Astronomy & Celestial Mechanics
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Timescales — the series

2 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A clock that has lost 5.7 hours in 2720 years. ΔT = TT − UT1, the accumulated difference between a uniform time scale and the Earth's own rotation, from 700 BC to 2020, on a logarithmic scale. The points are the published record; the smooth curve is the parabola 31.94·u² fitted to the entries at or before 1000, with u in centuries from 1820. It is a parabola and not a line because the Earth is not merely slow, it is slowing: a day lengthening at a constant rate makes a clock fall behind by the integral of the lag. That coefficient is a length-of-day rate and nothing else — 31.94 = ½ × r × 36525 days per century gives r = 1.75 ms per century, against the 1.78 ms computed from tidal angular momentum with no eclipse anywhere in the derivation. At 700 BC the offset is 5.7 hours, which is 85° of the Earth's rotation, and that is why the ancient measurements are records of the place a total eclipse was seen from rather than of the hour: the hour was never written down accurately enough to matter, and the shadow's track on the ground was.

    Six kinds of second

    The Earth is a clock that loses, and it has lost five and a half hours since 700 BC. That number was measured from the places ancient eclipses were seen from, not the times they were seen at — because the record carries a longitude and no clock.

    part 1 · sky
  2. TCB has gained 23.5 seconds on TT since 1977. How far four of the solar system's time scales have drifted apart, against years since they were set equal. TT is the time a clock on the Earth's geoid keeps, and it is flat here by definition. TCG is the time a clock at rest just outside the Earth's gravitational well would keep, and it runs faster by seven parts in ten thousand million — 0.0220 seconds a year. TCB is the time a clock at rest outside the Sun's well would keep, and it runs faster by fifteen parts in a thousand million, or 0.4893 seconds a year. TDB is TCB with that rate divided out so that it stays within milliseconds of TT, which is what makes it usable as an ephemeris argument and what makes it a coordinate rather than a proper time. The differences are constants and they are not corrections: an ephemeris tabulated against one of these and evaluated against another is wrong by the whole of this plot.

    A second that depends on where the clock is

    A clock in orbit runs fast, a clock at the bottom of a gravity well runs slow, and a clock at rest far from the Sun runs faster than anything on Earth by fifteen parts in a thousand million. Astronomy therefore has four different seconds, two of them longer than the others, and an ephemeris has to declare which one it is tabulated against.

    part 2 · sky

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