In

^{3}/11.86^{2}, is practically equal to that for Venus, 0.723^{3}/0.615^{2}, in accord with the relationship. Idealised orbits meeting these rules are known as

_{2} is the force acting on the mass ''m''_{2} caused by the gravitational attraction mass ''m''_{1} has for ''m''_{2}, ''G'' is the universal gravitational constant, and ''r'' is the distance between the two masses centers.
From Newton's Second Law, the summation of the forces acting on ''m''_{2} related to that body's acceleration:
: $F\_2\; =\; m\_2\; A\_2$
where ''A''_{2} is the acceleration of ''m''_{2} caused by the force of gravitational attraction ''F''_{2} of ''m''_{1} acting on ''m''_{2}.
Combining Eq. 1 and 2:
: $-\backslash frac\; =\; m\_2\; A\_2$
Solving for the acceleration, ''A''_{2}:
: $A\_2\; =\; \backslash frac\; =\; -\; \backslash frac\; \backslash frac\; =\; -\backslash frac$
where $\backslash mu\backslash ,$ is the _{2}, hence the subscripts can be dropped.
We assume that the central body is massive enough that it can be considered to be stationary and we ignore the more subtle effects of _{0} are arbitrary constants. This resulting equation of the orbit of the object is that of an ellipse#Polar form relative to focus, ellipse in Polar form relative to one of the focal points. This is put into a more standard form by letting $e\; \backslash equiv\; h^2\; A/\backslash mu$ be the eccentricity (orbit), eccentricity, letting $a\; \backslash equiv\; h^2/\backslash mu\backslash left(1\; -\; e^2\backslash right)$ be the semi-major axis. Finally, letting $\backslash theta\_0\; \backslash equiv\; 0$ so the long axis of the ellipse is along the positive ''x'' coordinate.
:$r(\backslash theta)\; =\; \backslash frac$
When the two-body system is under the influence of torque, the angular momentum ''h'' is not a constant. After the following calculation:
:$\backslash begin\; \backslash frac\; \&=\; -\backslash frac\; \backslash frac\; =\; -\backslash frac\; \backslash frac\; \backslash \backslash \; \backslash frac\; \&=\; -\backslash frac\; \backslash frac\; -\; \backslash frac\; \backslash frac\; \backslash frac\; \backslash \backslash \; \backslash left(\backslash frac\backslash right)^2\; r\; \&=\; \backslash frac\; \backslash end$
we will get the Sturm-Liouville equation of two-body system.

_{0}).
In principle, once the orbital elements are known for a body, its position can be calculated forward and backward indefinitely in time. However, in practice, orbits are affected or Perturbation (astronomy), perturbed, by other forces than simple gravity from an assumed point source (see the next section), and thus the orbital elements change over time.

periapsis
File:Periapsis_apoapsis.png, upright=1.15, The two-body system of interacting elliptic orbits: The smaller, satellite body (blue) orbits the primary (astronomy), primary body (yellow); both are in elliptic orbits around their center of mass, co ...

, the object experiences atmospheric drag, losing energy. Each time, the orbit grows less eccentric (more circular) because the object loses kinetic energy precisely when that energy is at its maximum. This is similar to the effect of slowing a pendulum at its lowest point; the highest point of the pendulum's swing becomes lower. With each successive slowing more of the orbit's path is affected by the atmosphere and the effect becomes more pronounced. Eventually, the effect becomes so great that the maximum kinetic energy is not enough to return the orbit above the limits of the atmospheric drag effect. When this happens the body will rapidly spiral down and intersect the central body.
The bounds of an atmosphere vary wildly. During a solar maximum, the Earth's atmosphere causes drag up to a hundred kilometres higher than during a solar minimum.
Some satellites with long conductive tethers can also experience orbital decay because of electromagnetic drag from the Earth's magnetic field. As the wire cuts the magnetic field it acts as a generator, moving electrons from one end to the other. The orbital energy is converted to heat in the wire.
Orbits can be artificially influenced through the use of rocket engines which change the kinetic energy of the body at some point in its path. This is the conversion of chemical or electrical energy to kinetic energy. In this way changes in the orbit shape or orientation can be facilitated.
Another method of artificially influencing an orbit is through the use of solar sails or magnetic sails. These forms of propulsion require no propellant or energy input other than that of the Sun, and so can be used indefinitely. See statite for one such proposed use.
Orbital decay can occur due to tidal forces for objects below the synchronous orbit for the body they're orbiting. The gravity of the orbiting object raises tidal bulges in the primary, and since below the synchronous orbit, the orbiting object is moving faster than the body's surface the bulges lag a short angle behind it. The gravity of the bulges is slightly off of the primary-satellite axis and thus has a component along with 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 (moon), Phobos is a prime example and is expected to either impact Mars' surface or break up into a ring within 50 million years.
Orbits can decay via the emission of gravitational waves. This mechanism is extremely weak for most stellar objects, only becoming significant in cases where there is a combination of extreme mass and extreme acceleration, such as with black holes or neutron stars that are orbiting each other closely.

^{−11} (kg/m^{3})^{−1}s^{−2}.
Thus the constant has dimension density^{−1} time^{−2}. This corresponds to the following properties.
Scale factor, Scaling of distances (including sizes of bodies, while keeping the densities the same) gives Similarity (geometry), 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 and other travel times related to gravity remain the same. For example, 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 adjusting 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 Orbital period#Small body orbiting a central body, formula for the orbital period)
:$GT^2\; \backslash rho\; =\; 3\backslash pi\; \backslash left(\; \backslash frac\; \backslash right)^3,$
for an elliptical orbit with

CalcTool: Orbital period of a planet calculator

Has wide choice of units. Requires JavaScript.

Requires Java.

includes (calculated) data on Earth orbit variations over the last 50 million years and for the coming 20 million years

Requires JavaScript.

(Rocket and Space Technology)

Orbital simulations by Varadi, Michael Ghil, 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 th

eccentricity data for Earth and Mercury

are available online.

Understand orbits using direct manipulation

Requires JavaScript and Macromedia * {{Authority control Orbits, Celestial mechanics Periodic phenomena Gravity Astrodynamics Concepts in astronomy

celestial mechanics
Celestial mechanics is the branch of astronomy
Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial objects and cel ...

