In celestial mechanics, an orbit is the curved

trajectory 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 tr ...

of an object such as the trajectory of a planet around a star, or of a

natural satellite A natural satellite is, in the most common usage, an astronomical body that orbits a planet, dwarf planet, or small Solar System body (or sometimes another natural satellite). Natural satellites are often colloquially referred to as ''moons'' ...

around a planet, or of an

artificial satellite A satellite or artificial satellite is an object intentionally placed into orbit in outer space. Except for passive satellites, most satellites have an electricity generation system for equipment on board, such as solar panels or radioisotop ...

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 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 motio ...

, which explains

gravity 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 str ...

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 unders ...

. However, Albert Einstein's

general theory of relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. G ...

, which accounts for gravity as due to curvature of spacetime, with orbits following geodesics, 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 The celestial spheres, or celestial orbs, were the fundamental entities of the cosmological models developed by Plato, Eudoxus, Aristotle, Ptolemy, Copernicus, and others. In these celestial models, the apparent motions of the fixed star ...

. This model posited the existence of perfect moving spheres or rings to which the stars and planets were attached. It assumed the heavens were fixed apart from the motion of the spheres and was developed without any understanding of gravity. After the planets' motions were more accurately measured, theoretical mechanisms such as

deferent and epicycle In the Hipparchian, Ptolemaic, and Copernican systems of astronomy, the epicycle (, meaning "circle moving on another circle") was a geometric model used to explain the variations in speed and direction of the apparent motion of the Moon, S ...

s were added. Although the model was capable of reasonably accurately predicting the planets' positions in the sky, more and more epicycles were required as the measurements became more accurate, hence the model became increasingly unwieldy. Originally

geocentric In astronomy, the geocentric model (also known as geocentrism, often exemplified specifically by the Ptolemaic system) is a superseded description of the Universe with Earth at the center. Under most geocentric models, the Sun, Moon, stars, an ...

, it was modified by

Copernicus Nicolaus Copernicus (; pl, Mikołaj Kopernik; gml, Niklas Koppernigk, german: Nikolaus Kopernikus; 19 February 1473 – 24 May 1543) was a Renaissance polymath, active as a mathematician, astronomer, and Catholic canon, who formulate ...

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 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 Circular may refer to: * The shape of a circle * ''Circular'' (album), a 2006 album by Spanish singer Vega * Circular letter (disambiguation) ** Flyer (pamphlet), a form of advertisement * Circular reasoning, a type of logical fallacy * 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 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 ...

Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a "natural philosopher"), widely recognised as one of the great ...

demonstrated that Kepler's laws were derivable from his theory of gravitation and that, in general, the orbits of bodies subject to gravity were

conic section In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...

s (this assumes that the force of gravity propagates instantaneously). Newton showed that, for a pair of bodies, the orbits' sizes are in inverse proportion to their masses, 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.

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 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 and removed Newton's assumption that changes propagate instantaneously. This led astronomers to recognize that Newtonian mechanics did not provide the highest accuracy in understanding orbits. In

relativity theory The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena in ...

, 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 bound non- stellar objects in or out of orbit around a star or star system. Generally speaking, systems with one or more planets constitute a planetary system, although such systems may also cons ...

, planets, dwarf planets, asteroids and other

minor planet According to the International Astronomical Union (IAU), a minor planet is an astronomical object in direct orbit around the Sun that is exclusively classified as neither a planet nor a comet. Before 2006, the IAU officially used the term ''minor ...

s, comets, and space debris orbit the system's

barycenter In astronomy, the barycenter (or barycentre; ) is the center of mass of two or more bodies that orbit one another and is the point about which the bodies orbit. A barycenter is a dynamical point, not a physical object. It is an important con ...

in elliptical orbits. A comet in a parabolic or

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

orbit about a barycenter is not gravitationally bound to the star and therefore is not considered part of the star's planetary system. Bodies that are gravitationally bound to one of the planets in a planetary system, either natural or

artificial satellites A satellite or artificial satellite is an object intentionally placed into orbit in outer space. Except for passive satellites, most satellites have an electricity generation system for equipment on board, such as solar panels or radioisotop ...

