# Orbit

In physics, an

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.

Historically, the apparent motions of the planets were described geometrically (without regard to gravity) in terms of epicycles, which are the sums of numerous circular motions.

The celestial spheres model is a geocentric model of the solar system originally used to explain the apparent motion of the planets in the sky. It posited the existence of perfect moving spheres or rings to which the stars and planets were attached. 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' positions in the sky, more and more epicycles were required as the measurements became more accurate, hence 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.

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 (this assumes that the force of gravity propagates instantaneously). Newton showed that, for a pair of bodies, the orbits' sizes are in inverse proportion to their masses, and that those bodies orbit their common center of mass. Where one body is much more massive than the other (as is the case of an artificial satellite orbiting a planet), it is a convenient approximation to take the center of mass as coinciding with the center of the more massive body.

Albert Einstein showed 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. The differences between the two theories are measurable. Essentially all the experimental evidence that can distinguish between the theories agrees with relativity theory to within experimental measurement accuracy. The differences between the two are usually very small (except where there are very strong gravity fields and very high speeds). The first calculation of the relativistic distortion came from measurements of Mercury's orbit period caused by the solar gravity field. However, Newton's solution is still used for most short term purposes since it is significantly easier to use and sufficiently accurate.

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 with 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. (More specific terms are used for specific bodies. For example,

In the case of planets orbiting a star, the mass of the star and all its satellites are calculated to be at a single point called the barycenter. The paths of all the star's satellites are elliptical orbits about that barycenter. Each satellite in that system will have its own elliptical orbit with the barycenter at one focal point of that ellipse. At any point along its orbit, any satellite will have a certain value of kinetic and potential energy with respect to the barycenter, and that energy is a constant value at every point along its orbit. As a result, as a planet approaches periapsis, the planet will increase in speed as its potential energy decreases; as a planet approaches apoapsis, its velocity will decrease as its potential energy increases.

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, dependent on the mass of the planet, an open orbit (E) is achieved that has a parabolic path. At even greater speeds 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" never to return.

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

Once in orbit, their speed keeps them in orbit above the atmosphere. If e.g., an elliptical orbit dips into dense air, the object will lose speed and re-enter (i.e. fall). Occasionally a space craft will intentionally intercept the atmosphere, in an act commonly referred to as an aerobraking maneuver

An open orbit will have a parabolic shape if it has velocity of exactly the escape velocity at that point in its trajectory, and it will have the shape of a hyperbola when its velocity is greater than the escape velocity. When bodies with escape velocity or greater approach each other, they will briefly curve around each other at the time of their closest approach, and then separate, forever. This may be the case with most comets if they come from outside the solar system.

All closed orbits have the shape of an ellipse. A circular orbit is a special case, wherein the foci of the ellipse coincide. The point where the orbiting body is closest to Earth is the called the perigee, and called periapsis (less properly, "perifocus" or "pericentron") when the orbit is about a body other than Earth. The point where the satellite is farthest from Earth is called the apogee, apoapsis, or sometimes apifocus or apocentron. A line drawn from periapsis to apoapsis is the

The following derivation applies to such an elliptical orbit. We start only with the Newtonian law of gravitation such that the gravitational acceleration towards the central body is related to the inverse of the square of the distance between them.

From Newton's Second Law, the summation of the forces acting on m

Combining Eq 1 and 2:

We assume that the central body is massive enough that it can be considered to be stationary and we ignore the more subtle effects of general relativity.

When a pendulum or an object attached to a spring swings in an ellipse, the inward acceleration/force is proportional to the distance Due to the way vectors add, the component of the force in the or in the directions are also proportionate to the respective components of the distances, . Hence, the entire analysis can be done separately in these dimensions. This results in the harmonic parabolic equations and of the ellipse. But with the decreasing relationship , the dimensions cannot be separated.

