Editing Perturbation (astronomy)

{{Short description|Classical approach to the many-body problem of astronomy}}
[[File:Moon perturbation diagram.svg|thumb|300px|alt=Vector diagram of the Sun's perturbations on the Moon. When the gravitational force of the Sun common to both the Earth and the Moon is subtracted, what is left is the perturbations.|The perturbing forces of the [[Sun]] on the [[Moon]] at two places in its [[orbit]]. The blue arrows represent the [[Euclidean vector|direction and magnitude]] of the gravitational force on the [[Earth]]. Applying this to both the Earth's and the Moon's position does not disturb the positions relative to each other. When it is subtracted from the force on the Moon (black arrows), what is left is the perturbing force (red arrows) on the Moon relative to the Earth. Because the perturbing force is different in direction and magnitude on opposite sides of the orbit, it produces a change in the shape of the orbit.]]
{{Astrodynamics}}
In [[astronomy]], '''perturbation''' is the complex motion of a [[astronomical object|massive body]] subjected to forces other than the [[gravity|gravitational]] attraction of a single other [[mass]]ive [[physical body|body]].<ref>{{harvp|Bate|Mueller|White|1971|at=ch. 9, p. 385}}</ref> The other forces can include a third (fourth, fifth, etc.) body, [[drag (physics)|resistance]], as from an [[atmosphere]], and the off-center attraction of an [[oblate spheroid|oblate]] or otherwise misshapen body.<ref name="moulton"/>

== 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]], at the time he formulated his laws of [[Newton's laws of motion|motion]] and of [[Newton's law of universal gravitation|gravitation]], applied them to the first analysis of perturbations,<ref name=moulton/> recognizing the complex difficulties of their calculation.{{efn|
Newton (1684) wrote:<br/>"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."<ref name=GE_Smith_3Lecs_nr1>
Newton quoted by Prof G.E. Smith (Tufts University), in<br/>
{{cite web
 |first=G.E. |last=Smith
 |title=Closing the loop: Testing Newtonian gravity, then and now
 |series=Three lectures on the role of theory in science |volume=1
 |type=symposium talk
 |place=Stanford University
 |url=https://stanford.edu/dept/cisst/SmithPowerpointTalk1.ppt
 |format=[[PowerPoint]]
}}
</ref>{{refn|
After quoting the same passage from Newton<ref name=GE_Smith_3Lecs_nr1/> Prof R.F. Egerton (Portland State University) concludes: "Here, Newton identifies the "many body problem" which remains unsolved analytically."<ref>
{{cite web
 |first=R.F. |last=Egerton
 |title=Newton
 |series=Physics&nbsp;311-12
 |type=course notes
 |place=Portland, OR
 |publisher=[[Portland State University]]
 |url=http://physics.pdx.edu/~egertonr/ph311-12/newton.htm
 |via=physics.pdx.edu
 |archive-url=https://web.archive.org/web/20050310192531/http://physics.pdx.edu/~egertonr/ph311-12/newton.htm
 |archive-date=2005-03-10
}}
</ref>
}}
}}
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 [[planet]]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]], and can be described in [[geometry|geometrical]] terms. This is called a [[two-body problem]], or an unperturbed [[Kepler orbit|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|{{mvar|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.<ref name=roy>{{harvp|Roy|1988|at=ch. 6–7}}</ref>

[[File:Mercury perturbation comparison.png|thumb|300px|alt=Plot of Mercury's position in its orbit, with and without perturbations from various planets. The perturbations cause Mercury to move in looping paths around its unperturbed position.|[[Mercury (planet)|Mercury]]'s orbital longitude and latitude, as perturbed by [[Venus]], [[Jupiter]], and all of the planets of the [[Solar System]], at intervals of 2.5&nbsp;days. Mercury would remain centered on the crosshairs if there were no perturbations.]]

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.

