In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a planet, moon, asteroid, or

Lagrange point In celestial mechanics, the Lagrange points (; also Lagrangian points or libration points) are points of equilibrium for small-mass objects under the influence of two massive orbiting bodies. Mathematically, this involves the solution of th ...

. 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 orbits, with the

center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...

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 heliocentric theory of Nicolaus Copernicus, replacing its circular orbits ...

. For most situations, orbital motion is adequately approximated by

Newtonian mechanics Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in motion ...

, which explains gravity as a force obeying an

inverse-square law In science, an inverse-square law is any scientific law stating that a specified physical quantity is inversely proportional to the square of the distance from the source of that physical quantity. The fundamental cause for this can be understo ...

. However, Albert Einstein's general theory of relativity, which accounts for gravity as due to curvature of spacetime, with orbits following

geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...

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 of celestial spheres. 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 epicycles 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, it was modified by Copernicus 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, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws ...

whose results are summarised in his three laws of planetary motion. First, he found that the orbits of the planets in our Solar System are elliptical, not circular (or epicyclic), as had previously been believed, and that the Sun is not located at the center of the orbits, but rather at one focus. Second, he found that the orbital speed of each planet is not constant, as had previously been thought, but rather that the speed depends on the planet's distance from the Sun. Third, Kepler found a universal relationship between the orbital properties of all the planets orbiting the Sun. For the planets, the cubes of their distances from the Sun are proportional to the squares of their orbital periods. Jupiter and Venus, for example, are respectively about 5.2 and 0.723 AU distant from the Sun, their orbital periods respectively about 11.86 and 0.615 years. The proportionality is seen by the fact that the ratio for Jupiter, 5.2³/11.86², is practically equal to that for Venus, 0.723³/0.615², 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 In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stron ...

and that, in general, the orbits of bodies subject to gravity were conic sections (this assumes that the force of gravity propagates instantaneously). Newton showed that, for a pair of bodies, the orbits' sizes are in inverse proportion to their masses, and that those bodies orbit their common

. 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.

Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangianew approach to

emphasizing energy more than force, and made progress on the three-body 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 Neptune is the eighth planet from the Sun and the farthest known planet in the Solar System. It is the fourth-largest planet in the Solar System by diameter, the third-most-massive planet, and the densest giant planet. It is 17 times ...

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

space-time In physics, spacetime is a mathematical model that combines the three-dimensional space, three dimensions of space and one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visualize S ...

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, 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 a

planetary system A planetary system is a set of gravitationally In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interacti ...

, planets, dwarf planets,

asteroid An asteroid is a minor planet of the inner Solar System. Sizes and shapes of asteroids vary significantly, ranging from 1-meter rocks to a dwarf planet almost 1000 km in diameter; they are rocky, metallic or icy bodies with no atmosphere. ...

s and other minor planets, comets, and

space debris Space debris (also known as space junk, space pollution, space waste, space trash, or space garbage) are defunct human-made objects in space—principally in Earth orbit—which no longer serve a useful function. These include derelict spacecr ...

orbit the system's barycenter in elliptical orbits. A comet in a parabolic or hyperbolic orbit about a barycenter is not gravitationally bound to the star and therefore is not considered part of the star's planetary system. Bodies that are gravitationally bound to one of the planets in a planetary system, either natural or artificial satellites, follow orbits about a barycenter near or within that planet. Owing to mutual

gravitational perturbations In astronomy, perturbation is the complex motion of a massive body subjected to forces other than the gravitational attraction of a single other massive body. The other forces can include a third (fourth, fifth, etc.) body, resistance, as from ...

, the eccentricities of the planetary orbits vary over time.

Mercury Mercury commonly refers to: * Mercury (planet), the nearest planet to the Sun * Mercury (element), a metallic chemical element with the symbol Hg * Mercury (mythology), a Roman god Mercury or The Mercury may also refer to: Companies * Merc ...

, 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

. 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.

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', 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). Newton Cannon

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 A circular orbit is an orbit with a fixed distance around the barycenter; that is, in the shape of a circle. Listed below is a circular orbit in astrodynamics or celestial mechanics under standard assumptions. Here the centripetal force is ...

