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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).

- Formulas
- From state vectors
- From the eccentric anomaly
- From the mean anomaly
- Radius from true anomaly
- See also
- References
- Further reading
- External links

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}).

As shown in the image, 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 2π − ν)

- (if

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
*r*< 0 then replace_{z}*u*by 2π −*u*)

- (if

where:

**n**is a vector pointing towards the ascending node (i.e. the*z*-component of**n**is zero).*r*is the_{z}*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
*v*> 0 then replace l by 2π − l)_{x}

- (if

where:

*r*is the_{x}*x*-component of the orbital position vector**r***v*is the_{x}*x*-component of the orbital velocity vector**v**.

The relation between the true anomaly ν and the eccentric anomaly *E* is:

or using the sine ^{ [1] } and tangent:

or equivalently:

so

Alternatively, a form of this equation was derived by ^{ [2] } 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:^{ [3] }

where the means that the omitted terms are all of order *e*^{4} or higher. Note that for reasons of accuracy this approximation is usually limited to orbits where the eccentricity (e) is small.

The expression is known as the equation of the center.

The radius (distance between the focus of attraction and the orbiting body) is related to the true anomaly by the formula

where *a* is the orbit's semi-major axis.

In electrodynamics, **elliptical polarization** is the polarization of electromagnetic radiation such that the tip of the electric field vector describes an ellipse in any fixed plane intersecting, and normal to, the direction of propagation. An elliptically polarized wave may be resolved into two linearly polarized waves in phase quadrature, with their polarization planes at right angles to each other. Since the electric field can rotate clockwise or counterclockwise as it propagates, elliptically polarized waves exhibit chirality.

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In geometry, the **elliptic coordinate system** is a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal ellipses and hyperbolae. The two foci and are generally taken to be fixed at and , respectively, on the -axis of the Cartesian coordinate system.

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In celestial mechanics, a **Kepler orbit** 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. It is thus said to be a solution of a special case of the two-body problem, known as the Kepler problem. As a theory in classical mechanics, it also does not take into account the effects of general relativity. Keplerian orbits can be parametrized into six orbital elements in various ways.

**Solution of triangles** is the main trigonometric problem of finding the characteristics of a triangle, when some of these are known. The triangle can be located on a plane or on a sphere. Applications requiring triangle solutions include geodesy, astronomy, construction, and navigation.

- ↑ Fundamentals of Astrodynamics and Applications by David A. Vallado
- ↑ Broucke, R., & Cefola, P. 1973, Celestial Mechanics, 7, 388
- ↑ Roy, A.E. (2005).
*Orbital Motion*(4 ed.). Bristol, UK; Philadelphia, PA: Institute of Physics (IoP). p. 84. ISBN 0750310154.

- Murray, C. D. & Dermott, S. F., 1999,
*Solar System Dynamics*, Cambridge University Press, Cambridge. ISBN 0-521-57597-4 - Plummer, H. C., 1960,
*An Introductory Treatise on Dynamical Astronomy*, Dover Publications, New York. OCLC 1311887 (Reprint of the 1918 Cambridge University Press edition.)

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