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Mean anomaly

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Specifies the orbit of an object in space

Area swept out per unit time by an object in anelliptical orbit, and by an imaginary object in acircular orbit (with the same orbital period). Both sweep out equal areas in equal times, but the angular rate of sweep varies for the elliptical orbit and is constant for the circular orbit. Shown aremean anomaly andtrue anomaly for two units of time. (Note that for visual simplicity, a non-overlapping circular orbit is diagrammed, thus this circular orbit with same orbital period is not shown in true scale with this elliptical orbit: for scale to be true for the two orbits of equal period, these orbits must intersect.)

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
Efficiency measures Gravity assist Oberth effect
Propulsive maneuvers Orbital maneuver Orbit insertion
v t e

Incelestial mechanics, themean anomaly is the fraction of anelliptical orbit's period that has elapsed since the orbiting body passedperiapsis, expressed as anangle which can be used in calculating the position of that body in the classicaltwo-body problem. It is the angular distance from thepericenter which a fictitious body would have if it moved in acircular orbit, with constantspeed, in the sameorbital period as the actual body in its elliptical orbit.^[1]^[2]

Definition

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DefineT as the time required for a particular body to complete one orbit. In timeT, theradius vector sweeps out 2π radians, or 360°. The average rate of sweep,n, is then

$n={\frac {\,2\,\pi \,}{T}}={\frac {\,360^{\circ }\,}{T}}~,$

which is called themean angular motion of the body, with dimensions of radians per unit time or degrees per unit time.

Defineτ as the time at which the body is at the pericenter. From the above definitions, a new quantity,M, themean anomaly can be defined

$M=n\,(t-\tau )~,$

which gives an angular distance from the pericenter at arbitrary timet^[3] with dimensions of radians or degrees.

Because the rate of increase,n, is a constant average, the mean anomaly increases uniformly (linearly) from 0 to 2π radians or 0° to 360° during each orbit. It is equal to 0 when the body is at the pericenter,π radians (180°) at theapocenter, and 2π radians (360°) after one complete revolution.^[4] If the mean anomaly is known at any given instant, it can be calculated at any later (or prior) instant by simply adding (or subtracting)n⋅δt whereδt represents the small time difference.

Mean anomaly does not measure an angle between any physical objects (except at pericenter or apocenter, or for a circular orbit). It is simply a convenient uniform measure of how far around its orbit a body has progressed since pericenter. The mean anomaly is one of three angular parameters (known historically as "anomalies") that define a position along an orbit, the other two being theeccentric anomaly and thetrue anomaly.

Mean anomaly at epoch

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Themean anomaly at epoch,M₀, is defined as the instantaneous mean anomaly at a givenepoch,t₀. This value is sometimes provided with other orbital elements to enable calculations of the object's past and future positions along the orbit. The epoch for whichM₀ is defined is often determined by convention in a given field or discipline. For example, planetary ephemerides often defineM₀ for the epochJ2000, while for earth orbiting objects described by atwo-line element set the epoch is specified as a date in the first line.^[5]

Formulae

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The mean anomalyM can be computed from theeccentric anomalyE and theeccentricitye withKepler's equation:

$M=E-e\,\sin E~.$

Mean anomaly is also frequently seen as

$M=M_{0}+n\left(t-t_{0}\right)~,$

whereM₀ is the mean anomaly at the epocht₀, which may or may not coincide withτ, the time of pericenter passage. The classical method of finding the position of an object in an elliptical orbit from a set of orbital elements is to calculate the mean anomaly by this equation, and then to solve Kepler's equation for the eccentric anomaly.

Defineϖ as thelongitude of the pericenter, the angular distance of the pericenter from a reference direction. Defineℓ as themean longitude, the angular distance of the body from the same reference direction, assuming it moves with uniform angular motion as with the mean anomaly. Thus mean anomaly is also^[6]

M=\ell -\varpi ~.

