Orbit

Overview

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, an

**orbit**is the gravitationally curved path of an object around a point in space, for example the orbit of a planet

Planet

A planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...

around the center of a star system, such as the Solar System

Solar System

The Solar System consists of the Sun and the astronomical objects gravitationally bound in orbit around it, all of which formed from the collapse of a giant molecular cloud approximately 4.6 billion years ago. The vast majority of the system's mass is in the Sun...

. Orbits of planets are typically elliptical.

Current understanding of the mechanics of orbital motion is based on Albert Einstein

Albert Einstein

Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

's general theory of relativity, which accounts for gravity as due to curvature of space-time, with orbits following geodesic

Geodesic

In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...

s.

Discussions

Encyclopedia

In physics

, an

around the center of a star system, such as the Solar System

. 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 geodesic

s. For ease of calculation, relativity is commonly approximated by the force-based theory of universal gravitation

based on Kepler's laws of planetary motion

.

was able to show that the motions of planets were in fact (at least approximately) elliptical motions.

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 epicycle

s 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

. 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

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 section

s, if the force of gravity propagated instantaneously. Newton showed that, for a pair of bodies, the orbits' sizes are in inverse proportion to their mass

es, 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.

, planets, dwarf planet

s, asteroid

s (a.k.a. minor planets), comet

s, 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 satellite

s, 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.

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.)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:

), 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

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

Orbiting bodies in closed orbits repeat their path 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:

Note that that while bound orbits around a point mass or around a spherical body with an Newtonian gravitational field are closed ellipse

s, 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 effect

s, changing the gravitational field's behavior with distance) will cause the orbit's shape to depart from the closed ellipse

s 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 point

s, 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:

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.

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.)

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:

and

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

After integrating, we have

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,

. It then follows that

where we have introduced the auxiliary variable

The radial force

) yields:

In the case of gravity, Newton's law of universal gravitation

states that the force is proportional to the inverse square of the distance:

where

,

So for the gravitational force — or, more generally, for

(up to a shift of origin of the dependent variable). The solution is:

where

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

where

In general, this can be recognized as the equation of a conic section

in polar coordinates (

If parameter

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.

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:

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.

, 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.

. 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 sail

s or magnetic sail

s. 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 force

s 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 wave

s. 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 hole

s or neutron star

s 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.

The general effect of this is to change the orbital parameters over time; predominantly this gives a rotation of the orbital plane around the rotational pole of the central body (it perturbs the argument of perigee) in a way that is dependent on the angle of orbital plane to the equator as well as altitude at perigee. This is termed nodal regression

.

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.

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 asteroid

s is particularly affected over large periods when the asteroids are rotating relative to the Sun.

and celestial mechanics

to the practical problems concerning the motion of rocket

s 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, planet

s, moon

s, and comet

s. Orbital mechanics focuses on spacecraft trajectories

, including orbital maneuver

s, 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).

Thus the constant has dimension density

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)

for an elliptical orbit with semi-major axis

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, an

**orbit**is the gravitationally curved path of an object around a point in space, for example the orbit of a planetPlanet

A planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...

around the center of a star system, such as the Solar System

Solar System

The Solar System consists of the Sun and the astronomical objects gravitationally bound in orbit around it, all of which formed from the collapse of a giant molecular cloud approximately 4.6 billion years ago. The vast majority of the system's mass is in the Sun...

. Orbits of planets are typically elliptical.

Current understanding of the mechanics of orbital motion is based on Albert Einstein

Albert Einstein

Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

's general theory of relativity, which accounts for gravity as due to curvature of space-time, with orbits following geodesic

Geodesic

In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...

s. For ease of calculation, relativity is commonly approximated by the force-based theory of universal gravitation

Newton's law of universal gravitation

Newton's law of universal gravitation states that every point mass in the universe attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them...

based on Kepler's laws of planetary motion

Kepler's laws of planetary motion

In astronomy, Kepler's laws give a description of the motion of planets around the Sun.Kepler's laws are:#The orbit of every planet is an ellipse with the Sun at one of the two foci....

.

## History

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. Theories of this kind predicted paths of the planets moderately well, until Johannes KeplerJohannes Kepler

Johannes Kepler was a German mathematician, astronomer and astrologer. A key figure in the 17th century scientific revolution, he is best known for his eponymous laws of planetary motion, codified by later astronomers, based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican...

was able to show that the motions of planets were in fact (at least approximately) elliptical motions.

