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In astronomy, **perturbation** is the complex motion of a massive body subject to forces other than the gravitational attraction of a single other massive body.^{ [1] } The other forces can include a third (fourth, fifth, etc.) body, resistance, as from an atmosphere, and the off-center attraction of an oblate or otherwise misshapen body.^{ [2] }

The study of perturbations began with the first attempts to predict planetary motions in the sky. In ancient times the causes were a mystery. Newton, at the time he formulated his laws of motion and of gravitation, applied them to the first analysis of perturbations,^{ [2] } recognizing the complex difficulties of their calculation.^{ [3] } Many of the great mathematicians since then have given attention to the various problems involved; throughout the 18th and 19th centuries there was demand for accurate tables of the position of the Moon and planets for marine navigation.

The complex motions of gravitational perturbations can be broken down. The hypothetical motion that the body follows under the gravitational effect of one other body only is a conic section, and can be readily described with the methods of geometry. This is called a two-body problem, or an unperturbed Keplerian orbit. The differences between that and the actual motion of the body are **perturbations** due to the additional gravitational effects of the remaining body or bodies. If there is only one other significant body then the perturbed motion is a three-body problem; if there are multiple other bodies it is an *n*-body problem. A general analytical solution (a mathematical expression to predict the positions and motions at any future time) exists for the two-body problem; when more than two bodies are considered analytic solutions exist only for special cases. Even the two-body problem becomes insoluble if one of the bodies is irregular in shape.^{ [4] }

Most systems that involve multiple gravitational attractions present one primary body which is dominant in its effects (for example, a star, in the case of the star and its planet, or a planet, in the case of the planet and its satellite). The gravitational effects of the other bodies can be treated as perturbations of the hypothetical unperturbed motion of the planet or satellite around its primary body.

In methods of **general perturbations**, general differential equations, either of motion or of change in the orbital elements, are solved analytically, usually by series expansions. The result is usually expressed in terms of algebraic and trigonometric functions of the orbital elements of the body in question and the perturbing bodies. This can be applied generally to many different sets of conditions, and is not specific to any particular set of gravitating objects.^{ [5] } Historically, general perturbations were investigated first. The classical methods are known as *variation of the elements*, * variation of parameters * or *variation of the constants of integration*. In these methods, it is considered that the body is always moving in a conic section, however the conic section is constantly changing due to the perturbations. If all perturbations were to cease at any particular instant, the body would continue in this (now unchanging) conic section indefinitely; this conic is known as the osculating orbit and its orbital elements at any particular time are what are sought by the methods of general perturbations.^{ [2] }

General perturbations takes advantage of the fact that in many problems of celestial mechanics, the two-body orbit changes rather slowly due to the perturbations; the two-body orbit is a good first approximation. General perturbations is applicable only if the perturbing forces are about one order of magnitude smaller, or less, than the gravitational force of the primary body.^{ [4] } In the Solar System, this is usually the case; Jupiter, the second largest body, has a mass of about 1/1000 that of the Sun.

General perturbation methods are preferred for some types of problems, as the source of certain observed motions are readily found. This is not necessarily so for special perturbations; the motions would be predicted with similar accuracy, but no information on the configurations of the perturbing bodies (for instance, an orbital resonance) which caused them would be available.^{ [4] }

In methods of **special perturbations**, numerical datasets, representing values for the positions, velocities and accelerative forces on the bodies of interest, are made the basis of numerical integration of the differential equations of motion.^{ [6] } In effect, the positions and velocities are perturbed directly, and no attempt is made to calculate the curves of the orbits or the orbital elements.^{ [2] }

Special perturbations can be applied to any problem in celestial mechanics, as it is not limited to cases where the perturbing forces are small.^{ [4] } Once applied only to comets and minor planets, special perturbation methods are now the basis of the most accurate machine-generated planetary ephemerides of the great astronomical almanacs.^{ [2] }^{ [7] } Special perturbations are also used for modeling an orbit with computers.

