In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an physical body such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an satellite around an object or position in space such as a planet, moon, asteroid, or Lagrange point. Normally, orbit refers to a regularly repeating trajectory, although it may also refer to a nonrepeating trajectory. To a close approximation, planets and satellites follow , with the barycenter being orbited at a focal point of the ellipse, as described by Kepler's laws of planetary motion.
For most situations, orbital motion is adequately approximated by Newtonian mechanics, which explains gravity as a force obeying an inversesquare law.Kuhn, The Copernican Revolution, pp. 238, 246–252 However, Albert Einstein's general theory of relativity, which accounts for gravity as due to curvature of spacetime, with orbits following , provides a more accurate calculation and understanding of the exact mechanics of orbital motion.
The basis for the modern understanding of orbits was first formulated by Johannes Kepler whose results are summarised in his three laws of planetary motion. First, he found that the orbits of the planets in our Solar System are elliptical, not circle (or epicycle), as had previously been believed, and that the Sun is not located at the center of the orbits, but rather at one focus. Second, he found that the orbital speed of each planet is not constant, as had previously been thought, but rather that the speed depends on the planet's distance from the Sun. Third, Kepler found a universal relationship between the orbital properties of all the planets orbiting the Sun. For the planets, the cubes of their distances from the Sun are proportional to the squares of their orbital periods. Jupiter and Venus, for example, are respectively about 5.2 and 0.723 AU distant from the Sun, their orbital periods respectively about 11.86 and 0.615 years. The proportionality is seen by the fact that the ratio for Jupiter, 5.2^{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 Kepler orbits.
Isaac Newton demonstrated that Kepler's laws were derivable from his theory of gravitation and that, in general, the orbits of bodies subject to gravity were (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 , and that those bodies orbit their common center of mass. Where one body is much more massive than the other (as is the case of an artificial satellite orbiting a planet), it is a convenient approximation to take the center of mass as coinciding with the center of the more massive body.
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. JosephLouis Lagrange developed a new approach to Newtonian mechanics emphasizing energy more than force, and made progress on the threebody problem, discovering the Lagrangian points. In a dramatic vindication of classical mechanics, in 1846 Urbain Le Verrier was able to predict the position of Neptune based on unexplained perturbations in the orbit of Uranus.
Albert Einstein in his 1916 paper The Foundation of the General Theory of Relativity explained that gravity was due to curvature of spacetime and removed Newton's assumption that changes in gravity propagate instantaneously. This led astronomers to recognize that Newtonian mechanics did not provide the highest accuracy in understanding orbits. In relativity theory, 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.
Owing to mutual gravitational perturbations, the eccentricities of the planetary orbits vary over time. Mercury, the smallest planet in the Solar System, has the most eccentric orbit. At the present epoch, Mars has the next largest eccentricity while the smallest orbital eccentricities are seen with Venus and Neptune.
As two objects orbit each other, the periapsis is that point at which the two objects are closest to each other and the apoapsis is that point at which they are the farthest. (More specific terms are used for specific bodies. For example, perigee and apogee are the lowest and highest parts of an orbit around Earth, while perihelion and aphelion are the closest and farthest points of an orbit around the Sun.)
In the case of planets orbiting a star, the mass of the star and all its satellites are calculated to be at a single point called the barycenter. The paths of all the star's satellites are elliptical orbits about that barycenter. Each satellite in that system will have its own elliptical orbit with the barycenter at one focal point of that ellipse. At any point along its orbit, any satellite will have a certain value of kinetic and potential energy with respect to the barycenter, and the sum of those two energies is a constant value at every point along its orbit. As a result, as a planet approaches periapsis, the planet will increase in speed as its potential energy decreases; as a planet approaches apoapsis, its velocity will decrease as its potential energy increases.
The velocity relationship of two moving objects with mass can thus be considered in four practical classes, with subtypes:
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 reenter (i.e. fall). Occasionally a space craft will intentionally intercept the atmosphere, in an act commonly referred to as an aerobraking maneuver.
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 noninterrupted 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, as shown in (C).
As the firing speed is increased beyond this, noninterrupted 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 noninterrupted 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, 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 parabolic path. At even greater speeds the object will follow a range of hyperbolic trajectories. In a practical sense, both of these trajectory types mean the object is "breaking free" of the planet's gravity, and "going off into space" never to return.
