Orbit 22498 226048801 2008-07-16T16:29:02Z 82.181.74.192 /* Understanding orbits */ {{otheruses1|orbits in celestial mechanics, due to gravity}} [[Image:orbit2.gif|thumb|200px|Two bodies with a slight difference in [[mass]] orbiting around a common [[barycenter]]. The sizes, and this particular type of orbit are similar to the [[Pluto]]-[[Charon (moon)|Charon]] system.]] In [[physics]], an orbit is the gravitationally curved path of one object around a point or another body, for example the gravitational orbit of a planet around a star.<ref>[http://www2.jpl.nasa.gov/basics/bsf3-1.html]</ref> Historically, orbits were first understood in terms of [[epicycles]], which are the sums of numerous circular motions. This predicted the path of the planets quite well, until [[Johannes Kepler]] was able to show that the motion of the planets were in fact elliptical motions. [[Sir Isaac Newton]] was able to prove that this was equivalent to an inverse square, instantaneously propagating force he called [[gravitation]]. [[Albert Einstein]] later was able to show that gravity is due to curvature of space-time, and that orbits lie upon [[geodesic]]s and this is the current understanding. ==History== In the [[geocentric model]] of the solar system, mechanisms such as the [[deferent and epicycle]] were supposed to explain the motion of the planets in terms of perfect spheres or rings. The basis for the modern understanding of orbits was first formulated by [[Johannes Kepler]] whose results are summarized in his three [[Kepler's laws of planetary motion|laws of planetary motion]]. First, he found that the orbits of the [[planet]]s in our [[solar system]] are [[ellipse|elliptical]], not [[circle|circular]] (or [[epicycle|epicyclic]]), as had previously been believed, and that the sun is not located at the center of the orbits, but rather at one [[focus (geometry)|focus]].<ref name="Kepler's Laws of Planetary Motion">{{cite web|url=http://physics.about.com/od/astronomy/p/keplerlaws.htm|title=Kepler's Laws of Planetary Motion|last=Jones|first=Andrew|publisher=[[about.com]]|language=en|accessdate=2008-06-01}}</ref> Second, he found that the orbital speed of each planet is not constant, as had previously been thought, but rather that the speed of the planet depends on the planet's distance from the sun. And third, Kepler found a universal relationship between the orbital properties of all the planets orbiting the sun. For each planet, the cube of the planet's distance from the sun, measured in [[astronomical units]] (AU), is equal to the square of the planet's orbital period, measured in Earth years. Jupiter, for example, is approximately 5.2 AU from the sun and its orbital period is 11.86 Earth years. So 5.2 cubed equals 11.86 squared, as predicted. [[Isaac Newton]] demonstrated that Kepler's laws were derivable from his theory of [[gravitation]] and that, in general, the orbits of bodies responding to an instantaneously propagating force of gravity were [[conic section]]s. Newton showed that a pair of bodies follow orbits of dimensions that are in inverse proportion to their [[mass]]es 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 was able to remove the assumption of Newton that changes propagate instantaneously. In [[relativity theory]] orbits follow [[geodesic]] trajectories which approximate very well to the Newtonian predictions. However there are differences and these can be used to determine which theory relativity agrees with. Essentially all experimental evidence agrees with relativity theory to within experimental measuremental accuracy. == Planetary orbits == Within a [[planetary system]]; [[planet]]s, [[dwarf planet]]s, [[asteroid]]s (a.k.a. minor planets), [[comet]]s, and [[space debris]] orbit the central [[star]] in [[elliptical orbit]]s. A comet in a [[Parabolic trajectory|parabolic]] or [[Hyperbolic trajectory|hyperbolic]] orbit about a central star is not gravitationally bound to the star and therefore is not considered part of the star's planetary system. To date, no comet has been observed in our [[solar system]] with a distinctly hyperbolic orbit. Bodies which are gravitationally bound to one of the planets in a planetary system, either [[natural satellite|natural]] or [[satellite|artificial satellite]]s, follow orbits about that planet. Owing to mutual [[Perturbation (astronomy)|gravitational perturbations]], the [[eccentricity (orbit)|eccentricities]] of the orbits of the planets in our solar system vary over time. [[Mercury (planet)|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 eccentricities are those of the orbits of [[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 Earth orbit, respectively.) In the elliptical orbit, the [[center of mass]] of the orbiting-orbited system will sit at one [[focus (geometry)|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, the planet will decrease in velocity. See also: *[[Kepler's laws of planetary motion]] *[[Secular variations of the planetary orbits]] === 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]] model may prove useful (see image below). Imagine a cannon sitting on top of a tall mountain, which fires a cannonball horizontally. The mountain needs to be very tall, so that the cannon will be above the Earth's atmosphere and the effects of air friction on the cannonball can be ignored. [[Image:Newton Cannon.svg|thumb|300px|The Newton Cannon, an illustration of how objects can "fall" in a curve.]] If the cannon fires its ball with a low initial velocity, the trajectory of the ball curves downward and hits the ground (A). As the firing velocity is increased, the cannonball hits the ground farther (B) away from the cannon, because while the ball is still falling towards the ground, the ground is increasingly curving away from it (see first point, above). All these motions are actually "orbits" in a technical sense — they are describing a portion of an [[elliptical path]] around the center of gravity — but the orbits are interrupted by striking the Earth. If the cannonball is fired with sufficient velocity, the ground curves away from the ball at least as much as the ball falls — so the ball never strikes the ground. It is now in what could be called a non-interrupted, or circumnavigating, orbit. For any specific combination of height above the center of gravity, and mass of the planet, there is one specific firing velocity that produces a [[circular orbit]], as shown in (C). As the firing velocity is increased beyond this, a range of [[elliptic orbit]]s 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 infinite orbit such as (E) is produced — a [[parabolic trajectory]]. At even faster velocities the object will follow a range of [[hyperbolic trajectory|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 objects with mass can thus be considered in four practical classes, with subtypes: # '''No orbit''' # '''Interrupted orbits''' #* Range of interrupted elliptical paths # '''Circumnavigating orbits''' #* Range of elliptical paths with closest point opposite firing point #* Circular path #* Range of elliptical paths with closest point at firing point # '''Infinite orbits''' #* 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. Then the acceleration of each body is equal to the sum of the gravitational forces on it, divided by its mass, and the gravitational force between each pair of bodies is proportional to the product of their masses and decreases inversely with the square of the distance between them. To this Newtonian approximation, for a system of two point masses or spherical bodies, only influenced by their mutual gravitation (the [[two-body problem]]), the orbits can be exactly calculated. If the heavier body is much more massive than the smaller, as for a satellite or small moon orbiting a planet or for the Earth orbiting the Sun, it is accurate and convenient to describe the motion in a [[coordinate system]] that is centered on the heavier body, and we can 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 massive bodies against the pull of gravity, their gravitational potential energy increases as they are separated, and decreases as they approach one another. For point masses the gravitational energy decreases without limit as they approach zero separation, and it is convenient and conventional to take the potential energy as zero when they are an infinite distance apart, and then negative (since it decreases from zero) for smaller finite distances. With two bodies, an orbit is a [[conic section]]. The orbit can be open (so the object never returns) or closed (returning), depending on the total [[kinetic energy|kinetic]] + [[potential energy|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 [[comet]]s if they come from outside the solar system. A closed orbit has the shape of an [[ellipse]]. In the special case that the orbiting body is always the same distance from the center, it is also the shape of a [[circle]]. Otherwise, the point where the orbiting body is closest to Earth is the [[perigee]], called periapsis (less properly, "perifocus" or "pericentron") when the orbit is around a body other than Earth. The point where the satellite is farthest from Earth is called [[apogee]], apoapsis, or sometimes apifocus or apocentron. A line drawn from periapsis to apoapsis is the '''line-of-apsides'''. This is the major axis of the ellipse, the line through its longest part. Orbiting bodies in closed orbits repeat their path after a constant period of time. This motion is described by the empirical laws of [[Johannes Kepler|Kepler]], which can be mathematically derived from Newton's laws. These