, 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, 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. 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, and Newton's laws give a sufficiently accurate description of motion. The acceleration of a body is equal to the sum of the forces acting on it, divided by its mass, and the gravitational force acting on a body is proportional to the product of the masses of the two attracting bodies and decreases inversely with the square of the distance between them. To this Newtonian approximation, for a system of two-point masses or spherical bodies, only influenced by their mutual gravitation (called a two-body problem), their trajectories can be exactly calculated. If the heavier body is much more massive than the smaller, as in the case of a satellite or small moon orbiting a planet or for the Earth orbiting the Sun, it is accurate enough and convenient to describe the motion in terms of a

coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...

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 with

gravitational field In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational phenome ...

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 a conic section. The orbit can be open (implying the object never returns) or closed (returning). Which it is depends on the total energy ( kinetic +

) of the system. In the case of an open orbit, the speed at any position of the orbit is at least the escape velocity for that position, in the case of a closed orbit, the speed is always less than the escape velocity. Since the kinetic energy is never negative if the common convention is adopted of taking the potential energy as zero at infinite separation, the bound orbits will have negative total energy, the parabolic trajectories zero total energy, and hyperbolic orbits positive total energy. An open orbit will have a parabolic shape if it has 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 In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...

. 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 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, 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 the Sun 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">Sun-planet system; for simplicity, this explanation assumes the Sun's mass is infinitely larger than that planet's.] The planet's orbit lies in a plane, called the orbital plane. The point on the orbit closest to the attracting body is the periapsis. The point farthest from the attracting body is called the apoapsis. There are also specific terms for orbits about particular bodies; things orbiting the Sun have a perihelion and aphelion, things orbiting the Earth have a perigee and apogee, and things orbiting the Moon have a perilune and apolune (or periselene and aposelene respectively). An orbit around any

star A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...

, not just the Sun, has a periastron and an apastron. # As the planet moves in its orbit, the line from the Sun to the planet sweeps a constant area of the orbital plane for a given period of time, regardless of which part of its orbit the planet traces during that period of time. This means that the planet moves faster near its perihelion than near its aphelion, because at the smaller distance it needs to trace a greater arc to cover the same area. This law is usually stated as "equal areas in equal time." # For a given orbit, the ratio of the cube of its

semi-major axis In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the long ...

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 closed

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, or by relativistic effects, thereby changing the gravitational field's behavior with distance) will cause the orbit's shape to depart from the closed

s characteristic of Newtonian two-body motion. The two-body solutions were published by Newton in Principia in 1687. In 1912,

Karl Fritiof Sundman Karl Frithiof Sundman (28 October 1873, in Kaskinen – 28 September 1949, in Helsinki) was a Finnish mathematician who used analytic methods to prove the existence of a convergent infinite series solution to the three-body problem in two pa ...

developed a converging infinite series that solves the three-body problem; however, it converges too slowly to be of much use. Except for special cases like the

Lagrangian point In celestial mechanics, the Lagrange points (; also Lagrangian points or libration points) are points of equilibrium for small-mass objects under the influence of two massive orbiting bodies. Mathematically, this involves the solution of 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 adds perturbation terms to account for the gravitational influence of multiple bodies. This is convenient for calculating the positions of astronomical bodies. The equations of motion of the moons, planets, and other bodies are known with great accuracy, and are used to generate tables for

celestial navigation Celestial navigation, also known as astronavigation, is the practice of position fixing using stars and other celestial bodies that enables a navigator to accurately determine their actual current physical position in space (or on the surface of ...

. Still, there are secular phenomena that have to be dealt with by

post-Newtonian In general relativity, the post-Newtonian expansions (PN expansions) are used for finding an approximate solution of the Einstein field equations for the metric tensor. The approximations are expanded in small parameters which express orders of ...

methods. :The differential equation form is used for scientific or mission-planning purposes. According to Newton's laws, the sum of all the forces acting on a body will equal the mass of the body times its acceleration (''F = ma''). Therefore accelerations can be expressed in terms of positions. The perturbation terms are much easier to describe in this form. Predicting subsequent positions and velocities from initial values of position and velocity corresponds to solving an initial value problem. Numerical methods calculate the positions and velocities of the objects a short time in the future, then repeat the calculation ad 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 the Newtonian 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 = -\frac

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: :

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: :

-\frac  = m_2  A_2

Solving for the acceleration, ''A''₂: :

A_2 = \frac = - \frac \frac =  -\frac

where

\mu\,

is the

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 bodies. For two bodies the parameter may be expressed as G(m1+m2), or as GM when ...

