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Concept# Mean anomaly

Summary

In celestial mechanics, the mean anomaly is the fraction of an elliptical orbit's period that has elapsed since the orbiting body passed periapsis, expressed as an angle which can be used in calculating the position of that body in the classical two-body problem. It is the angular distance from the pericenter which a fictitious body would have if it moved in a circular orbit, with constant speed, in the same orbital period as the actual body in its elliptical orbit.
Define T as the time required for a particular body to complete one orbit. In time T, the radius vector sweeps out 2pi radians, or 360°. The average rate of sweep, n, is then
which is called the mean angular motion of the body, with dimensions of radians per unit time or degrees per unit time.
Define τ as the time at which the body is at the pericenter. From the above definitions, a new quantity, M, the mean anomaly can be defined
which gives an angular distance from the pericenter at arbitrary time t. 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 2pi radians or 0° to 360° during each orbit. It is equal to 0 when the body is at the pericenter, pi radians (180°) at the apocenter, and 2pi radians (360°) after one complete revolution. 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 where δt 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 the eccentric anomaly and the true anomaly.

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Kepler orbit

In celestial mechanics, a Kepler orbit (or Keplerian orbit, named after the German astronomer Johannes Kepler) is the motion of one body relative to another, as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space. A Kepler orbit can also form a straight line. It considers only the point-like gravitational attraction of two bodies, neglecting perturbations due to gravitational interactions with other objects, atmospheric drag, solar radiation pressure, a non-spherical central body, and so on.

Semi-major and semi-minor axes

In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the longest semidiameter or one half of the major axis, and thus runs from the centre, through a focus, and to the perimeter. The semi-minor axis (minor semiaxis) of an ellipse or hyperbola is a line segment that is at right angles with the semi-major axis and has one end at the center of the conic section.

Mean anomaly

In celestial mechanics, the mean anomaly is the fraction of an elliptical orbit's period that has elapsed since the orbiting body passed periapsis, expressed as an angle which can be used in calculating the position of that body in the classical two-body problem. It is the angular distance from the pericenter which a fictitious body would have if it moved in a circular orbit, with constant speed, in the same orbital period as the actual body in its elliptical orbit. Define T as the time required for a particular body to complete one orbit.

Discusses mean anomaly in circular orbits, exploring its relationship with velocity, period, and angular velocity.

Explores true anomaly in elliptical orbits and the transformation between Cartesian and polar coordinates.

Explains the relationship between eccentric anomaly and mean anomaly in orbits.