Six kinds of second

There is a question with no answer, and the fact that it has no answer is the whole subject: how long is a second?

Six candidate answers are in daily use, and one of them is not even the rotation everybody means — the day against the stars and the day against the Sun differ by four minutes before any of this begins. One is a count of caesium transitions. One is the Earth’s rotation. One is a definition chosen to match a scale that was itself defined to match the Moon. One is the first with an integer number of leap seconds subtracted. Two more are relativistic corrections for where the clock is sitting. They agree to nine or ten decimal places and they disagree in the eleventh, and the disagreements are not errors — they are the objects.

The Earth is not a clock #

Until 1955 the second was defined as $1/86{,}400$ of a mean solar day, and until 1935 nobody could show that this was a problem.

The trouble had been visible for two centuries in a different form. Halley noticed in 1695 that ancient eclipse records did not fit a Moon moving at its present rate; the Moon appeared to have been slower in the past, which came to be called the secular acceleration. Laplace explained most of it in 1787 as a consequence of the slow change in the Earth’s orbital eccentricity, and the explanation was correct in mechanism and wrong by half in size — a discrepancy that sat unresolved for another century and a half.

The resolution is that the Moon’s acceleration was never the whole of it. Part of the discrepancy was the clock. Timing an ancient eclipse means comparing a computed position of the Moon at some instant with a record of where the shadow fell, and the computation runs on uniform time while the record is indexed by the Earth’s rotation. If the Earth has slowed, the two indexes drift apart, and the drift appears in the comparison as an apparent acceleration of everything in the sky at once — including the saros itself, whose predicted repeats fall progressively west of where the record puts them.

The tell is that the effect is the same for the Sun, the Moon and the planets, in proportion to each one’s own angular speed. A physical acceleration of the Moon would be a fact about the Moon. A slowing Earth is a fact about the observer, and it shows up on everything.

A definition, a count, and a measurement #

The modern scales stack in a specific way, and the three kinds of number in the stack behave completely differently.

TAI is International Atomic Time, the weighted average of some 450 caesium and rubidium standards in about 80 laboratories, steered to realise the SI second on the rotating geoid. It is the reference and it has no offset from itself.

TT, Terrestrial Time, is $\mathrm{TAI} + 32.184\ \mathrm{s}$. Exactly. That constant was chosen in 1976 so that Terrestrial Time would continue Ephemeris Time without a step, and Ephemeris Time had been defined from the Moon’s orbital motion. The number is therefore a fossil: it encodes the offset that happened to exist between atomic time and the lunar ephemeris at one moment in the 1950s, and it will never change.

UTC is $\mathrm{TAI}$ minus an integer number of seconds — 37, since 2017. That integer is a count, incremented by hand when the International Earth Rotation Service decides one is needed. It is not a physical quantity and it does not vary continuously; it steps.

UT1 is the Earth’s rotation angle, expressed as a time. It is measured, weekly, by very long baseline interferometry on quasars and by satellite laser ranging, and it is not uniform: it wobbles at the millisecond level with the seasons as the atmosphere’s angular momentum shifts, and it has drifted by the amount the first figure plots.

The leap second is where the three kinds collide. It exists to keep UTC — the scale civil clocks run on — within 0.9 seconds of UT1, so that noon stays approximately noon. It does so by inserting a discontinuity into a scale that is otherwise uniform, which is exactly the property computers rely on. A time scale cannot be both uniform and tied to the Earth, and the leap second is the cost of pretending otherwise for as long as possible.

The measurement is a place, not a time #

Here is the part that makes ΔT an astronomical measurement rather than a bookkeeping exercise.

The Babylonian records on cuneiform tablets, the Chinese dynastic histories, the Arab observations from Cairo and Baghdad — these give dates and, at best, an hour estimated from the position of the Sun or from a water clock. An hour is not enough. To measure a five-hour offset one needs the time an eclipse happened to a precision far better than the offset itself, which no ancient record provides.

What the records do give, exactly and unambiguously, is a place. A total solar eclipse is total along a track a hundred kilometres wide and nowhere else; an observer who records totality is recording that they were inside that track. Where the track falls depends on the Earth’s rotation angle at that instant, and so on ΔT. A computed eclipse with $\Delta T = 0$ puts the 15 April 136 BC totality in the Atlantic; the record says Babylon. Sliding ΔT until the track covers Babylon gives $\Delta T \approx 11{,}900$ s for that date, and the width of the track gives the error bar.

