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In celestial mechanics
Celestial mechanics

Celestial mechanics is the branch of astronomy that deals with the motion s of celestial objects. The field applies principles of physics, historically classical mechanics, to astronomical objects such as stars and planets to produce ephemeris data....
, mean anomaly is one of the orbital elements
Orbital elements

In celestial mechanics, the elements of an orbit are the parameters needed to specify that orbit uniquely. Orbital elements are generally considered in classical mechanics two-body systems, where a Kepler orbit is used ....
 that defines a Kepler orbit
Kepler orbit

In celestial mechanics, a Kepler orbit describes the motion of an orbiting body as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space....
. It specifies the position of the orbiting objects along the ellipse defined by the other elements, but does not correspond to an actual geometric angle. True anomaly
True anomaly

In astronomy, the true anomaly is the angle between the direction z-s of periapsis and the current position p of an object on its orbit, measured at the focus s of the ellipse ....
 specifies the real geometric angle between periapsis (closest approach to the central body) and the position of the orbiting body at any given time. Mean anomaly is similar, but unlike the real position of the orbiting body, always varies linearly with time (which is more mathematically convenient).

In the two-body problem
Two-body problem

In classical mechanics, the two-body problem is to determine the motion of two point particles that interact only with each other. Common examples include a satellite orbiting a planet, a planet orbiting a star, two stars orbiting each other , and a classical electron orbiting an atomic nucleus....
 for the case corresponding to an elliptic Kepler orbit, the relation between the Eccentric anomaly
Eccentric anomaly

The definition of eccentric anomaly for an ellipse as a geometric figure directly applies for an elliptic Kepler orbit. The definitions of the true anomaly and the eccentric anomaly for an ellipse and the relations between these entities are all in Ellipse#True anomaly and Ellipse#Eccentric anomaly....
  and the time is:

   (1)

where is the eccentricity
Orbital eccentricity

In astrodynamics, under standard assumptions in astrodynamics, any orbit must be of conic section shape. The eccentricity of this conic section, the orbit's eccentricity, is an important parameter of the orbit that defines its absolute shape....
 and is time counted from a pericentre
Apsis

In celestial mechanics, an apsis, plural apsides is the point of greatest or least distance of the elliptical orbit of an object from its center of attraction, which is generally the center of mass of the system....
 passage.






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In celestial mechanics
Celestial mechanics

Celestial mechanics is the branch of astronomy that deals with the motion s of celestial objects. The field applies principles of physics, historically classical mechanics, to astronomical objects such as stars and planets to produce ephemeris data....
, mean anomaly is one of the orbital elements
Orbital elements

In celestial mechanics, the elements of an orbit are the parameters needed to specify that orbit uniquely. Orbital elements are generally considered in classical mechanics two-body systems, where a Kepler orbit is used ....
 that defines a Kepler orbit
Kepler orbit

In celestial mechanics, a Kepler orbit describes the motion of an orbiting body as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space....
. It specifies the position of the orbiting objects along the ellipse defined by the other elements, but does not correspond to an actual geometric angle. True anomaly
True anomaly

In astronomy, the true anomaly is the angle between the direction z-s of periapsis and the current position p of an object on its orbit, measured at the focus s of the ellipse ....
 specifies the real geometric angle between periapsis (closest approach to the central body) and the position of the orbiting body at any given time. Mean anomaly is similar, but unlike the real position of the orbiting body, always varies linearly with time (which is more mathematically convenient).

In the two-body problem
Two-body problem

In classical mechanics, the two-body problem is to determine the motion of two point particles that interact only with each other. Common examples include a satellite orbiting a planet, a planet orbiting a star, two stars orbiting each other , and a classical electron orbiting an atomic nucleus....
 for the case corresponding to an elliptic Kepler orbit, the relation between the Eccentric anomaly
Eccentric anomaly

The definition of eccentric anomaly for an ellipse as a geometric figure directly applies for an elliptic Kepler orbit. The definitions of the true anomaly and the eccentric anomaly for an ellipse and the relations between these entities are all in Ellipse#True anomaly and Ellipse#Eccentric anomaly....
  and the time is:

   (1)

where is the eccentricity
Orbital eccentricity

In astrodynamics, under standard assumptions in astrodynamics, any orbit must be of conic section shape. The eccentricity of this conic section, the orbit's eccentricity, is an important parameter of the orbit that defines its absolute shape....
 and is time counted from a pericentre
Apsis

In celestial mechanics, an apsis, plural apsides is the point of greatest or least distance of the elliptical orbit of an object from its center of attraction, which is generally the center of mass of the system....
 passage. The mean anomaly is then defined as:

   (2)

This fictive "angle" without direct geometrical meaning increases uniformly with time:

   (3)

To find the position of the object in an elliptic Kepler orbit at a certain time, the corresponding mean anomaly is determined with (3) and then the corresponding Eccentric anomaly
Eccentric anomaly

The definition of eccentric anomaly for an ellipse as a geometric figure directly applies for an elliptic Kepler orbit. The definitions of the true anomaly and the eccentric anomaly for an ellipse and the relations between these entities are all in Ellipse#True anomaly and Ellipse#Eccentric anomaly....
 is found from (2) using the Newton-Raphson algorithm.

See also

  • Kepler orbit
    Kepler orbit

    In celestial mechanics, a Kepler orbit describes the motion of an orbiting body as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space....
  • Ellipse
    Ellipse

    In mathematics, an ellipse is the apparent shape of a circle viewed obliquely from outside it, as distinct from a hyperbola which is the shape seen from inside....
  • Eccentric anomaly
    Eccentric anomaly

    The definition of eccentric anomaly for an ellipse as a geometric figure directly applies for an elliptic Kepler orbit. The definitions of the true anomaly and the eccentric anomaly for an ellipse and the relations between these entities are all in Ellipse#True anomaly and Ellipse#Eccentric anomaly....
  • True anomaly
    True anomaly

    In astronomy, the true anomaly is the angle between the direction z-s of periapsis and the current position p of an object on its orbit, measured at the focus s of the ellipse ....