, follow orbits about a barycenter near or within that planet. Owing to mutual gravitational perturbations, the eccentricities of the planetary orbits vary over time. Mercury, the smallest planet in the Solar System, has the most eccentric orbit. At the present

epoch In chronology and periodization, an epoch or reference epoch is an instant in time chosen as the origin of a particular calendar era. The "epoch" serves as a reference point from which time is measured. The moment of epoch is usually decided by ...

, 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 An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, the apsides of the Earth are called the aphelion and perihelion. General description There are two apsides in any elli ...

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

, its velocity will decrease as its potential energy increases.

Understanding orbits

There are a few common ways of understanding orbits: * A force, such as gravity, pulls an object into a curved path as it attempts to fly off in a straight line. * As the object is pulled toward the massive body, it falls toward that body. However, if it has enough tangential velocity it will not fall into the body but will instead continue to follow the curved trajectory caused by that body indefinitely. The object is then said to be orbiting the body. As an illustration of an orbit around a planet, the Newton's cannonball model may prove useful (see image below). This is a '

thought experiment A thought experiment is a hypothetical situation in which a hypothesis, theory, or principle is laid out for the purpose of thinking through its consequences. History The ancient Greek ''deiknymi'' (), or thought experiment, "was the most anc ...

', 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, 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 In celestial mechanics, escape velocity or escape speed is the minimum speed needed for a free, non- propelled object to escape from the gravitational influence of a primary body, thus reaching an infinite distance from it. It is typically ...

, 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 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 motio ...

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 s ...

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 fields. A stationary body far from another can do external work if it is pulled towards it, and therefore has gravitational '' potential energy''. Since work is required to separate two bodies against the pull of gravity, their gravitational potential energy increases as they are separated, and decreases as they approach one another. For point masses, the gravitational energy decreases 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

. The orbit can be open (implying the object never returns) or closed (returning). Which it is depends on the total energy ( kinetic + potential energy) of the system. In the case of an open orbit, the speed at any position of the orbit is at least the

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 In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, cal ...

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 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.html" ;"title="barycenter.html" ;"title="his focal point is actually the barycenter">his focal point is actually the barycenter of the Solar System">Sun-planet system; for simplicity, this explanation assumes the Sun's mass is infinitely larger than that planet's.] The planet's orbit lies in a plane, called the orbital plane. The 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 An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, the apsides of the Earth are called the aphelion and perihelion. General description There are two apsides in any ell ...

, 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 held together by its gravity. The nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night, but their immense distances from Earth ma ...

, 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

, 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.

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 ellipses, which repeat the same path exactly and indefinitely, any non-spherical or non-Newtonian effects (such as caused by the slight oblateness of the

Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large list of largest lakes and seas in the Solar System, volumes of water can be found throughout the Solar System, only water distributi ...

, or by relativistic effects, thereby changing the gravitational field's behavior with distance) will cause the orbit's shape to depart from the closed ellipses characteristic of Newtonian two-body motion. The two-body solutions were published by Newton in Principia in 1687. In 1912, Karl Fritiof Sundman developed a converging infinite series that solves the three-body problem; however, it converges too slowly to be of much use. Except for special cases like the

Lagrangian point 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 orbit, orbiting bodies. Mathematically, this involves the solutio ...

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. Still, there are secular phenomena that have to be dealt with by post-Newtonian methods. :The

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...

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

. 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 (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 ellipse 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 Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off-center, in geometry * Eccentricity (graph theory) of a ...

, 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

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

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

Inclination Orbital inclination measures the tilt of an object's orbit around a celestial body. It is expressed as the angle between a reference plane and the orbital plane or axis of direction of the orbiting object. For a satellite orbiting the Earth ...

(''i'') * Longitude of the ascending node (Ω) * Argument of periapsis (ω) *

Eccentricity Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off-center, in geometry * Eccentricity (graph theory) of a ...

(''e'') *

Semimajor 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 ...

