Put the question another way: Suppose you want to determine the moment of midnight to within 0.5s without using any clock more accurate than counting the number

Put the question another way: Suppose you want to determine the moment of midnight to within 0.5s without using any clock more accurate than counting the number of days. You are, however, allowed to use telescopes, reference materials and perform arbitrary amounts of calculations. And you're allowed to wait for ideal observing conditions. The general kind of instrument you want is the https://en.wikipedia.org/wiki/Transit_instrument https://en.wikipedia.org/wiki/Transit_instrument -- if you can measure the transit of a star to 1 arcsecond, that's better than 1 second/day. That article says there are transit instruments that work down to .01 arcsecond. Unless you have to go a very long time just counting days (enough for a half a leap second to accumulate), it seems pretty clear you could get the correct UTC second. You mention computer glitches; there also exist non-transistorized clocks that are accurate to much more than 1s/day. In fact, the Wikipedia article about the Elgin Observatory talks about how use of a transit instrument to discipline a mechanical clock was accurate to .01s https://en.wikipedia.org/wiki/Elgin_National_Watch_Company_Observatory https://en.wikipedia.org/wiki/Elgin_National_Watch_Company_O... -- though I doubt such a system is in continuous use today.