Cowell's formulation (so named for Philip H. Cowell, who, with A.C.D. Cromellin, used a similar method to predict the return of Halley's comet) is perhaps the simplest of the special perturbation methods.^{ [8] } In a system of mutually interacting bodies, this method mathematically solves for the Newtonian forces on body by summing the individual interactions from the other bodies:

where is the acceleration vector of body , is the gravitational constant, is the mass of body , and are the position vectors of objects and respectively, and is the distance from object to object , all vectors being referred to the barycenter of the system. This equation is resolved into components in , , and and these are integrated numerically to form the new velocity and position vectors. This process is repeated as many times as necessary. The advantage of Cowell's method is ease of application and programming. A disadvantage is that when perturbations become large in magnitude (as when an object makes a close approach to another) the errors of the method also become large.^{ [9] } However, for many problems in celestial mechanics, this is never the case. Another disadvantage is that in systems with a dominant central body, such as the Sun, it is necessary to carry many significant digits in the arithmetic because of the large difference in the forces of the central body and the perturbing bodies, although with modern computers this is not nearly the limitation it once was.^{ [10] }

Encke's method begins with the osculating orbit as a reference and integrates numerically to solve for the variation from the reference as a function of time.^{ [11] } Its advantages are that perturbations are generally small in magnitude, so the integration can proceed in larger steps (with resulting lesser errors), and the method is much less affected by extreme perturbations. Its disadvantage is complexity; it cannot be used indefinitely without occasionally updating the osculating orbit and continuing from there, a process known as *rectification*.^{ [9] } Encke's method is similar to the general perturbation method of variation of the elements, except the rectification is performed at discrete intervals rather than continuously.^{ [12] }

Letting be the radius vector of the osculating orbit, the radius vector of the perturbed orbit, and the variation from the osculating orbit,

**(1)**

- .
**(2)**

and are just the equations of motion of and

- for the perturbed orbit and
**(3)**

- for the unperturbed orbit,
**(4)**

where is the gravitational parameter with and the masses of the central body and the perturbed body, is the perturbing acceleration, and and are the magnitudes of and .

Substituting from equations (** 3 **) and (** 4 **) into equation (** 2 **),

**(5)**

which, in theory, could be integrated twice to find . Since the osculating orbit is easily calculated by two-body methods, and are accounted for and can be solved. In practice, the quantity in the brackets, , is the difference of two nearly equal vectors, and further manipulation is necessary to avoid the need for extra significant digits.^{ [13] }^{ [14] } Encke's method was more widely used before the advent of modern computers, when much orbit computation was performed on mechanical calculating machines.

In the Solar System, many of the disturbances of one planet by another are periodic, consisting of small impulses each time a planet passes another in its orbit. This causes the bodies to follow motions that are periodic or quasi-periodic – such as the Moon in its strongly perturbed orbit, which is the subject of lunar theory. This periodic nature led to the discovery of Neptune in 1846 as a result of its perturbations of the orbit of Uranus.

On-going mutual perturbations of the planets cause long-term quasi-periodic variations in their orbital elements, most apparent when two planets' orbital periods are nearly in sync. For instance, five orbits of Jupiter (59.31 years) is nearly equal to two of Saturn (58.91 years). This causes large perturbations of both, with a period of 918 years, the time required for the small difference in their positions at conjunction to make one complete circle, first discovered by Laplace.^{ [2] } Venus currently has the orbit with the least eccentricity, i.e. it is the closest to circular, of all the planetary orbits. In 25,000 years' time, Earth will have a more circular (less eccentric) orbit than Venus. It has been shown that long-term periodic disturbances within the Solar System can become chaotic over very long time scales; under some circumstances one or more planets can cross the orbit of another, leading to collisions.^{ [15] }

The orbits of many of the minor bodies of the Solar System, such as comets, are often heavily perturbed, particularly by the gravitational fields of the gas giants. While many of these perturbations are periodic, others are not, and these in particular may represent aspects of chaotic motion. For example, in April 1996, Jupiter's gravitational influence caused the period of Comet Hale–Bopp's orbit to decrease from 4,206 to 2,380 years, a change that will not revert on any periodic basis.^{ [16] }

- Formation and evolution of the Solar System
- Frozen orbit
- Molniya orbit
- Nereid one of the outer moons of Neptune with a high orbital eccentricity of ~0.75 and is frequently perturbed
- Osculating orbit
- Orbit modeling
- Orbital resonance
- Proper orbital elements
- Stability of the Solar System

In physics, **angular momentum** is the rotational analog of linear momentum. It is an important quantity in physics because it is a conserved quantity—the total angular momentum of a closed system remains constant. Angular momentum has both a direction and a magnitude, and both are conserved. Motorcycles, frisbees and rifled bullets all owe their useful properties to conservation of angular momentum. Conservation of angular momentum is also why hurricanes have spirals and neutron stars have high rotational rates. In general, conservation limits the possible motion of a system but does not uniquely determine it.

