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title	summary	tags
Earth Rotation Angle	The angle between the inertial and Earth-fixed frames and the linear expression that gives it.	frames, earth-rotation, iers, ut1

Earth Rotation Angle

The Earth Rotation Angle is the angle through which the Earth has turned. It is the quantity that relates the inertial and Earth-fixed frames, and it is simple enough to be computed from one expression. This document gives the definition, the expression, and its limits.

Definition

The Earth Rotation Angle, written theta or ERA, is the angle measured along the true equator from the Celestial Intermediate Origin to the Terrestrial Intermediate Origin. In practice it is the angle between the inertial and Earth-fixed frames, and that is how it is used here.

It is a function of UT1, the time scale defined by the rotation of the Earth.

Mathematical meaning

The angle is computed from the interval in days of UT1 since the J2000.0 epoch by a linear expression:


\theta = 0.779\,057\,273\,2640 + 1.002\,737\,909\,350\,795 \, d

where d is the number of days of UT1 since 2000-01-01 12:00 UT1, and the result is in turns. The constant term is the angle at the epoch itself, which is about 280.46 degrees. The rate is slightly more than one turn per day because the Earth rotates once in a sidereal day, which is shorter than a solar day.

The angle is not reduced to one turn. The value returned is a continuous function of time, so the difference between two angles is meaningful even when the interval is longer than a day. A caller that needs an angle in [0, 2 pi) reduces it explicitly.

The rotation between the frames is a clockwise turn about the third axis:


R_3(\theta): \text{inertial} \to \text{Earth-fixed}, \qquad
R_3(\theta)^{\mathsf{T}}: \text{Earth-fixed} \to \text{inertial}

Unit convention

The angle is stored in turns and offered in turns and in radians.
The time argument is a number of days of UT1 since J2000.0, or a Julian Date on the UT1 scale, from which the interval is computed.

Applicability

The linear expression is accurate to the precision of the formulation. Its residual against the full IAU 2000 model is below a milliarcsecond over decades, which is below a millimeter at the Earth's surface, so for the positioning accuracy these kernels target it is exact.

The limitation is not the formula but the argument. UT1 differs from UTC by the rotational irregularity that the leap second system approximates, and the difference is called UT1 - UTC or DUT1. It is published with a resolution of a day and can reach 0.9 seconds, which is about 400 meters at the equator. A caller that has a UTC instant and no DUT1 value must decide what to assume; using UTC in place of UT1 makes an error bounded by the same 0.9 seconds.

Polar motion, the motion of the rotation axis within the Earth, is not part of the Earth Rotation Angle. It is below 0.5 arcseconds, about 15 meters on the ground, and is neglected here.

Comparison of approaches

Approach	Accuracy	Cost	Input needed	Used here
IAU 2000 linear expression	Below a milliarcsecond over decades	One multiply and one add	UT1	Yes
Full IAU 2000/2006 CIO-based transformation	Microarcsecond	Precession and nutation series	UT1 and TT	No
Greenwich Mean Sidereal Time	Arcsecond over decades	Apparent sidereal time series	UT1 and TT	No
Fixed rotation rate from a reference epoch	Grows without bound	One multiply	Epoch only	No

Greenwich Mean Sidereal Time and the Earth Rotation Angle differ by a rotation about the pole by the accumulated precession in right ascension. The Earth Rotation Angle is the quantity the IERS conventions define and the one this implementation uses.

Source

IERS Conventions (2010), equation 8.32, as implemented by apsis-frames.

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