Mean angular motion can also be expressed,

$n={\sqrt {{\frac {\mu }{\;a^{3}\,}}\,}}~,$

whereμ is thegravitational parameter, which varies with the masses of the objects, anda is thesemi-major axis of the orbit. Mean anomaly can then be expanded,

$M={\sqrt {{\frac {\mu }{\;a^{3}\,}}\,}}\,\left(t-\tau \right)~,$

and here mean anomaly represents uniform angular motion on a circle of radiusa.^[7]

Mean anomaly can be calculated from the eccentricity and thetrue anomalyv by finding the eccentric anomaly and then using Kepler's equation. This gives, in radians: $M=\operatorname {atan2} \left({\sqrt {1-e^{2}}}\sin \nu ,e+\cos \nu \right)-e{\frac {{\sqrt {1-e^{2}}}\sin \nu }{1+e\cos \nu }}$ whereatan2(y, x) is the angle from the x-axis of the ray from (0, 0) to (x, y), having the same sign as y.

For parabolic and hyperbolic trajectories the mean anomaly is not defined, because they don't have a period. But in those cases, as with elliptical orbits, the area swept out by a chord between the attractor and the object following the trajectory increases linearly with time. For the hyperbolic case, there is a formula similar to the above giving the elapsed time as a function of the angle (the true anomaly in the elliptic case), as explained in the articleKepler orbit. For the parabolic case there is a different formula, the limiting case for either the elliptic or the hyperbolic case as the distance between the foci goes to infinity – seeParabolic trajectory#Barker's equation.

Mean anomaly can also be expressed as aseries expansion:^[8] $M=\nu +2\sum _{n=1}^{\infty }(-1)^{n}\left[{\frac {1}{n}}+{\sqrt {1-e^{2}}}\right]\beta ^{n}\sin {n\nu }$

with $\beta ={\frac {1-{\sqrt {1-e^{2}}}}{e}}$

$M=\nu -2\,e\sin \nu +\left({\frac {3}{4}}e^{2}+{\frac {1}{8}}e^{4}\right)\sin 2\nu -{\frac {1}{3}}e^{3}\sin 3\nu +{\frac {5}{32}}e^{4}\sin 4\nu +\operatorname {\mathcal {O}} \left(e^{5}\right)$

A similar formula gives the true anomaly directly in terms of the mean anomaly:^[9]

$\nu =M+\left(2\,e-{\frac {1}{4}}e^{3}\right)\sin M+{\frac {5}{4}}e^{2}\sin 2M+{\frac {13}{12}}e^{3}\sin 3M+\operatorname {\mathcal {O}} \left(e^{4}\right)$

A general formulation of the above equation can be written as theequation of the center:^[10]

$\nu =M+2\sum _{s=1}^{\infty }{\frac {1}{s}}\left[J_{s}(se)+\sum _{p=1}^{\infty }\beta ^{p}{\big (}J_{s-p}(se)+J_{s+p}(se){\big )}\right]\sin(sM)$

References

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^Montenbruck, Oliver (1989).Practical Ephemeris Calculations.Springer-Verlag. p. 44.ISBN 0-387-50704-3.
^Meeus, Jean (1991).Astronomical Algorithms. Willmann-Bell, Inc., Richmond, VA. p. 182.ISBN 0-943396-35-2.
^Smart, W. M. (1977).Textbook on Spherical Astronomy (sixth ed.). Cambridge University Press, Cambridge. p. 113.ISBN 0-521-29180-1.
^Meeus (1991), p. 183
^"Space-Track.org".www.space-track.org. Retrieved2024-08-19.
^Smart (1977), p. 122
^Vallado, David A. (2001).Fundamentals of Astrodynamics and Applications (2nd ed.). El Segundo, California: Microcosm Press. pp. 53–54.ISBN 1-881883-12-4.
^Smart, W. M. (1953).Celestial Mechanics. London, UK: Longmans, Green, and Co. p. 38.
^Roy, A. E. (1988).Orbital Motion (1st ed.). Bristol, UK; Philadelphia, Pennsylvania: A. Hilger.ISBN 0852743602.
^Brouwer, Dirk (1961).Methods of celestial mechanics. Elsevier. pp. e.g. 77.

External links

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Glossary entryanomaly, mean Archived 2017-12-23 at theWayback Machine at the US Naval Observatory'sAstronomical Almanac Online Archived 2015-04-20 at theWayback Machine

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