== Mathematical analysis ==

=== General perturbations ===
In methods of '''general perturbations''', general differential equations, either of motion or of change in the [[orbital elements]], are solved analytically, usually by [[series expansion]]s. 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.<ref>{{harvp|Bate|Mueller|White|1971|at=p. 387; p. 410 §9.4.3}}</ref> 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.<ref name="moulton"/>

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.<ref name=roy/> In the [[Solar System]], this is usually the case; [[Jupiter]], the second largest body, has a mass of about {{small|{{sfrac| 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.<ref name=roy/>

=== 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]] of the differential [[equations of motion]].<ref>{{harvp|Bate|Mueller|White|1971|pp=387–409}}</ref> 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]].<ref name=moulton>{{harvp|Moulton|1914|at=ch. IX}}</ref>

Special perturbations can be applied to any problem in [[celestial mechanics]], as it is not limited to cases where the perturbing forces are small.<ref name=roy/> Once applied only to comets and minor planets, special perturbation methods are now the basis of the most accurate machine-generated [[Fundamental ephemeris|planetary ephemerides]] of the great astronomical almanacs.<ref name="moulton"/>{{efn|
See, for instance, the Wikipedia article on the [[Jet Propulsion Laboratory Development Ephemeris]].
}} Special perturbations are also used for [[Orbit Modeling|modeling]] an orbit with computers.

==== Cowell's formulation ====
[[File:Cowells method.svg|thumb|Cowell's method. Forces from all perturbing bodies (black and gray) are summed to form the total force on body <math>\ i\ </math> (red), and this is numerically integrated starting from the initial position (the ''epoch of osculation'').]]

Cowell's formulation (so named for [[Philip Herbert Cowell|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.<ref>
{{cite journal
 |last1=Cowell    |first1=P.H.
 |last2=Crommelin |first2=A.C.D.
 |year=1910
 |title=Investigation of the motion of Halley's comet from 1759 to 1910
 |journal = Greenwich Observations in Astronomy
 |publisher=Neill & Co.
 |location=Bellevue, for His Majesty's Stationery Office
 |volume = 71 |pages = O1
 |bibcode=1911GOAMM..71O...1C
}}
</ref> In a system of <math>\ n\ </math> mutually interacting bodies, this method mathematically solves for the [[Newton's law of universal gravitation|Newtonian]] forces on body <math>\ i\ </math> by summing the individual interactions from the other <math>j</math> bodies:
:<math>\mathbf{\ddot{r}}_i = \sum_{\underset{j \ne i}{j=1}}^n \ G\ m_j \frac{\ (\mathbf{r}_j-\mathbf{r}_i)\ }{\ \| \mathbf{r}_j-\mathbf{r}_i \|^3 }</math>

where <math>\ \mathbf{\ddot{r}}_i\ </math> is the [[acceleration]] vector of body <math>i</math>, <math>G</math> is the [[gravitational constant]], <math>\ m_j\ </math> is the [[mass]] of body <math>j</math>, <math>\ \mathbf{r}_i\ </math> and <math>\ \mathbf{r}_j\ </math> are the [[position vector]]s of objects <math>\ i\ </math> and <math>\ j\ </math> respectively, and <math>\ r_{ij} \equiv \| \mathbf{r}_j-\mathbf{r}_i \|\ </math> is the distance from object <math>i</math> to object <math>\ j\ </math>, all [[Euclidean vector#Physics|vectors]] being referred to the [[Center of mass#Astronomy|barycenter]] of the system. This equation is resolved into components in <math>\ x\ ,</math> <math>\ y\ ,</math> and <math>\ z\ ,</math> 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.<ref name=danby>
{{cite book 
 |last = Danby |first = J.M.A.
 |year=1988
 |title = Fundamentals of Celestial Mechanics |edition = 2nd
 |at = chapter&nbsp;11
 |publisher = Willmann-Bell, Inc. 
 |isbn = 0-943396-20-4 
}}
</ref> 
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 figures|significant digits]] in the [[arithmetic]] because of the large difference in the forces of the central body and the perturbing bodies, although with [[double-precision floating-point format|high precision numbers]] built into modern [[computer]]s this is not as much of a limitation as it once was.<ref>
{{cite book 
 |last = Herget |first = Paul
 |year = 1948
 |title = The Computation of Orbits
 |publisher = self-published 
 |page=91&nbsp;ff
}}
</ref>

==== Encke's method ====
[[File:Enckes method-vector.svg|thumb|Encke's method. Greatly exaggerated here, the small difference δ'''r''' (blue) between the osculating, unperturbed orbit (black) and the perturbed orbit (red), is numerically integrated starting from the initial position (the ''epoch of osculation'').]]