, an orbit is the curved trajectory
A trajectory or flight path is the path that an with in follows through as a function of time. In , a trajectory is defined by via ; hence, a complete trajectory is defined by position and momentum, simultaneously.
The mass might be a or ...

of an object
Object may refer to:
General meanings
* Object (philosophy), a thing, being, or concept
** Entity, something that is tangible and within the grasp of the senses
** Object (abstract), an object which does not exist at any particular time or pl ...

such as the trajectory of a planet
A planet is an astronomical body orbiting a star or Stellar evolution#Stellar remnants, stellar remnant that is massive enough to be Hydrostatic equilibrium, rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and ...

around a star, or of a natural satellite
A natural satellite is in the most common usage, an astronomical body
Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science
Natural science is a branch ...

around a planet, or of an artificial satellite
alt=, A full-size model of the Earth observation satellite ERS 2 ">ERS_2.html" ;"title="Earth observation satellite ERS 2">Earth observation satellite ERS 2
In the context of spaceflight, a satellite is an object that has been intentionally ...

around an object or position in space such as a planet, moon, asteroid, or Lagrange point
In celestial mechanics
Celestial mechanics is the branch of astronomy
Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical ob ...

. In an atom
An atom is the smallest unit of ordinary matter
In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of ato ...

, electron
The electron is a subatomic particle (denoted by the symbol or ) whose electric charge is negative one elementary charge. Electrons belong to the first generation (particle physics), generation of the lepton particle family,
and are general ...

s follow similar curved paths, or orbits, around a nucleus
''Nucleus'' (plural nuclei) is a Latin word for the seed inside a fruit. It most often refers to:
*Atomic nucleus, the very dense central region of an atom
*Cell nucleus, a central organelle of a eukaryotic cell, containing most of the cell's DNA ...

. Normally, orbit refers to a regularly repeating trajectory, although it may also refer to a non-repeating trajectory. To a close approximation, planets and satellites follow elliptic orbit
In astrodynamics
Orbital mechanics or astrodynamics is the application of ballistics
Ballistics is the field of mechanics concerned with the launching, flight behavior and impact effects of projectiles, especially ranged weapon munitio ...

s, with the center of mass
In physics, the center of mass of a distribution of mass
Mass is the physical quantity, quantity of ''matter'' in a physical body. It is also a measure (mathematics), measure of the body's ''inertia'', the resistance to acceleration (change ...

being orbited at a focal point of the ellipse, as described by Kepler's laws of planetary motion
In astronomy, Kepler's laws of planetary motion, published by Johannes Kepler between 1609 and 1619, describe the orbits of planets around the Sun. The laws modified the Copernican heliocentrism, heliocentric theory of Nicolaus Copernicus, repl ...

.
For most situations, orbital motion is adequately approximated by Newtonian mechanics, which explains gravity
Gravity (), or gravitation, is a by which all things with or —including s, s, , and even —are attracted to (or ''gravitate'' toward) one another. , gravity gives to s, and the causes the s of the oceans. The gravitational attracti ...

as a force obeying an inverse-square law
420px, S represents the light source, while r represents the measured points. The lines represent the flux emanating from the sources and fluxes.
The total number of flux lines depends on the strength of the light source and is constant with in ...

. However, Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest physicists of all time. Einstein is known for developing the theory of relativity
The theo ...

's general theory of relativity
General relativity, also known as the general theory of relativity, is the differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern ph ...

, which accounts for gravity as due to curvature of spacetime
In , spacetime is any which fuses the and the one of into a single . can be used to visualize effects, such as why different observers perceive differently where and when events occur.
Until the 20th century, it was assumed that the three ...

, with orbits following geodesic
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

s, provides a more accurate calculation and understanding of the exact mechanics of orbital motion.
History

Historically, the apparent motions of the planets were described by European and Arabic philosophers using the idea ofcelestial spheres
The celestial spheres, or celestial orbs, were the fundamental entities of the cosmological
Cosmology (from Greek κόσμος, ''kosmos'' "world" and -λογία, ''-logia'' "study of") is a branch of astronomy
Astronomy (from ...

. This model posited the existence of perfect moving spheres or rings to which the stars and planets were attached. It assumed the heavens were fixed apart from the motion of the spheres and was developed without any understanding of gravity. After the planets' motions were more accurately measured, theoretical mechanisms such as deferent and epicycle
In the Hipparchian, Ptolemaic, and Copernican systems of astronomy
Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, c ...

s were added. Although the model was capable of reasonably accurately predicting the planets' positions in the sky, more and more epicycles were required as the measurements became more accurate, hence the model became increasingly unwieldy. Originally geocentric
In , the geocentric model (also known as geocentrism, often exemplified specifically by the Ptolemaic system) is a description of the with at the center. Under the geocentric model, the , , s, and all Earth. The geocentric model was the pre ...

, it was modified by Copernicus
Nicolaus Copernicus (; pl, Mikołaj Kopernik; german: link=no, Niclas Koppernigk, modern: ''Nikolaus Kopernikus''; 19 February 1473 – 24 May 1543) was a Renaissance
The Renaissance ( , ) , from , with the same meanings. was a ...

to place the Sun at the centre to help simplify the model. The model was further challenged during the 16th century, as comets were observed traversing the spheres.
The basis for the modern understanding of orbits was first formulated by Johannes Kepler
Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer
An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth. They observe as ...

whose results are summarised in his three laws of planetary motion. First, he found that the orbits of the planets in our Solar System
The Solar SystemCapitalization
Capitalization ( North American English) or capitalisation ( British English) is writing a word with its first letter as a capital letter (uppercase letter) and the remaining letters in lower case, in writin ...

are elliptical, not circular
Circular may refer to:
* The shape of a circle
A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre; equivalently it is ...

(or epicyclic
An epicyclic gear train (also known as a planetary gearset) consists of two gear
Cast iron mortise wheel with wooden cogs (powered by an external water wheel) meshing with a cast iron gear wheel, connected to a pulley with drive belt. ...

), as had previously been believed, and that the Sun is not located at the center of the orbits, but rather at one focus
FOCUS is a fourth-generation programming language (4GL) computer programming programming language, language and development environment that is used to build database queries. Produced by Information Builders Inc., it was originally developed for d ...

. 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 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, 5.2Kepler orbits
In celestial mechanics, a Kepler orbit (or Keplerian orbit, named after the German astronomer Johannes Kepler) is the motion of one body relative to another, as an ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (ge ...

.
Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics a ...

demonstrated that Kepler's laws were derivable from his theory of gravitation
Gravity (), or gravitation, is a natural phenomenon
Types of natural phenomena include:
Weather, fog, thunder, tornadoes; biological processes, decomposition, germination
seedlings, three days after germination.
Germination is th ...

and that, in general, the orbits of bodies subject to gravity were conic section
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s (this assumes that the force of gravity propagates instantaneously). Newton showed that, for a pair of bodies, the orbits' sizes are in inverse proportion to their mass
Mass is the quantity
Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value ...

es, and that those bodies orbit their common center of mass
In physics, the center of mass of a distribution of mass
Mass is the physical quantity, quantity of ''matter'' in a physical body. It is also a measure (mathematics), measure of the body's ''inertia'', the resistance to acceleration (change ...

