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Concept# True anomaly

Summary

In celestial mechanics, true anomaly is an angular parameter that defines the position of a body moving along a Keplerian orbit. It is the angle between the direction of periapsis and the current position of the body, as seen from the main focus of the ellipse (the point around which the object orbits).
The true anomaly is usually denoted by the Greek letters ν or θ, or the Latin letter f, and is usually restricted to the range 0–360° (0–2π^c).
The true anomaly f is one of three angular parameters (anomalies) that defines a position along an orbit, the other two being the eccentric anomaly and the mean anomaly.
For elliptic orbits, the true anomaly ν can be calculated from orbital state vectors as:
(if r ⋅ v < 0 then replace ν by 2pi − ν)
where:
v is the orbital velocity vector of the orbiting body,
e is the eccentricity vector,
r is the orbital position vector (segment FP in the figure) of the orbiting body.
For circular orbits the true anomaly is undefined, because circular orbits do not have a uniquely determined periapsis. Instead the argument of latitude u is used:
(if rz < 0 then replace u by 2pi − u)
where:
n is a vector pointing towards the ascending node (i.e. the z-component of n is zero).
rz is the z-component of the orbital position vector r
For circular orbits with zero inclination the argument of latitude is also undefined, because there is no uniquely determined line of nodes. One uses the true longitude instead:
(if vx > 0 then replace l by 2pi − l)
where:
rx is the x-component of the orbital position vector r
vx is the x-component of the orbital velocity vector v.
The relation between the true anomaly ν and the eccentric anomaly is:
or using the sine and tangent:
or equivalently:
so
Alternatively, a form of this equation was derived by that avoids numerical issues when the arguments are near , as the two tangents become infinite. Additionally, since and are always in the same quadrant, there will not be any sign problems.
where
so
The true anomaly can be calculated directly from the mean anomaly via a Fourier expansion:
with Bessel functions and parameter .

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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.

True anomaly

In celestial mechanics, true anomaly is an angular parameter that defines the position of a body moving along a Keplerian orbit. It is the angle between the direction of periapsis and the current position of the body, as seen from the main focus of the ellipse (the point around which the object orbits). The true anomaly is usually denoted by the Greek letters ν or θ, or the Latin letter f, and is usually restricted to the range 0–360° (0–2π^c).

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.

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True Anomaly in Elliptical Orbits

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

Mean Anomaly (circular orbit)

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

Orbital Parameters and Calendars

Explains classical orbital parameters, spacecraft's state vector, and Gregorian vs. Julian days.