can be formulated as follows: # The orbit of a planet around the Sun is an ellipse, with the Sun in one of the focal points of the ellipse. Therefore the orbit lies in a plane, called the '''orbital plane'''. The point on the orbit closest to the attracting body is the periapsis. The point farthest from the attracting body is called the apoapsis. There are also specific terms for orbits around particular bodies; things orbiting the Sun have a perihelion and aphelion, things orbiting the Earth have a perigee and apogee, and things orbiting the [[Moon]] have a perilune and apolune (or, synonymously, periselene and aposelene). An orbit around any [[star]], 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 Sun to planet sweeps a constant area of the orbital plane, regardless of which part of its orbit the planet traces during that period of time. This means that the planet moves faster near its perihelion than near its aphelion, because at the smaller distance it needs to trace a greater arc to cover the same area. This law is usually stated as "equal areas in equal time." # For a given orbit, the ratio of the cube of its semi-major axis to the square of its period is constant. Note that that while the bound orbits around a point mass, or a spherical body with an ideal Newtonian gravitational field, are all closed ellipses, which repeat the same path exactly and indefinitely, any non-spherical or non-Newtonian effects (as caused, for example, by the slight oblateness of the Earth, or by [[Theory of relativity|relativistic effect]]s, changing the gravitational field's behavior with distance) will cause the orbit's shape to depart to a greater or lesser extent from the closed ellipses characteristic of Newtonian two body motion. The 2-body solutions were published by Newton in [[Philosophiae Naturalis Principia Mathematica|Principia]] in 1687. In 1912, [[Karl Fritiof Sundman]] developed a converging infinite series that solves the 3-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. One form takes the pure elliptic motion as a basis, and adds [[perturbation (astronomy)|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 moon, planets and other bodies are known with great accuracy, and are used to generate [[ephemeris|tables]] for [[celestial navigation]]. Still there are [[secular phenomena]] that have to be dealt with by [[Parameterized post-Newtonian formalism|post-newtonian]] methods. The [[differential equation]] form is used for scientific or mission-planning purposes. According to Newton's laws, the sum of all the forces will equal the mass times its acceleration (''F = ma''). Therefore accelerations can be expressed in terms of positions. The perturbation terms are much easier to describe in this form. Predicting subsequent positions and velocities from initial ones corresponds to solving an [[initial value problem]]. Numerical methods calculate the positions and velocities of the objects a tiny time in the future, then repeat this. However, tiny arithmetic errors from the limited accuracy of a computer's math accumulate, limiting 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]], [[orbit equation]] and [[Kepler's laws of planetary motion#Kepler.27s first law|Kepler's first law]].'') Please note that the following is a classical ([[Newtonian]]) analysis of orbital mechanics, which assumes the more subtle effects of [[general relativity]] (like [[frame dragging]] and [[gravitational time dilation]]) are negligible. General relativity does, however, need to be considered for some applications such as analysis of extremely massive heavenly bodies, precise prediction of a system's state after a long period of time, and in the case of interplanetary travel, where fuel economy, and thus precision, is paramount. 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: :<math>a_r = \frac{d^2r}{dt^2} - r\left( \frac{d\theta}{dt} \right)^2</math> and :<math>a_{\theta} = \frac{1}{r}\frac{d}{dt}\left( r^2\frac{d\theta}{dt} \right)</math>. Since the force is entirely radial, and since acceleration is proportional to force, it follows that the transverse acceleration is zero. As a result, :<math> \frac{d}{dt}\left( r^2\frac{d\theta}{dt} \right) = 0</math>. After integrating, we have :<math> r^2\frac{d\theta}{dt} = {\rm const.