102 lines
4.1 KiB
Markdown
102 lines
4.1 KiB
Markdown
---
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title: Earth Rotation Angle
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summary: The angle between the inertial and Earth-fixed frames and the linear expression that gives it.
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tags: frames, earth-rotation, iers, ut1
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---
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# Earth Rotation Angle
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The Earth Rotation Angle is the angle through which the Earth has turned. It
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is the quantity that relates the inertial and Earth-fixed frames, and it is
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simple enough to be computed from one expression. This document gives the
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definition, the expression, and its limits.
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## Definition
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The Earth Rotation Angle, written $\theta$ or ERA, is the angle measured
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along the true equator from the Celestial Intermediate Origin to the
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Terrestrial Intermediate Origin. In practice it is the angle between the
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inertial and Earth-fixed frames, and that is how it is used here.
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It is a function of UT1, the time scale defined by the rotation of the Earth.
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## Mathematical meaning
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The angle is computed from the interval in days of UT1 since the J2000.0
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epoch by a linear expression:
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$$
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\theta = 0.779\,057\,273\,2640 + 1.002\,737\,909\,350\,795 \, d
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$$
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where `d` is the number of days of UT1 since 2000-01-01 12:00 UT1, and the
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result is in turns. The constant term is the angle at the epoch itself,
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which is about 280.46 degrees. The rate is slightly more than one turn per
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day because the Earth rotates once in a sidereal day, which is shorter than
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a solar day.
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The angle is not reduced to one turn. The value returned is a continuous
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function of time, so the difference between two angles is meaningful even
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when the interval is longer than a day. A caller that needs an angle in
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$[0, 2\pi)$ reduces it explicitly.
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The rotation between the frames is a clockwise turn about the third axis:
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$$
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R_3(\theta): \text{inertial} \to \text{Earth-fixed}, \qquad
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R_3(\theta)^{\mathsf{T}}: \text{Earth-fixed} \to \text{inertial}
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$$
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## Unit convention
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- The angle is stored in **turns** and offered in **turns** and in
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**radians**.
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- The time argument is a number of **days of UT1 since J2000.0**, or a
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**Julian Date on the UT1 scale**, from which the interval is computed.
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## Applicability
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The linear expression is accurate to the precision of the formulation. Its
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residual against the full IAU 2000 model is below a milliarcsecond over
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decades, which is below a millimeter at the Earth's surface, so for the
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positioning accuracy these kernels target it is exact.
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The limitation is not the formula but the argument. UT1 differs from UTC by
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the rotational irregularity that the leap second system approximates, and
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the difference is called `UT1 - UTC` or DUT1. It is published with a
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resolution of a day and can reach 0.9 seconds, which is about 400 meters at
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the equator. A caller that has a UTC instant and no DUT1 value must decide
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what to assume; using UTC in place of UT1 makes an error bounded by the same
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0.9 seconds.
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Polar motion, the motion of the rotation axis within the Earth, is not part
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of the Earth Rotation Angle. It is below 0.5 arcseconds, about 15 meters on
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the ground, and is neglected here.
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## Comparison of approaches
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| Approach | Accuracy | Cost | Input needed | Used here |
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|----------|----------|------|--------------|-----------|
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| IAU 2000 linear expression | Below a milliarcsecond over decades | One multiply and one add | UT1 | Yes |
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| Full IAU 2000/2006 CIO-based transformation | Microarcsecond | Precession and nutation series | UT1 and TT | No |
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| Greenwich Mean Sidereal Time | Arcsecond over decades | Apparent sidereal time series | UT1 and TT | No |
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| Fixed rotation rate from a reference epoch | Grows without bound | One multiply | Epoch only | No |
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Greenwich Mean Sidereal Time and the Earth Rotation Angle differ by a
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rotation about the pole by the accumulated precession in right ascension.
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The Earth Rotation Angle is the quantity the IERS conventions define and the
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one this implementation uses.
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## Source
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- *IERS Conventions (2010)*, equation 8.32, as implemented by
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`apsis-frames`.
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## See also
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- [[reference-frames]] for the frames the rotation relates
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- [[time-scales]] for UT1 against UTC
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- [[calendar-and-mjd]] for the days since J2000.0
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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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