The location of the orbiting object at the current time is located in the plane using Vector calculus in polar coordinates both with the standard Euclidean basis and with the polar basis with the origin coinciding with the center of force. Let be the distance between the object and the center and be the angle it has rotated. Let and be the standard Euclidean bases and let and be the radial and transverse polar basis with the first being the unit vector pointing from the central body to the current location of the orbiting object and the second being the orthogonal unit vector pointing in the direction that the orbiting object would travel if orbiting in a counter clockwise circle. Then the vector to the orbiting object is

In order to get an equation for the orbit from equation (1), we need to eliminate time.

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

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:

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.

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.

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.

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)

**orbit**is the gravitationally curved path of an object about a point in space, for example the orbit of a planet about a star.^{[1]}^{[2]}Orbits of planets are typically elliptical, and the central mass being orbited is at a focal point of the ellipse.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]}## Contents

[hide]- 1 History
- 2 Planetary orbits
- 3 Newton's laws of motion
- 4 Newtonian analysis of orbital motion
- 5 Relativistic orbital motion
- 6 Orbital planes
- 7 Orbital period
- 8 Specifying orbits
- 9 Orbital perturbations
- 10 Astrodynamics
- 11 Earth orbits
- 12 Scaling in gravity
- 13 Patents
- 14 See also
- 15 References
- 16 Further reading
- 17 External links

## History[edit]

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^{[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 elliptical motions.^{[5]}The celestial spheres model is a geocentric model of the solar system originally used to explain the apparent motion of the planets in the sky. It posited the existence of perfect moving spheres or rings to which the stars and planets were attached. 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' positions in the sky, more and more epicycles were required as the measurements became more accurate, hence 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, 5.2^{3}/11.86^{2}, is practically equal to that for Venus, 0.723^{3}/0.615^{2}, in accord with the relationship.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 (this assumes that the force of gravity propagates instantaneously). Newton showed that, for a pair of bodies, the orbits' sizes are in inverse proportion to their masses, and that those bodies orbit their common center of mass. Where one body is much more massive than the other (as is the case of an artificial satellite orbiting a planet), it is a convenient approximation to take the center of mass as coinciding with the center of the more massive body.

Albert Einstein showed 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. The differences between the two theories are measurable. Essentially all the experimental evidence that can distinguish between the theories agrees with relativity theory to within experimental measurement accuracy. The differences between the two are usually very small (except where there are very strong gravity fields and very high speeds). The first calculation of the relativistic distortion came from measurements of Mercury's orbit period caused by the solar gravity field. However, Newton's solution is still used for most short term purposes since it is significantly easier to use and sufficiently accurate.

## Planetary orbits[edit]

Within a planetary system, planets, dwarf planets, asteroids and other minor planets, comets, and space debris orbit the system's 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 or within 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 with 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. (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 case of planets orbiting a star, the mass of the star and all its satellites are calculated to be at a single point called the barycenter. The paths of all the star's satellites are elliptical orbits about that barycenter. Each satellite in that system will have its own elliptical orbit with the barycenter at one focal point of that ellipse. At any point along its orbit, any satellite will have a certain value of kinetic and potential energy with respect to the barycenter, and that energy is a constant value at every point along its orbit. As a result, as a planet approaches periapsis, the planet will increase in speed as its potential energy decreases; as a planet approaches apoapsis, its velocity will decrease as its potential energy increases.

### Understanding orbits[edit]

There are a few common ways of understanding orbits:- A force, such as gravity, pulls an object into a curved path as it attempts to fly off in a straight line.
- As the object is pulled toward the massive body, it falls toward that body. However, if it has enough tangential velocity it will not fall into the body but will instead continue to follow the curved trajectory caused by that body indefinitely. The object is then said to be orbiting the body.

^{[7]}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, dependent on the mass of the planet, an open orbit (E) is achieved that has a parabolic path. At even greater speeds 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" never to return.