. Where one body is much more massive than the other (as is the case of an artificial satellite orbiting a planet), it is a convenient approximation to take the center of mass as coinciding with the center of the more massive body.
Advances in Newtonian mechanics were then used to explore variations from the simple assumptions behind Kepler orbits, such as the perturbations due to other bodies, or the impact of spheroidal rather than spherical bodies. Lagrange
Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaRoman_Forum.html" ;"title="Curia Julia in the Roman Forum">Curia Julia in the Roman Forum
A senate is a deliberative assembly, often the upper house or Debating chamber, chamber of a bicame ...

(1736–1813) developed a new approach
New Approach (foaled 18 February 2005) is a retired Irish Thoroughbred racehorse and active stallion. In a racing career which lasted from July 2007 to October 2008 he ran eleven times and won eight races. He was undefeated in five races as a two- ...

to Newtonian mechanics emphasizing energy more than force, and made progress on the three body problem
In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momentum, momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and ...

, discovering the Lagrangian points
In celestial mechanics, the Lagrange points (also Lagrangian points, L-points, or libration points) are orbital points near two large orbit, co-orbiting bodies. Normally, the two objects exert an unbalanced gravitational force at a point, a ...

. In a dramatic vindication of classical mechanics, in 1846 Urbain Le Verrier
Urbain Jean Joseph Le Verrier FRS (FOR) HFRSE
Fellowship of the Royal Society of Edinburgh (FRSE) is an award granted to individuals that the Royal Society of Edinburgh, Scotland's national academy of science and Literature, letters, judged ...

was able to predict the position of Neptune
Neptune is the eighth and farthest-known Solar planet from the Sun. In the Solar System, it is the fourth-largest planet by diameter, the third-most-massive planet, and the densest giant planet. It is 17 times the mass of Earth, slightly mor ...

based on unexplained perturbations in the orbit of Uranus
Uranus is the seventh planet from the Sun. Its name is a reference to the Greek god of the sky, Uranus, who, according to Greek mythology
Greek mythology is the body of myths originally told by the Ancient Greece, ancient Greeks, and ...

.
Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest physicists of all time. Einstein is known for developing the theory of relativity
The theo ...

(1879-1955) in his 1916 paper ''The Foundation of the General Theory of Relativity'' explained that gravity was due to curvature of space-time
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "P ...

and removed Newton's assumption that changes propagate instantaneously. This led astronomers to recognize that Newtonian mechanics did not provide the highest accuracy in understanding orbits. In relativity theory
The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity
General relativity, also known as the general theory of relativity, is the geometric theory of gravita ...

, orbits follow geodesic trajectories which are usually approximated very well by the Newtonian predictions (except where there are very strong gravity fields and very high speeds) but the differences are measurable. Essentially all the experimental evidence that can distinguish between the theories agrees with relativity theory to within experimental measurement accuracy. The original vindication of general relativity is that it was able to account for the remaining unexplained amount in precession of Mercury's perihelion first noted by Le Verrier. However, Newton's solution is still used for most short term purposes since it is significantly easier to use and sufficiently accurate.
Planetary orbits

Within aplanetary system
A planetary system is a set of gravity, gravitationally bound non-Star, stellar objects in or out of orbit around a star or star system. Generally speaking, systems with one or more planets constitute a planetary system, although such systems m ...

, planets, dwarf planet
A dwarf planet is a small planetary-mass object that is in direct orbit of the Sun – something smaller than any of the eight classical planets, but still a world in its own right. The prototypical dwarf planet is Pluto. The interest of d ...

s, asteroid
An asteroid is a minor planet of the Solar System#Inner solar system, inner Solar System. Historically, these terms have been applied to any astronomical object orbiting the Sun that did not resolve into a disc in a telescope and was not observ ...

s and other minor planet
A minor planet is an astronomical object in direct orbit around the Sun (or more broadly, any star with a planetary system) that is neither a planet nor exclusively classified as a comet. Before 2006, the International Astronomical Union (IAU) o ...

s, comet
A comet is an icy, small Solar System body
A small Solar System body (SSSB) is an object in the Solar System
The Solar SystemCapitalization of the name varies. The International Astronomical Union, the authoritative body regarding astr ...

s, and space debris
Space debris (also known as space junk, space pollution, space waste, space trash, or space garbage) is defunct artificial objects in space—principally in Earth orbit
A geocentric orbit or Earth orbit involves any object orbiting the Earth, ...

orbit the system's barycenter
In astronomy
Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial objects and celestial event, phenomena. It uses ...

in elliptical orbit
In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an orbital eccentricity, eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. In a ...

s. A comet in a or hyperbolic
Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry.
The following phenomena are described as ''hyperbolic'' because they ...

orbit about a barycenter is not gravitationally bound to the star and therefore is not considered part of the star's planetary system. Bodies that are gravitationally bound to one of the planets in a planetary system, either natural
Nature, in the broadest sense, is the natural, physical, material world or universe
The universe ( la, universus) is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and ...

or artificial satellites
In the context of spaceflight
Spaceflight (or space flight) is an application of astronautics
Astronautics (or cosmonautics) is the theory and practice of travel beyond atmosphere of Earth, Earth's atmosphere into outer space. Spaceflight ...

, follow orbits about a barycenter near or within that planet.
Owing to mutual gravitational perturbations, the eccentricities of the planetary orbits vary over time. Mercury
Mercury usually refers to:
* Mercury (planet)
Mercury is the smallest planet in the Solar System and the closest to the Sun. Its orbit around the Sun takes 87.97 Earth days, the shortest of all the Sun's planets. It is named after the Roman g ...

, the smallest planet in the Solar System, has the most eccentric orbit. At the present epoch
In chronology
222px, Joseph Scaliger's ''De emendatione temporum'' (1583) began the modern science of chronology
Chronology (from Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-E ...

, Mars
Mars is the fourth planet
A planet is an astronomical body orbiting a star or Stellar evolution#Stellar remnants, stellar remnant that is massive enough to be Hydrostatic equilibrium, rounded by its own gravity, is not massive enough to ...

has the next largest eccentricity while the smallest orbital eccentricities are seen with Venus
Venus is the second planet from the Sun. It is named after the Venus (mythology), Roman goddess of love and beauty. As List of brightest natural objects in the sky, the brightest natural object in Earth's night sky after the Moon, Venus can ...

and Neptune
Neptune is the eighth and farthest-known Solar planet from the Sun. In the Solar System, it is the fourth-largest planet by diameter, the third-most-massive planet, and the densest giant planet. It is 17 times the mass of Earth, slightly mor ...