In the geocentric model

Geocentric model

In astronomy, the geocentric model , is the superseded theory that the Earth is the center of the universe, and that all other objects orbit around it. This geocentric model served as the predominant cosmological system in many ancient civilizations such as ancient Greece...

of the solar system, the celestial spheres

Celestial spheres

The celestial spheres, or celestial orbs, were the fundamental entities of the cosmological models developed by Plato, Eudoxus, Aristotle, Ptolemy, Copernicus and others...

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 epicycle

Deferent and epicycle

In the Ptolemaic system of astronomy, the epicycle was a geometric model used to explain the variations in speed and direction of the apparent motion of the Moon, Sun, and planets...

s 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

Johannes Kepler

Johannes Kepler was a German mathematician, astronomer and astrologer. A key figure in the 17th century scientific revolution, he is best known for his eponymous laws of planetary motion, codified by later astronomers, based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican...

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

Circle

A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

(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

Focus (geometry)

In geometry, the foci are a pair of special points with reference to which any of a variety of curves is constructed. For example, foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola...

. 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

Astronomical unit

An astronomical unit is a unit of length equal to about or approximately the mean Earth–Sun distance....

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

Isaac Newton

Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

demonstrated that Kepler's laws were derivable from his theory of gravitation

Gravitation

Gravitation, or gravity, is a natural phenomenon by which physical bodies attract with a force proportional to their mass. Gravitation is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped...

and that, in general, the orbits of bodies subject to gravity were conic section

Conic section

In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...

s, if the force of gravity propagated instantaneously. Newton showed that, for a pair of bodies, the orbits' sizes are in inverse proportion to their mass

Mass

Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

es, and that the bodies revolve about their common center of mass

Center of mass

In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...

. 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

Within a planetary systemPlanetary system

A planetary system consists of the various non-stellar objects orbiting a star such as planets, dwarf planets , asteroids, meteoroids, comets, and cosmic dust...

, planets, dwarf planet

Dwarf planet

A dwarf planet, as defined by the International Astronomical Union , is a celestial body orbiting the Sun that is massive enough to be spherical as a result of its own gravity but has not cleared its neighboring region of planetesimals and is not a satellite...

s, asteroid

Asteroid

Asteroids are a class of small Solar System bodies in orbit around the Sun. They have also been called planetoids, especially the larger ones...

s (a.k.a. minor planets), comet

Comet

A comet is an icy small Solar System body that, when close enough to the Sun, displays a visible coma and sometimes also a tail. These phenomena are both due to the effects of solar radiation and the solar wind upon the nucleus of the comet...

s, and space debris

Space debris

Space debris, also known as orbital debris, space junk, and space waste, is the collection of objects in orbit around Earth that were created by humans but no longer serve any useful purpose. These objects consist of everything from spent rocket stages and defunct satellites to erosion, explosion...

orbit the barycenter

Barycentric coordinates (astronomy)

In astronomy, barycentric coordinates are non-rotating coordinates with origin at the center of mass of two or more bodies.The barycenter is the point between two objects where they balance each other. For example, it is the center of mass where two or more celestial bodies orbit each other...

in elliptical orbits. A comet in a parabolic

Parabolic trajectory

In astrodynamics or celestial mechanics a parabolic trajectory is a Kepler orbit with the eccentricity equal to 1. When moving away from the source it is called an escape orbit, otherwise a capture orbit...

or hyperbolic

Hyperbolic trajectory

In astrodynamics or celestial mechanics a hyperbolic trajectory is a Kepler orbit with the eccentricity greater than 1. Under standard assumptions a body traveling along this trajectory will coast to infinity, arriving there with hyperbolic excess velocity relative to the central body. Similarly to...

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

Natural satellite

A natural satellite or moon is a celestial body that orbits a planet or smaller body, which is called its primary. The two terms are used synonymously for non-artificial satellites of planets, of dwarf planets, and of minor planets....

or artificial satellite

Satellite

In the context of spaceflight, a satellite is an object which has been placed into orbit by human endeavour. Such objects are sometimes called artificial satellites to distinguish them from natural satellites such as the Moon....

s, follow orbits about a barycenter near that planet.

Owing to mutual gravitational perturbations

Perturbation (astronomy)

Perturbation is a term used in astronomy in connection with descriptions of the complex motion of a massive body which is subject to appreciable gravitational effects from more than one other massive body....