**Continuum mechanics** is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such models in the 19th century.

A **centripetal force** is a force that makes a body follow a curved path. Its direction is always orthogonal to the motion of the body and towards the fixed point of the instantaneous center of curvature of the path. Isaac Newton described it as "a force by which bodies are drawn or impelled, or in any way tend, towards a point as to a centre". In Newtonian mechanics, gravity provides the centripetal force causing astronomical orbits.

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 and epicycles with elliptical trajectories, and explaining how planetary velocities vary. The three laws state that:

- The orbit of a planet is an ellipse with the Sun at one of the two foci.
- A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
- The square of a planet's orbital period is proportional to the cube of the length of the semi-major axis of its orbit.

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. 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 physics, the **Navier–Stokes equations** are certain partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842–1850 (Stokes).

**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 orbit, but certain schemes, each consisting of a set of six parameters, are commonly used in astronomy and orbital mechanics.

**Celestial mechanics** is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics to astronomical objects, such as stars and planets, to produce ephemeris data.

In physics, the **center of mass** of a distribution of mass in space 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 be applied to cause a linear acceleration without an angular acceleration. Calculations in mechanics are often simplified when formulated with respect to the center of mass. It is a hypothetical point where the entire mass of an object may be assumed to be concentrated to visualise its motion. In other words, the center of mass is the particle equivalent of a given object for application of Newton's laws of motion.

A **Sun-synchronous orbit** (**SSO**), also called a **heliosynchronous orbit**, is a nearly polar orbit around a planet, in which the satellite passes over any given point of the planet's surface at the same local mean solar time. More technically, it is an orbit arranged so that it precesses through one complete revolution each year, so it always maintains the same relationship with the Sun.

In astrodynamics and celestial dynamics, the **orbital state vectors** of an orbit are Cartesian vectors of position and velocity that together with their time (epoch) uniquely determine the trajectory of the orbiting body in space.

In physics and classical mechanics, the **three-body problem** is the problem of taking the initial positions and velocities of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's law of universal gravitation. The three-body problem is a special case of the n-body problem. Unlike two-body problems, no general closed-form solution exists, as the resulting dynamical system is chaotic for most initial conditions, and numerical methods are generally required.

A **classical field theory** is a physical theory that predicts how one or more physical fields interact with matter through **field equations**, without considering effects of quantization; theories that incorporate quantum mechanics are called quantum field theories. In most contexts, 'classical field theory' is specifically meant to describe electromagnetism and gravitation, two of the fundamental forces of nature.

In astronomy, and in particular in astrodynamics, the **osculating orbit** of an object in space at a given moment in time is the gravitational Kepler orbit that it would have around its central body if perturbations were absent. That is, it is the orbit that coincides with the current orbital state vectors.

In physics and astronomy, an ** N-body simulation** is a simulation of a dynamical system of particles, usually under the influence of physical forces, such as gravity.

The **Cauchy momentum equation** is a vector partial differential equation put forth by Cauchy that describes the non-relativistic momentum transport in any continuum.

In celestial mechanics, a **Kepler orbit** is the motion of one body relative to another, as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space. A Kepler orbit can also form a straight line. It considers only the point-like gravitational attraction of two bodies, neglecting perturbations due to gravitational interactions with other objects, atmospheric drag, solar radiation pressure, a non-spherical central body, and so on. It is thus said to be a solution of a special case of the two-body problem, known as the Kepler problem. As a theory in classical mechanics, it also does not take into account the effects of general relativity. Keplerian orbits can be parametrized into six orbital elements in various ways.

In physics, the **n-body problem** is the problem of predicting the individual motions of a group of celestial objects interacting with each other gravitationally. Solving this problem has been motivated by the desire to understand the motions of the Sun, Moon, planets, and visible stars. In the 20th century, understanding the dynamics of globular cluster star systems became an important n-body problem. The n-body problem in general relativity is considerably more difficult to solve due to additional factors like time and space distortions.

