Astronomy & Celestial Mechanics
the sky, computed
The observed sky

A clock allowed to leave the Sun

Twenty-seven leap seconds have kept civil time within a second of the Earth's rotation since 1972. By 2035 they will stop, and the clock will be allowed to drift from the Sun. How far it drifts, and how soon a minute accumulates, depends on a length of day that no one can forecast.

Assumes Timescales and Tides.

15 min read 6 figures The sky as a clockSmall perturbations, large consequences

Civil time is a compromise between two clocks. One is the Earth, turning once a day and marking noon when the Sun crosses the meridian. The other is the caesium atom, whose hyperfine transition defines the second and ticks at a rate that is the same everywhere and does not wander. The two do not agree, because the Earth is a clock that loses: its day is slightly longer than 86,400 of the atom’s seconds and irregularly so. Since 1972 the compromise has been Coordinated Universal Time — atomic seconds, with a whole second inserted whenever the Earth has fallen far enough behind that the two would otherwise differ by more than 0.9 seconds. Twenty-seven leap seconds have been inserted that way.

In 2022 the General Conference on Weights and Measures resolved to stop. By 2035 the tolerance between UTC and the Earth’s rotation will be raised to a value large enough that no leap second will be needed for at least a century, and the clock will be allowed to drift away from the Sun. The question this essay asks is how far and how fast — and the answer is that the size of the drift is well understood and its timing is not, because it depends on a quantity the Earth does not reveal in advance.

How far the clock drifts from the Sun once leap seconds stop. The accumulated difference between the Earth's rotation and atomic clock time, in seconds, that builds up after leap seconds are abandoned, against years since then, for a day that now exceeds 86,400 SI seconds by −0.3 ms, 0.5 ms, 1.5 ms, each lengthening by 1.7 milliseconds a century as the tides brake the Earth. The drift is the integral of the excess: 20 s after 100 years, a minute after 158 for −0.3 ms; 49 s after 100 years, a minute after 113 for 0.5 ms; 86 s after 100 years, a minute after 76 for 1.5 ms. The present excess is close to zero — the day has recently been slightly shorter than 86,400 seconds — so the drift starts slowly or even backwards, and the tidal term then takes over and bends every curve upward. The time for the difference to reach a minute is set far more by the next century's core-driven fluctuations, which have swung the excess by two or three milliseconds within a few decades before, than by anything that can be predicted.
Fig. 1 The accumulated difference between the Earth’s rotation and atomic time after leap seconds stop, for a day that now exceeds 86,400 SI seconds by −0.3, 0.5 and 1.5 milliseconds, each lengthening by 1.7 milliseconds a century under tidal braking. The drift is the integral of the excess: after a century it is 20, 49 and 86 seconds, and a full minute accumulates after 158, 113 and 76 years. The present excess is close to zero, so the drift starts slowly or even backwards before the tidal term bends every curve upward.

What twenty-seven seconds measured

A leap second is inserted when the accumulated lag of the Earth’s rotation behind atomic time has grown by about a second, so the rate at which they have been inserted is a direct measurement of how much longer than 86,400 seconds the average day has been.

Every leap second there has been. The number of leap seconds added to Coordinated Universal Time since 1972, when UTC was set ten seconds behind atomic time and the rule adopted that it would be held within 0.9 seconds of the Earth's rotation. Twenty-seven have been inserted, bringing the offset to 37 seconds; none since the last day of 2016. The rate is a direct measure of how long the day was: 1.13 a year in the 1970s, a day 3.1 milliseconds longer than 86,400 SI seconds on average; 0.65 a year from 1980 to 2000 (1.8 ms); 0.25 a year from 2000 to 2020 (0.7 ms). The staircase flattens because the Earth's day has been getting shorter again over these decades, through exchanges of angular momentum with its liquid core, against the steady tidal lengthening underneath.
Fig. 2 Every leap second added to UTC since 1972, when it was set ten seconds behind atomic time. Twenty-seven have been inserted, bringing the offset to 37 seconds; none since 31 December 2016. The rate measures the day: 1.13 a year in the 1970s, a day 3.1 milliseconds longer than 86,400 seconds; 0.65 a year from 1980 to 2000, 1.8 milliseconds; 0.25 a year from 2000 to 2020, 0.7 milliseconds.