, the eccentricities of the planetary orbits vary over time. Mercury

Mercury (planet)

Mercury is the innermost and smallest planet in the Solar System, orbiting the Sun once every 87.969 Earth days. The orbit of Mercury has the highest eccentricity of all the Solar System planets, and it has the smallest axial tilt. It completes three rotations about its axis for every two orbits...

, the smallest planet in the Solar System, has the most eccentric orbit. At the present epoch

Epoch (astronomy)

In astronomy, an epoch is a moment in time used as a reference point for some time-varying astronomical quantity, such as celestial coordinates, or elliptical orbital elements of a celestial body, where these are subject to perturbations and vary with time...

, Mars

Mars

Mars is the fourth planet from the Sun in the Solar System. The planet is named after the Roman god of war, Mars. It is often described as the "Red Planet", as the iron oxide prevalent on its surface gives it a reddish appearance...

has the next largest eccentricity while the smallest orbital eccentricities are seen in Venus

Venus

Venus is the second planet from the Sun, orbiting it every 224.7 Earth days. The planet is named after Venus, the Roman goddess of love and beauty. After the Moon, it is the brightest natural object in the night sky, reaching an apparent magnitude of −4.6, bright enough to cast shadows...

and Neptune

Neptune

Neptune is the eighth and farthest planet from the Sun in the Solar System. Named for the Roman god of the sea, it is the fourth-largest planet by diameter and the third largest by mass. Neptune is 17 times the mass of Earth and is slightly more massive than its near-twin Uranus, which is 15 times...

.

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

Velocity

In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

. As a planet approaches apoapsis, its velocity will decrease.

### Understanding orbits

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

Newton's cannonball

Newton's cannonball was a thought experiment Isaac Newton used to hypothesize that the force of gravity was universal, and it was the key force for planetary motion...

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.)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

Circular orbit

A circular orbit is the orbit at a fixed distance around any point by an object rotating around a fixed axis.Below we consider a circular orbit in astrodynamics or celestial mechanics under standard assumptions...

, 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

Escape velocity

In physics, escape velocity is the speed at which the kinetic energy plus the gravitational potential energy of an object is zero gravitational potential energy is negative since gravity is an attractive force and the potential is defined to be zero at infinity...

, again dependent on the firing height and mass of the planet, an open orbit such as (E) results — a parabolic trajectory

Parabolic trajectory

In astrodynamics or celestial mechanics a parabolic trajectory is a Kepler orbit with the eccentricity equal to 1. When moving away from the source it is called an escape orbit, otherwise a capture orbit...

. At even faster velocities the object will follow a range of hyperbolic trajectories

Hyperbolic trajectory

In astrodynamics or celestial mechanics a hyperbolic trajectory is a Kepler orbit with the eccentricity greater than 1. Under standard assumptions a body traveling along this trajectory will coast to infinity, arriving there with hyperbolic excess velocity relative to the central body. Similarly to...

. 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:

**No orbit****Suborbital trajectories**Sub-orbital spaceflightA sub-orbital space flight is a spaceflight in which the spacecraft reaches space, but its trajectory intersects the atmosphere or surface of the gravitating body from which it was launched, so that it does not complete one orbital revolution....- 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

## Newton's laws of motion

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 problemTwo-body problem

In classical mechanics, the two-body problem is to determine the motion of two point particles that interact only with each other. Common examples include a satellite orbiting a planet, a planet orbiting a star, two stars orbiting each other , and a classical electron orbiting an atomic nucleus In...

), 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

Coordinate system

In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...

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.Potential energy

In physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...

With two bodies, an orbit is a conic section

Conic section

In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...

. The orbit can be open (so the object never returns) or closed (returning), depending on the total energy

Energy

In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

(kinetic

Kinetic energy

The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...

+ potential

Potential energy

In physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...

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

Escape velocity

In physics, escape velocity is the speed at which the kinetic energy plus the gravitational potential energy of an object is zero gravitational potential energy is negative since gravity is an attractive force and the potential is defined to be zero at infinity...

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

Hyperbola

In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...

(when the velocity is greater than the escape velocity), or a parabola

Parabola

In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...

(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

Ellipse

In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...

. 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

Perigee

Perigee is the point at which an object makes its closest approach to the Earth.. Often the term is used in a broader sense to define the point in an orbit where the orbiting body is closest to the body it orbits. The opposite is the apogee, the farthest or highest point.The Greek prefix "peri"...

