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.