So the length of the day two thousand years ago is measured in units of longitude. The observation is that a scribe in a known city saw the Sun disappear entirely, and the quantity extracted is the number of seconds by which the Earth’s rotation has fallen behind. No clock is involved anywhere in the chain.

Why a parabola, and what its coefficient is #

If the day lengthens at a constant rate $r$, then after $n$ centuries the day is $86{,}400 + rn$ seconds long. A clock reading out the Earth’s rotation therefore falls behind by the accumulated excess, which is the integral:

$$\Delta T = \int_0^n r u \times 36{,}525 \, du = \tfrac12 r \times 36{,}525 \times n^2.$$

At $r = 1.78$ ms per century that is $32.5\,n^2$ seconds, and the fit to the pre-AD-1000 record in the first figure gives 31.94. The two agree to two per cent.

That agreement is the essay’s point. The 1.78 ms comes from angular momentum accounting on the Earth–Moon system — the Moon’s measured recession of 3.83 cm a year, times the mass ratio, times the geometry, with no historical record anywhere in it. The 31.94 comes from eclipse tracks on clay tablets. Two chains of inference sharing no data arrive at the same number.

They also disagree, in a way that is itself a result. The observed long-term rate is about 1.75 ms per century, and the rate that tidal friction alone predicts is about 2.3. The Earth is slowing more slowly than the tides demand, and the difference is attributed to glacial isostatic adjustment: the mantle is still rebounding from the ice sheets that melted 12,000 years ago, mass is moving from the equator towards the poles, and the moment of inertia is falling. The planet is spinning up as it re-rounds, by about a third as much as the tides slow it down.

Both of the last two sections describe a unit chasing the phenomena it was meant to measure, and losing. The second was tied to the Earth’s rotation until the rotation was shown to be irregular; to the Earth’s orbit until an atomic transition beat it; and to caesium until optical transitions beat that. Each redefinition kept the numerical value and changed what realises it, which is the only way a unit can be improved without invalidating every measurement previously expressed in it.

Where the model stops #

The rate is not constant. A parabola fits the last three thousand years because the continental configuration has not changed much over that interval. Over geological time the tidal dissipation depends strongly on the shapes of the ocean basins — most of it happens in shallow seas, and the present epoch has an unusually resonant configuration — so extrapolating the parabola backwards past a few million years gives absurd answers. Run it back far enough and the Moon is inside the Roche limit 1.5 billion years ago, which is a refutation of the constant rate rather than a date.

UT1 is not smooth even now. Superimposed on the secular slowing are seasonal variations of about a millisecond, driven by the atmosphere’s angular momentum; decadal variations of several milliseconds, thought to come from core–mantle coupling; and steps at the level of microseconds after large earthquakes. None of them is a perturbation of the orbit; all of them are perturbations of the clock. The 2004 Sumatra earthquake shortened the day by about 2.7 microseconds by moving mass towards the axis. Since about 2020 the Earth has been running slightly fast, so that the next leap second may need to be negative — which has never been done and which most timekeeping software would not survive.

And the scales themselves are relativistic. TT is defined on the rotating geoid, and a clock at altitude runs faster; TCG and TCB — Geocentric and Barycentric Coordinate Time — are the scales that remove those rate offsets, and TDB is TCB rescaled to stay close to TT. A planetary ephemeris is computed in TDB and observations are timestamped in UTC, so every reduction contains a chain of four conversions before any astronomy happens. The differences are periodic, of amplitude 1.7 milliseconds, and for pulsar timing they are the dominant systematic — which is what makes a period derivative read out of pulse arrival times a statement about four decades of clock conversions as much as about an orbit.

What the picture cannot show #

The error bars. The ΔT points before about AD 1600 carry uncertainties of tens of minutes at the far end, set by the widths of eclipse tracks and by the reliability of the records. On a logarithmic axis spanning four decades those uncertainties are invisible, and they are the reason the fitted coefficient is quoted to two figures rather than four.

The gaps. The record is dense in Babylon from about 700 to 50 BC, dense in China across most of the last two millennia, dense in the Arab world from the ninth to the twelfth centuries, and thin everywhere else. A smooth curve drawn through those points is an interpolation across centuries with no data at all.