, 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 path 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:

- The orbit of a planet around the SunSunThe Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...

is an ellipse, with the Sun in one of the focal points of the ellipse. 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 perigeeOrbital planeOrbital plane may refer to:*Orbital plane *In anatomy, it refers to a specific area of the maxilla...PerigeePerigee is the point at which an object makes its closest approach to the Earth.. Often the term is used in a broader sense to define the point in an orbit where the orbiting body is closest to the body it orbits. The opposite is the apogee, the farthest or highest point.The Greek prefix "peri"...

and apogee, and things orbiting the MoonMoonThe Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...

have a perilune and apolune (or periselene and aposelene respectively). An orbit around any starStarA star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...

, not just the Sun, has a periastron and an apastron. - 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 planeOrbital planeOrbital plane may refer to:*Orbital plane *In anatomy, it refers to a specific area of the maxilla...

, 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 axisSemi-major axisThe major axis of an ellipse is its longest diameter, a line that runs through the centre and both foci, its ends being at the widest points of the shape...

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 ellipse

Ellipse

In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...

s, 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

Earth

Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...

, or by relativistic effect

Theory of relativity

The theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....

s, changing the gravitational field's behavior with distance) will cause the orbit's shape to depart from the closed ellipse

Ellipse

In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...

s characteristic of Newtonian two-body motion. The two-body solutions were published by Newton in Principia

Philosophiae Naturalis Principia Mathematica

Philosophiæ Naturalis Principia Mathematica, Latin for "Mathematical Principles of Natural Philosophy", often referred to as simply the Principia, is a work in three books by Sir Isaac Newton, first published 5 July 1687. Newton also published two further editions, in 1713 and 1726...

in 1687. In 1912 Karl Fritiof Sundman

Karl Fritiof Sundman

Karl Frithiof Sundman was a Finnish mathematician who used analytic methods to prove the existence of a convergent infinite series solution to the three-body problem in 1906 and 1909....

developed a converging infinite series that solves the three-body problem

Three-body problem

Three-body problem has two distinguishable meanings in physics and classical mechanics:# In its traditional sense the three-body problem is the problem of taking an initial set of data that specifies the positions, masses and velocities of three bodies for some particular point in time and then...

; however, it converges too slowly to be of much use. Except for special cases like the Lagrangian point

Lagrangian point

The Lagrangian points are the five positions in an orbital configuration where a small object affected only by gravity can theoretically be stationary relative to two larger objects...

s, 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 perturbationPerturbation (astronomy)Perturbation is a term used in astronomy in connection with descriptions of the complex motion of a massive body which is subject to appreciable gravitational effects from more than one other massive body....

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 tablesEphemerisAn ephemeris is a table of values that gives the positions of astronomical objects in the sky at a given time or times. Different kinds of ephemerides are used for astronomy and astrology...

for celestial navigationCelestial navigationCelestial navigation, also known as astronavigation, is a position fixing technique that has evolved over several thousand years to help sailors cross oceans without having to rely on estimated calculations, or dead reckoning, to know their position...

. Still, there are secular phenomenaSecular phenomenaIn astronomy, secular phenomena are contrasted with phenomena observed to repeat periodically. In particular, astronomical ephemerides use secular to label the longest-lasting or non-oscillatory perturbations in the motion of planets, as opposed to periodic perturbations which exhibit repetition...

that have to be dealt with by post-NewtonianParameterized post-Newtonian formalismPost-Newtonian formalism is a calculational tool that expresses Einstein's equations of gravity in terms of the lowest-order deviations from Newton's theory. This allows approximations to Einstein's equations to be made in the case of weak fields...

methods. - The differential equationDifferential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

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 problemInitial value problemIn mathematics, in the field of differential equations, an initial value problem is an ordinary differential equation together with a specified value, called the initial condition, of the unknown function at a given point in the domain of the solution...

. 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

*(See also Kepler orbit*)Kepler orbitIn celestial mechanics, a Kepler orbit describes the motion of an orbiting body as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space...

, orbit equationOrbit equationIn astrodynamics an orbit equation defines the path of orbiting body m_2\,\! around central body m_1\,\! relative to m_1\,\!, without specifying position as a function of time...

and Kepler's first law.

Note that the following is a classical (Newtonian

Classical mechanics

In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...