, 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

\hat

or in the

\hat

directions are also proportionate to the respective components of the distances,

r''_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 \cos(t)

and

y = B \sin(t)

of the ellipse. In contrast, with the decreasing relationship

A = \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 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

\theta

be the angle it has rotated. Let

\hat

and

\hat

be the standard Euclidean bases and let

\hat = \cos(\theta)\hat + \sin(\theta)\hat

and

\hat = - \sin(\theta)\hat + \cos(\theta)\hat

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 :

\hat = r \cos(\theta)\hat + r \sin(\theta)\hat = r \hat

We use

\dot r

and

\dot \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 + \delta t

and dividing by

\delta t

. The result is also a vector. Because our basis vector

\hat

moves as the object orbits, we start by differentiating it. From time

t

t + \delta t

, the vector

\hat

keeps its beginning at the origin and rotates from angle

\theta

\theta  + \dot \theta\ \delta t

which moves its head a distance

\dot \theta\ \delta t

in the perpendicular direction

\hat

giving a derivative of

\dot \theta \hat

. :

\begin
  \hat
    &= \cos(\theta)\hat + \sin(\theta)\hat \\
  \frac = \dot
    &= -\sin(\theta)\dot \theta \hat + \cos(\theta)\dot \theta \hat = \dot \theta \hat \\
  \hat
    &= -\sin(\theta)\hat + \cos(\theta)\hat \\
  \frac = \dot
    &= -\cos(\theta)\dot \theta \hat - \sin(\theta) \dot \theta \hat = -\dot \theta \hat
\end

We can now find the velocity and acceleration of our orbiting object. :

\hat \end

The coefficients of

\hat

and

\hat

give the accelerations in the radial and transverse directions. As said, Newton gives this first due to gravity is

-\mu/r^2

and the second is zero. Equation (2) can be rearranged using integration by parts. :

r \ddot\theta + 2 \dot r \dot\theta = \frac\frac\left( r^2 \dot \theta \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 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 heliocentric theory of Nicolaus Copernicus, replacing its circular orbits ...

(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

\theta

. However, it is easier to introduce the auxiliary variable

u = 1/r

and to express

u

as a function of

\theta

. Derivatives of

r

with respect to time may be rewritten as derivatives of

u

with respect to angle. :

u =

\dot\theta = \frac = hu^2

(reworking (3)) :

\begin
      \frac &= \frac\left(\frac\right)\frac = -\frac = -\frac \\
  \frac &= -\frac\frac\frac = -\frac = -\frac
  \ \ \ \text \ \ \  \ddot r = - h^2 u^2 \frac
\end

Plugging these into (1) gives :

\begin
                                                          \ddot r - r\dot\theta^2 &= -\frac \\
   -h^2 u^2 \frac - \frac \left(h u^2\right)^2 &= -\mu u^2
\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 equation (up to a shift of origin of the dependent variable). The solution is: :

u(\theta) = \frac\mu  - A \cos(\theta - \theta_0)

where ''A'' and ''θ''₀ are arbitrary constants. This resulting equation of the orbit of the object is that of an

in Polar form relative to one of the focal points. This is put into a more standard form by letting

e \equiv h^2 A/\mu

be the eccentricity, letting

a \equiv h^2/\mu\left(1 - e^2\right)

be the semi-major axis. Finally, letting

\theta_0 \equiv 0

so the long axis of the ellipse is along the positive ''x'' coordinate. :

r(\theta) = \frac

When the two-body system is under the influence of torque, the angular momentum ''h'' is not a constant. After the following calculation: :

\begin
                   \frac &= -\frac \frac = -\frac \frac \\
               \frac &= -\frac \frac - \frac \frac \frac \\
  \left(\frac\right)^2 r &= \frac
\end

we will get the Sturm-Liouville equation of two-body system.

