("How exactly is UT1 measured/calculated?" -- with "interest in fine details about how UT1 time is measured.")
(edited 2023/Mch/27 to add references including text+image extract of IAU 2009 explanation showing continuing relation of UT1 to the sun.)
"UT1" is a standard measure of time currently close to mean solar time at longitude 0, and has been (re)defined in the last ~40 years successively by several IAU standards (1982, 2000, 2006). Effectively these standards are much-modified conventional descendants of results of older methods from optical astronomy for accurate determination of mean solar time, dating largely from the 18th and 19th centuries. The modifications have effectively discarded some part of the physical basis and rationale that originally underlay determinations of time (see also below).
Although UT1 is usually obtained now in practice from the difference with the uniform time scale UTC by using the quantity UT1 − UTC (available from the IERS), fundamental redefinitions for UT1 as a standard over the last ~40 years have been offered as follows:
1982 standard (in official use from 1984): S Aoki et al (1982) "New Definition of Universal Time", Astronomy & Astrophysics 105: 359-361
2000 standard (in official use from 2003): N Capitaine et al (2003) "Expressions to implement the IAU 2000 definition of UT1", Astronomy & Astrophysics 406: 1135-1149.
The IAU 2000 standard introduced an officially-preferred alternative to the traditional equinox-based calculations, involving a 'non-rotating origin' (NRO) and an 'earth rotation angle' (ERA). Full details, too long to reproduce here, are given in the paper of Capitaine et al (2003) linked above. The 2000 standard has often been found since to be in need of explanation, and the following extract is from a presentation by N Capitaine & P Wallace, "Implementation of the IAU2000 definition of UT1 in astronomy" at the 2009 27th General Assembly of the IAU (JD06, IAU Gen. Assembly 2009):
The presentation showed that the sun's mean position (in hour angle) is still intended to be represented by UT1, also in the context of use of the ERA formulae (constants shown above in their post-2006 revised values). The details of this representation have previously been discussed in Astronomy SE, at (Does Universal Time really track mean solar time?).
The UT1 definition underwent indirect modification effected by IAU 2006 resolution incorporating latest precession-nutation data in the standard calculation of earth-rotation angle and hence of UT1: see P Wallace & N Capitaine (2006), "Precession-nutation procedures consistent with IAU 2006 resolutions", Astronomy & Astrophysics 459: 981-985.
The standard definitions are effectively available as C and Fortran algorithms at the IAU SOFA ('Standards of Fundamental Astronomy') webpages at (https://www.iausofa.org/), where there is also a guide to 'SOFA Time Scale and Calendar Tools' .
A summary of the recent position on time standards including UT1 was given for example in October 2019 at the conference "Journees 2019" by N Capitaine (https://syrte.obspm.fr/astro/journees2019/journees_pdf/SessionII_1/CAPITAINE_Nicole.pdf).
Further information about implementation of current standard UT1 can be found via:
SOFA earth-rotation page.
These UT1 standards are no longer strictly in accord with the classical physical basis that a mean solar time naturally depends on two independent variables, related respectively to angular rates of the earth's axial rotation, and of the earth's orbital motion around the sun. The standard UT1 calculations have effectively condensed these two dependencies into a single conventional constant, thus losing one of the independent variables of the natural physical model (discussed on Astronomy SE at (Does Universal Time really track mean solar time?)); but the adopted conventional relationship is expected to hold to adequate closeness for an extended period of time into the future.
(The redefinition of UT1, in some ways physically obscure, has also had some critiques of principle, e.g. Xu et al (1986) "Discussion of Meaning and Definition of UT" (IAU Symposia 109 "Astrometric Techniques", 13-17).)
In view of the wider-ranging matters asked in this question, it may be helpful to add the following:-
A good source on the physical foundations underlying determinations of mean time, especially in the 19th-century period shortly before the adoption of UT, is still F Brünnow's 'Spherical astronomy' (1865, New York).
Essential tools for the classical determination of mean solar time at the local meridian of an observatory were, in the 19th century --
- (a) an observatory clock keeping sidereal time as closely as possible, subject to observational corrections,
- (b) a transit instrument, i.e. a telescope aligned to the local meridian (meaning for stations at northern latitudes, pointing due south), and mounted on horizontal bearings for adjustable pointing to objects at various altitudes but always in the meridian direction, and
- (c) tables giving positions in right ascension (expressed as time) of well-characterized "clock stars" and their slow precessional motions, plus tables of the motion of the 'mean sun' (expressed as mean longitude from the equinox, given in units of time, where 360 degrees converts to 24 hr), taken from a theory of solar motion then currently accepted as most accurate. (During the early 19th-c. this usually meant the solar theory of Delambre as arranged by Carlini, displaced in the 1860s by LeVerrier's improved solar theory).
A bare bones summary of the practice was then to:--
- (i) record the times of observed transit of some "clock stars" past the central 'wire' in the viewing field of the transit instrument, using the observatory sidereal-time clock;
- (ii) evaluate the clock-correction needed for the meridian transit of each observed star to occur at a (corrected) clock sidereal time equal to the current right ascension of the star: the so-corrected clock then gave (local) sidereal time to the accuracy of the observations;
- (iii) compute the local mean solar time at the meridian of the observatory, for any instant, as the difference "local sidereal time minus sun's mean longitude in terms of time". The principle of this conversion is described in: (Does Universal Time really track mean solar time?) in the section headed "Classical method for determining mean solar time (Nautical Almanac, 19th-c.)". After adoption of the conventional standards, GMT or UT could then be derived from local mean solar time by applying the difference between the longitudes of Greenwich and of the local observatory expressed as time, up to the accuracy of the longitude determinations.
Greenwich Mean Time used to be the local mean solar time at Greenwich derived by those methods; it was renamed in the 1880s to Universal Time ('UT') and then renamed again in the 1950s to 'UT0' to make room for further versions, especially UT1, which originally was UT0 with corrections intended to remove small fluctuations due to the effect of polar motions on the longitude of the observatory.
The mean sun's position in celestial longitude used to be considered as what would result from observations after calculation to free them from all discoverable periodic variations (see e.g. J C Adams (1884) "On the definition of mean solar time", Observatory 7: 42-44, and A Cayley (1884), Monthly Notices R Astron Soc 77: 84-5).