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astronomy Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos. It uses mathematics, physics, and chemistry in order to explain their origin and their overall evolution. Objects of interest includ ...
, perturbation is the complex motion of a massive body subjected to forces other than the gravitational attraction of a single other
mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...
ive body. The other forces can include a third (fourth, fifth, etc.) body, resistance, as from an
atmosphere An atmosphere () is a layer of gases that envelop an astronomical object, held in place by the gravity of the object. A planet retains an atmosphere when the gravity is great and the temperature of the atmosphere is low. A stellar atmosph ...
, and the off-center attraction of an oblate or otherwise misshapen body.


Introduction

The study of perturbations began with the first attempts to predict planetary motions in the sky. In ancient times the causes were unknown.
Isaac Newton Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Age of Enlightenment, Enlightenment that followed ...
, at the time he formulated his laws of
motion In physics, motion is when an object changes its position with respect to a reference point in a given time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed, and frame of reference to an o ...
and of
gravitation In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
, applied them to the first analysis of perturbations, recognizing the complex difficulties of their calculation. 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 The Moon is Earth's only natural satellite. It Orbit of the Moon, orbits around Earth at Lunar distance, an average distance of (; about 30 times Earth diameter, Earth's diameter). The Moon rotation, rotates, with a rotation period (lunar ...
and
planet A planet is a large, Hydrostatic equilibrium, rounded Astronomical object, astronomical body that is generally required to be in orbit around a star, stellar remnant, or brown dwarf, and is not one itself. The Solar System has eight planets b ...
s 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 A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, tho ...
, and can be described in geometrical terms. 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 In physics, specifically classical mechanics, the three-body problem is to take the initial positions and velocities (or momenta) of three point masses orbiting each other in space and then calculate their subsequent trajectories using Newton' ...
; if there are multiple other bodies it is an ‑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. Most systems that involve multiple gravitational attractions present one primary body which is dominant in its effects (for example, a
star A star is a luminous spheroid of plasma (physics), plasma held together by Self-gravitation, self-gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night sk ...
, 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 A satellite or an artificial satellite is an object, typically a spacecraft, placed into orbit around a celestial body. They have a variety of uses, including communication relay, weather forecasting, navigation ( GPS), broadcasting, scient ...
around its primary body.


Mathematical analysis


General perturbations

In methods of general perturbations, general differential equations, either of motion or of change in 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 o ...
, 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. 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 A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, tho ...
, 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 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 ...
at any particular time are what are sought by the methods of general perturbations. General perturbations takes advantage of the fact that in many problems of
celestial 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 (classical mechanics) to astronomical objects, such as stars and planets, to ...
, 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. In the
Solar System The Solar SystemCapitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Sola ...
, this is usually the case;
Jupiter Jupiter is the fifth planet from the Sun and the List of Solar System objects by size, largest in the Solar System. It is a gas giant with a Jupiter mass, mass more than 2.5 times that of all the other planets in the Solar System combined a ...
, the second largest body, has a mass of about 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.


Special perturbations

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 In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral. The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for "numerical integr ...
of the differential equations of motion. 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 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 ...
. Special perturbations can be applied to any problem in
celestial 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 (classical mechanics) to astronomical objects, such as stars and planets, to ...
, as it is not limited to cases where the perturbing forces are small. 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. Special perturbations are also used for modeling an orbit with computers.


Cowell's formulation

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. In a system of \ n\ mutually interacting bodies, this method mathematically solves for the Newtonian forces on body \ i\ by summing the individual interactions from the other j bodies: :\mathbf_i = \sum_^n \ G\ m_j \frac where \ \mathbf_i\ is the
acceleration In mechanics, acceleration is the Rate (mathematics), rate of change of the velocity of an object with respect to time. Acceleration is one of several components of kinematics, the study of motion. Accelerations are Euclidean vector, vector ...
vector of body i, G is the gravitational constant, \ m_j\ is the
mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...
of body j, \ \mathbf_i\ and \ \mathbf_j\ are the position vectors of objects \ i\ and \ j\ respectively, and \ r_ \equiv \, \mathbf_j-\mathbf_i \, \ is the distance from object i to object \ j\ , all vectors being referred to the barycenter of the system. This equation is resolved into components in \ x\ , \ y\ , and \ z\ , 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. However, for many problems in
celestial 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 (classical mechanics) to astronomical objects, such as stars and planets, to ...
, 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 Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms. ...
because of the large difference in the forces of the central body and the perturbing bodies, although with high precision numbers built into modern
computer A computer is a machine that can be Computer programming, programmed to automatically Execution (computing), carry out sequences of arithmetic or logical operations (''computation''). Modern digital electronic computers can perform generic set ...
s this is not as much of a limitation as it once was.