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

*No orbit**Suborbital trajectories*- Range of interrupted elliptical paths

*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

*Open (or escape) trajectories*- Parabolic paths
- Hyperbolic paths

Once in orbit, their speed keeps them in orbit above the atmosphere. If e.g., an elliptical orbit dips into dense air, the object will lose speed and re-enter (i.e. fall). Occasionally a space craft will intentionally intercept the atmosphere, in an act commonly referred to as an aerobraking maneuver

## Newton's laws of motion[edit]

### Newton's law of gravitation and laws of motion for two-body problems[edit]

In most situations relativistic effects can be neglected, and Newton's laws give a sufficiently accurate description of motion. The acceleration of a body is equal to the sum of the forces acting on it, divided by its mass, and the gravitational force acting on a body is proportional to the product of the masses of the two attracting bodies 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 (called a two-body problem), their trajectories can be exactly calculated. If the heavier body is much more massive than the smaller, as in the case of a satellite or small moon orbiting a planet or for the Earth orbiting the Sun, it is accurate enough and convenient to describe the motion in terms of 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 sufficient and can be had by placing the coordinate system at the center of mass of the system.### Defining gravitational potential energy[edit]

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 to zero as they approach zero separation. It is convenient and conventional to assign the potential energy as having zero value when they are an infinite distance apart, and hence it has a negative value (since it decreases from zero) for smaller finite distances.### Orbital energies and orbit shapes[edit]

When only two gravitational bodies interact, their orbits follow a conic section. The orbit can be open (implying the object never returns) or closed (returning). Which it is depends 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, the speed is always less than the escape velocity. 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 will have negative total energy, the parabolic trajectories zero total energy, and hyperbolic orbits positive total energy.An open orbit will have a parabolic shape if it has velocity of exactly the escape velocity at that point in its trajectory, and it will have the shape of a hyperbola when its velocity is greater than the escape velocity. When bodies with escape velocity or greater approach each other, they will briefly curve around each other at the time of their closest approach, and then separate, forever. This may be the case with most comets if they come from outside the solar system.

All closed orbits have the shape of an ellipse. A circular orbit is a special case, wherein the foci of the ellipse coincide. The point where the orbiting body is closest to Earth is the called the perigee, and called periapsis (less properly, "perifocus" or "pericentron") when the orbit is about a body other than Earth. The point where the satellite is farthest from Earth is called the 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.### Kepler's laws[edit]

Bodies following closed orbits repeat their paths with a certain time called the period. 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:- The orbit of a planet around the Sun is an ellipse, with the Sun in one of the focal points of that 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 planet's 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 about 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. - As the planet moves in its orbit, the line from the Sun to planet sweeps a constant area of the orbital plane for a given period of time, 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."
- For a given orbit, the ratio of the cube of its semi-major axis to the square of its period is constant.

### Limitations of Newton's law of gravitation[edit]

Note that while bound orbits of a point mass or a spherical body with a Newtonian gravitational field are closed ellipses, which repeat the same path exactly and indefinitely, any non-spherical or non-Newtonian effects (such as caused by the slight oblateness of the Earth, or by relativistic effects, thereby 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.### Approaches to many-body problems[edit]

Rather than an exact closed form solution, 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 acting on a body will equal the mass of the body 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 of position and velocity 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 ad nauseum. However, tiny arithmetic errors from the limited accuracy of a computer's math are cumulative, which limits the accuracy of this approach.

^{[citation needed]}## Newtonian analysis of orbital motion[edit]

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

The following derivation applies to such an elliptical orbit. We start only with the Newtonian law of gravitation such that the gravitational acceleration towards the central body is related to the inverse of the square of the distance between them.

- eq 1.

_{2}is the force acting on the mass m_{2}caused by the gravitational attraction mass m_{1}has for m_{2}, G is the universal gravitational constant, and r is the distance between the two masses centers.From Newton's Second Law, the summation of the forces acting on m

_{2}related to that bodies acceleration:- eq 2.

_{2}is the acceleration of m_{2}caused by the force of gravitational attraction F_{2}of m_{1}acting on m_{2}.Combining Eq 1 and 2:

_{2}:_{2}, hence the subscripts can be dropped.We assume that the central body is massive enough that it can be considered to be stationary and we ignore the more subtle effects of general relativity.

When a pendulum or an object attached to a spring swings in an ellipse, the inward acceleration/force is proportional to the distance Due to the way vectors add, the component of the force in the or in the directions are also proportionate to the respective components of the distances, . Hence, the entire analysis can be done separately in these dimensions. This results in the harmonic parabolic equations and of the ellipse. But with the decreasing relationship , the dimensions cannot be separated.