) analysis of orbital mechanics, which assumes that the more subtle effects of general relativity

General relativity

General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

, such as frame dragging and gravitational time dilation

Gravitational time dilation

Gravitational time dilation is the effect of time passing at different rates in regions of different gravitational potential; the lower the gravitational potential, the more slowly time passes...

are negligible. Relativistic effects cease to be negligible when near very massive bodies (as with the precession of Mercury's orbit

Kepler problem in general relativity

The two-body problem in general relativity is to determine the motion and gravitational field of two bodies interacting with one another by gravitation, as described by the field equations of general relativity. Solving the Kepler problem is essential to calculate the bending of light by gravity...

about the Sun), or when extreme precision is needed (as with calculations of the orbital elements

Orbital elements

Orbital elements are the parameters required to uniquely identify a specific orbit. In celestial mechanics these elements are generally considered in classical two-body systems, where a Kepler orbit is used...

and time signal references for GPS satellites.)

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

Acceleration

In physics, acceleration is the rate of change of velocity with time. In one dimension, acceleration is the rate at which something speeds up or slows down. However, since velocity is a vector, acceleration describes the rate of change of both the magnitude and the direction of velocity. ...

are, respectively:

and

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

After integrating, we have

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 massSpecific relative angular momentum

The specific relative angular momentum is also known as the areal momentum .In astrodynamics, the specific relative angular momentum of two orbiting bodies is the vector product of the relative position and the relative velocity. Equivalently, it is the total angular momentum divided by the...

. It then follows that

where we have introduced the auxiliary variable

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(See also Binet equationBinet equation

The Binet equation, derived by Jacques Philippe Marie Binet, provides the form of a central force given the shape of the orbital motion in plane polar coordinates. The equation can also be used to derive the shape of the orbit for a given force law, but this usually involves the solution to a...

) yields:

In the case of gravity, Newton's law of universal gravitation

Newton's law of universal gravitation

Newton's law of universal gravitation states that every point mass in the universe attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them...

states that the force is proportional to the inverse square of the distance:

where

*G*is the constant of universal gravitationGravitational constant

The gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...

,

*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 haveSo 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 equationHarmonic oscillator

In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x: \vec F = -k \vec x \, where k is a positive constant....

(up to a shift of origin of the dependent variable). The solution is:

where

*A*and*θ*_{0}are arbitrary constants.The equation of the orbit described by the particle is thus:

where

*e*is:In general, this can be recognized as the equation of a conic section

Conic section

In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...

in polar coordinates (

*r*,*θ*). We can make a further connection with the classic description of conic section with:If parameter

*e*is smaller than one,*e*is the eccentricity and*a*the semi-major axis of an ellipse.## Orbital planes

The analysis so far has been two dimensional; it turns out that an unperturbedPerturbation theory

Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...

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

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

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

Orbital elements

Orbital elements are the parameters required to uniquely identify a specific orbit. In celestial mechanics these elements are generally considered in classical two-body systems, where a Kepler orbit is used...

, after Johannes Kepler and his laws. The Keplerian elements are six:

- InclinationInclinationInclination in general is the angle between a reference plane and another plane or axis of direction.-Orbits:The inclination is one of the six orbital parameters describing the shape and orientation of a celestial orbit...

(*i*) - Longitude of the ascending nodeLongitude of the ascending nodeThe longitude of the ascending node is one of the orbital elements used to specify the orbit of an object in space. It is the angle from a reference direction, called the origin of longitude, to the direction of the ascending node, measured in a reference plane...

(Ω) - Argument of periapsisArgument of periapsisThe argument of periapsis , symbolized as ω, is one of the orbital elements of an orbiting body...

(ω) - Eccentricity (
*e*) - Semimajor axis (
*a*) - Mean anomalyMean anomalyIn celestial mechanics, the mean anomaly is a parameter relating position and time for a body moving in a Kepler orbit. It is based on the fact that equal areas are swept at the focus in equal intervals of time....

at epochEpoch (astronomy)In astronomy, an epoch is a moment in time used as a reference point for some time-varying astronomical quantity, such as celestial coordinates, or elliptical orbital elements of a celestial body, where these are subject to perturbations and vary with time...

(*M*_{0})

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

Perturbation (astronomy)

Perturbation is a term used in astronomy in connection with descriptions of the complex motion of a massive body which is subject to appreciable gravitational effects from more than one other massive body....

, 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

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

A small radial impulse given to a body in orbit changes the eccentricityEccentricity (mathematics)

In mathematics, the eccentricity, denoted e or \varepsilon, is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.In particular,...

