# How is Earth's Rotation Angle (ERA) defined and measured

UTC typically ticks with TAI. But to keep UTC from drifting far from the rotation angle of the Earth we occasionally add leap seconds to ensure that UTC does not drift too far from the UT1 timescale. The UT1 timescale is defined by a simple linear relation with the Earth Rotation Angle (ERA)

ERA = 2π(0.7790572732640 + 1.00273781191135448 · Tu) radians

where Tu = (Julian UT1 date - 2451545.0)

How is ERA measured? The following is my guess. Please let me know if it is correct or if I am missing something.

I understand that we have an International Celestial Reference Frame (ICRF) which is realized as a catalog of many celestial objects whose positions have been measured and catalogued. It seems like, once the positions are measured and the catalog is published we actually take these positions to be fixed by definition. That is, in analogy to how the second is defined as 9 192 631 770 oscillations of radiation resonant with the Cs hyperfine transition, the positions of some objects are just taken as being perfectly accurate by definition. I think the current ICRF3 catalog appears here. You can see some observed objects are marked as "defining" objects and their declinations and right ascension with respect to some celestial origin are tabulated.

By making terrestrial measurements of these reference stars from observatories around the world we can determine the orientation of a terrestrial reference frame with respect to the celestial reference frame. I guess this orientation would be described either by 3 Euler angles evolving over time or by an instantaneous rotation axis and rotation angle. The latter seems immediately amenable to a definition of the Earth's Rotation Angle. Naively it seems like a simple observatory would suffice in principle by monitoring how quickly celestial objects pass its meridian over time. But if the Earth is tilting and wobbling with respect to the celestial reference frame then I think you might find that you observe different "rotation rates" from different observatories. e.g. if the Earth's rotation axis is tilting towards one observatory in one hemisphere but away from another in another hemisphere observatories in one hemisphere might see a "speed up" compared to the other. So to get the full 3D picture of the Earth's rotation axis you require measurements from around the globe which you then analyze to get the full picture.

update: ok, actually 3 Euler angles is enough to specify that rotation axis of the telescope network relative to the celestial reference frame. But there is an additional degree of freedom which is the orientation of the telescope network relative to the networks rotation axis. In other words, the rotation axis of the Earth can move relative to the crust of the Earth. This requires 2 more angles to describe. Given these 5 parameters (2 parameterizing the rotation axis relative to the crust, and 3 parametrizing the orientation of the Earth relative to the celestial reference frame), how do we define the Earth's rotation angle?

Also a finer detail but: I think the positions of the celestial objects are determined using VLBI. in this case there are many radio telescopes and we locate an object by tuning the temporal delays used when analyzing all the data together. The exact combination of N delays used for N antennas in the total array to get a signal from a specific celestial object corresponds to a particular angle between the rigid orientation on Earth of the Antenna array and the specific celestial object.

Is this an accurate assessment of how the ERA is measured? Am I close and just missing some details? Or am I just totally off base?

Also bonus question: Lots of time when I look up or ask about UT1 I get answers relating to the equation of time, or new modern models for precession and nutation. I don't see how these things can be relevant if ERA is based as much as possible on celestial measurements like I described above. Why do these comments about various models pertaining to details about the Sun and Earth's dynamics always come up? My guesses are (1) these types of things were critical to older definitions of universal time or (2) these models feed into the exact numerical parameters that relate UT1 to ERA shown at the top of this question, but they do not in fact feed into the calculation of ERA at all.

• There's also plate tectonics, which causes the observatories to move relative to each other. Also, the use of the ERA as you've described it is a fairly new concept, adopted as part of the ICRS. The old "equinox" system used Greenwich Sidereal Time. So any references you see to old uses of ERA are talking about something else. The definitions of UT and GMT have also varied over the years, so it's important to know which version the reference is using. Mar 27, 2023 at 4:57

Disclaimer: This is the answer I've come up with myself after a few days of research involving asking some Stack Exchange questions and reading lots of IERS and related documents and papers. Also some textbooks on historical astronomy. I post it less because I think it's right and more because I think it pulls together a number of the important pieces and can act as a target for people to poke holes in to help refine the complete understanding. So please, point out all the places I go wrong in this answer in the comments!

The answer to this question is complicated because of General Relativity. I will first give an answer as if relativity could be neglected, then I will wave my hands to try to explain how general relativity changes the picture.