^{[citation needed]}The location of the orbiting object at the current time is located in the plane using Vector calculus in polar coordinates both with the standard Euclidean basis and with the polar basis with the origin coinciding with the center of force. Let be the distance between the object and the center and be the angle it has rotated. Let and be the standard Euclidean bases and let and be the radial and transverse polar basis with the first being the unit vector pointing from the central body to the current location of the orbiting object and the second being the orthogonal unit vector pointing in the direction that the orbiting object would travel if orbiting in a counter clockwise circle. Then the vector to the orbiting object is

**(1)**

**(2)**

**(3)**

*h*, is the angular momentum per unit mass.In order to get an equation for the orbit from equation (1), we need to eliminate time.

^{[8]}(See also Binet equation.) In polar coordinates, this would express the distance of the orbiting object from the center as a function of its angle . However, it is easier to introduced the auxiliary variable and to express as a function of . Derivatives of with respect to time may be rewritten as derivatives of with respect to angle.- (reworking (3))

*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:*A*and*θ*_{0}are arbitrary constants. This resulting equation of the orbit of the object is that of an ellipse in Polar form relative to one of the focal points. This is put into a more standard form by letting be the eccentricity, letting be the semi-major axis. Finally, letting so the long axis of the ellipse is along the positive*x*coordinate.## Relativistic orbital motion[edit]

The above classical (Newtonian) analysis of orbital mechanics 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.^{[9]})## Orbital planes[edit]

Main article: Orbital plane (astronomy)

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[edit]

Main article: Orbital period

The orbital period is simply how long an orbiting body takes to complete one orbit.## Specifying orbits[edit]

Main article: Orbital trajectory

See also: Keplerian elements

Six parameters are required to specify a Keplerian orbit about a body. For example, the three numbers that specify the body's initial position, and the three values that specify its velocity will define a unique orbit that can be calculated forwards (or backwards) in time. 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 (
*M*_{0}).

## Orbital perturbations[edit]

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[edit]

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 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[edit]

Main article: Orbital decay

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[edit]

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[edit]

Main article: n-body problem

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. One approximate result is that bodies will usually have reasonably stable orbits around a heavier planet or moon, in spite of these perturbations, provided they are orbiting well within the heavier body's Hill sphere.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[edit]

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[edit]

Main article: Orbital mechanics

**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[edit]

Main article: List of orbits

- Low Earth orbit (LEO): Geocentric orbits with altitudes up to 2,000 km (0–1,240 miles).
^{[10]} - Medium Earth orbit (MEO): Geocentric orbits ranging in altitude from 2,000 km (1,240 miles) to just below geosynchronous orbit at 35,786 kilometers (22,236 mi). Also known as an intermediate circular orbit. These are "most commonly at 20,200 kilometers (12,600 mi), or 20,650 kilometers (12,830 mi), with an orbital period of 12 hours."
^{[11]} - Both Geosynchronous orbit (GSO) and Geostationary orbit (GEO) are orbits around Earth matching Earth's sidereal rotation period. All geosynchronous and geostationary orbits have a semi-major axis of 42,164 km (26,199 mi).
^{[12]}All geostationary orbits are also geosynchronous, but not all geosynchronous orbits are geostationary. A geostationary orbit stays exactly above the equator, whereas a geosynchronous orbit may swing north and south to cover more of the Earth's surface. Both complete one full orbit of Earth per sidereal day (relative to the stars, not the Sun). - High Earth orbit: Geocentric orbits above the altitude of geosynchronous orbit 35,786 km (22,240 miles).
^{[11]}

## Scaling in gravity[edit]

The gravitational constant*G*has been calculated as:- (6.6742 ± 0.001) × 10
^{−11}(kg/m^{3})^{−1}s^{−2}.

^{−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)

*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.## Patents[edit]

The application of certain orbits or orbital maneuvers to specific useful purposes have been the subject of patents.^{[13]}