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title	summary	tags
Keplerian elements	The six classical orbital elements, the two-body solution, and the anomalies.	orbit, kepler, elements, anomalies

Keplerian elements

Six numbers describe an elliptical orbit around a central body when only the gravity of that body acts. They are the classical orbital elements, and they are the form in which an orbit is easiest to inspect and to plan a maneuver against. This document defines them and the conversions.

Definition

Element	Symbol	Meaning	Unit
Semi-major axis	`a`	Size of the ellipse	km
Eccentricity	`e`	Shape, `0` for a circle, below `1` for an ellipse	dimensionless
Inclination	`i`	Tilt of the orbit plane against the equator	rad
Right ascension of the ascending node	`Omega`	Where the orbit crosses the equator going north	rad
Argument of perigee	`\omega`	Angle from the ascending node to perigee, in the orbit plane	rad
True anomaly	`nu`	Angle from perigee to the satellite	rad

The first two give the shape and size. The second two give the orientation of the plane. The last two give the position of the perigee within the plane and of the satellite along the orbit.

The orbit is an ellipse with one focus at the center of the Earth, as Kepler's first law states. The point closest to the Earth is perigee and the farthest is apogee.

Mathematical meaning

The radius as a function of the true anomaly follows from the conic equation:


r = \frac{p}{1 + e \cos\nu}, \qquad p = a (1 - e^2)

where p is the semi-latus rectum. The velocity follows from the vis-viva equation:


v^2 = \mu \left( \frac{2}{r} - \frac{1}{a} \right)

where \mu is the gravitational parameter of the Earth.

Two derived quantities appear often:


\begin{aligned}
r_p &= a (1 - e) && \text{(perigee radius)} \\
r_a &= a (1 + e) && \text{(apogee radius)} \\
T &= 2 \pi \sqrt{\frac{a^3}{\mu}} && \text{(Kepler's third law)}
\end{aligned}

The conversion to a Cartesian state places the satellite in the perifocal frame, whose first axis points to perigee and whose third axis is along the orbit normal:


\begin{aligned}
r &= \frac{p}{1 + e \cos\nu} \\
x &= r \cos\nu, \quad y = r \sin\nu, \quad z = 0 \\
v_x &= -\sqrt{\frac{\mu}{p}} \sin\nu \\
v_y &=  \sqrt{\frac{\mu}{p}} \left( e + \cos\nu \right)
\end{aligned}

and then rotates by the argument of perigee, the inclination, and the right ascension of the ascending node.

The three anomalies

Three angles name a position along the orbit. They agree at perigee and at apogee and differ in between.

Anomaly	Symbol	Defined as	Geometric meaning
True anomaly	`nu`	Angle at the focus from perigee	Actual direction of the satellite
Eccentric anomaly	`E`	Angle at the center of the ellipse	Auxiliary angle of the projecting circle
Mean anomaly	`M`	`M = E - e\sin E`	Angle a fictitious body on a circular orbit would have

The relations are


\tan\frac{\nu}{2} = \sqrt{\frac{1 + e}{1 - e}} \tan\frac{E}{2}, \qquad M = E - e \sin E

M = E - e\sin E is Kepler's equation. It is transcendental, so E is recovered from M by iteration; the implementation uses Newton's method with a fixed iteration budget and a tolerance, which converges quickly for the eccentricities an Earth orbit has. The equation has no closed form solution, and any implementation that claims one is approximating.

The eccentric anomaly is the quantity most computations use internally, because both conversions above are simple in it.

Unit convention

The semi-major axis is in kilometers.
Angles in the interface are in radians. A tool that accepts degrees converts on the input boundary and reports both.
The eccentricity is dimensionless.
The gravitational parameter \mu is in cubic kilometers per second squared.

Applicability

The two-body solution assumes a point mass and a point mass, with no other force. An Earth orbit is affected by the oblateness of the Earth, the drag of the atmosphere, the gravity of the Sun and the Moon, and the pressure of sunlight. Each of these moves the satellite away from the ellipse.

The Keplerian elements computed from a state vector at one instant are therefore the elements of the osculating orbit, the ellipse that touches the true trajectory at that instant. They change along the orbit. They are not the mean elements of a two-line element set, and converting one to the other is a different operation from propagating either.

The two-body solution is exact for its assumptions and is the right model for:

Inspecting an instantaneous state.
Planning an impulsive maneuver, where the change is applied to the osculating state.
Approximate period and geometry estimates.

It is the wrong model for reproducing a catalogue position. Use sgp4.

Comparison of element descriptions

Description	Input	Model	Changes along the orbit	Used here
Mean elements	Two-line element set	SGP4	No, until refit	As the propagator input
Osculating elements	A state vector	Two-body	Yes	As an output of the propagator
Position and velocity	Any propagator	Any	Yes	Internally
Equinoctial elements	A state vector	Two-body	Yes	No
Position on a time grid	Any propagator	Any	Yes	Interpolated, see ephemeris-interpolation

Equinoctial elements avoid the singularity that the classical elements have at zero inclination and at zero eccentricity. The classical form is used here because that is the form the two-line element format and the published literature use, and because an Earth orbit almost always has a nonzero inclination.

Source

The two-body solution and the anomalies follow the standard treatment in Vallado, Fundamentals of Astrodynamics and Applications.
The element conversion follows the formulation in the same reference.

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