Start ignoring relativity

We have a celestial reference system. Historically the north pole of this reference system passed through the point in the sky about which Earth rotated (near the Polaris star). There is a North and a South pole to this reference system and azimuthal angles are referred to as declination angles away from the equator. The polar angles were referenced to the location in the sky of the vernal equinox, the point where the ecliptic intersects the equator and the sun passes through in the spring. Time used to be defined by the polar angle (measured along the equator) of the sun along. As the polar angle increased so did the time.

In the 20th century time was redefined to be based on oscillations of electromagnetic radiation resonant with a particular atomic transition in a particular atom and time is now kept by atomic clocks that monitor these oscillations. But it is still important to know how Earth is oriented with respect to this celestial sphere.

Nowadays Earth's rotation is monitored as follows. There is a technology called Very Long Baseline Interferometry (VLBI). It is a radio astronomy technique in which radio antennas are located around the globe. Radio signals from extragalactic sources reach Earth and are recorded on all of these antennas. Because the antennas are located many microwave wavelengths apart from each other, the signals arrive out of sync with each other at the different detectors. At the site of each antenna is an atomic clock, and the clocks at all of the antennas are synchronized. In this way the radio signals at all antennas can be recorded with atomic time timestamps. The positions of the observation sites are also mapped out very carefully using satellite GPS and possibly other surveying techniques. The radio signals from many antennas can then be replayed but with differing time delays between the antennas to re-sync the radio signals from a particular source in the sky. The set of delays which results in a signal between all of the different signals can be mapped very precisely onto a particular location in the sky. In this VLBI is able to map out the polar and azimuthal angles (over time) on the celestial sphere of extra-galactic radio sources it is able to observe.

At some point in the late 20th and early 21st century catalogs were made of the celestial coordinates of sources measured using VLBI. Eventually the coordinates of these sources were taken to be true by definition to define a so-called reference frame. This is analogous to how the frequency of the Cs hyperfine transition for atomic clocks was taken to be a particular value by definition. The origin of this reference frame is such that it is close the historical origin of the celestial reference frame: the vernal equinox. Likewise for the pole. With this reference frame catalog in place we can monitor the orientation of Earth in space with respect the stars in this catalog.

Essentially a measurement of the positions of the sources of this catalog at some moment in time tells us the relative orientation between the array of radio antennas and the celestial reference frame. Because the array is on Earth and (in-as-much as the relative positions of the antennas is kept up-to-date) we can take the orientation of the array to be a proxy for the orientation of Earth. We can decompose this orientation into a rotation axis and a rotation angle.

This rotation angle is essentially the Earth Rotation Angle (ERA) which is used in the definition of the UT1 timescale.

End ignoring relativity

Now, what did we miss in the above picture with respect to general relativity? The observations we are making now are at high enough precision that general relativistic effects must be taken into account in our observations. For this reason, astronomers have defined a number of GR reference systems to which astronomical measurements can be referred. These reference systems are oriented either with the historical pole and vernal equinox or with the geometric positions on Earth's crust.

The first and "cleanest" reference system is the Barycentric Celestial Reference System (BCRS) which is closely related to the International Celestial Reference System (ICRS). This system has its origin located at the barycenter of the solar system. I believe it is significant because the GR metric at that location can somehow be quite simple, so it is the best local reference system we can come up with.

The next reference system is the Geocentric Celestial Reference System (GCRS). This reference system has its origin at Earth's center of mass and it's coordinate axes aligned with those of the ICRS. But note that to get from the ICRS to GCRS coordinates you must perform a non-trivial GR coordinate transformation because the GCRS is translating and accelerating relative to the ICRS.

The next reference system is the International Terrestrial Reference System (ITRS). This system has its origin at the Earth's center of mass but it's axes are aligned with some fixed geometry on Earth's crust. In other words this reference system rotates, precesses, nutates, etc. along with Earth. ITRS is related to GCRS by a rotation matrix.

When we make astronomical measurements (especially at this level of precision) and record the results we always try to refer them to the ICRS (or International Celestial Reference Frame [ICRF], the realization of that reference system -- this is the source catalog mentioned above). The rub is that our measurements are made at some particular time and location probably using the ITRS to initially record measurement results. This means we must convert GTRS coordinates to ICRS coordinates and back when it comes to reading and writing historical measurements. But ITRS is translating and rotating in a time-varying way with respect to ICRS. The means all the funny wobbles of Earth will need to be taken into account in the change of coordinates algorithms.