(''a'') * Mean anomaly at

(''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

, 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

, 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. Particularly at each periapsis, the object experiences atmospheric drag, losing energy. Each time, the orbit grows less eccentric (more circular) because the object loses kinetic energy precisely when that energy is at its maximum. This is similar to the effect of slowing a pendulum at its lowest point; the highest point of the pendulum's swing becomes lower. With each successive slowing more of the orbit's path is affected by the atmosphere and the effect becomes more pronounced. Eventually, the effect becomes so great that the maximum kinetic energy is not enough to return the orbit above the limits of the atmospheric drag effect. When this happens the body will rapidly spiral down and intersect the central body. The bounds of an atmosphere vary wildly. During a

solar maximum Solar maximum is the regular period of greatest solar activity during the Sun's 11-year solar cycle. During solar maximum, large numbers of sunspots appear, and the solar irradiance output grows by about 0.07%. On average, the solar cycle tak ...

, the Earth's atmosphere causes drag up to a hundred kilometres higher than during a solar minimum. Some satellites with long conductive tethers can also experience orbital decay because of electromagnetic drag from the Earth's magnetic field. As the wire cuts the magnetic field it acts as a generator, moving electrons from one end to the other. The orbital energy is converted to heat in the wire. Orbits can be artificially influenced through the use of rocket engines which change the kinetic energy of the body at some point in its path. This is the conversion of chemical or electrical energy to kinetic energy. In this way changes in the orbit shape or orientation can be facilitated. Another method of artificially influencing an orbit is through the use of solar sails or magnetic sails. These forms of propulsion require no propellant or energy input other than that of the Sun, and so can be used indefinitely. See statite for one such proposed use. Orbital decay can occur due to

tidal force The tidal force is a gravitational effect that stretches a body along the line towards the center of mass of another body due to a gradient (difference in strength) in gravitational field from the other body; it is responsible for diverse phenomen ...

s for objects below the synchronous orbit for the body they're orbiting. The gravity of the orbiting object raises tidal bulges in the primary, and since below the synchronous orbit, the orbiting object is moving faster than the body's surface the bulges lag a short angle behind it. The gravity of the bulges is slightly off of the primary-satellite axis and thus has a component along 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 gravity is so strong that nothing, including light or other electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts that a sufficiently compact mass can def ...

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 The Moon orbits Earth in the prograde direction and completes one revolution relative to the Vernal Equinox and the stars in about 27.32 days (a tropical month and sidereal month) and one revolution relative to the Sun in about 29.53 days (a ...

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 The Hill sphere of an astronomical body is the region in which it dominates the attraction of satellites. To be retained by a planet, a moon must have an orbit that lies within the planet's Hill sphere. That moon would, in turn, have a Hill s ...

. 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 In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th r ...

, 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. ...

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 asteroids 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 equivalent to the edges of a

cuboctahedron A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it ...

. 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

rocket A rocket (from it, rocchetto, , bobbin/spool) is a vehicle that uses jet propulsion to Acceleration, accelerate without using the surrounding Atmosphere of Earth, air. A rocket engine produces thrust by Reaction (physics), reaction to exhaust ...

s and other spacecraft. The motion of these objects is usually calculated from

Newton's laws of motion 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 ...

and

Newton's law of universal gravitation Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distanc ...

. 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

, including spacecraft and natural astronomical bodies such as star systems, planets, moons, and comets. Orbital mechanics focuses on spacecraft trajectories, including orbital maneuvers, orbit plane changes, and interplanetary transfers, and is used by mission planners to predict the results of propulsive maneuvers.

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

is a more exact theory than Newton's laws for calculating orbits, and is sometimes necessary for greater accuracy or in high-gravity situations (such as orbits close to the Sun).

Earth orbits

* Low Earth orbit (LEO):

Geocentric orbit A geocentric orbit or Earth orbit involves any object orbiting Earth, such as the Moon or artificial satellites. In 1997, NASA estimated there were approximately 2,465 artificial satellite payloads orbiting Earth and 6,216 pieces of space deb ...

s 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 Englis ...

s). *

Medium Earth orbit A medium Earth orbit (MEO) is an Earth-centered orbit with an altitude above a low Earth orbit (LEO) and below a high Earth orbit (HEO) – between above sea level.

(MEO):

s 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 (GEO) are orbits around Earth matching Earth's sidereal rotation period. All geosynchronous and geostationary orbits have a semi-major axis 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 motio ...

s above the altitude of

35,786 km (22,240

s).

Scaling in gravity

The gravitational constant ''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 semi-major 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.

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