.
As two objects orbit each other, the periapsis
File:Periapsis_apoapsis.png, upright=1.15, The two-body system of interacting elliptic orbits: The smaller, satellite body (blue) orbits the primary (astronomy), primary body (yellow); both are in elliptic orbits around their center of mass, co ...

is that point at which the two objects are closest to each other and the apoapsis
upright=1.15, The two-body system of interacting primary body (yellow); both are in elliptic orbits around their center of mass">common center of mass (or barycenter), (red +). ∗Periapsis and apoapsis as distances: The smallest and largest ...

is that point at which they are the farthest. (More specific terms are used for specific bodies. For example, ''perigee'' and ''apogee'' are the lowest and highest parts of an orbit around Earth, while ''perihelion'' and ''aphelion'' are the closest and farthest points of an orbit around the Sun.)
In the case of planets orbiting a star, the mass of the star and all its satellites are calculated to be at a single point called the barycenter. The paths of all the star's satellites are elliptical orbits about that barycenter. Each satellite in that system will have its own elliptical orbit with the barycenter at one focal point of that ellipse. At any point along its orbit, any satellite will have a certain value of kinetic and potential energy with respect to the barycenter, and that energy is a constant value at every point along its orbit. As a result, as a planet approaches periapsis
File:Periapsis_apoapsis.png, upright=1.15, The two-body system of interacting elliptic orbits: The smaller, satellite body (blue) orbits the primary (astronomy), primary body (yellow); both are in elliptic orbits around their center of mass, co ...

, the planet will increase in speed as its potential energy decreases; as a planet approaches apoapsis
upright=1.15, The two-body system of interacting primary body (yellow); both are in elliptic orbits around their center of mass">common center of mass (or barycenter), (red +). ∗Periapsis and apoapsis as distances: The smallest and largest ...

, its velocity will decrease as its potential energy increases.
Understanding orbits

There are a few common ways of understanding orbits: * A force, such as gravity, pulls an object into a curved path as it attempts to fly off in a straight line. * As the object is pulled toward the massive body, it falls toward that body. However, if it has enough tangential velocity it will not fall into the body but will instead continue to follow the curved trajectory caused by that body indefinitely. The object is then said to be orbiting the body. 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
A thought experiment is a hypothetical situation in which a hypothesis
A hypothesis (plural hypotheses) is a proposed explanation for a phenomenon. For a hypothesis to be a scientific hypothesis, the scientific method requires that one can t ...

', in which a cannon on top of a tall mountain is able to fire a cannonball horizontally at any chosen muzzle speed. The effects of air friction on the cannonball are ignored (or perhaps the mountain is high enough that the cannon is above the Earth's atmosphere, which is the same thing).
If the cannon fires its ball with a low initial speed, the trajectory of the ball curves downward and hits the ground (A). As the firing speed 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 speed, 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 speed (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
File:counterintuitive_orbital_mechanics.svg, 250px, At the top of the diagram, a satellite in a clockwise circular orbit (yellow spot) launches objects of negligible mass:(1 - blue) towards Earth,(2 - red) away from Earth,(3 - grey) in the direct ...

, as shown in (C).
As the firing speed is increased beyond this, non-interrupted 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 non-interrupted elliptical orbits at slower firing speed; these will come closest to the Earth at the point half an orbit beyond, and directly opposite the firing point, below the circular orbit.
At a specific horizontal firing speed called escape velocity #REDIRECT Escape velocity#REDIRECT Escape velocity
In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matt ...

, dependent on the mass of the planet and the distance of the object from the barycenter, an open orbit (E) is achieved that has a . At even greater speeds the object will follow a range of hyperbolic trajectories. In a practical sense, both of these trajectory types mean the object is "breaking free" of the planet's gravity, and "going off into space" never to return.
The velocity relationship of two moving objects with mass can thus be considered in four practical classes, with subtypes:
; No orbit:
; Suborbital trajectories: Range of interrupted elliptical paths
; Orbital trajectories (or simply, orbits):
; Open (or escape) trajectories:
It is worth noting that orbital rockets are launched vertically at first to lift the rocket above the atmosphere (which causes frictional drag), and then slowly pitch over and finish firing the rocket engine parallel to the atmosphere to achieve orbit speed.
Once in orbit, their speed keeps them in orbit above the atmosphere. If e.g., an elliptical orbit dips into dense air, the object will lose speed and re-enter (i.e. fall). Occasionally a space craft will intentionally intercept the atmosphere, in an act commonly referred to as an aerobraking maneuver.
Newton's laws of motion

Newton's law of gravitation and laws of motion for two-body problems

In most situations, relativistic effects can be neglected, andNewton's laws
In classical mechanics, Newton's laws of motion are three law
Law is a system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole.
A system, surround ...

give a sufficiently accurate description of motion. The acceleration of a body is equal to the sum of the forces acting on it, divided by its mass, and the gravitational force acting on a body is proportional to the product of the masses of the two attracting bodies and decreases inversely with the square of the distance between them. To this Newtonian approximation, for a system of two-point masses or spherical bodies, only influenced by their mutual gravitation (called a two-body problem
In classical mechanics, the two-body problem is to predict the motion of two massive objects which are abstractly viewed as point particle
A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an ideal ...

), their trajectories can be exactly calculated. If the heavier body is much more massive than the smaller, as in the case of a satellite or small moon orbiting a planet or for the Earth orbiting the Sun, it is accurate enough and convenient to describe the motion in terms of a coordinate system
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

that is centered on the heavier body, and we say that the lighter body is in orbit around the heavier. For the case where the masses of two bodies are comparable, an exact Newtonian solution is still sufficient and can be had by placing the coordinate system at the center of the mass of the system.
Defining gravitational potential energy

Energy is associated withgravitational field
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "P ...

s. A stationary body far from another can do external work if it is pulled towards it, and therefore has gravitational ''potential energy
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.
Common types of potential energy include the gravitational potentia ...

''. Since work is required to separate two bodies against the pull of gravity, their gravitational potential energy increases as they are separated, and decreases as they approach one another. For point masses, the gravitational energy decreases to zero as they approach zero separation. It is convenient and conventional to assign the potential energy as having zero value when they are an infinite distance apart, and hence it has a negative value (since it decreases from zero) for smaller finite distances.
Orbital energies and orbit shapes

When only two gravitational bodies interact, their orbits follow aconic section
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

. The orbit can be open (implying the object never returns) or closed (returning). Which it is depends on the total energy
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regula ...

( kinetic + potential energy
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.
Common types of potential energy include the gravitational potentia ...

