. ISBN 0-88385-703-0

vep:Orbit

This biographical article needs additional citations for verification. Please help by adding reliable sources. Contentious material about living persons that is unsourced or poorly sourced must be removed immediately, especially if potentially libelous or harmful. (March 2011)

William Orbit (born William Mark Wainwright, 15 December 1956) is an English musician, composer and record producer, perhaps best known to most for his work on Madonna's album Ray of Light. He has won three Grammy Awards for his work with Madonna. He produced 13 by Blur and remixed some of the songs on the album.

Biography[link]

1956–1980: Early life & first work[link]

William Orbit was born William Wainwright in London's East End in 1956. His mother, of Italian and Swedish descent, and his father, from Yorkshire, were both schoolteachers.

He does not have a formal education, having left school at 15 and being entirely self taught. Aged 16, and living in a squat in Norwich and working as a strawberry picker in the summer, he started to devote himself to guitar playing.

Living in squats around Europe and working at various jobs, hotel kitchen worker, motorbike messenger, at a ladies shoes factory, as an accounts clerk (in the days before computers), a night shift worker at the old Heineken Brewery in Amsterdam, junior draftsman at a North Sea oil rig design company (he was 'creative' with his CV') and so on.

In these early years he was creatively frustrated, with no means to bring his ideas to life. But an opportunity to work as an assistant at Basing Street Studios just off the Portobello Road in West London gave him a first glimpse into the world of recording.

Aged 23, he met Laurie Mayer who had just moved into his squat in an abandoned primary school in Paddington. This semi-derelict Victorian edifice was the perfect environment for live music to flourish, with bands such as Throbbing Gristle frequently performing there. Laurie and William held mini-festivals in the deserted classrooms and workshops of the old school. The building was also home to the Spanish Anarchist collective Centro Iberico.

It was around that time, in 1980, that William was asked to model clothes for the designer Takeo Kikuchi in Tokyo and this experience opened his eyes to the world of fashion and the visual arts. On that same trip was also the young Hamish Bowles, now International Editor at Large for Vogue magazine.

1981–1993: Torch Song, production & remix work[link]

Laurie Mayer and William Orbit formed the band Torch Song. An experiment in audio cassette soundscapes that also included the graphic designer Neville Brody. Laurie and William have been close friends and creative collaborators ever since, with Laurie this year writing the melodic top line for Madonna's track "Falling Free" from the album 'MDNA'.

Encouraged by Laurie's creative zeal, he went on to build a commercially viable recording studio in Little Venice called Guerilla Studios (later situated in Crouch End, North London) A variety of clients booked the facility, including: The Cocteau Twins, Colour Box,Stewart Copeland, Belinda Carlisle, Etienne Daho, Howard Devoto, Erasure, The Fall, The Frank Chickens, Goldie and Metalheadz, Martin Gore, Kemistry and Storm, Laibach, Lords of the New Church, Daniel Miller ^{[disambiguation needed ]}, S'Express, Gary Numan, Renegade Soundwave, Les Rita Mitsouko, Sting, The Swans, 23 Skidoo, The Waterboys, Jah Wobble, World Party / Karl Wallinger

He began to create remixes for a long list of artists. At first for free, then later on, as word spread, for considerable sums. Remixed artists include: Prince, The Cure, Seal, Nitzer Ebb, Peter Gabriel, Depeche Mode, Seal & The Human League

1990–1992: Guerilla & Bassomatic[link]

William founded Guerilla Records at the beginning of the 90s, releasing vinyl discs from artists such as Leftfield, Underworld, Felix Da Housecat and DJ Pierre in the then-new genre of 'Progressive House', and hosting illegal rave parties in disused warehouses in Central London, before forming the band Bassomatic, Fascinating Rhythm being their biggest hit.

1993 onwards: Strange Cargo III & Madonna[link]

He then moved to California, living in the surfer haven of Hermosa Beach, and had a radio program on the Southern Californian radio station KCRW (he had always had a fascination for radio in all its forms) and in this pioneering station he found a true voice)

He also began recording his album Strange Cargo III, from which the track 'Water From A Vine Leaf', written with his ex-girlfriend Beth Orton, became an underground classic.

After traveling the world, and encountering different cultures and worldviews, he met Madonna in New York in 1997 and they recorded the multi Grammy Award winning album 'Ray of Light'. The two have remained friends since, and created tracks such as 'Beautiful Stranger' (circa 2000) and this year William co-wrote and co-produced six tracks for her album 'MDNA'.

He has said on many occasions that he remains truly grateful to Madonna for seeing what nobody else saw at the time, and giving him a global platform for his work.

Since then he has written and recorded with a wide range of artists, from Andreas Scholl in the classical world, to Beck, Pink, The Cure, Kraftwerk (a project with Ralf Hütter), Blur (the album '13'), All Saints ('Pure Shores' 'Black Coffee' and other songs), Katie Melua, Robbie Williams, Sugababes, Ricky Martin and the late Malcolm Mclaren.

He lived at the Leonard Hotel just behind Oxford Street in London for three years where he recorded most of Bono's vocals for 'Electrical Storm'. A track that he produced and performed on for U2

He has created a series of classical renditions 'Pieces in a Modern Style', from which came the club classic 'Adagio For Strings, Ferry Corsten mix'

In 2007 Orbit composed an orchestral suite for full orchestra. and choir. In nine movements. It debuted on 8 July 2007 with the BBC Philharmonic Orchestra at the Bridgewater Hall in Manchester.