, but not the orbital period

Orbital period

The orbital period is the time taken for a given object to make one complete orbit about another object.When mentioned without further qualification in astronomy this refers to the sidereal period of an astronomical object, which is calculated with respect to the stars.There are several kinds of...

(to first order). A prograde or retrograde

Retrograde motion

Retrograde motion is motion in the direction opposite to the movement of something else, and is the contrary of direct or prograde motion. This motion can be the orbit of one body about another body or about some other point, or the rotation of a single body about its axis, or other phenomena such...

impulse (i.e. an impulse applied along the orbital motion) changes both the eccentricity and the orbital period

Orbital period

The orbital period is the time taken for a given object to make one complete orbit about another object.When mentioned without further qualification in astronomy this refers to the sidereal period of an astronomical object, which is calculated with respect to the stars.There are several kinds of...

. 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

Orbital plane

Orbital plane may refer to:*Orbital plane *In anatomy, it refers to a specific area of the maxilla...

without changing the period

Orbit (dynamics)

In mathematics, in the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. The orbit is a subset of the phase space and the set of all orbits is a partition of the phase space, that is different orbits do not intersect in the...

or eccentricity. In all instances, a closed orbit will still intersect the perturbation point.

### Orbital decay

If an orbit is about a planetary body with significant atmosphere, its orbit can decay because of dragDrag (physics)

In fluid dynamics, drag refers to forces which act on a solid object in the direction of the relative fluid flow velocity...

. 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

Solar maximum

Solar maximum or solar max is the period of greatest solar activity in the solar cycle of the sun. During solar maximum, sunspots appear....

, 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

Earth's magnetic field

Earth's magnetic field is the magnetic field that extends from the Earth's inner core to where it meets the solar wind, a stream of energetic particles emanating from the Sun...

. 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 sail

Solar sail

Solar sails are a form of spacecraft propulsion using the radiation pressure of light from a star or laser to push enormous ultra-thin mirrors to high speeds....

s or magnetic sail

Magnetic sail

A magnetic sail or magsail is a proposed method of spacecraft propulsion which would use a static magnetic field to deflect charged particles radiated by the Sun as a plasma wind, and thus impart momentum to accelerate the spacecraft...

s. These forms of propulsion require no propellant or energy input other than that of the Sun, and so can be used indefinitely. See statite

Statite

A statite is a hypothetical type of artificial satellite that employs a solar sail to continuously modify its orbit in ways that gravity alone would not allow. Typically, a statite would use the solar sail to "hover" in a location that would not otherwise be available as a stable geosynchronous...

for one such proposed use.

Orbital decay can occur due to tidal force

Tidal force

The tidal force is a secondary effect of the force of gravity and is responsible for the tides. It arises because the gravitational force per unit mass exerted on one body by a second body is not constant across its diameter, the side nearest to the second being more attracted by it than the side...

s for objects below the synchronous orbit

Synchronous orbit

A synchronous orbit is an orbit in which an orbiting body has a period equal to the average rotational period of the body being orbited , and in the same direction of rotation as that body.-Properties:...

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

Torque

Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....

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

Phobos (moon)

Phobos is the larger and closer of the two natural satellites of Mars. Both moons were discovered in 1877. With a mean radius of , Phobos is 7.24 times as massive as Deimos...

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 wave

Gravitational wave

In physics, gravitational waves are theoretical ripples in the curvature of spacetime which propagates as a wave, traveling outward from the source. Predicted to exist by Albert Einstein in 1916 on the basis of his theory of general relativity, gravitational waves theoretically transport energy as...

s. 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 hole

Black hole

A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...

s or neutron star

Neutron star

A neutron star is a type of stellar remnant that can result from the gravitational collapse of a massive star during a Type II, Type Ib or Type Ic supernova event. Such stars are composed almost entirely of neutrons, which are subatomic particles without electrical charge and with a slightly larger...

s that are orbiting each other closely.

### Oblateness

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.

The general effect of this is to change the orbital parameters over time; predominantly this gives a rotation of the orbital plane around the rotational pole of the central body (it perturbs the argument of perigee) in a way that is dependent on the angle of orbital plane to the equator as well as altitude at perigee. This is termed nodal regression

Nodal regression

Nodal precession is the precession of an orbital plane around the rotation axis of an astronomical body such as the Earth. This precession is due to the non spherical nature of a spinning body which creates a non spherical gravitational field....

.