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 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. 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, and when orbiting a body other than earth it is called the periapsis (less properly, "perifocus" or "pericentron"). The point where the satellite is farthest from Earth is called the apogee, apoapsis, or sometimes apifocus or apocentron. A line drawn from periapsis to apoapsis is the lineofapsides. This is the major axis of the ellipse, the line through its longest part.
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.
where F_{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:
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:
Solving for the acceleration, A_{2}:
where $\backslash mu\backslash ,$ is the standard gravitational parameter, in this case $G\; m\_1$. It is understood that the system being described is m_{2}, hence the subscripts can be dropped.
We assume that the central body is massive enough that it can be considered to be stationary and we ignore the more subtle effects of general relativity.
When a pendulum or an object attached to a spring swings in an ellipse, the inward acceleration/force is proportional to the distance $A\; =\; F/m\; =\; \; k\; r.$ Due to the way vectors add, the component of the force in the $\backslash hat\{\backslash mathbf\{x\}\}$ or in the $\backslash hat\{\backslash mathbf\{y\}\}$ 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.
The location of the orbiting object at the current time $t$ is located in the plane using vector calculus in polar coordinates both with the standard Euclidean basis and with the polar basis with the origin coinciding with the center of force. Let $r$ be the distance between the object and the center and $\backslash theta$ be the angle it has rotated. Let $\backslash hat\{\backslash mathbf\{x\}\}$ and $\backslash hat\{\backslash mathbf\{y\}\}$ be the standard Euclidean space bases and let $\backslash hat\{\backslash mathbf\{r\}\}\; =\; \backslash cos(\backslash theta)\backslash hat\{\backslash mathbf\{x\}\}\; +\; \backslash sin(\backslash theta)\backslash hat\{\backslash mathbf\{y\}\}$ and $\backslash hat\{\backslash boldsymbol\; \backslash theta\}\; =\; \; \backslash sin(\backslash theta)\backslash hat\{\backslash mathbf\{x\}\}\; +\; \backslash cos(\backslash theta)\backslash hat\{\backslash mathbf\{y\}\}$ be the radial and transverse polar basis with the first being the unit vector pointing from the central body to the current location of the orbiting object and the second being the orthogonal unit vector pointing in the direction that the orbiting object would travel if orbiting in a counter clockwise circle. Then the vector to the orbiting object is
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 $\backslash hat\{\backslash mathbf\{r\}\}$ moves as the object orbits, we start by differentiating it. From time $t$ to $t\; +\; \backslash delta\; t$, the vector $\backslash hat\{\backslash mathbf\{r\}\}$ 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 $\backslash hat\{\backslash boldsymbol\; \backslash theta\}$ giving a derivative of $\backslash dot\; \backslash theta\; \backslash hat\{\backslash boldsymbol\; \backslash theta\}$.
\hat{\mathbf{r}} &= \cos(\theta)\hat{\mathbf{x}} + \sin(\theta)\hat{\mathbf{y}} \\ \frac{\delta \hat{\mathbf{r}}}{\delta t} = \dot{\mathbf r} &= \sin(\theta)\dot \theta \hat{\mathbf{x}} + \cos(\theta)\dot \theta \hat{\mathbf{y}} = \dot \theta \hat{\boldsymbol \theta} \\ \hat{\boldsymbol \theta} &= \sin(\theta)\hat{\mathbf{x}} + \cos(\theta)\hat{\mathbf{y}} \\ \frac{\delta \hat{\boldsymbol \theta}}{\delta t} = \dot{\boldsymbol \theta} &= \cos(\theta)\dot \theta \hat{\mathbf{x}}  \sin(\theta) \dot \theta \hat{\mathbf{y}} = \dot \theta \hat{\mathbf r}\end{align}
We can now find the velocity and acceleration of our orbiting object.
\hat{\mathbf{O}} &= r \hat{\mathbf{r}} \\ \dot{\mathbf{O}} &= \frac{\delta r}{\delta t} \hat{\mathbf{r}} + r \frac{\delta \hat{\mathbf{r}}}{\delta t} = \dot r \hat {\mathbf r} + r \left[ \dot \theta \hat {\boldsymbol \theta} \right] \\ \ddot{\mathbf{O}} &= \left[\ddot r \hat {\mathbf r} + \dot r \dot \theta \hat {\boldsymbol \theta}\right] + \left[\dot r \dot \theta \hat {\boldsymbol \theta} + r \ddot \theta \hat {\boldsymbol \theta}  r \dot \theta^2 \hat {\mathbf r} \right] \\ &= \left[\ddot r  r\dot\theta^2\right]\hat{\mathbf{r}} + \left[r \ddot\theta + 2 \dot r \dot\theta\right] \hat{\boldsymbol \theta}\end{align}
The coefficients of $\backslash hat\{\backslash mathbf\{r\}\}$ and $\backslash hat\{\backslash boldsymbol\; \backslash theta\}$ 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.