**Orbit modeling** is the process of creating mathematical models to simulate motion of a massive body as it moves in orbit around another massive body due to gravity. Other forces such as gravitational attraction from tertiary bodies, air resistance, solar pressure, or thrust from a propulsion system are typically modeled as secondary effects. Directly modeling an orbit can push the limits of machine precision due to the need to model small perturbations to very large orbits. Because of this, perturbation methods are often used to model the orbit in order to achieve better accuracy.

In orbital mechanics, **Gauss's method** is used for preliminary orbit determination from at least three observations of the orbiting body of interest at three different times. The required information are the times of observations, the position vectors of the observation points, the direction cosine vector of the orbiting body from the observation points and general physical data.

- Bibliography

- Bate, Roger R.; Mueller, Donald D.; White, Jerry E. (1971).
*Fundamentals of Astrodynamics*. New York: Dover Publications. ISBN 0-486-60061-0. - Moulton, Forest Ray (1914).
*An Introduction to Celestial Mechanics*(2nd revised ed.). Macmillan. - Roy, A. E. (1988).
*Orbital Motion*(3rd ed.). Institute of Physics Publishing. ISBN 0-85274-229-0.

- Footnotes

- ↑ Bate, Mueller, White (1971): ch. 9, p. 385.
- 1 2 3 4 5 6 Moulton (1914): ch. IX
- ↑ Newton in 1684 wrote: "By reason of the deviation of the Sun from the center of gravity, the centripetal force does not always tend to that immobile center, and hence the planets neither move exactly in ellipses nor revolve twice in the same orbit. Each time a planet revolves it traces a fresh orbit, as in the motion of the Moon, and each orbit depends on the combined motions of all the planets, not to mention the action of all these on each other. But to consider simultaneously all these causes of motion and to define these motions by exact laws admitting of easy calculation exceeds, if I am not mistaken, the force of any human mind." (quoted by Prof G E Smith (Tufts University), in "Three Lectures on the Role of Theory in Science" 1. Closing the loop: Testing Newtonian Gravity, Then and Now); and Prof R F Egerton (Portland State University, Oregon) after quoting the same passage from Newton concluded: "Here, Newton identifies the "many body problem" which remains unsolved analytically." Archived 2005-03-10 at the Wayback Machine
- 1 2 3 4 Roy (1988): ch. 6, 7.
- ↑ Bate, Mueller, White (1971): p. 387; sec. 9.4.3, p. 410.
- ↑ Bate, Mueller, White (1971), pp. 387–409.
- ↑ See, for instance, Jet Propulsion Laboratory Development Ephemeris.
- ↑ Cowell, P. H.; Crommelin, A. C. D. (1910). "Investigation of the Motion of Halley's Comet from 1759 to 1910".
*Greenwich Observations in Astronomy*. Bellevue, for His Majesty's Stationery Office: Neill & Co.**71**: O1. Bibcode:1911GOAMM..71O...1C. - 1 2 Danby, J.M.A. (1988).
*Fundamentals of Celestial Mechanics*(second ed.). Willmann-Bell, Inc. ISBN 0-943396-20-4., chapter 11. - ↑ Herget, Paul (1948).
*The Computation of Orbits*. privately published by the author., p. 91 ff. - ↑ Encke, J. F. (1854).
*Über die allgemeinen Störungen der Planeten*.*Berliner Astronomisches Jahrbuch für 1857*. pp. 319–397. - ↑ Battin (1999), sec. 10.2.
- ↑ Bate, Mueller, White (1971), sec. 9.3.
- ↑ Roy (1988), sec. 7.4.
- ↑ see references at Stability of the Solar System
- ↑ Don Yeomans (1997-04-10). "Comet Hale–Bopp Orbit and Ephemeris Information". JPL/NASA. Retrieved 2008-10-23.

- P.E. El'Yasberg: Introduction to the Theory of Flight of Artificial Earth Satellites

- Solex (by Aldo Vitagliano) predictions for the position/orbit/close approaches of Mars
- Gravitation Sir George Biddell Airy's 1884 book on gravitational motion and perturbations, using little or no math.(at Google books)

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