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Perturbation (astronomy)

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Classical approach to the many-body problem of astronomy

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 theSun on theMoon at two places in itsorbit. The blue arrows represent thedirection and magnitude of the gravitational force on theEarth. 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
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Orbital mechanics
Orbital elements Apsis Argument of periapsis Eccentricity Inclination Mean anomaly Orbital nodes Semi-major axis True anomaly
Types oftwo-body orbits by eccentricity Circular orbit Elliptic orbit Transfer orbit (Hohmann transfer orbit Bi-elliptic transfer orbit) Parabolic orbit Hyperbolic orbit Radial orbit Decaying orbit
Equations Dynamical friction Escape velocity Kepler's equation Kepler's laws of planetary motion Orbital period Orbital velocity Surface gravity Specific orbital energy Vis-viva equation
Celestial mechanics
Gravitational influences Barycenter Hill sphere Perturbations Sphere of influence
N-body orbits Lagrangian points (Halo orbits) Lissajous orbits Lyapunov orbits
Engineering and efficiency
Preflight engineering Mass ratio Payload fraction Propellant mass fraction Tsiolkovsky rocket equation
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Inastronomy,perturbation is the complex motion of amassive body subjected to forces other than thegravitational attraction of a single othermassive body.^[1] The other forces can include a third (fourth, fifth, etc.) body,resistance, as from anatmosphere, and the off-center attraction of anoblate or otherwise misshapen body.^[2]

Introduction

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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 ofmotion and ofgravitation, applied them to the first analysis of perturbations,^[2] recognizing the complex difficulties of their calculation.^[a]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 theMoon andplanets formarine 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 aconic section, and can be described ingeometrical terms. This is called atwo-body problem, or an unperturbedKeplerian 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 athree-body problem; if there are multiple other bodies it is ann‑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.^[6]

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's orbital longitude and latitude, as perturbed byVenus,Jupiter, and all of the planets of theSolar System, at intervals of 2.5 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, astar, 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 orsatellite around its primary body.

Mathematical analysis

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General perturbations

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In methods ofgeneral perturbations, general differential equations, either of motion or of change in theorbital elements, are solved analytically, usually byseries 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.^[7] Historically, general perturbations were investigated first. The classical methods are known asvariation of the elements,variation of parameters orvariation of the constants of integration. In these methods, it is considered that the body is always moving in aconic 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 theosculating orbit and itsorbital 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 ofcelestial 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.^[6] In theSolar System, this is usually the case;Jupiter, the second largest body, has a mass of about⁠1/ 1000 ⁠ that of theSun.

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, anorbital resonance) which caused them would be available.^[6]

Special perturbations

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In methods ofspecial perturbations, numerical datasets, representing values for the positions, velocities and accelerative forces on the bodies of interest, are made the basis ofnumerical integration of the differentialequations of motion.^[8] In effect, the positions and velocities are perturbed directly, and no attempt is made to calculate the curves of the orbits or theorbital elements.^[2]

Special perturbations can be applied to any problem incelestial mechanics, as it is not limited to cases where the perturbing forces are small.^[6] Once applied only to comets and minor planets, special perturbation methods are now the basis of the most accurate machine-generatedplanetary ephemerides of the great astronomical almanacs.^[2]^[b] Special perturbations are also used formodeling an orbit with computers.

Cowell's formulation

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Cowell's method. Forces from all perturbing bodies (black and gray) are summed to form the total force on body $\ i\$ (red), and this is numerically integrated starting from the initial position (theepoch of osculation).

{\displaystyle \ i\ } — Cowell's method. Forces from all perturbing bodies (black and gray) are summed to form the total force on body $\ i\$ (red), and this is numerically integrated starting from the initial position (theepoch of osculation).

