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In astronomy
Astronomy

Astronomy is the science of Astronomical object and Phenomenon that originate outside the Earth's atmosphere . It is concerned with the evolution, physics, chemistry, meteorology, and motion of celestial objects, as well as the physical cosmology....
, the true anomaly (Greek nu, also written or ) is the angle between the direction z-s of periapsis and the current position p of an object on its orbit
ORBit

ORBit is a Common Object Request Broker Architecture 2.4 compliant Object Request Broker . It features mature C , C++ and Python bindings, and less developed bindings for Perl, Lisp , Pascal , Ruby , and Tcl....
, measured at the focus s of the 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....
 (the point around which the object orbits). In the diagram below, true anomaly is the angle z-s-p.

elliptic orbits true anomaly can be calculated from orbital state vectors
Orbital state vectors

In astrodynamics or celestial dynamics orbital state vectors are vectors of position and velocity that together with their time uniquely determine the state of an orbiting body....
 as:

  (if then replace by )

where:
For circular orbit
Circular orbit

In astrodynamics or celestial mechanics a circular orbit is an elliptic orbit with the eccentricity equal to 0. It is an example of a rotation around a fixed axis: this axis is the line through the center of mass perpendicular to the plane of motion....
s this can be simplified to:

  (if then replace by )

where:
For circular orbit
Circular orbit

In astrodynamics or celestial mechanics a circular orbit is an elliptic orbit with the eccentricity equal to 0. It is an example of a rotation around a fixed axis: this axis is the line through the center of mass perpendicular to the plane of motion....
s with the inclination of zero this can be simplified further to:

  (if then replace by )

where:

relation between ν and E, 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....
, is:

or equivalently

The relations between the radius (position vector magnitude) and the anomalies are:

and

where a is the orbit's semi-major axis
Semi-major axis

In geometry, the semi-major axis is used to describe the dimensions of ellipses and hyperbolae....
 (segment cz).






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In astronomy
Astronomy

Astronomy is the science of Astronomical object and Phenomenon that originate outside the Earth's atmosphere . It is concerned with the evolution, physics, chemistry, meteorology, and motion of celestial objects, as well as the physical cosmology....
, the true anomaly (Greek nu, also written or ) is the angle between the direction z-s of periapsis and the current position p of an object on its orbit
ORBit

ORBit is a Common Object Request Broker Architecture 2.4 compliant Object Request Broker . It features mature C , C++ and Python bindings, and less developed bindings for Perl, Lisp , Pascal , Ruby , and Tcl....
, measured at the focus s of the 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....
 (the point around which the object orbits). In the diagram below, true anomaly is the angle z-s-p.

Calculation from state vectors

For elliptic orbits true anomaly can be calculated from orbital state vectors
Orbital state vectors

In astrodynamics or celestial dynamics orbital state vectors are vectors of position and velocity that together with their time uniquely determine the state of an orbiting body....
 as:

  (if then replace by )

where:
  • is orbital velocity vector of the orbiting body,
  • is eccentricity vector
    Eccentricity vector

    In astrodynamics, the eccentricity vector of a conic section orbit is the vector pointing towards the periapsis and with magnitude equal to the orbit's scalar eccentricity ....
    ,
  • is orbital position vector (segment sp) of the orbiting body.
For circular orbit
Circular orbit

In astrodynamics or celestial mechanics a circular orbit is an elliptic orbit with the eccentricity equal to 0. It is an example of a rotation around a fixed axis: this axis is the line through the center of mass perpendicular to the plane of motion....
s this can be simplified to:

  (if then replace by )

where:
  • is vector pointing towards the ascending node (i.e. the z-component of is zero).
For circular orbit
Circular orbit

In astrodynamics or celestial mechanics a circular orbit is an elliptic orbit with the eccentricity equal to 0. It is an example of a rotation around a fixed axis: this axis is the line through the center of mass perpendicular to the plane of motion....
s with the inclination of zero this can be simplified further to:

  (if then replace by )

where:
  • is x-component of orbital position vector ,
  • is x-component of orbital velocity vector .


Other relations

The relation between ν and E, 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....
, is:

or equivalently

The relations between the radius (position vector magnitude) and the anomalies are:

and

where a is the orbit's semi-major axis
Semi-major axis

In geometry, the semi-major axis is used to describe the dimensions of ellipses and hyperbolae....
 (segment cz). Note that z is the closest approach to the focus s or object being orbited but also the furthest point from the center c.

See also

  • Kepler's laws of planetary motion
    Kepler's laws of planetary motion

    In astronomy, Kepler's three laws of planetary motion are*"The orbit of every planet is an ellipse with the sun at a Focus ."*"A line joining a planet and the sun sweeps out equal areas during equal intervals of time."...
  • 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....
  • Mean anomaly
    Mean anomaly

    In celestial mechanics, mean anomaly is one of the orbital elements that defines a Kepler orbit. 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....


Mathematical definitions:
  • Ellipse#True anomaly
    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....
  • Ellipse#Eccentric anomaly
    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....
  • Hyperbola#The true anomaly
    Hyperbola

    In mathematics a hyperbola is a smooth function planar curve having two connected components or branches, each a mirror image of the other and resembling two infinite bow aimed at each other....