) of the system. In the case of an open orbit, the speed at any position of the orbit is at least the escape velocity #REDIRECT Escape velocity#REDIRECT Escape velocity
In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matt ...

for that position, in the case of a closed orbit, the speed is always less than the escape velocity. Since the kinetic energy is never negative if the common convention is adopted of taking the potential energy as zero at infinite separation, the bound orbits will have negative total energy, the parabolic trajectories zero total energy, and hyperbolic orbits positive total energy.
An open orbit will have a parabolic shape if it has the velocity of exactly the escape velocity at that point in its trajectory, and it will have the shape of a hyperbola
File:Hyperbel-def-ass-e.svg, 300px, Hyperbola (red): features
In mathematics, a hyperbola () (adjective form hyperbolic, ) (plural ''hyperbolas'', or ''hyperbolae'' ()) is a type of smooth function, smooth plane curve, curve lying in a plane, defi ...

when its velocity is greater than the escape velocity. When bodies with escape velocity or greater approach each other, they will briefly curve around each other at the time of their closest approach, and then separate, forever.
All closed orbits have the shape of an ellipse
In , an ellipse is a surrounding two , such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a , which is the special type of ellipse in which the two focal points are t ...

. A circular orbit is a special case, wherein the foci of the ellipse coincide. The point where the orbiting body is closest to Earth is called the perigee
Apsis ( el, ἀψίς; plural apsides , Greek: ἀψῖδες; "orbit") denotes either of the two extreme points (i.e., the farthest or nearest point) in the orbit
In celestial mechanics, an orbit is the curved trajectory of an phys ...

, and is called the periapsis (less properly, "perifocus" or "pericentron") when the orbit is about a body other than Earth. The point where the satellite is farthest from Earth is called the apogee
Apsis ( el, ἀψίς; plural apsides , Greek: ἀψῖδες; "orbit") denotes either of the two extreme points (i.e., the farthest or nearest point) in the of a about its (or simply, "the primary"). The plural term, "apsides," usuall ...

, apoapsis, or sometimes apifocus or apocentron. A line drawn from periapsis to apoapsis is the line-of-apsides. This is the major axis of the ellipse, the line through its longest part.
Kepler's laws

Bodies following closed orbits repeat their paths with a certain time called the period. This motion is described by the empirical laws of Kepler, which can be mathematically derived from Newton's laws. These can be formulated as follows: # The orbit of a planet around theSun
The Sun is the star
A star is an astronomical object consisting of a luminous spheroid of plasma (physics), plasma held together by its own gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many othe ...

is an ellipse, with the Sun in one of the focal points of that ellipse. his_focal_point_is_actually_the_barycenter_of_the_Solar_System.html" ;"title="barycenter.html" ;"title="his focal point is actually the barycenter">his focal point is actually the barycenter of the Solar System">Sun-planet system; for simplicity, this explanation assumes the Sun's mass is infinitely larger than that planet's.] The planet's orbit lies in a plane, called the orbital plane
The orbital plane of a revolving body is the geometric plane in which its orbit
In physics, an orbit is the gravitationally curved trajectory of an physical body, object, such as the trajectory of a planet around a star or a natural satel ...

. The point on the orbit closest to the attracting body is the periapsis. The point farthest from the attracting body is called the apoapsis. There are also specific terms for orbits about particular bodies; things orbiting the Sun have a perihelion
upright=1.15, The two-body system of interacting primary body (yellow); both are in elliptic orbits around their center of mass">common center of mass (or barycenter), (red +). ∗Periapsis and apoapsis as distances: The smallest and largest ...

and aphelion
upright=1.15, The two-body system of interacting primary body
A primary (also called a gravitational primary, primary body, or central body) is the main physical body of a gravity, gravitationally bound, multi-object system. This object consti ...

, things orbiting the Earth have a perigee
Apsis ( el, ἀψίς; plural apsides , Greek: ἀψῖδες; "orbit") denotes either of the two extreme points (i.e., the farthest or nearest point) in the orbit
In celestial mechanics, an orbit is the curved trajectory of an phys ...

and apogee
Apsis ( el, ἀψίς; plural apsides , Greek: ἀψῖδες; "orbit") denotes either of the two extreme points (i.e., the farthest or nearest point) in the of a about its (or simply, "the primary"). The plural term, "apsides," usuall ...

, and things orbiting the Moon
The Moon is Earth's only natural satellite. At about one-quarter the diameter of Earth (comparable to the width of Australia (continent), Australia), it is the largest natural satellite in the Solar System relative to the size of its plane ...

have a perilune
upright=1.15, The two-body system of interacting primary body (yellow); both are in elliptic orbits around their center of mass">common center of mass (or barycenter), (red +). ∗Periapsis and apoapsis as distances: The smallest and largest ...

and apolune
Apsis ( el, ἀψίς; plural apsides , Greek: ἀψῖδες; "orbit") denotes either of the two extreme points (i.e., the farthest or nearest point) in the orbit of a planetary body about its primary (astronomy), primary body (or simply ...

(or periselene and aposelene respectively). An orbit around any star
A star is an astronomical object consisting of a luminous spheroid of plasma
Plasma or plasm may refer to:
Science
* Plasma (physics), one of the four fundamental states of matter
* Plasma (mineral) or heliotrope, a mineral aggregate
* Quark ...

, not just the Sun, has a periastron
upright=1.15, The two-body system of interacting primary body (yellow); both are in elliptic orbits around their center of mass">common center of mass (or barycenter), (red +). ∗Periapsis and apoapsis as distances: The smallest and largest ...

and an apastron
File:Periapsis_apoapsis.png, upright=1.15, The two-body system of interacting elliptic orbits: The smaller, satellite body (blue) orbits the primary (astronomy), primary body (yellow); both are in elliptic orbits around their center of mass, co ...

.
# As the planet moves in its orbit, the line from the Sun to the planet sweeps a constant area of the orbital plane
The orbital plane of a revolving body is the geometric plane in which its orbit
In physics, an orbit is the gravitationally curved trajectory of an physical body, object, such as the trajectory of a planet around a star or a natural satel ...

for a given period of time, regardless of which part of its orbit the planet traces during that period of time. This means that the planet moves faster near its perihelion
upright=1.15, The two-body system of interacting primary body (yellow); both are in elliptic orbits around their center of mass">common center of mass (or barycenter), (red +). ∗Periapsis and apoapsis as distances: The smallest and largest ...

than near its aphelion
upright=1.15, The two-body system of interacting primary body
A primary (also called a gravitational primary, primary body, or central body) is the main physical body of a gravity, gravitationally bound, multi-object system. This object consti ...

, because at the smaller distance it needs to trace a greater arc to cover the same area. This law is usually stated as "equal areas in equal time."
# For a given orbit, the ratio of the cube of its semi-major axis
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...

to the square of its period is constant.
Limitations of Newton's law of gravitation

Note that while bound orbits of a point mass or a spherical body with a Newtonian gravitational field are closedellipse
In , an ellipse is a surrounding two , such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a , which is the special type of ellipse in which the two focal points are t ...

s, which repeat the same path exactly and indefinitely, any non-spherical or non-Newtonian effects (such as caused by the slight oblateness of the Earth
Earth is the third planet from the Sun and the only astronomical object known to harbour and support life. 29.2% of Earth's surface is land consisting of continents and islands. The remaining 70.8% is Water distribution on Earth, covered wi ...