Which second. Every axis on this page is labelled in seconds, and the whole essay is about the fact that “second” names at least six things. The seconds on the vertical axis of the ΔT plot are SI seconds; the ones the ancient observers were unknowingly counting were 1/86,400 of their own days, and those were shorter than a modern day by about forty milliseconds.

The second that was defined twice #

The constant of 32.184 seconds mentioned above is a fossil, and following it back gives the strangest fact in this subject: the SI second is calibrated on a nineteenth-century measurement of the Earth’s orbit.

Once it became clear that the Earth’s rotation was not uniform, the second had to be redefined against something that was, and the choice made in 1956 was the Earth’s orbit rather than its spin. The ephemeris second was defined as a specific fraction of the tropical year 1900 — a particular year, chosen because Newcomb’s tables of the Sun, published in 1895 and fitted to observations spanning the eighteenth and nineteenth centuries, defined the Sun’s mean motion at that epoch.

So for eleven years the international unit of time was a fraction of a year that had already passed, defined by a table fitted to meridian observations made with visual instruments.

In 1967 the second was redefined again, as a number of periods of a caesium transition. The number chosen — 9,192,631,770 — was not derived from anything about caesium. It was measured, by comparing a caesium standard against the ephemeris second over several years, and then adopted so that the new second would be as nearly as possible equal to the old one.

The chain is therefore intact and slightly absurd. The modern second is defined by caesium; the caesium number was chosen to match the ephemeris second; the ephemeris second was a fraction of the tropical year 1900; and that year’s length came from Newcomb’s fit to observations of the Sun made before anyone knew the Earth’s rotation was irregular.

Every atomic clock in the world is ultimately calibrated against the Earth’s orbital motion as it was understood in the 1890s, to the precision of the 1950s comparison, and the 32.184-second offset is the arithmetic residue of the join.

Two further consequences of that history are worth noting before leaving it, and both are still live rather than historical.

The clocks that have outrun the definition #

The caesium second has a realisation uncertainty of a few parts in $10^{16}$, which is the floor set by how well the transition frequency can be measured in a working standard.

Optical clocks are better by two orders of magnitude. They use a transition at optical rather than microwave frequency, which gives a far higher quality factor, and the best of them reach parts in $10^{18}$ — a discrepancy of one second over the age of the universe.

That creates an awkward position. A quantity can be measured better than the unit it is expressed in, so comparisons between optical clocks are now limited by the definition of the second rather than by the clocks. A redefinition is expected, and the practical obstacle is choosing which transition, since several are comparable and switching means recalibrating everything.

There is a consequence for the rest of this essay that is not merely metrological. At a part in $10^{18}$ a clock is sensitive to the gravitational redshift over a height difference of about a centimetre. Two such clocks on the same bench, one raised by a hand’s breadth, run at measurably different rates.

That makes a clock a gravimeter, and the technique has a name — chronometric levelling — and a use. Height above the geoid is currently measured by combining satellite positioning with a model of the gravity field, and the model is the weak part; a network of optical clocks compared by fibre would measure the geoid directly, at centimetre accuracy, from the rates alone.

A clock accurate enough stops being a clock and becomes an instrument for measuring where it is, and the scales this essay is about — which are all defined on a surface of constant gravitational potential — become quantities that have to be measured rather than assumed.

Where the ladder goes next #

Later rungs on this anchor: the leap second’s abolition, agreed in 2022 for implementation by 2035, and what replaces it. The proposed leap hour. Pulsar timing as an independent clock, and the millisecond pulsars whose stability rivals a maser over decades. The relativistic scales in full — TCG, TCB, TDB — and why a planetary ephemeris cannot be computed in any scale a clock on Earth realises. Universal Time’s subdivisions, UT0 and UT2, and what polar motion has to do with them. And the reverse inference: using ΔT as an input to constrain the history of ocean-basin geometry, which is one of very few observational handles on the deep past of tidal dissipation.

The last object in this essay is a clay tablet. It records that on a particular day the Sun was covered, in a city whose latitude and longitude are known to a kilometre, and it does not say what time. That absence is what makes it a measurement: a recorded hour would have been a scribe’s estimate, and the recorded place is a fact about the geometry of a shadow that nobody had any reason to falsify and nobody could have understood.