Part of a series on |

Astrodynamics |
---|

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}).

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 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] }

with Bessel functions and parameter .

Omitting all terms of order or higher (indicated by ), it can be written as^{ [3] }^{ [4] }^{ [5] }

Note that for reasons of accuracy this approximation is usually limited to orbits where the eccentricity is small.

The expression is known as the equation of the center, where more details about the expansion are given.

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 mathematical physics and mathematics, the **Pauli matrices** are a set of three 2 × 2 complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma, they are occasionally denoted by tau when used in connection with isospin symmetries.

In mathematics, a **spherical coordinate system** is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the *radial distance* of that point from a fixed origin; its *polar angle* measured from a fixed polar axis or zenith direction; and the *azimuthal angle* of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the fixed axis, measured from another fixed reference direction on that plane. When radius is fixed, the two angular coordinates make a coordinate system on the sphere sometimes called **spherical polar coordinates**.

In particle physics, ** bremsstrahlung** is electromagnetic radiation produced by the deceleration of a charged particle when deflected by another charged particle, typically an electron by an atomic nucleus. The moving particle loses kinetic energy, which is converted into radiation, thus satisfying the law of conservation of energy. The term is also used to refer to the process of producing the radiation.

Integration is the basic operation in integral calculus. While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful. This page lists some of the most common antiderivatives.

In mathematics, the **inverse trigonometric functions** are the inverse functions of the trigonometric functions. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.

In physics, a **wave vector** is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave, and its direction is perpendicular to the wavefront. In isotropic media, this is also the direction of wave propagation.

In astrodynamics or celestial mechanics a **parabolic trajectory** is a Kepler orbit with the eccentricity equal to 1 and is an unbound orbit that is exactly on the border between elliptical and hyperbolic. When moving away from the source it is called an **escape orbit**, otherwise a **capture orbit**. It is also sometimes referred to as a **C _{3} = 0 orbit** (see Characteristic energy).

In astrodynamics or celestial mechanics, an **elliptic orbit** or **elliptical orbit** is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. In a stricter sense, it is a Kepler orbit with the eccentricity greater than 0 and less than 1. In a wider sense, it is a Kepler orbit with negative energy. This includes the **radial elliptic orbit**, with eccentricity equal to 1.

In mathematics, the **Legendre chi function** is a special function whose Taylor series is also a Dirichlet series, given by

In relativistic physics, a **velocity-addition formula** is an equation that specifies how to combine the velocities of objects in a way that is consistent with the requirement that no object's speed can exceed the speed of light. Such formulas apply to successive Lorentz transformations, so they also relate different frames. Accompanying velocity addition is a kinematic effect known as Thomas precession, whereby successive non-collinear Lorentz boosts become equivalent to the composition of a rotation of the coordinate system and a boost.

A **theoretical motivation for general relativity**, including the motivation for the geodesic equation and the Einstein field equation, can be obtained from special relativity by examining the dynamics of particles in circular orbits about the Earth. A key advantage in examining circular orbits is that it is possible to know the solution of the Einstein Field Equation *a priori*. This provides a means to inform and verify the formalism.

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.

**Oblate spheroidal coordinates** are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the non-focal axis of the ellipse, i.e., the symmetry axis that separates the foci. Thus, the two foci are transformed into a ring of radius in the *x*-*y* plane. Oblate spheroidal coordinates can also be considered as a limiting case of ellipsoidal coordinates in which the two largest semi-axes are equal in length.

In geometry, various **formalisms** exist to express a rotation in three dimensions as a mathematical transformation. In physics, this concept is applied to classical mechanics where rotational kinematics is the science of quantitative description of a purely rotational motion. The orientation of an object at a given instant is described with the same tools, as it is defined as an imaginary rotation from a reference placement in space, rather than an actually observed rotation from a previous placement in space.

There are several equivalent ways for defining trigonometric functions, and the proof of the trigonometric identities between them depend on the chosen definition. The oldest and somehow the most elementary definition is based on the geometry of right triangles. The proofs given in this article use this definition, and thus apply to non-negative angles not greater than a right angle. For greater and negative angles, see Trigonometric functions.

**Orbit determination** is the estimation of orbits of objects such as moons, planets, and spacecraft. One major application is to allow tracking newly observed asteroids and verify that they have not been previously discovered. The basic methods were discovered in the 17th century and have been continuously refined.

**Lateral earth pressure** is the pressure that soil exerts in the horizontal direction. The lateral earth pressure is important because it affects the consolidation behavior and strength of the soil and because it is considered in the design of geotechnical engineering structures such as retaining walls, basements, tunnels, deep foundations and braced excavations.

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). "A Note on the Relations between True and Eccentric Anomalies in the Two-Body Problem".
*Celestial Mechanics*.**7**(3): 388–389. Bibcode:1973CeMec...7..388B. doi:10.1007/BF01227859. ISSN 0008-8714. S2CID 122878026. - 1 2 Battin, R.H. (1999).
*An Introduction to the Mathematics and Methods of Astrodynamics*. AIAA Education Series. American Institute of Aeronautics & Astronautics. p. 212 (Eq. (5.32)). ISBN 978-1-60086-026-3 . Retrieved 2022-08-02. - ↑ Smart, W. M. (1977).
*Textbook on Spherical Astronomy*(PDF). p. 120 (Eq. (87)). Bibcode:1977tsa..book.....S. - ↑ Roy, A.E. (2005).
*Orbital Motion*(4 ed.). Bristol, UK; Philadelphia, PA: Institute of Physics (IoP). p. 78 (Eq. (4.65)). Bibcode:2005ormo.book.....R. ISBN 0750310154. Archived from the original on 2021-05-15. Retrieved 2020-08-29.

- 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.)

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.