In
celestial mechanics
Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, ...
, true anomaly is an angular
parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
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
In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
(the point around which the object orbits).
The true anomaly is usually denoted by the
Greek letters or , or the
Latin letter , and is usually restricted to the range 0–360° (0–2π).
As shown in the image, the true anomaly 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.
Formulas
From state vectors
For elliptic orbits, the true anomaly can be calculated from
orbital state vectors
In astrodynamics and celestial dynamics, the orbital state vectors (sometimes state vectors) of an orbit are
Cartesian vectors of position (\mathbf) and velocity (\mathbf) that together with their time (epoch) (t) uniquely determine the trajector ...
as:
:
::(if then replace by )
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.
Circular orbit
For
circular orbit
A circular orbit is an orbit with a fixed distance around the barycenter; that is, in the shape of a circle.
Listed below is a circular orbit in astrodynamics or celestial mechanics under standard assumptions. Here the centripetal force is ...
s the true anomaly is undefined, because circular orbits do not have a uniquely determined periapsis. Instead the
argument of latitude In celestial mechanics, the argument of latitude ( u ) is an angular parameter that defines the position of a body moving along a Kepler orbit. It is the angle between the ascending node and the body.
It is the sum of the more commonly used true ...
''u'' is used:
:
::(if then replace )
where:
* n is a vector pointing towards the ascending node (i.e. the ''z''-component of n is zero).
* ''r
z'' is the ''z''-component of the
orbital position vector r
Circular orbit with zero inclination
For
circular orbit
A circular orbit is an orbit with a fixed distance around the barycenter; that is, in the shape of a circle.
Listed below is a circular orbit in astrodynamics or celestial mechanics under standard assumptions. Here the centripetal force is ...
s 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 then replace by )
where:
* ''r
x'' is the ''x''-component of the
orbital position vector r
* ''v
x'' is the ''x''-component of the
orbital velocity vector v.
From the eccentric anomaly
The relation between the true anomaly and the
eccentric anomaly is:
:
or using the
sine
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opp ...
and
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
:
:
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
:
From the mean anomaly
The true anomaly can be calculated directly from the
mean anomaly via a
Fourier expansion:
:
with
Bessel functions
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrar ...
and parameter
.
Omitting all terms of order
or higher (indicated by
), it can be written as
:
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
In two-body, Keplerian orbital mechanics, the equation of the center is the angular difference between the actual position of a body in its elliptical orbit and the position it would occupy if its motion were uniform, in a circular orbit of the ...
, where more details about the expansion are given.
Radius from true anomaly
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 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 lon ...
.
See also
*
Kepler's laws of planetary motion
In astronomy, Kepler's laws of planetary motion, published by Johannes Kepler between 1609 and 1619, describe the orbits of planets around the Sun. The laws modified the heliocentric theory of Nicolaus Copernicus, replacing its circular orb ...
*
Eccentric anomaly
*
Mean anomaly
*
Ellipse
In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
*
Hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...
References
Further reading
* Murray, C. D. & Dermott, S. F., 1999, ''Solar System Dynamics'', Cambridge University Press, Cambridge.
* Plummer, H. C., 1960, ''An Introductory Treatise on Dynamical Astronomy'', Dover Publications, New York. (Reprint of the 1918 Cambridge University Press edition.)
External links
Federal Aviation Administration - Describing Orbits{{orbits
Orbits