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.<ref>{{cite book | last = Encke | first = J.F. |author-link = Johann Franz Encke | publication-date = 1854 |year = 1857 | title = Über die allgemeinen Störungen der Planeten | series = [[Berliner Astronomisches Jahrbuch]] für 1857 | pages = 319–397 | url = https://books.google.com/books?id=t5VCAQAAMAAJ }}</ref>
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''.<ref name=danby/> 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.<ref>{{harvp|Battin|1999|at=§10.2}}</ref>

Letting <math>\boldsymbol{\rho}</math> be the [[Position vector|radius vector]] of the [[osculating orbit]], <math>\mathbf{r}</math> the radius vector of the perturbed orbit, and <math>\delta \mathbf{r}</math> the variation from the osculating orbit,

{{NumBlk|:|<math>\delta \mathbf{r} = \mathbf{r} - \boldsymbol{\rho}</math>, and the [[Equations of motion|equation of motion]] of <math>\delta \mathbf{r}</math> is simply|{{EquationRef|1}}}}

{{NumBlk|:|<math>\delta \ddot{\mathbf{r}} = \mathbf{\ddot{r}} - \boldsymbol{\ddot{\rho}}</math>.|{{EquationRef|2}}}}

<math>\mathbf{\ddot{r}}</math> and <math>\boldsymbol{\ddot{\rho}}</math> are just the equations of motion of <math>\mathbf{r}</math> and <math>\boldsymbol{\rho},</math>

{{NumBlk|:|<math>\mathbf{\ddot{r}} = \mathbf{a}_{\text{per}} - {\mu \over r^3} \mathbf{r}</math> for the perturbed orbit and |{{EquationRef|3}}}}

{{NumBlk|:|<math>\boldsymbol{\ddot{\rho}} = - {\mu \over \rho^3} \boldsymbol{\rho}</math> for the unperturbed orbit,|{{EquationRef|4}}}}

where <math>\mu = G(M+m)</math> is the [[Standard gravitational parameter|gravitational parameter]] with <math>M</math> and <math>m</math> the [[mass]]es of the central body and the perturbed body, <math>\mathbf{a}_{\text{per}}</math> is the perturbing [[acceleration]], and <math>r</math> and <math>\rho</math> are the magnitudes of <math>\mathbf{r}</math> and <math>\boldsymbol{\rho}</math>.

Substituting from equations ({{EquationNote|3}}) and ({{EquationNote|4}}) into equation ({{EquationNote|2}}),

{{NumBlk|:|<math>\delta \ddot{\mathbf{r}} = \mathbf{a}_{\text{per}} + \mu \left( {\boldsymbol{\rho} \over \rho^3} - {\mathbf{r} \over r^3} \right),</math> |{{EquationRef|5}}}}

which, in theory, could be integrated twice to find <math>\delta \mathbf{r}</math>. Since the osculating orbit is easily calculated by two-body methods, <math>\boldsymbol{\rho}</math> and <math>\delta \mathbf{r}</math> are accounted for and <math>\mathbf{r}</math> can be solved. In practice, the quantity in the brackets, <math> {\boldsymbol{\rho} \over \rho^3} - {\mathbf{r} \over r^3} </math>, is the difference of two nearly equal vectors, and further manipulation is necessary to avoid the need for extra [[Significant figures|significant digits]].<ref>{{harvp|Bate|Mueller|White|1971|at=§9.3}}</ref><ref>{{harvp|Roy|1988|at=§7.4}}</ref> Encke's method was more widely used before the advent of modern [[computer]]s, when much orbit computation was performed on [[Calculating machine|mechanical calculating machines]].

== Periodic nature ==
[[File:Eccentricity rocky planets.jpg|thumb|300px|[http://www.orbitsimulator.com/gravity/articles/what.html Gravity Simulator] plot of the changing [[orbital eccentricity]] of [[Mercury (planet)|Mercury]], [[Venus]], [[Earth]], and [[Mars]] over the next 50,000&nbsp;years. The zero-point on this plot is the year 2007.]]