### Multiple gravitating bodies

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

N-body problem

The n-body problem is the problem of predicting the motion of a group of celestial objects that interact with each other gravitationally. Solving this problem has been motivated by the need to understand the motion of the Sun, planets and the visible stars...

. Most n-body problems

N-body problem

The n-body problem is the problem of predicting the motion of a group of celestial objects that interact with each other gravitationally. Solving this problem has been motivated by the need to understand the motion of the Sun, planets and the visible stars...

have no closed form solution, although some special cases have been formulated.

### Light radiation and stellar wind

For smaller bodies particularly, light and stellar windStellar wind

A stellar wind is a flow of neutral or charged gas ejected from the upper atmosphere of a star. It is distinguished from the bipolar outflows characteristic of young stars by being less collimated, although stellar winds are not generally spherically symmetric.Different types of stars have...

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 asteroid

Asteroid

Asteroids are a class of small Solar System bodies in orbit around the Sun. They have also been called planetoids, especially the larger ones...

s is particularly affected over large periods when the asteroids are rotating relative to the Sun.

## Astrodynamics

**Orbital mechanics**or**astrodynamics**is the application of ballisticsBallistics

Ballistics is the science of mechanics that deals with the flight, behavior, and effects of projectiles, especially bullets, gravity bombs, rockets, or the like; the science or art of designing and accelerating projectiles so as to achieve a desired performance.A ballistic body is a body which is...

and celestial mechanics

Celestial mechanics

Celestial mechanics is the branch of astronomy that deals with the motions of celestial objects. The field applies principles of physics, historically classical mechanics, to astronomical objects such as stars and planets to produce ephemeris data. Orbital mechanics is a subfield which focuses on...

to the practical problems concerning the motion of rocket

Rocket

A rocket is a missile, spacecraft, aircraft or other vehicle which obtains thrust from a rocket engine. In all rockets, the exhaust is formed entirely from propellants carried within the rocket before use. Rocket engines work by action and reaction...

s and other spacecraft

Spacecraft

A spacecraft or spaceship is a craft or machine designed for spaceflight. Spacecraft are used for a variety of purposes, including communications, earth observation, meteorology, navigation, planetary exploration and transportation of humans and cargo....

. The motion of these objects is usually calculated from Newton's laws of motion

Newton's laws of motion

Newton's laws of motion are three physical laws that form the basis for classical mechanics. They describe the relationship between the forces acting on a body and its motion due to those forces...

and Newton's law of universal gravitation

Newton's law of universal gravitation

Newton's law of universal gravitation states that every point mass in the universe attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them...

. 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, planet

Planet

A planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...

s, moon

Moon

The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...

s, and comet

Comet

A comet is an icy small Solar System body that, when close enough to the Sun, displays a visible coma and sometimes also a tail. These phenomena are both due to the effects of solar radiation and the solar wind upon the nucleus of the comet...

s. Orbital mechanics focuses on spacecraft trajectories

Trajectory

A trajectory is the path that a moving object follows through space as a function of time. The object might be a projectile or a satellite, for example. It thus includes the meaning of orbit—the path of a planet, an asteroid or a comet as it travels around a central mass...

, including orbital maneuver

Orbital maneuver

In spaceflight, an orbital maneuver is the use of propulsion systems to change the orbit of a spacecraft.For spacecraft far from Earth—for example those in orbits around the Sun—an orbital maneuver is called a deep-space maneuver .-delta-v:...

s, orbit plane changes, and interplanetary transfers, and is used by mission planners to predict the results of propulsive maneuvers

Spacecraft propulsion

Spacecraft propulsion is any method used to accelerate spacecraft and artificial satellites. There are many different methods. Each method has drawbacks and advantages, and spacecraft propulsion is an active area of research. However, most spacecraft today are propelled by forcing a gas from the...

. General relativity

General relativity

General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

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).

## Scaling in gravity

The gravitational constantGravitational constant

The gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...

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

Thus the constant has dimension density

^{−1}time^{−2}. This corresponds to the following properties.Scaling

Scale factor

A scale factor is a number which scales, or multiplies, some quantity. In the equation y=Cx, C is the scale factor for x. C is also the coefficient of x, and may be called the constant of proportionality of y to x...

of distances (including sizes of bodies, while keeping the densities the same) gives similar

Similarity (geometry)

Two geometrical objects are called similar if they both have the same shape. More precisely, either one is congruent to the result of a uniform scaling of the other...