Cowell's formulation (so named forPhilip 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.^[9] In a system of $\ n\$ mutually interacting bodies, this method mathematically solves for theNewtonian forces on body $\ i\$ by summing the individual interactions from the other $j {\displaystyle j}$ bodies:

\mathbf {\ddot {r}} _{i}=\sum _{\underset {j\neq i}{j=1}}^{n}\ G\ m_{j}{\frac {\ (\mathbf {r} _{j}-\mathbf {r} _{i})\ }{\ \|\mathbf {r} _{j}-\mathbf {r} _{i}\|^{3}}}

where $\ \mathbf {\ddot {r}} _{i}\$ is theacceleration vector of body $i {\displaystyle i}$ , $G {\displaystyle G}$ is thegravitational constant, $\ m_{j}\$ is themass of body $j {\displaystyle j}$ , $\ \mathbf {r} _{i}\$ and $\ \mathbf {r} _{j}\$ are theposition vectors of objects $\ i\$ and $\ j\$ respectively, and $\ r_{ij}\equiv \|\mathbf {r} _{j}-\mathbf {r} _{i}\|\$ is the distance from object $i {\displaystyle i}$ to object $\ j\$ , allvectors being referred to thebarycenter 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.^[10] However, for many problems incelestial mechanics, this is never the case. Another disadvantage is that in systems with a dominant central body, such as theSun, it is necessary to carry manysignificant digits in thearithmetic because of the large difference in the forces of the central body and the perturbing bodies, although withhigh precision numbers built into moderncomputers this is not as much of a limitation as it once was.^[11]

Encke's method

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Encke's method begins with theosculating orbit as a reference and integrates numerically to solve for the variation from the reference as a function of time.^[12]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 asrectification.^[10] 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.^[13]

Letting ${\boldsymbol {\rho }}$ be theradius vector of theosculating orbit, $\mathbf {r}$ the radius vector of the perturbed orbit, and $\delta \mathbf {r}$ the variation from the osculating orbit,

\delta \mathbf {r} =\mathbf {r} -{\boldsymbol {\rho }}

, and theequation of motion of

\delta \mathbf {r}

is simply

\delta {\ddot {\mathbf {r} }}=\mathbf {\ddot {r}} -{\boldsymbol {\ddot {\rho }}}

$\mathbf {\ddot {r}}$ and ${\boldsymbol {\ddot {\rho }}}$ are just the equations of motion of $\mathbf {r}$ and ${\boldsymbol {\rho }},$

\mathbf {\ddot {r}} =\mathbf {a} _{\text{per}}-{\mu  \over r^{3}}\mathbf {r}

for the perturbed orbit and

{\boldsymbol {\ddot {\rho }}}=-{\mu  \over \rho ^{3}}{\boldsymbol {\rho }}

for the unperturbed orbit,

where $\mu =G(M+m)$ is thegravitational parameter with $M {\displaystyle M}$ and $m {\displaystyle m}$ themasses of the central body and the perturbed body, $\mathbf {a} _{\text{per}}$ is the perturbingacceleration, and $r {\displaystyle r}$ and $\rho$ are the magnitudes of $\mathbf {r}$ and ${\boldsymbol {\rho }}$ .

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

\delta {\ddot {\mathbf {r} }}=\mathbf {a} _{\text{per}}+\mu \left({{\boldsymbol {\rho }} \over \rho ^{3}}-{\mathbf {r}  \over r^{3}}\right),

which, in theory, could be integrated twice to find $\delta \mathbf {r}$ . Since the osculating orbit is easily calculated by two-body methods, ${\boldsymbol {\rho }}$ and $\delta \mathbf {r}$ are accounted for and $\mathbf {r}$ can be solved. In practice, the quantity in the brackets, ${{\boldsymbol {\rho }} \over \rho ^{3}}-{\mathbf {r} \over r^{3}}$ , is the difference of two nearly equal vectors, and further manipulation is necessary to avoid the need for extrasignificant digits.^[14]^[15] Encke's method was more widely used before the advent of moderncomputers, when much orbit computation was performed onmechanical calculating machines.

Periodic nature

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Gravity Simulator plot of the changingorbital eccentricity ofMercury,Venus,Earth, andMars over the next 50,000 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 perturbedorbit, which is the subject oflunar theory. This periodic nature led to thediscovery of Neptune in 1846 as a result of its perturbations of the orbit ofUranus.

