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title	summary	tags
Reference frames	The inertial and Earth-fixed frames, the typed positions that carry their frame, and the rotation between them.	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.

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