} </math> which is actually the theoretical proof of [[Kepler's laws of planetary motion#Kepler.27s second law|Kepler's 2nd 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]]. It then follows that :<math>\frac{d\theta}{dt} = { h \over r^2 } = hu^2</math> where we have introduced the auxiliary variable :<math> u = { 1 \over r } </math>. The radial force is ''f(r)'' per unit is <math>a_r</math>, then the elimination of the time variable from the radial component of the equation of motion yields: :<math>\frac{d^2u}{d\theta^2} + u = -\frac{f(1 / u)}{h^2u^2}</math>. In the case of [[gravity]], Newton's [[Gravity#Newton's law of universal gravitation|law of universal gravitation]] states that the force is proportional to the inverse square of the distance: :<math>f(1/u) = a_r = { -GM \over r^2 } = -GM u^2 </math> where ''G'' is the [[gravitational constant|constant of universal gravitation]], ''m'' is the mass of the orbiting body (planet), and ''M'' is the mass of the central body (the Sun). Substituting into the prior equation, we have :<math>\frac{d^2u}{d\theta^2} + u = \frac{ GM }{h^2}</math>. So for the gravitational force &ndash; or, more generally, for ''any'' inverse square force law &ndash; 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: :<math> u(\theta) = \frac{ GM }{h^2} + A \cos(\theta-\theta_0)</math> where <math>A</math> and <math>\theta_0</math> are arbitrary constants. The equation of the orbit described by the particle is thus: :<math>r = \frac{1}{u} = \frac{ h^2 / GM }{1 + e \cos (\theta - \theta_0)} </math>, where <math>e</math> is: :<math> e \equiv \frac{h^2A}{G M}\ . </math> In general, this can be recognized as the equation of a [[conic section]] in [[polar coordinates]] (<math>r</math>, <math>\theta</math>). We can make a further connection with the classic description of conic section with: :<math> \frac{h^2}{GM} = a(1-e^2) </math> If parameter <math>e</math> is smaller than one, <math>e</math> is the [[Eccentricity (orbit)|eccentricity]] and <math>a</math> the [[semi-major axis]] of an [[ellipse]]. ==Orbital planes== {{main|Orbital plane (astronomy)}} 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 == {{main|Orbital period}} The orbital period is simply how long an orbiting body takes to complete one orbit. ==Specifying orbits== {{main|orbital elements}} It turns out that it takes a minimum 6 numbers to specify an orbit about a body, and this can be done in several ways. For example, specifying the 3 numbers specifying location and 3 specifying the [[velocity]] of a body gives a unique orbit that can be calculated forwards (or backwards). However, traditionally the parameters used are slightly different. The traditionally used set of orbital elements is called the set of '''Keplerian elements''', after [[Johannes Kepler]] and his [[Kepler's laws]]. The Keplerian elements are six: *[[Inclination]] (<math>i\,\!</math>) *[[Longitude of the ascending node]] <!--(☊-->(<math>\Omega\,\!</math>) *[[Argument of periapsis]] (<math>\omega\,\!</math>) *[[Eccentricity (orbit)|Eccentricity]] (<math>e\,\!</math>) *[[Semimajor axis]] (<math>a\,\!</math>) *[[Mean anomaly]] at [[Epoch (astronomy)|epoch]] (<math>M_o\,\!</math>) 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, ''perturbed'', by forces other than gravity due to the central body 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=== It can be shown that a radial impulse given to a body in orbit doesn't change the orbital period (since it doesn't affect the angular momentum), but changes the eccentricity. This means that the orbit still intersects the original orbit in two places. For a prograde or retrograde impulse (i.e. an impulse applied along the orbital motion), this changes both the eccentricity as well as the orbital period, but any closed orbit will still intersect the perturbation point. Notably, a prograde impulse given at [[periapsis]] raises the altitude at [[apoapsis]], and vice versa, and a retrograde impulse does the opposite. A transverse force out of the orbital plane causes rotation of the orbital plane. === Orbital decay === {{main|Orbital decay}} If some part of a body's orbit enters an atmosphere, its orbit can decay because of [[drag (physics)|drag]]. Particularly at each periapsis, the object scrapes the air, 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 [[solar maximum|solar maxima]], the Earth's atmosphere causes drag up to a hundred kilometres higher than during solar minima. Some satellites with long conductive tethers can also decay because of electromagnetic drag from the [[Earth's magnetic field]]. Basically, the wire cuts the magnetic field, and acts as a generator. The wire moves electrons from the near vacuum on one end to the near-vacuum on the other end. The orbital energy is converted to heat in the wire. Orbits can be artificially influenced through the use of rocket motors 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 also 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 bulge]]s 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 (moon)|Phobos]] is a prime example, and is expected to either impact Mars' surface or break up into a ring within 50 million years. Finally, 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. ===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 [[Quadropole#Gravitational_quadrupole|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 a central planet (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. ===Other gravitating bodies=== The effects of other gravitating bodies can be very large. For example, the [[orbit of the Moon]] cannot be in any way accurately described without allowing for the action of the Sun's gravity as well as the Earth's. ==Earth orbits== {{main|Geocentric orbit}} == Scaling in gravity == The [[gravitational constant]] ''G'' is measured to be: * (6.6742 ± 0.001) × 10<sup>−11</sup> N·m²/kg² * (6.6742 ± 0.001) × 10<sup>−11</sup> m³/(kg·s²) * (6.6742 ± 0.001) × 10<sup>−11</sup> (kg/m³)<sup>-1</sup>s<sup>-2</sup>. Thus the constant has dimension density<sup>-1</sup> time<sup>-2</sup>. This corresponds to the following properties. [[Scale factor|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 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. When all densities are multiplied by four, orbits are the same, but with orbital velocities doubled. When all densities are multiplied by four, and all sizes are halved, orbits are similar, with the same orbital velocities. These properties are illustrated in the formula :<math> GT^2 \sigma = 3\pi \left( \frac{a}{r} \right)^3, </math> for an elliptical orbit with [[semi-major axis]] ''a'', of a small body around a spherical body with radius ''r'' and average density σ, where ''T'' is the orbital period. == See also == {{MultiCol}} Types of orbit: * [[Artificial satellite orbit]] * [[Box orbit]] * [[Circular orbit]] * [[Elliptic orbit]] * [[Geostationary orbit|Geostationary or Clarke orbit]] * [[Geosynchronous orbit]] * [[Halo orbit]] * [[Hohmann transfer orbit]] * [[Kepler orbit]] * [[Lissajous orbit]] * [[List of orbits]] * [[Lyapunov exponent|Lyapunov orbit]] * [[Molniya orbit]] * [[Rosetta (orbit)|Rosetta orbit]] * [[Tundra orbit]] {{ColBreak}} Elements of an orbit:<ref name="basic">[http://www2.jpl.nasa.gov/basics/bsf5-1.html Basics of Spaceflight] a [[JPL]] primer used by its employees.</ref> * [[Orbital elements]] * [[Semi-major axis]] * [[Orbital eccentricity|Eccentricity]] * [[Inclination]] * [[Argument of periapsis]] * [[periapsis|Time of periapsis passage]] * [[ascending node|Celestial longitude of the ascending node]] {{ColBreak}} Related concepts: * [[Gravity]], [[Gravitational slingshot]], and [[Escape velocity]] * [[Interplanetary Transport Network]] * [[Kepler's laws of planetary motion]] * [[N-body problem]] * [[Orbit (dynamics)]] * [[Orbit equation]] * [[Orbital spaceflight]]/[[Sub-orbital spaceflight]] * [[Orbital maneuver]], [[Retrograde motion]] * [[Orbital mechanics]] * [[Specific orbital energy]] * [[Orbital period]] * [[Orbital speed]] * [[Trajectory]], [[Hyperbolic trajectory|Hyperbolic]] and [[Parabolic trajectory]] {{EndMultiCol}} {{wiktionary}} == References == * {{cite book | author = Abell, Morrison, and Wolff | title = Exploration of the Universe | edition = fifth edition | year = 1987 | publisher = Saunders College Publishing}} <references /> == External links == * [http://www.lri.fr/~dragice/gravity/ Understand orbits using direct manipulation] * An on-line orbit plotter: http://www.bridgewater.edu/~rbowman/ISAW/PlanetOrbit.html * [http://www.braeunig.us/space/orbmech.htm Orbital Mechanics] (Rocket and Space Technology) * [http://www.ncdc.noaa.gov/paleo/forcing.html The 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 * [http://astrobiology.ucla.edu/OTHER/SSO/ The 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<ref>{{cite journal | author=F. Varadi, B. Runnegar, M. Ghil | title=Successive Refinements in Long-Term Integrations of Planetary Orbits | journal=The Astrophysical Journal | volume=592 | year=2003 | pages=620–630}} {{doi|10.1086/375560}}</ref>, but only the [http://www.astrobiology.ucla.edu/OTHER/SSO/Misc/ eccentricity data for Earth and Mercury] are available online. * [http://www.phy.hk/wiki/englishhtm/Motion.htm Java simulation on orbital motion] {{orbits}} <!--Categories--> [[Category:Celestial mechanics]] [[Category:Astrological aspects]] <!--Interwiki--> [[af:Wentelbaan]] [[als:Umlaufbahn]] [[ast:Órbita]] [[bn:কক্ষপথ (গ্রহ)]] [[bs:Planetarna putanja]] [[bg:Орбита]] [[ca:Òrbita]] [[cs:Oběžná dráha]] [[da:Omløbsbane]] [[de:Umlaufbahn]] [[et:Orbiit]] [[es:Órbita]] [[eo:Orbito]] [[fa:مدار (سیاره)]] [[fr:Orbite]] [[gd:Reul-chuairt]] [[gl:Órbita]] [[ko:궤도]] [[io:Orbito]] [[id:Orbit]] [[it:Orbita]] [[lv:Orbīta]] [[lb:Ëmlafbunn]] [[lt:Orbita]] [[hu:Pálya (csillagászat)]] [[nl:Baan (hemellichaam)]] [[ja:軌道 (力学)]] [[no:Bane]] [[nn:Bane]] [[uz:Mehvar]] [[nds:Ümloopbahn]] [[pl:Orbita]] [[pt:Órbita]] [[ru:Орбита]] [[scn:Òrbita]] [[simple:Orbit]] [[sk:Obežná dráha]] [[sl:Tir]] [[sr:Орбита]] [[fi:Kiertorata]] [[sv:Omloppsbana]] [[th:วงโคจร]] [[vi:Quỹ đạo]] [[tr:Yörünge]] [[uk:Орбіта]] [[vec:Òrbita]] [[zh:轨道 (力学)]]