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Every reference I find says that they are "essentially" the same, which we all know really means that they are not the same, but different only by a some small amount that someone else other than me decides is negligible.

They also like to say that UT1 and UTC are the same, but if you look into it you find that they differ by as much as 0.9 seconds. This is, admittedly, insignificant to me. But the practicality of it is not my concern here. The technical reality of it is.

I want specifics and exactness where applicable here.

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UT1 is a specific "flavor" of Universal Time, which is a measure of Earth's rotation relative to the mean sun, a fictitious "prime mover" upon which all our clocks are based. UT1 is related to sidereal time (Earth's rotation relative to the fixed background stars) by a rather long mathematical expression usually expressed as a polynomial function of mean solar time. Earth's rotation is not uniform though; it varies. UTC (Coordinated Universal Time) is a specific "flavor" of Universal Time intended to smooth out these variations by staying within 0.9 s of UT1. The difference between the two is called $\Delta UT1$ and is adjusted as necessary, but can only be adjusted after observation. Larger adjustments come in the form of leap seconds. The current approximate value of $\Delta UT$ is called $DUT1$ and is encoded in the standard time signals broadcast by stations such as WWV (in America) and CHU (in Canada). Listen for doubling of pips at the beginning of each minute.

UT0 is an observational approximation to UT1 based on meridian observations of standard stars. UT0 must be corrected for polar motion, which varies from observatory to observatory.

Both UT1 and UTC are generically referred to as Universal Time and the distinction between them is important only if the that maximum discrepancy of 0.9 s is important for your application.

There is another "flavor" of Universal Time called UT1R, which is intended to account for tidal variations in Earth's rotation.

There is another "flavor" of Universal Time called UT2, which is intended to account for seasonal variations in Earth's rotation. UT2 isn't used any more.

GMT, Greenwich Mean Time, is a sometimes deprecated historic term equivalent to UT, but is no longer used in astronomical applications. It used still used in many civil applications though (and remains the legal time standard (outside the period of daylight-saving summer time) in the UK).

By far your best reference on this topic is the latest edition of The Explanatory Supplement to the Astronomical Almanac, edited by Sean Urban and Kenneth Seidelmann. The third edition was just published in November by University Science Books. My own book, Fundamental Ephemeris Computations (Willmann-Bell, 2000) also discusses this topic and includes computer code.

Be aware that since FEC was published, certain astronomical terminology has changed and now the term "Earth rotation angle" is now used to mean roughly what sidereal time previously meant. The new terminology is reflected in the new Explanatory Supplement referenced above.

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Simpler answer:

UT1 is the time used for most common purposes. It follows the sun and the earth's rotation. There is always 86400 seconds in a day and always either 365 or 366 days in a year. Unfortunately the length of a second or day varies by a tiny amount from one year to another. In computers, it's usually modeled as an integer number of seconds since the epoch, which, for unix, is Jan 1, 1970, midnight GMT/UTC/UT1 timezone. Exact hours are always even multiples of 3600 seconds.

UTC is time as measured by atomic clocks. The second is always the exact same amount of time. Unfortunately the earth's movement isn't so exact. So, the powers that be insert a 'leap second' every so often on Dec 31 or June 30, and the time is then 23:59:60 for just that second. (They're inserted irregularly but there's been about two dozen since the 1970s.) So, the count of seconds for UTC is rarely an even number at the end of an hour. But, at least you know the time exactly.

Most of us can set our clocks to either, as the slide of a second isn't noticed for most practical purposes. If you're late to a meeting, you can't blame it on leap seconds.

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