Encke's method

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. 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''. 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. Letting \boldsymbol be the radius vector of the osculating orbit, \mathbf the radius vector of the perturbed orbit, and \delta \mathbf the variation from the osculating orbit, \mathbf and \boldsymbol are just the equations of motion of \mathbf and \boldsymbol, where \mu = G(M+m) is the gravitational parameter with M and m the
mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...
es of the central body and the perturbed body, \mathbf_ is the perturbing
acceleration In mechanics, acceleration is the Rate (mathematics), rate of change of the velocity of an object with respect to time. Acceleration is one of several components of kinematics, the study of motion. Accelerations are Euclidean vector, vector ...
, and r and \rho are the magnitudes of \mathbf and \boldsymbol. Substituting from equations () and () into equation (), which, in theory, could be integrated twice to find \delta \mathbf. Since the osculating orbit is easily calculated by two-body methods, \boldsymbol and \delta \mathbf are accounted for and \mathbf 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. Encke's method was more widely used before the advent of modern
computer A computer is a machine that can be Computer programming, programmed to automatically Execution (computing), carry out sequences of arithmetic or logical operations (''computation''). Modern digital electronic computers can perform generic set ...
s, when much orbit computation was performed on mechanical calculating machines.


Periodic nature

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 In celestial mechanics, an orbit (also known as orbital revolution) 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 ...
, 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 Uranus is the seventh planet from the Sun. It is a gaseous cyan-coloured ice giant. Most of the planet is made of water, ammonia, and methane in a Supercritical fluid, supercritical phase of matter, which astronomy calls "ice" or Volatile ( ...
. 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 Jupiter is the fifth planet from the Sun and the List of Solar System objects by size, largest in the Solar System. It is a gas giant with a Jupiter mass, mass more than 2.5 times that of all the other planets in the Solar System combined a ...
(59.31 years) is nearly equal to two of
Saturn Saturn is the sixth planet from the Sun and the second largest in the Solar System, after Jupiter. It is a gas giant, with an average radius of about 9 times that of Earth. It has an eighth the average density of Earth, but is over 95 tim ...
(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.
Venus Venus is the second planet from the Sun. It is often called Earth's "twin" or "sister" planet for having almost the same size and mass, and the closest orbit to Earth's. While both are rocky planets, Venus has an atmosphere much thicker ...
currently has the orbit with the least
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-Centre (geometry), center, in geometry * Eccentricity (g ...
, i.e. it is the closest to circular, of all the planetary orbits. In 25,000 years' time,
Earth Earth is the third planet from the Sun and the only astronomical object known to Planetary habitability, harbor life. This is enabled by Earth being an ocean world, the only one in the Solar System sustaining liquid surface water. Almost all ...
will have a more circular (less eccentric) orbit than Venus. It has been shown that long-term periodic disturbances within the
Solar System The Solar SystemCapitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Sola ...
can become chaotic over very long time scales; under some circumstances one or more
planet A planet is a large, Hydrostatic equilibrium, rounded Astronomical object, astronomical body that is generally required to be in orbit around a star, stellar remnant, or brown dwarf, and is not one itself. The Solar System has eight planets b ...
s can cross the orbit of another, leading to collisions. The orbits of many of the minor bodies of the Solar System, such as
comet A comet is an icy, small Solar System body that warms and begins to release gases when passing close to the Sun, a process called outgassing. This produces an extended, gravitationally unbound atmosphere or Coma (cometary), coma surrounding ...
s, are often heavily perturbed, particularly by the gravitational fields of the
gas giant A gas giant is a giant planet composed mainly of hydrogen and helium. Jupiter and Saturn are the gas giants of the Solar System. The term "gas giant" was originally synonymous with "giant planet". However, in the 1990s, it became known that Uranu ...
s. 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 Jupiter is the fifth planet from the Sun and the List of Solar System objects by size, largest in the Solar System. It is a gas giant with a Jupiter mass, mass more than 2.5 times that of all the other planets in the Solar System combined a ...
's gravitational influence caused the period of
Comet Hale–Bopp Comet Hale–Bopp (formally designated C/1995 O1) is a long-period comet that was one of the most widely observed of the 20th century and one of the brightest seen for many decades. Alan Hale (astronomer), Alan Hale and Thomas Bopp disc ...
's orbit to decrease from 4,206 to 2,380 years, a change that will not revert on any periodic basis.


See also

* Formation and evolution of the Solar System * Frozen orbit * Molniya orbit * Nereid one of the outer moons of Neptune with a high
orbital eccentricity In astrodynamics, the orbital eccentricity of an astronomical object is a dimensionless parameter that determines the amount by which its orbit around another body deviates from a perfect circle. A value of 0 is a circular orbit, values be ...
of ~0.75 and is frequently perturbed * Osculating orbit * Orbit modeling * Orbital resonance * Perturbation theory * Proper orbital elements * Stability of the Solar System


References

;Footnotes ;Citations ;Bibliography * * * *


Further reading

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


External links



(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.(a
Google books
{{Portal bar, Astronomy, Stars, Spaceflight, Outer space, Solar System * Dynamical systems Dynamics of the Solar System Celestial mechanics