Mean anomaly

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In the study of orbital dynamics the mean anomaly of an orbiting body is the angle the body would have traveled about the center of the orbit's auxiliary circle. Unlike other measures of anomaly, the mean anomaly grows linearly with time. Because of the linear growth with time, the mean anomaly makes calculating the time of flight between two points on the orbit very easy. The mean anomalies for the two points are calculated and their difference is found. Knowing this, the ratio of this difference relative to the entire <math>2\pi</math> encompassing one orbit is simply equal to ratio of the time of flight to the period of one whole orbit (i.e. <math>\frac{M_2 - M_1}{2\pi} = \frac{t}{T}</math>).

When measured in radians the mean anomaly has a value of <math>0</math> at the initial crossing of the orbit's point of periapsis, and a multiple of <math>2\pi</math> at any later crossing of the point periapsis. In the diagram below, the mean anomaly of point <math>p</math> on the orbit around <math>s</math> is given by angle <math>M</math> (the angle <math>\angle zcy</math>).

Image:Kepler's equation scheme.svg

The point y is defined such that the circular sector area z-c-y is equal to the elliptic sector area z-s-p, scaled up by the ratio of the major to minor axes of the ellipse. Or, in other words, the circular sector area z-c-y is equal to the area x-s-z.

[edit] Calculation

In astrodynamics mean anomaly <math>M\,\!</math> can be calculated as follows:

where:

<math>M_0\,\!</math> is the mean anomaly at time <math>t_0\,\!</math>,
<math>t_0\,\!</math> is the start time,
<math>t\,\!</math> is the time of interest, and
<math>n\,\!</math> is the mean motion.

Alternatively:

where:

<math>E\,\!</math> is orbit's eccentric anomaly,
<math>e\,\!</math> is orbit's eccentricity.

[edit] See also

v • d • e

Orbits

Types

General	Box · Circular · Non-inclined · Elliptic (Highly Elliptical) · Graveyard · Hyperbolic trajectory · Inclined · Osculating · Parabolic trajectory · Capture · Escape · Semi-synchronous · Subsynchronous · Synchronous · Parking
Geocentric	Geosynchronous · Geostationary · Sun-synchronous · Low Earth · Medium Earth · Molniya · Near equatorial · Moon · Polar · Tundra
Other	Areosynchronous · Areostationary · Halo · Lissajous · Lunar · Heliocentric · Heliosynchronous

v • d • e

Parameters

Classical orbital elements
<math>i\,\!</math> Inclination
<math>\Omega\,\!</math> Longitude of the ascending node
<math>e\,\!</math> Eccentricity

<math>\omega\,\!</math> Argument of periapsis
<math>a\,\!</math> Semi-major axis
<math>M_o\,\!</math> Mean anomaly at epoch

Other parameters
<math>v\,</math> True anomaly
<math>b\,</math> Semi-minor axis
<math>\epsilon\,</math> Linear eccentricity
<math>E\,</math> Eccentric anomaly

<math>L\,</math> Mean longitude
<math>l\,</math> True longitude
<math>T\,</math> Orbital period

Maneuvers

Bi-elliptic transfer · Geostationary transfer · Gravity assist · Hohmann transfer · Inclination change · Phasing · Rendezvous