DJ & Radio[link]

William deejays occasionally, with a hardcore techno set, in various locations around the world such as Paris, Tokyo, Thailand and Novosibirsk, Siberia, He has described his mission as 'to get everyone on their feet, despite what they usually listen to'.

He also conducts lectures and masterclasses. And appears on BBC Radio and the BBC Worldservice discussing his work and guest editing music and arts programmes.

Recent work[link]

After a recent first-time trip to Hong Kong, as part of a cultural festival with his friend Sir V S Naipaul, Pharrell Williams and screenwriter Paul Shrader, he is currently in discussions about a regular club event in Hong Kong

He has recently been in the studio, writing and producing with Kid Cudi, Mika, St. Vincent, Kreayshawn, Roger Taylor from Queen & Anoushka Shankar

And has recently co-written and co-produced a track with Chris Brown, 'Don't Wake Me Up', for his forthcoming album 'Fortune'.

He has just completed the fifth album in the 'Strange Cargo' series, which is due for release later in 2012.

He is currently half way through his first (autobiographical) book

Location & Interests[link]

He primarily records in London, Los Angeles and at The Palms Casino's in-house studio in Las Vegas.

Orbit now lives in North London and Los Angeles and has his office and studio at Hoxton Square, London.

His interests in addition to composing and producing music are: photography and design, Cinema and the dramatic arts, Collecting rare books, Performance art and audio visual installations, Radio controlled helicopters, Unorthodox deejay events.

Discography[link]

More details and information about his discography can be found when exploring external links

Torch Song[link]

Bassomatic[link]

Solo[link]

Studio albums

Compilation albums

• The Best of Strange Cargos (1996, I.R.S.) — A compilation from Orbit's first three Strange Cargo albums, highlighted by two versions of "Water From A Vine Leaf", another collaboration with Beth Orton
• Strange Cargo Vol.1-3 (2000, EMI Records) — 3 CD Box Set
• William / Orbit (2002, Universal) — U.S. two-disc publishing CD set to support William Orbit's publishing catalogue ^[5]

Singles

Other Online, Production & Live Performances[link]

He often DJs at various locations. Additional information concerning the dates and places of his live performances appears^[7] on his webpage.

• William Orbit Odyssey^[8] by Ministry of Sound

‘Odyssey’ is a compilation of 42 tracks, including some of William Orbit’s own exclusives. The album was released in January 2010 and is divided into 3 discs; Discs 1 and 2 are unmixed, with many tracks reinterpreted to a greater or lesser extent, while Disc 3 is a continuous DJ mix by William Orbit.

• Orbit Streamcast

Orbit has recently enabled an internet radio (streamcast) which allow listeners to hear more than 10 hours of his music, completely for free. In the playlist are variations of his well-known tracks, as well as released material. In addition, and what's more appreciated by fans, unreleased and rare material can also be found in this streamcast.^[9]

• Latest productions

He produced Katie Melua's fourth album, called The House. It was released on 24 May 2010 and had a remarkable chart performance (Gold in Germany^[10] and United Kingdom,^[11] as well as Platinum in Poland^[12]).

He also produced a track on Nadine Coyle's debut solo album "Insatiable" released on 8 November 2010.

In September 2011 it was confirmed that Orbit will collaborate on Madonna's 12th studio album MDNA. On 3 February 2012 Orbit confirmed on his Twitter account that he produced six tracks - "Gang Bang", "Some Girls", "I'm a Sinner", "Masterpiece", "Falling Free" and "Love Spent".^{[citation needed]}

• Solo career

In August 2010 Orbit released Pieces in a Modern Style 2, on Decca, which is follow-up a to the successful Pieces in a Modern Style. The two CD version containing some notable remixes of interpretations of Vaughan Williams Violin Concerto - Lark Ascending.

Orbit is working as part of the art collective Luxor, with former English National Ballet dancer Anna-Mi Fredriksson and the artist Pauline Amos.

Miscellaneous[link]

• Stereo Odyssey — 4-CDR set of KCRW radio show William produced circa 1996
• Rare Orbit Vol.1 — A 3-CDR collection of rare & unreleased material & productions. 2008, source unknown. Contained unreleased sessions with Madonna, and some new material.
• Essential Mix — 2-CDR set of UK BBC Radio 1 Essential Mix, 6 February 2000
• XFM 2003 — 30 minute CDR of UK XFM Radio 'Superchunk', a mix/mashup collection of special rare and unreleased material, 17 August 2003
• XFM 2006 — 30 minute CDR of UK XFM Radio 'Superchunk', another mix/mashup collection, 12 March 2006
• World Odyssey Mix — 25 minute CDR of UK BBC Asia Network Radio 'Pathaans Musical Rickshaw', a diversified mix/mashup collection, 17 May 2009
• KCRW 2009 — 40 minute CDR of Los Angeles KCRW Radio Station show 'Morning Becomes Eclectic', a two piece live session with Laurie Mayer and interviews with Jason Bentley - 2 July 2009
• In 2010, some of the rare tracks were made public on www.orbitstreamcast.com.^[13]

References[link]

External links[link]

• Official website