, or by relativistic effects
Relativistic quantum chemistry combines relativistic mechanics
In physics, relativistic mechanics refers to mechanics compatible with special relativity (SR) and general relativity (GR). It provides a non-quantum mechanics, quantum mechanical desc ...

, thereby changing the gravitational field's behavior with distance) will cause the orbit's shape to depart from the closed ellipse
In , an ellipse is a surrounding two , such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a , which is the special type of ellipse in which the two focal points are t ...

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
In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through S ...

; however, it converges too slowly to be of much use. Except for special cases like the Lagrangian point
In celestial mechanics, the Lagrange points (also Lagrangian points, L-points, or libration points) are points of equilibrium for small-mass objects under the influence of two massive orbit, orbiting bodies. Mathematically, this involves th ...

s, no method is known to solve the equations of motion for a system with four or more bodies.
Approaches to many-body problems

Rather than an exact closed form solution, orbits with many bodies can be approximated with arbitrarily high accuracy. These approximations take two forms: :One form takes the pure elliptic motion as a basis and addsperturbation
Perturbation or perturb may refer to:
* Perturbation theory, mathematical methods that give approximate solutions to problems that cannot be solved exactly
* Perturbation (geology), changes in the nature of alluvial deposits over time
* Perturbation ...

terms to account for the gravitational influence of multiple bodies. This is convenient for calculating the positions of astronomical bodies. The equations of motion of the moons, planets, and other bodies are known with great accuracy, and are used to generate tables
Table may refer to:
* Table (information)
A table is an arrangement of information or data, typically in rows and columns, or possibly in a more complex structure. Tables are widely used in communication, research, and data analysis. Tables ap ...

for celestial navigation
Celestial navigation, also known as astronavigation, is the ancient and modern practice of position fixing that enables a navigator to transition through a space without having to rely on estimated calculations, or dead reckoning, to know their p ...

. Still, there are secular phenomena that have to be dealt with by post-Newtonian methods.
:The differential equation
In mathematics, a differential equation is an functional equation, equation that relates one or more function (mathematics), functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives ...

form is used for scientific or mission-planning purposes. According to Newton's laws, the sum of all the forces acting on a body will equal the mass of the body times its acceleration (''F = ma''). Therefore accelerations can be expressed in terms of positions. The perturbation terms are much easier to describe in this form. Predicting subsequent positions and velocities from initial values of position and velocity corresponds to solving an initial value problem
In multivariable calculus
Multivariable calculus (also known as multivariate calculus) is the extension of calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal
In mathematics, infinitesimals or infi ...

. Numerical methods calculate the positions and velocities of the objects a short time in the future, then repeat the calculation ad nauseam. 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 assemblages of objects have been simulated.
Newtonian analysis of orbital motion

The following derivation applies to such an elliptical orbit. We start only with theNewtonianNewtonian refers to the work of Isaac Newton, in particular:
* Newtonian mechanics, i.e. classical mechanics
* Newtonian telescope, a type of reflecting telescope
* Newtonian cosmology
* Newtonian dynamics
* Newtonianism, the philosophical principle ...

law of gravitation stating that the gravitational acceleration towards the central body is related to the inverse of the square of the distance between them, namely
: $F\_2\; =\; -\backslash frac$
where ''F''standard gravitational parameter
In celestial mechanics, the standard gravitational parameter ''μ'' of a celestial body is the product of the gravitational constant ''G'' and the mass ''M'' of the body.
:\mu=GM \
For several objects in the Solar System
The Solar Syste ...

, in this case $G\; m\_1$. It is understood that the system being described is ''m''general relativity
General relativity, also known as the general theory of relativity, is the geometric
Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...

.
When a pendulum or an object attached to a spring swings in an ellipse, the inward acceleration/force is proportional to the distance $A\; =\; F/m\; =\; -\; k\; r.$ Due to the way vectors add, the component of the force in the $\backslash hat$ or in the $\backslash hat$ directions are also proportionate to the respective components of the distances, $r\text{'}\text{'}\_x\; =\; A\_x\; =\; -\; k\; r\_x$. Hence, the entire analysis can be done separately in these dimensions. This results in the harmonic parabolic equations $x\; =\; A\; \backslash cos(t)$ and $y\; =\; B\; \backslash sin(t)$ of the ellipse. In contrast, with the decreasing relationship $A\; =\; \backslash mu/r^2$, the dimensions cannot be separated.
The location of the orbiting object at the current time $t$ is located in the plane using vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Product differentiation, in ...

in polar coordinates
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

both with the standard Euclidean basis and with the polar basis with the origin coinciding with the center of force. Let $r$ be the distance between the object and the center and $\backslash theta$ be the angle it has rotated. Let $\backslash hat$ and $\backslash hat$ be the standard Euclidean
Euclidean (or, less commonly, Euclidian) is an adjective derived from the name of Euclid, an ancient Greek mathematician. It is the name of:
Geometry
*Euclidean space, the two-dimensional plane and three-dimensional space of Euclidean geometry a ...

bases and let $\backslash hat\; =\; \backslash cos(\backslash theta)\backslash hat\; +\; \backslash sin(\backslash theta)\backslash hat$ and $\backslash hat\; =\; -\; \backslash sin(\backslash theta)\backslash hat\; +\; \backslash cos(\backslash theta)\backslash hat$ be the radial and transverse polar
Polar may refer to:
Geography
Polar may refer to:
* Geographical pole, either of two fixed points on the surface of a rotating body or planet, at 90 degrees from the equator, based on the axis around which a body rotates
*Polar climate, the clim ...