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)

for an elliptical orbit with semi-major axis

Semi-major axis

The major axis of an ellipse is its longest diameter, a line that runs through the centre and both foci, its ends being at the widest points of the shape...

*a*, 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

- 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

- List of orbits
- Escape velocityEscape velocityIn physics, escape velocity is the speed at which the kinetic energy plus the gravitational potential energy of an object is zero gravitational potential energy is negative since gravity is an attractive force and the potential is defined to be zero at infinity...
- Gravity
- Kepler orbitKepler orbitIn celestial mechanics, a Kepler orbit describes the motion of an orbiting body as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space...
- Kepler's laws of planetary motionKepler's laws of planetary motionIn astronomy, Kepler's laws give a description of the motion of planets around the Sun.Kepler's laws are:#The orbit of every planet is an ellipse with the Sun at one of the two foci....
- Molniya orbitMolniya orbitMolniya orbit is a type of highly elliptical orbit with an inclination of 63.4 degrees, an argument of perigee of -90 degree and an orbital period of one half of a sidereal day...
- Orbit (dynamics)Orbit (dynamics)In mathematics, in the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. The orbit is a subset of the phase space and the set of all orbits is a partition of the phase space, that is different orbits do not intersect in the...
- Orbital spaceflightOrbital spaceflightAn orbital spaceflight is a spaceflight in which a spacecraft is placed on a trajectory where it could remain in space for at least one orbit. To do this around the Earth, it must be on a free trajectory which has an altitude at perigee above...

/Sub-orbital spaceflightSub-orbital spaceflightA sub-orbital space flight is a spaceflight in which the spacecraft reaches space, but its trajectory intersects the atmosphere or surface of the gravitating body from which it was launched, so that it does not complete one orbital revolution.... - Perifocal coordinate systemPerifocal coordinate systemThe perifocal frame is the two-dimensional frame of reference for an orbit. It is centered at the focus of the orbit, i.e. the planet about which the orbit is centered. Its plane is given the coordinates p and q, where the p axis is directed toward the periapsis of the elliptical orbit...
- Lagrangian pointLagrangian pointThe Lagrangian points are the five positions in an orbital configuration where a small object affected only by gravity can theoretically be stationary relative to two larger objects...
- Rosetta (orbit)Rosetta (orbit)A Rosetta orbit is a complex type of orbit. Theoretically, an object approaching a black hole with an intermediate velocity will enter a complex orbit pattern, bounded by a near and far distance to the hole and tracing an oscillating pattern known as a hypotrochoid.In quantum mechanics, the...
- Klemperer rosetteKlemperer rosetteA Klemperer rosette is a gravitational system of heavier and lighter bodies orbiting in a regular repeating pattern around a common barycenter. It was first described by W. B...
- TrajectoryTrajectoryA trajectory is the path that a moving object follows through space as a function of time. The object might be a projectile or a satellite, for example. It thus includes the meaning of orbit—the path of a planet, an asteroid or a comet as it travels around a central mass...

, Hyperbolic trajectoryHyperbolic trajectoryIn astrodynamics or celestial mechanics a hyperbolic trajectory is a Kepler orbit with the eccentricity greater than 1. Under standard assumptions a body traveling along this trajectory will coast to infinity, arriving there with hyperbolic excess velocity relative to the central body. Similarly to...

, Parabolic trajectoryParabolic trajectoryIn astrodynamics or celestial mechanics a parabolic trajectory is a Kepler orbit with the eccentricity equal to 1. When moving away from the source it is called an escape orbit, otherwise a capture orbit...

and Radial trajectoryRadial trajectoryIn astrodynamics and celestial mechanics a radial trajectory is a Kepler orbit with zero angular momentum. Two objects in a radial trajectory move directly towards or away from each other in a straight line.- Classification :... - Polar OrbitPolar orbitA polar orbit is an orbit in which a satellite passes above or nearly above both poles of the body being orbited on each revolution. It therefore has an inclination of 90 degrees to the equator...

s - Secular variations of the planetary orbitsSecular variations of the planetary orbitsThe secular variations of the planetary orbits is a concept describing long-term changes in the orbits of the planets Mercury to Neptune. If one ignores the gravitational attraction between the planets and only models the attraction between the Sun and the planets, then with some further...

## External links

- 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 , 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 0521827507 - Swetz, Frank; et al. (1997).
*Learn from the Masters!*. Mathematical Association of America. ISBN 0883857030