Relativistic orbital motion

The above classical ( Newtonian) analysis of orbital mechanics assumes that the more subtle effects of general relativity, such as

frame dragging Frame-dragging is an effect on spacetime, predicted by Albert Einstein's general theory of relativity, that is due to non-static stationary distributions of mass–energy. A stationary field is one that is in a steady state, but the masses cau ...

and gravitational time dilation are negligible. Relativistic effects cease to be negligible when near very massive bodies (as with the precession of Mercury's orbit about the Sun), or when extreme precision is needed (as with calculations of the

orbital elements Orbital elements are the parameters required to uniquely identify a specific orbit. In celestial mechanics these elements are considered in two-body systems using a Kepler orbit. There are many different ways to mathematically describe the same ...

and time signal references for

GPS The Global Positioning System (GPS), originally Navstar GPS, is a Radionavigation-satellite service, satellite-based radionavigation system owned by the United States government and operated by the United States Space Force. It is one of t ...

satellites.).

Orbital planes

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

Orbital period

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

Keplerian elements Orbital elements are the parameters required to uniquely identify a specific orbit. In celestial mechanics these elements are considered in two-body systems using a Kepler orbit. There are many different ways to mathematically describe the same o ...

, after Johannes Kepler and his laws. The Keplerian elements are six: * Inclination (''i'') *

Longitude of the ascending node The longitude of the ascending node (☊ or Ω) is one of the orbital elements used to specify the orbit of an object in space. It is the angle from a specified reference direction, called the ''origin of longitude'', to the direction of the asce ...

(Ω) *

Argument of periapsis The argument of periapsis (also called argument of perifocus or argument of pericenter), symbolized as ''ω'', is one of the orbital elements of an orbiting body. Parametrically, ''ω'' is the angle from the body's ascending node to its periapsi ...

(ω) * Eccentricity (''e'') * Semimajor axis (''a'') *

Mean anomaly In celestial mechanics, the mean anomaly is the fraction of an elliptical orbit's period that has elapsed since the orbiting body passed periapsis, expressed as an angle which can be used in calculating the position of that body in the classical ...

at epoch (''M''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 perturbed, by other forces than simple gravity from an assumed point source (see the next section), and thus the orbital elements change over time.

Orbital perturbations

An orbital perturbation is when a force or impulse which is much smaller than the overall force or average impulse of the main gravitating body and which is external to the two orbiting bodies causes an acceleration, which changes the parameters of the orbit over time.

Radial, prograde and transverse perturbations

A small radial impulse given to a body in orbit changes the eccentricity, but not the orbital period (to first order). A prograde or retrograde impulse (i.e. an impulse applied along the orbital motion) changes both the eccentricity and the orbital period. Notably, a prograde impulse at periapsis raises the altitude at apoapsis, and vice versa and a retrograde impulse does the opposite. A transverse impulse (out of the orbital plane) causes rotation of the orbital plane without changing the period or eccentricity. In all instances, a closed orbit will still intersect the perturbation point.

Orbital decay

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

drag Drag or The Drag may refer to: Places * Drag, Norway, a village in Tysfjord municipality, Nordland, Norway * ''Drág'', the Hungarian name for Dragu Commune in Sălaj County, Romania * Drag (Austin, Texas), the portion of Guadalupe Street adj ...

. Particularly at each periapsis, the object experiences atmospheric drag, losing energy. Each time, the orbit grows less eccentric (more circular) because the object loses kinetic energy precisely when that energy is at its maximum. This is similar to the effect of slowing a pendulum at its lowest point; the highest point of the pendulum's swing becomes lower. With each successive slowing more of the orbit's path is affected by the atmosphere and the effect becomes more pronounced. Eventually, the effect becomes so great that the maximum kinetic energy is not enough to return the orbit above the limits of the atmospheric drag effect. When this happens the body will rapidly spiral down and intersect the central body. The bounds of an atmosphere vary wildly. During a solar maximum, the Earth's atmosphere causes drag up to a hundred kilometres higher than during a solar minimum. Some satellites with long conductive tethers can also experience orbital decay because of electromagnetic drag from the Earth's magnetic field. As the wire cuts the magnetic field it acts as a generator, moving electrons from one end to the other. The orbital energy is converted to heat in the wire. Orbits can be artificially influenced through the use of rocket engines which change the kinetic energy of the body at some point in its path. This is the conversion of chemical or electrical energy to kinetic energy. In this way changes in the orbit shape or orientation can be facilitated. Another method of artificially influencing an orbit is through the use of

solar sail Solar sails (also known as light sails and photon sails) are a method of spacecraft propulsion using radiation pressure exerted by sunlight on large mirrors. A number of spaceflight missions to test solar propulsion and navigation have been p ...

s 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 is a prime example and is expected to either impact Mars' surface or break up into a ring within 50 million years. Orbits can decay via the emission of

gravitational wave Gravitational waves are waves of the intensity of gravity generated by the accelerated masses of an orbital binary system that propagate as waves outward from their source at the speed of light. They were first proposed by Oliver Heaviside in 1 ...

