114 lines
5.1 KiB
Markdown
114 lines
5.1 KiB
Markdown
---
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title: Reference frames
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summary: The inertial and Earth-fixed frames, the typed positions that carry their frame, and the rotation between them.
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tags: frames, eci, ecef, geodetic, rotation
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---
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# Reference frames
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A position is a set of numbers only when the frame they are measured in is
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known. Two frames are used in the kernels: one fixed in inertial space and
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one fixed to the rotating Earth. This document defines them and the
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transformation between them.
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## Definition
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| Frame | Origin | Axes | Rotates with the Earth |
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|-------|--------|------|------------------------|
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| ECI, Earth-centered inertial | Earth center | Mean equator and equinox of J2000.0 | No |
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| ECEF, Earth-centered Earth-fixed | Earth center | Rotates with the Earth | Yes |
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| Geodetic | On the ellipsoid | Latitude, longitude, height | Yes |
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A point fixed to the ground has constant coordinates in the Earth-fixed
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frame and circles the origin in the inertial frame. A vector fixed in
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inertial space, such as the direction to a distant star, has constant
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coordinates in the inertial frame.
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The kernels attach the frame to the type. `Eci`, `Ecef`, and `Geodetic` are
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distinct types, so a position expressed in one frame cannot be passed where
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another is expected. A conversion is an explicit method call.
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## Mathematical meaning
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The transformation between the inertial and Earth-fixed frames is a
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rotation about the third axis by the Earth Rotation Angle:
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$$
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\mathbf{r}_{\text{ecef}} = R_3(\theta) \, \mathbf{r}_{\text{eci}}, \qquad \mathbf{r}_{\text{eci}} = R_3(\theta)^{\mathsf{T}} \mathbf{r}_{\text{ecef}}
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$$
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where `theta` is the Earth Rotation Angle and `R3` is the rotation matrix
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$$
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R_3(\theta) = \begin{bmatrix} \cos\theta & \sin\theta & 0 \\ -\sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix}
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$$
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The rotation is orthogonal, so it preserves lengths and angles. This is a
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useful check: a conversion that changes the length of a vector by more than
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rounding is wrong.
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The transformation between Earth-fixed Cartesian coordinates and geodetic
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coordinates is not a rotation, because the ellipsoid is not a sphere. It is
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solved by the standard closed form described in [[geodesy-wgs84]].
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## Unit convention
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- Cartesian positions in `Eci` and `Ecef` are in **meters**.
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- Velocities in the inertial frames of the propagators are in **kilometers
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per second** and positions in **kilometers**; the orbit layer works in the
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units the element sets and the SGP4 theory use.
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- Geodetic latitude and longitude are in **radians**, and height is in
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**meters** above the ellipsoid.
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The difference between the meter convention of the coordinate layer and the
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kilometer convention of the orbit layer is deliberate rather than an
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oversight: each follows the convention of the theory it implements. A
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conversion between the layers is an explicit multiplication, not an implicit
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one.
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## Applicability
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An inertial frame is an idealization. The frame used here is aligned with
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the mean equator and equinox of J2000.0 and treated as inertial, which
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neglects precession and nutation. Over the intervals a two-line element set
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is valid, which is days, the neglected motion is far below the accuracy of
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the propagation. Over years it is not, and a caller that needs it must use a
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frame implementation that models precession and nutation.
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The Earth Rotation Angle defines the rotation between the frames. It does
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not include polar motion, which is the small motion of the rotation axis
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within the Earth. Polar motion is below 0.5 arcseconds, which is about 15
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meters on the ground, and is neglected here.
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## Comparison of frame conventions
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| Convention | Axes defined by | Includes precession | Includes nutation | Used here |
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|------------|-----------------|---------------------|-------------------|-----------|
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| J2000.0 inertial | Mean equator and equinox of 2000-01-01 12:00 TT | No | No | Yes |
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| True of date | True equator and equinox of the date | Yes | Yes | No |
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| Mean of date | Mean equator and equinox of the date | Yes | No | No |
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| TEME, True Equator Mean Equinox | True equator, mean equinox | Implicitly | Yes for the pole | Yes, as the output of SGP4 |
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| Earth-fixed | Rotates with the Earth | Not applicable | Not applicable | Yes |
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| ITRF | A realization of the Earth-fixed frame | Not applicable | Not applicable | As WGS 84, approximated |
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TEME is the frame a two-line element set is written for. It is the output of
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SGP4 and is not the same as the J2000.0 inertial frame. Converting from TEME
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to the Earth-fixed frame requires the Earth rotation and the equation of the
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equinoxes, which is the small angle between the mean and true equinoxes.
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## Source
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- Earth Rotation Angle: *IERS Conventions (2010)*, equation 8.32, as
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implemented by `apsis-frames`.
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- The frame definitions follow the usage in Vallado, *Fundamentals of
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Astrodynamics and Applications*, and in the *Explanatory Supplement to the
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Astronomical Almanac*.
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## See also
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- [[earth-rotation-angle]] for the rotation angle
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- [[geodesy-wgs84]] for the ellipsoid and the Cartesian conversion
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- [[sgp4]] for the frame the propagator returns
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- Try the rotation in the online tool: *Rotate between inertial and
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Earth-fixed frames*
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