Basically that is as much as I've gathered at this point: Basically because of GR, and the inclusion of Earth's wobbling in the GR change-of-coordinates between reference system, we end up needing complicated models of the Earth's motion to be able to measure.... Earth's motion. However, once we can use these models, my guess/hope, is we can always transform reference ICRS readings (such as those found in the ICRF catalog) into ITRS using the conversion scheme described above (and possibly recent measurements of Earth's orientation) and vice-versa: when we take new measurements we can record them with respect to ICRS by using the reverse of the conversion scheme above (and, again, possibly recent measurements of Earth's orientation). There are further complicated details such as the introduction of an intermediate reference system between the GCRS and ITRS that helps with precession/nutation models, but I am definitely not close to understanding these details yet.

A very brief summary: Yes, the Earth's rotation angle is measured by using multiple observatories to measure angles of the celestial objects in the sky. Were it not for GR this would be the end of the story. But, because of GR, we must make a number of coordinate transformations to make sure all of our measurements are referred to a coordinate system located at the solar system's barycenter. This requires sort of a bootstrapping technique where we need to use models for the Earth's orientation. to perform these coordinate transformations so that we can continue to monitor Earth's orientation. I'm sure I'm getting something wildly wrong here!

• This answer is looking pretty good. This sentence is a bit weird, though: "Time used to be defined by the polar angle (measured along the equator) of the sun along." FWIW, Meuss gives a nice definition of mean solar time, which I quote here. Mar 27, 2023 at 7:19
• The true SSB (Solar System Barycentre) is a complicated place, gravitationally, since it moves in & out of the Sun. We don't really know its true location, we use the nominal SSB of the JPL DE, which shifted by ~100 km a few years ago, as I mentioned here. The Barycentric Dynamical Time scale is based at the SSB, but it's defined in terms of an imaginary comoving clock outside the Solar System's gravity well. Mar 27, 2023 at 7:30
• Barycentric Coordinate Time has replaced Barycentric Dynamic Time. Mar 27, 2023 at 16:26
• @GregMiller Yes, but JPL still use their version of TDB. See en.wikipedia.org/wiki/… & The JPL Planetary and Lunar Ephemerides DE440 and DE441 Mar 27, 2023 at 19:51
• It's usual to offer sources/bases for statements such as those made in this answer. Mar 27, 2023 at 20:07

From IERS Rapid Service / Prediction Center, What is Earth Orientation? has the following to say:

How Do We Observe Earth Orientation?

To determine Earth orientation, observations must be made from the Earth of objects located in space. Objects that are used include stars, artificial satellites, the Moon, and distant radio sources called quasars. These provide useful reference directions with which to measure the Earth's orientation. Stars had been observed photographically for decades to determine the motion of the pole and the rotation of the Earth. Recently, more accurate methods have been devised including the use of lasers and radio telescopes. Laser bursts can be reflected off of artificial satellites or the Moon. This provides information on exactly where the object is at a particular time which, in turn, can be used to determine the Earth's orientation in space.

Radio telescopes are used in a technique called Very Long Baseline Interferometry (VLBI). By having several radio telescopes observing the same quasar at the same time, and recording the information that is seen at each telescope, the Earth's orientation can be determined. The recorded information is then processed further before the final results can be determined. The reduction procedure involves the use of a highly specialized computer called a correlator. Once the data from a VLBI observing session has been correlated, it can be processed further to produce information on Earth orientation and other useful quantities. USNO participates in correlating and analyzing observations from international and domestic VLBI networks for the purpose of measuring Earth orientation.

Astronomical observations are made routinely by a number of observatories located around the world for this purpose. The IERS is the international organization responsible for coordinating observations of polar motion and nutation as well as rotation angle. This organization consists of product centers that provide specific services to users, such as rapid service/predictions of Earth orientation, data related to the motions of geophysical fluids (e.g., atmosphere and oceans), and the celestial and terrestrial reference frames. Observations are contributed to the IERS by individual observing techniques, which in turn receive results from numerous observatories, laboratories, and analysis centers around the world. The IERS Rapid Service/Prediction Center then combines these data into a series of polar motion x, polar motion y, UT1-UTC, and celestial pole offsets. This information is recomputed at least daily and disseminated on our website, the IERS website, and CDDIS.

• Sorry, but this answer is just a copy and paste and doesn't answer any questions I didn't already know. I knew that observe celestial object to determine ERA (and that we now use radio more than optical), I knew that we used the VLBI technique which involves correlating results from many observatories and I knew that IERS is responsible for monitoring and reporting many of these metrics. Mar 26, 2023 at 15:06