On-going mutual perturbations of the planets cause long-term quasi-periodic variations in theirorbital elements, most apparent when two planets' orbital periods are nearly in sync. For instance, five orbits ofJupiter (59.31 years) is nearly equal to two ofSaturn (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 atconjunction to make one complete circle, first discovered byLaplace.^[2]Venus currently has the orbit with the leasteccentricity, i.e. it is the closest tocircular, 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 theSolar System can become chaotic over very long time scales; under some circumstances one or moreplanets can cross the orbit of another, leading to collisions.^[c]

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

References

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Footnotes

^Newton (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."^[3]^[5]
^See, for instance, the Wikipedia article on theJet Propulsion Laboratory Development Ephemeris.
^See references for the Wikipedia articleStability of the Solar System.

Citations

^Bate, Mueller & White (1971), ch. 9, p. 385
^^a ^b ^c ^d ^e ^fMoulton (1914), ch. IX
^^a ^bNewton quoted by Prof G.E. Smith (Tufts University), in
Smith, G.E."Closing the loop: Testing Newtonian gravity, then and now"(PowerPoint) (symposium talk). Three lectures on the role of theory in science. Stanford University.
^Egerton, R.F."Newton" (course notes). Physics 311-12. Portland, OR:Portland State University. Archived fromthe original on 2005-03-10 – via physics.pdx.edu.
^After quoting the same passage from Newton^[3] Prof R.F. Egerton (Portland State University) concludes: "Here, Newton identifies the "many body problem" which remains unsolved analytically."^[4]
^^a ^b ^c ^dRoy (1988), ch. 6–7
^Bate, Mueller & White (1971), p. 387; p. 410 §9.4.3
^Bate, Mueller & White (1971), pp. 387–409
^Cowell, P.H.; Crommelin, A.C.D. (1910). "Investigation of the motion of Halley's comet from 1759 to 1910".Greenwich Observations in Astronomy.71. Bellevue, for His Majesty's Stationery Office: Neill & Co.: O1.Bibcode:1911GOAMM..71O...1C.
^^a ^bDanby, J.M.A. (1988).Fundamentals of Celestial Mechanics (2nd ed.). Willmann-Bell, Inc. chapter 11.ISBN 0-943396-20-4.
^Herget, Paul (1948).The Computation of Orbits. self-published. p. 91 ff.
^Encke, J.F. (1857).Über die allgemeinen Störungen der Planeten.Berliner Astronomisches Jahrbuch für 1857 (published 1854). pp. 319–397.
^Battin (1999), §10.2
^Bate, Mueller & White (1971), §9.3
^Roy (1988), §7.4
^Yeomans, Don (10 April 1997)."Comet Hale–Bopp orbit and ephemeris information". Pasadena, CA: NASAJet Propulsion Laboratory. Retrieved23 October 2008.

Bibliography

Bate, Roger R.; Mueller, Donald D.;White, Jerry E. (1971).Fundamentals of Astrodynamics. New York, NY:Dover Publications.ISBN 0-486-60061-0.
Battin, Richard H. (1999).An Introduction to the Mathematics and Methods of Astrodynamics, Revised Edition. American Institute of Aeronautics and Astronautics, Inc.ISBN 1-56347-342-9.
Moulton, F.R. (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.

External links

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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.(atGoogle books)

Gravitationalorbits

Types

General	Box Circular Elliptical /Highly elliptical Horseshoe Hyperbolic trajectory Inclined /Non-inclined Kepler Lagrange point Osculating Parabolic / Escape / Capture Parking Prograde / Retrograde Radial Synchronous semi sub Transfer orbit
Geocentric	Geosynchronous Geostationary Geostationary transfer Graveyard High Earth Low Earth Medium Earth Molniya Near-equatorial Orbit of the Moon Polar Sun-synchronous Transatmospheric Tundra Very low Earth
About other points	Mars Areocentric Areosynchronous Areostationary Lagrange points Distant retrograde Halo Lissajous Libration Lunar Sun Heliocentric Earth's orbit Mars cycler Heliosynchronous Other Lunar cycler

Parameters

Shape Size	e Eccentricity a Semi-major axis b Semi-minor axis Q, q Apsides
Orientation	i Inclination Ω Longitude of the ascending node ω Argument of periapsis ϖ Longitude of the periapsis
Position	M Mean anomaly ν,θ,f True anomaly E Eccentric anomaly L Mean longitude l True longitude
Variation	T Orbital period n Mean motion v Orbital speed t₀ Epoch