basis with the first being the unit vector pointing from the central body to the current location of the orbiting object and the second being the orthogonal unit vector pointing in the direction that the orbiting object would travel if orbiting in a counter clockwise circle. Then the vector to the orbiting object is
: $\backslash hat\; =\; r\; \backslash cos(\backslash theta)\backslash hat\; +\; r\; \backslash sin(\backslash theta)\backslash hat\; =\; r\; \backslash hat$
We use $\backslash dot\; r$ and $\backslash dot\; \backslash theta$ to denote the standard derivatives of how this distance and angle change over time. We take the derivative of a vector to see how it changes over time by subtracting its location at time $t$ from that at time $t\; +\; \backslash delta\; t$ and dividing by $\backslash delta\; t$. The result is also a vector. Because our basis vector $\backslash hat$ moves as the object orbits, we start by differentiating it. From time $t$ to $t\; +\; \backslash delta\; t$, the vector $\backslash hat$ keeps its beginning at the origin and rotates from angle $\backslash theta$ to $\backslash theta\; +\; \backslash dot\; \backslash theta\backslash \; \backslash delta\; t$ which moves its head a distance $\backslash dot\; \backslash theta\backslash \; \backslash delta\; t$ in the perpendicular direction $\backslash hat$ giving a derivative of $\backslash dot\; \backslash theta\; \backslash hat$.
: $\backslash begin\; \backslash hat\; \&=\; \backslash cos(\backslash theta)\backslash hat\; +\; \backslash sin(\backslash theta)\backslash hat\; \backslash \backslash \; \backslash frac\; =\; \backslash dot\; \&=\; -\backslash sin(\backslash theta)\backslash dot\; \backslash theta\; \backslash hat\; +\; \backslash cos(\backslash theta)\backslash dot\; \backslash theta\; \backslash hat\; =\; \backslash dot\; \backslash theta\; \backslash hat\; \backslash \backslash \; \backslash hat\; \&=\; -\backslash sin(\backslash theta)\backslash hat\; +\; \backslash cos(\backslash theta)\backslash hat\; \backslash \backslash \; \backslash frac\; =\; \backslash dot\; \&=\; -\backslash cos(\backslash theta)\backslash dot\; \backslash theta\; \backslash hat\; -\; \backslash sin(\backslash theta)\; \backslash dot\; \backslash theta\; \backslash hat\; =\; -\backslash dot\; \backslash theta\; \backslash hat\; \backslash end$
We can now find the velocity and acceleration of our orbiting object.
: $\backslash begin\; \backslash hat\; \&=\; r\; \backslash hat\; \backslash \backslash \; \backslash dot\; \&=\; \backslash frac\; \backslash hat\; +\; r\; \backslash frac\; =\; \backslash dot\; r\; \backslash hat\; +\; r\; \backslash left;\; href="/html/ALL/s/\backslash dot\_\backslash theta\_\backslash hat\_\_\backslash right.html"\; ;"title="\backslash dot\; \backslash theta\; \backslash hat\; \backslash right">\backslash dot\; \backslash theta\; \backslash hat\; \backslash right$
The coefficients of $\backslash hat$ and $\backslash hat$ give the accelerations in the radial and transverse directions. As said, Newton gives this first due to gravity is $-\backslash mu/r^2$ and the second is zero.
Equation (2) can be rearranged using integration by parts.
: $r\; \backslash ddot\backslash theta\; +\; 2\; \backslash dot\; r\; \backslash dot\backslash theta\; =\; \backslash frac\backslash frac\backslash left(\; r^2\; \backslash dot\; \backslash theta\; \backslash right)\; =\; 0$
We can multiply through by $r$ because it is not zero unless the orbiting object crashes.
Then having the derivative be zero gives that the function is a constant.
which is actually the theoretical proof of Kepler's second law (A line joining a planet and the Sun sweeps out equal areas during equal intervals of time). The constant of integration, ''h'', is the specific relative angular momentum, angular momentum per unit mass.
In order to get an equation for the orbit from equation (1), we need to eliminate time. (See also Binet equation.)
In polar coordinates, this would express the distance $r$ of the orbiting object from the center as a function of its angle $\backslash theta$. However, it is easier to introduce the auxiliary variable $u\; =\; 1/r$ and to express $u$ as a function of $\backslash theta$. Derivatives of $r$ with respect to time may be rewritten as derivatives of $u$ with respect to angle.
: $u\; =$
: $\backslash dot\backslash theta\; =\; \backslash frac\; =\; hu^2$ (reworking (3))
: $\backslash begin\; \backslash frac\; \&=\; \backslash frac\backslash left(\backslash frac\backslash right)\backslash frac\; =\; -\backslash frac\; =\; -\backslash frac\; \backslash \backslash \; \backslash frac\; \&=\; -\backslash frac\backslash frac\backslash frac\; =\; -\backslash frac\; =\; -\backslash frac\; \backslash \; \backslash \; \backslash \; \backslash text\; \backslash \; \backslash \; \backslash \; \backslash ddot\; r\; =\; -\; h^2\; u^2\; \backslash frac\; \backslash end$
Plugging these into (1) gives
: $\backslash begin\; \backslash ddot\; r\; -\; r\backslash dot\backslash theta^2\; \&=\; -\backslash frac\; \backslash \backslash \; -h^2\; u^2\; \backslash frac\; -\; \backslash frac\; \backslash left(h\; u^2\backslash right)^2\; \&=\; -\backslash mu\; u^2\; \backslash end$
So for the gravitational force – or, more generally, for ''any'' inverse square force law – the right hand side of the equation becomes a constant and the equation is seen to be the harmonic oscillator, harmonic equation (up to a shift of origin of the dependent variable). The solution is:
:$u(\backslash theta)\; =\; \backslash frac\backslash mu\; -\; A\; \backslash cos(\backslash theta\; -\; \backslash theta\_0)$
where ''A'' and ''θ''Relativistic orbital motion

The above classical (NewtonianNewtonian refers to the work of Isaac Newton, in particular:
* Newtonian mechanics, i.e. classical mechanics
* Newtonian telescope, a type of reflecting telescope
* Newtonian cosmology
* Newtonian dynamics
* Newtonianism, the philosophical principle ...

) analysis of orbital mechanics assumes that the more subtle effects of general relativity
General relativity, also known as the general theory of relativity, is the geometric
Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...

, such as frame dragging and gravitational time dilation are negligible. Relativistic effects cease to be negligible when near very massive bodies (as with the Kepler problem in general relativity, 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 Global Positioning System#Relativity, GPS satellites.).
Orbital planes

The analysis so far has been two dimensional; it turns out that an perturbation theory, unperturbed orbit is two-dimensional in a plane fixed in space, and thus the extension to three dimensions requires simply rotating the two-dimensional plane into the required angle relative to the poles of the planetary body involved. The rotation to do this in three dimensions requires three numbers to uniquely determine; traditionally these are expressed as three angles.Orbital period

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

Six parameters are required to specify a Keplerian orbit about a body. For example, the three numbers that specify the body's initial position, and the three values that specify its velocity will define a unique orbit that can be calculated forwards (or backwards) in time. However, traditionally the parameters used are slightly different. The traditionally used set of orbital elements is called the set of Orbital elements, Keplerian elements, after Johannes Kepler and his laws. The Keplerian elements are six: * Inclination (''i'') * Longitude of the ascending node (Ω) * Argument of periapsis (ω) * orbital eccentricity, Eccentricity (''e'') * Semimajor axis (''a'') * Mean anomaly atepoch
In chronology
222px, Joseph Scaliger's ''De emendatione temporum'' (1583) began the modern science of chronology
Chronology (from Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-E ...