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 A black hole is a region of spacetime where gravitation, gravity is so strong that nothing, including light or other Electromagnetic radiation, electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts t ...

s or neutron stars 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 Flattening is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution (spheroid) respectively. Other terms used are ellipticity, or oblateness. The usual notation for flattening is ...

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

can cause significant perturbations to the attitude and direction of motion of the body, and over time can be significant. Of the planetary bodies, the motion of

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 moving bodies. Further studies have discovered that nonplanar orbits are also possible, including one involving 12 masses moving in 4 roughly circular, interlocking orbits

topologically In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...

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 Ballistics is the field of mechanics concerned with the launching, flight behaviour and impact effects of projectiles, especially ranged weapon munitions such as bullets, unguided bombs, rockets or the like; the science or art of designing and a ...

and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and Newton's law of universal gravitation. It is a core discipline within space mission design and control. Celestial mechanics treats more broadly the orbital dynamics of systems under the influence of gravity, including spacecraft and natural astronomical bodies such as star systems, planets, moons, and comets. Orbital mechanics focuses on spacecraft

trajectories A trajectory or flight path is the path that an object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete traj ...

, including

orbital maneuver In spaceflight, an orbital maneuver (otherwise known as a burn) is the use of propulsion systems to change the orbit of a spacecraft. For spacecraft far from Earth (for example those in orbits around the Sun) an orbital maneuver is called a ' ...

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

Earth orbits

* Low Earth orbit (LEO): Geocentric orbits with altitudes up to 2,000 km (0–1,240

mile The mile, sometimes the international mile or statute mile to distinguish it from other miles, is a British imperial unit and United States customary unit of distance; both are based on the older English unit of length equal to 5,280 English ...

s). * Medium Earth orbit (MEO): Geocentric orbits ranging in altitude from 2,000 km (1,240

s) to just below

geosynchronous orbit A geosynchronous orbit (sometimes abbreviated GSO) is an Earth-centered orbit with an orbital period that matches Earth's rotation on its axis, 23 hours, 56 minutes, and 4 seconds (one sidereal day). The synchronization of rotation and orbital ...

at . Also known as an intermediate circular orbit. These are "most commonly at , or , with an orbital period of 12 hours." * Both

(GSO) and

geostationary orbit A geostationary orbit, also referred to as a geosynchronous equatorial orbit''Geostationary orbit'' and ''Geosynchronous (equatorial) orbit'' are used somewhat interchangeably in sources. (GEO), is a circular geosynchronous orbit in altitud ...

(GEO) are orbits around Earth matching Earth's

sidereal rotation The rotation period of a celestial object (e.g., star, gas giant, planet, moon, asteroid) may refer to its sidereal rotation period, i.e. the time that the object takes to complete a single revolution around its axis of rotation relative to the b ...

period. All geosynchronous and geostationary orbits have a

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 A high Earth orbit is a geocentric orbit with an altitude entirely above that of a geosynchronous orbit (). The orbital periods of such orbits are greater than 24 hours, therefore satellites in such orbits have an apparent retrograde motion ...

: Geocentric orbits above the altitude of

35,786 km (22,240

s).

Scaling in gravity

The

gravitational constant The gravitational constant (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant), denoted by the capital letter , is an empirical physical constant involved in ...

''G'' has been calculated as: * (6.6742 ± 0.001) × 10⁻¹¹ (kg/m³)⁻¹s⁻². Thus the constant has dimension density⁻¹ time⁻². This corresponds to the following properties. Scaling of distances (including sizes of bodies, while keeping the densities the same) gives similar orbits without scaling the time: if for example distances are halved, masses are divided by 8, gravitational forces by 16 and gravitational accelerations by 2. Hence velocities are halved and orbital periods 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) :

GT^2 \rho = 3\pi \left( \frac \right)^3,

for an elliptical orbit with

''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 has 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 and Pluto-Charon system.

References

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, 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 Celestial mechanics Periodic phenomena Gravity Astrodynamics Concepts in astronomy