163 lines
6.2 KiB
Markdown
163 lines
6.2 KiB
Markdown
---
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title: Keplerian elements
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summary: The six classical orbital elements, the two-body solution, and the anomalies.
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tags: orbit, kepler, elements, anomalies
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---
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# Keplerian elements
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Six numbers describe an elliptical orbit around a central body when only the
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gravity of that body acts. They are the classical orbital elements, and they
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are the form in which an orbit is easiest to inspect and to plan a maneuver
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against. This document defines them and the conversions.
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## Definition
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| Element | Symbol | Meaning | Unit |
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|---------|--------|---------|------|
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| Semi-major axis | $a$ | Size of the ellipse | km |
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| Eccentricity | `e` | Shape, `0` for a circle, below `1` for an ellipse | dimensionless |
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| Inclination | $i$ | Tilt of the orbit plane against the equator | rad |
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| Right ascension of the ascending node | `Omega` | Where the orbit crosses the equator going north | rad |
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| Argument of perigee | $\omega$ | Angle from the ascending node to perigee, in the orbit plane | rad |
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| True anomaly | `nu` | Angle from perigee to the satellite | rad |
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The first two give the shape and size. The second two give the orientation
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of the plane. The last two give the position of the perigee within the plane
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and of the satellite along the orbit.
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The orbit is an ellipse with one focus at the center of the Earth, as
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Kepler's first law states. The point closest to the Earth is perigee and the
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farthest is apogee.
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## Mathematical meaning
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The radius as a function of the true anomaly follows from the conic
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equation:
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$$
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r = \frac{p}{1 + e \cos\nu}, \qquad p = a (1 - e^2)
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$$
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where $p$ is the semi-latus rectum. The velocity follows from the vis-viva
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equation:
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$$
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v^2 = \mu \left( \frac{2}{r} - \frac{1}{a} \right)
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$$
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where $\mu$ is the gravitational parameter of the Earth.
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Two derived quantities appear often:
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$$
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\begin{aligned}
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r_p &= a (1 - e) && \text{(perigee radius)} \\
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r_a &= a (1 + e) && \text{(apogee radius)} \\
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T &= 2 \pi \sqrt{\frac{a^3}{\mu}} && \text{(Kepler's third law)}
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\end{aligned}
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$$
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The conversion to a Cartesian state places the satellite in the perifocal
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frame, whose first axis points to perigee and whose third axis is along the
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orbit normal:
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$$
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\begin{aligned}
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r &= \frac{p}{1 + e \cos\nu} \\
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x &= r \cos\nu, \quad y = r \sin\nu, \quad z = 0 \\
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v_x &= -\sqrt{\frac{\mu}{p}} \sin\nu \\
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v_y &= \sqrt{\frac{\mu}{p}} \left( e + \cos\nu \right)
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\end{aligned}
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$$
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and then rotates by the argument of perigee, the inclination, and the right
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ascension of the ascending node.
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### The three anomalies
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Three angles name a position along the orbit. They agree at perigee and at
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apogee and differ in between.
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| Anomaly | Symbol | Defined as | Geometric meaning |
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|---------|--------|------------|-------------------|
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| True anomaly | `nu` | Angle at the focus from perigee | Actual direction of the satellite |
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| Eccentric anomaly | $E$ | Angle at the center of the ellipse | Auxiliary angle of the projecting circle |
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| Mean anomaly | $M$ | $M = E - e\sin E$ | Angle a fictitious body on a circular orbit would have |
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The relations are
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$$
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\tan\frac{\nu}{2} = \sqrt{\frac{1 + e}{1 - e}} \tan\frac{E}{2}, \qquad M = E - e \sin E
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$$
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$M = E - e\sin E$ is Kepler's equation. It is transcendental, so $E$ is
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recovered from $M$ by iteration; the implementation uses Newton's method
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with a fixed iteration budget and a tolerance, which converges quickly for
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the eccentricities an Earth orbit has. The equation has no closed form
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solution, and any implementation that claims one is approximating.
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The eccentric anomaly is the quantity most computations use internally,
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because both conversions above are simple in it.
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## Unit convention
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- The semi-major axis is in **kilometers**.
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- Angles in the interface are in **radians**. A tool that accepts degrees
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converts on the input boundary and reports both.
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- The eccentricity is **dimensionless**.
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- The gravitational parameter $\mu$ is in **cubic kilometers per second
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squared**.
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## Applicability
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The two-body solution assumes a point mass and a point mass, with no other
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force. An Earth orbit is affected by the oblateness of the Earth, the drag
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of the atmosphere, the gravity of the Sun and the Moon, and the pressure of
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sunlight. Each of these moves the satellite away from the ellipse.
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The Keplerian elements computed from a state vector at one instant are
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therefore the elements of the **osculating** orbit, the ellipse that touches
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the true trajectory at that instant. They change along the orbit. They are
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not the mean elements of a two-line element set, and converting one to the
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other is a different operation from propagating either.
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The two-body solution is exact for its assumptions and is the right model
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for:
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- Inspecting an instantaneous state.
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- Planning an impulsive maneuver, where the change is applied to the
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osculating state.
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- Approximate period and geometry estimates.
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It is the wrong model for reproducing a catalogue position. Use [[sgp4]].
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## Comparison of element descriptions
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| Description | Input | Model | Changes along the orbit | Used here |
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|-------------|-------|-------|-------------------------|-----------|
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| Mean elements | Two-line element set | SGP4 | No, until refit | As the propagator input |
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| Osculating elements | A state vector | Two-body | Yes | As an output of the propagator |
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| Position and velocity | Any propagator | Any | Yes | Internally |
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| Equinoctial elements | A state vector | Two-body | Yes | No |
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| Position on a time grid | Any propagator | Any | Yes | Interpolated, see [[ephemeris-interpolation]] |
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Equinoctial elements avoid the singularity that the classical elements have
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at zero inclination and at zero eccentricity. The classical form is used
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here because that is the form the two-line element format and the published
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literature use, and because an Earth orbit almost always has a nonzero
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inclination.
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## Source
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- The two-body solution and the anomalies follow the standard treatment in
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Vallado, *Fundamentals of Astrodynamics and Applications*.
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- The element conversion follows the formulation in the same reference.
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## See also
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- [[tle-format]] for the mean elements the propagator consumes
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- [[sgp4]] for the propagator and its relation to these elements
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- [[perturbations]] for what the two-body model leaves out
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- Try the conversion in the online tool: *Keplerian elements to a state
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vector*
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