(''M''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 Eccentricity (mathematics), eccentricity, but not the orbital period (to first order). A Direct motion, prograde or Retrograde motion, retrograde impulse (i.e. an impulse applied along the orbital motion) changes both the eccentricity and the orbital period. Notably, a prograde impulse atperiapsis
File:Periapsis_apoapsis.png, upright=1.15, The two-body system of interacting elliptic orbits: The smaller, satellite body (blue) orbits the primary (astronomy), primary body (yellow); both are in elliptic orbits around their center of mass, co ...

raises the altitude at apoapsis
upright=1.15, The two-body system of interacting primary body (yellow); both are in elliptic orbits around their center of mass">common center of mass (or barycenter), (red +). ∗Periapsis and apoapsis as distances: The smallest and largest ...

, and vice versa and a retrograde impulse does the opposite. A transverse impulse (out of the orbital plane) causes rotation of the orbital plane
The orbital plane of a revolving body is the geometric plane in which its orbit
In physics, an orbit is the gravitationally curved trajectory of an physical body, object, such as the trajectory of a planet around a star or a natural satel ...

without changing the Orbit (dynamics), period 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 a significant atmosphere, its orbit can decay because of drag (physics), drag. Particularly at eachOblateness

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 Quadropole#Gravitational quadrupole, quadrupole moment to the gravitational field which is significant at distances comparable to the radius of the body. In the general case, the gravitational potential of a rotating body such as, e.g., a planet is usually expanded in multipoles accounting for the departures of it from spherical symmetry. From the point of view of satellite dynamics, of particular relevance are the so-called even zonal harmonic coefficients, or even zonals, since they induce secular orbital perturbations which are cumulative over time spans longer than the orbital period. They do depend on the orientation of the body's symmetry axis in the space, affecting, in general, the whole orbit, with the exception of the semimajor axis.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. One approximate result is that bodies will usually have reasonably stable orbits around a heavier planet or moon, in spite of these perturbations, provided they are orbiting well within the heavier body's Hill sphere. When there are more than two gravitating bodies it is referred to as an n-body problem. Most n-body problems have no closed form solution, although some special cases have been formulated.Light radiation and stellar wind

For smaller bodies particularly, light and stellar wind can cause significant perturbations to the attitude (geometry), attitude and direction of motion of the body, and over time can be significant. Of the planetary bodies, the motion ofasteroid
An asteroid is a minor planet of the Solar System#Inner solar system, inner Solar System. Historically, these terms have been applied to any astronomical object orbiting the Sun that did not resolve into a disc in a telescope and was not observ ...

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

Mathematicians have discovered that it is possible in principle to have multiple bodies in non-elliptical orbits that repeat periodically, although most such orbits are not stable regarding small perturbations in mass, position, or velocity. However, some special stable cases have been identified, including a planar figure-eight orbit occupied by Three-body problem, three moving bodies. Further studies have discovered that nonplanar orbits are also possible, including one involving 12 masses moving in 4 roughly circular, interlocking orbits topology, topologically equivalent to the edges of a cuboctahedron. Finding such orbits naturally occurring in the universe is thought to be extremely unlikely, because of the improbability of the required conditions occurring by chance.Astrodynamics

Orbital mechanics or astrodynamics is the application of ballistics andcelestial mechanics
Celestial mechanics is the branch of astronomy
Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial objects and cel ...

to the practical problems concerning the motion of rockets and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and Newton's law of universal gravitation. It is a core discipline within space mission design and control. Celestial mechanics treats more broadly the orbital dynamics of systems under the influence of gravity, including spacecraft and natural astronomical bodies such as star systems, planet
A planet is an astronomical body orbiting a star or Stellar evolution#Stellar remnants, stellar remnant that is massive enough to be Hydrostatic equilibrium, rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and ...

s, Natural satellite, moons, and comet
A comet is an icy, small Solar System body
A small Solar System body (SSSB) is an object in the Solar System
The Solar SystemCapitalization of the name varies. The International Astronomical Union, the authoritative body regarding astr ...

s. Orbital mechanics focuses on spacecraft trajectory, trajectories, including orbital maneuvers, orbit plane changes, and interplanetary transfers, and is used by mission planners to predict the results of Spacecraft propulsion, propulsive maneuvers. General relativity is a more exact theory than Newton's laws for calculating orbits, and is sometimes necessary for greater accuracy or in high-gravity situations (such as orbits close to the Sun).
Earth orbits

* Low Earth orbit (LEO): Geocentric orbits with altitudes up to 2,000 km (0–1,240 miles). * Medium Earth orbit (MEO): Geocentric orbits ranging in altitude from 2,000 km (1,240 miles) to just below geosynchronous orbit at . Also known as an intermediate circular orbit. These are "most commonly at , or , with an orbital period of 12 hours." * Both geosynchronous orbit (GSO) and geostationary orbit (GEO) are orbits around Earth matching Earth's sidereal rotation period. All geosynchronous and geostationary orbits have asemi-major axis
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...

of . All geostationary orbits are also geosynchronous, but not all geosynchronous orbits are geostationary. A geostationary orbit stays exactly above the equator, whereas a geosynchronous orbit may swing north and south to cover more of the Earth's surface. Both complete one full orbit of Earth per sidereal day (relative to the stars, not the Sun).
* High Earth orbit: Geocentric orbits above the altitude of geosynchronous orbit 35,786 km (22,240 miles).
Scaling in gravity

The gravitational constant ''G'' has been calculated as: * (6.6742 ± 0.001) × 10semi-major axis
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...

''a'', of a small body around a spherical body with radius ''r'' and average density ''ρ'', where ''T'' is the orbital period. See also Kepler's Third Law.
Patents

The application of certain orbits or orbital maneuvers to specific useful purposes have been the subject of patents.Tidal locking

Some bodies are tidally locked with other bodies, meaning that one side of the celestial body is permanently facing its host object. This is the case for Earth-Moon
The Moon is Earth's only natural satellite. At about one-quarter the diameter of Earth (comparable to the width of Australia (continent), Australia), it is the largest natural satellite in the Solar System relative to the size of its plane ...

and Pluto-Charon system.
See also

* Ephemeris is a compilation of positions of naturally occurring astronomical objects as well as artificial satellites in the sky at a given time or times. * Free drift * Klemperer rosette * List of orbits * Molniya orbit * Orbit determination * Orbital spaceflight * Perifocal coordinate system * Polar Orbits * Radial trajectory * Rosetta (orbit) * VSOP (planets)Notes

References

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.External links

CalcTool: Orbital period of a planet calculator

Has wide choice of units. Requires JavaScript.

Requires Java.

includes (calculated) data on Earth orbit variations over the last 50 million years and for the coming 20 million years

Requires JavaScript.

(Rocket and Space Technology)

Orbital simulations by Varadi, Michael Ghil, 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 th

eccentricity data for Earth and Mercury

are available online.

Understand orbits using direct manipulation

Requires JavaScript and Macromedia * {{Authority control Orbits, Celestial mechanics Periodic phenomena Gravity Astrodynamics Concepts in astronomy