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 of the Moon|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 element]]s, most apparent when two planets' orbital periods are nearly in sync. For instance, five orbits of [[Jupiter]] (59.31&nbsp;years) is nearly equal to two of [[Saturn]] (58.91&nbsp;years). This causes large perturbations of both, with a period of 918&nbsp;years, the time required for the small difference in their positions at [[Conjunction (astronomy and astrology)|conjunction]] to make one complete circle, first discovered by [[Pierre-Simon Laplace|Laplace]].<ref name=moulton/> [[Venus]] currently has the orbit with the least [[Orbital eccentricity|eccentricity]], i.e. it is the closest to [[Circle|circular]], of all the planetary orbits. In 25,000&nbsp;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 [[planet]]s can cross the orbit of another, leading to collisions.{{efn|
See references for the Wikipedia article [[Stability of the Solar System]].
}}

The orbits of many of the minor bodies of the Solar System, such as [[comet]]s, are often heavily perturbed, particularly by the gravitational fields of the [[gas giant]]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&nbsp;1996, [[Jupiter]]'s gravitational influence caused the [[Orbital period|period]] of [[Comet Hale–Bopp]]'s orbit to decrease from 4,206 to 2,380&nbsp;years, a change that will not revert on any periodic basis.<ref name=perturb>
{{cite web
 |first=Don |last=Yeomans
 |date=1997-04-10 |df=dmy-all
 |title=Comet Hale–Bopp orbit and ephemeris information
 |publisher=NASA [[Jet Propulsion Laboratory]]
 |place=Pasadena, CA
 |url=http://www2.jpl.nasa.gov/comet/ephemjpl8.html
 |access-date=2008-10-23
}}
</ref>

== See also ==
{{div col begin|colwidth=18em}}
* [[Formation and evolution of the Solar System]]
* [[Frozen orbit]]
* [[Molniya orbit]]
* [[Nereid (moon)|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]]
* [[Perturbation theory]]
* [[Proper orbital elements]]
* [[Stability of the Solar System]]
{{div col end}}

== References ==
;Footnotes
{{notelist}}

;Citations
{{reflist}}

;Bibliography
{{refbegin|25em|small=yes}}
* {{cite book
 |last1=Bate    |first1=Roger R.   |author1-link=Roger R. Bate
 |last2=Mueller |first2=Donald D.  |author2-link=
 |last3=White   |first3=Jerry E.   |author3-link=Jerry White (Navigators)
 |year=1971
 |title=Fundamentals of Astrodynamics
 |publisher=[[Dover Publications]]
 |location=New York, NY
 |isbn=0-486-60061-0
 |url=https://archive.org/details/fundamentalsofas00bate
 |url-access=registration
}}
* {{cite book 
|last = Battin 
|first = Richard H. 
| title = An Introduction to the Mathematics and Methods of Astrodynamics, Revised Edition 
|publisher = American Institute of Aeronautics and Astronautics, Inc. 
|isbn = 1-56347-342-9 
|date=1999
}}
* {{cite book
 |last=Moulton |first=F.R.  |author-link=Forest Ray Moulton
 |year=1914
 |title=An Introduction to Celestial Mechanics
 |edition=2nd, revised
 |publisher=Macmillan
 |url=https://archive.org/details/anintroductiont04moulgoog
}}
* {{cite book
 |last=Roy |first=A.E.  |author-link=Archie Roy
 |year=1988
 |title=Orbital Motion |edition=3rd
 |publisher=Institute of Physics Publishing
 |isbn=0-85274-229-0
}}
{{refend}}

==Further reading==
* P.E. El'Yasberg: [https://archive.org/details/nasa_techdoc_19670020827 Introduction to the Theory of Flight of Artificial Earth Satellites]

== External links ==
*[http://www.solexorb.it/SolexOld/MarsDist.html Solex] (by Aldo Vitagliano) predictions for the position/orbit/close approaches of Mars
*[https://books.google.com/books?id=snK4AAAAIAAJ Gravitation] Sir George Biddell Airy's 1884 book on gravitational motion and perturbations, using little or no math.(at [https://books.google.com/books Google books])

{{Orbits}}
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[[Category:Orbital perturbations|*]]
[[Category:Dynamical systems]]
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[[Category:Celestial mechanics]]