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 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.
\frac{\delta u}{\delta \theta} &= \frac{\delta}{\delta t}\left(\frac{1}{r}\right)\frac{\delta t}{\delta \theta } = \frac{\dot{r}}{r^2\dot{\theta }} = \frac{\dot{r}}{h} \\ \frac{\delta^2 u}{\delta \theta^2} &= \frac{1}{h}\frac{\delta \dot{r}}{\delta t}\frac{\delta t}{\delta \theta } = \frac{\ddot{r}}{h\dot{\theta}} = \frac{\ddot{r}}{h^2 u^2} \ \ \ \text{ or } \ \ \ \ddot r =  h^2 u^2 \frac{\delta^2 u}{\delta \theta^2}\end{align}
Plugging these into (1) gives
\ddot r  r\dot\theta^2 &= \frac{\mu}{r^2} \\ h^2 u^2 \frac{\delta^2 u}{\delta \theta^2}  \frac{1}{u} \left(h u^2\right)^2 &= \mu u^2\end{align}
So for the gravitational force – or, more generally, for any inverse square force law – the right hand side of the equation becomes a constant and the equation is seen to be the harmonic equation (up to a shift of origin of the dependent variable). The solution is:
where A and θ_{0} are arbitrary constants. This resulting equation of the orbit of the object is that of an ellipse in Polar form relative to one of the focal points. This is put into a more standard form by letting $e\; \backslash equiv\; h^2\; A/\backslash mu$ be the eccentricity, which when rearranged we see:
Note that by letting $a\; \backslash equiv\; h^2/\backslash mu\backslash left(1\; \; e^2\backslash right)$ be the semimajor axis and letting $\backslash theta\_0\; \backslash equiv\; 0$ so the long axis of the ellipse is along the positive x coordinate we yield:
When the twobody system is under the influence of torque, the angular momentum h is not a constant. After the following calculation:
\frac{\delta r}{\delta \theta} &= \frac{1}{u^2} \frac{\delta u}{\delta \theta} = \frac{h}{m} \frac{\delta u}{\delta \theta} \\ \frac{\delta^2 r}{\delta \theta^2} &= \frac{h^2u^2}{m^2} \frac{\delta^2 u}{\delta \theta^2}  \frac{hu^2}{m^2} \frac{\delta h}{\delta \theta} \frac{\delta u}{\delta \theta} \\ \left(\frac{\delta \theta}{\delta t}\right)^2 r &= \frac{h^2 u^3}{m^2}\end{align}
we will get the SturmLiouville equation of twobody system.
The rotation to do this in three dimensions requires three numbers to uniquely determine; traditionally these are expressed as three angles.
The traditionally used set of orbital elements is called the set of Orbital elements, after Johannes Kepler and his laws. The Keplerian elements are six:
In principle, once the orbital elements are known for a body, its position can be calculated forward and backward indefinitely in time. However, in practice, orbits are affected or perturbed, by other forces than simple gravity from an assumed point source (see the next section), and thus the orbital elements change over time.
Note that, unless the eccentricity is zero, a is not the average orbital radius. The timeaveraged orbital distance is given by:
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 or . 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 for objects below the synchronous orbit for the body they're orbiting. The gravity of the orbiting object raises 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 primarysatellite 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 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 or that are orbiting each other closely.
However, in the real world, many bodies rotate, and this introduces oblateness and distorts the gravity field, and gives a quadrupole moment to the gravitational field which is significant at distances comparable to the radius of the body. 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 socalled 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.
When there are more than two gravitating bodies it is referred to as an nbody problem. Most nbody problems have no closed form solution, although some special cases have been formulated.
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.
Thus the constant has dimension density^{−1} time^{−2}. This corresponds to the following properties.
Scale factor of distances (including sizes of bodies, while keeping the densities the same) gives similar orbits without scaling the time: if for example distances are halved, masses are divided by 8, gravitational forces by 16 and gravitational accelerations by 2. Hence velocities are halved and orbital periods 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 formula for the orbital period)
for an elliptical orbit with semimajor axis 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.

