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