The staircase flattens, and the reason is not the tides. Tidal friction, the Moon’s grip on the oceans transferring angular momentum from the Earth’s spin to the Moon’s orbit, lengthens the day steadily, by about 2.3 milliseconds a century, partly offset by the slow rebound of the land from the weight of the last ice age, which makes the Earth rounder and speeds it up by about 0.6. The net trend is a lengthening of about 1.7 milliseconds a century, and over two and a half thousand years it has accumulated into the hours of difference recorded in ancient eclipse observations.

On top of that trend sit fluctuations of a few milliseconds over decades, and those are what the leap-second staircase has been recording. They come mostly from the liquid outer core, which exchanges angular momentum with the solid mantle through magnetic and gravitational coupling at the boundary between them; when the core speeds up, the mantle slows, and the day lengthens, and vice versa. The day was unusually long in the early 1970s — by about three milliseconds — which is why the first years of the system needed a leap second nearly every year. It has shortened since, and by the early 2020s the excess was slightly negative: the Earth was turning a little faster than 86,400 seconds, and the day in 2020 included the shortest ever measured by atomic clocks.

The leap second that ran backwards

A negative excess makes the Earth’s rotation run ahead of atomic time rather than behind it, and UTC would then need a negative leap second — a minute of fifty-nine seconds — to stay within 0.9 seconds. None has ever been inserted, and the software that runs the world’s networks, much of which handles positive leap seconds badly, has never been tested against one.

The prospect became concrete in the early 2020s. Extrapolations of the Earth’s rotation suggested that UT1 − UTC would reach the threshold for a negative leap second around 2029. An analysis published in 2024 by Duncan Agnew added a factor nobody had weighed in the forecast: the melting of the Greenland and Antarctic ice sheets moves mass from high latitudes towards the equator, which increases the Earth’s moment of inertia and slows its rotation, and the acceleration of melting since the 1990s had delayed the expected negative leap second by about three years. Human-caused climate change had, measurably, moved the timing of a correction to civil time — which is an unexpected thing for a clock to be sensitive to, and an accurate one.

The question whether the negative leap second would be needed before 2035 is now moot as policy but not as physics. It showed plainly that the length of the day, on the timescale of a decade, is not forecastable to the precision the leap second requires, and it made the case for stopping more urgent.

Why they are being stopped

The arguments against leap seconds are about computers, not about astronomy. A leap second is announced six months in advance, is inserted at the end of a day as a 61st second of the last minute, and must be handled correctly by every system that keeps time in UTC. Many have not. The leap second of 30 June 2012 crashed servers running a widely used operating system kernel and delayed flights at an airline whose booking system failed. Large technology companies adopted “leap smearing” — spreading the extra second over the surrounding hours by running their clocks very slightly slow — which keeps their systems working and makes their clocks disagree with UTC by up to half a second during the smear. Satellite navigation systems keep their own time scales: GPS time has not had a leap second since 1980 and is now eighteen seconds ahead of UTC, and receivers convert between them with a broadcast offset.

The atomic time scale that UTC is built from has no leap seconds at all. International Atomic Time, TAI, is an average of hundreds of clocks, corrected after the fact, and it has run uniformly since 1958. Everything that needs uniform time could use it, and some systems do; the difficulty is that UTC is written into law and into communications standards in most countries, and changing what those standards refer to is harder than changing how UTC is defined. So the decision was to keep the name and change the rule.

The arguments for leap seconds are about the Sun. Civil time was originally solar time, and a century of drift would let noon move by a minute or more. That is small beside what civil time already does — time zones are an hour wide and some span three hours of solar time; daylight saving time moves noon by an hour twice a year; the Sun itself is an irregular clock, running up to sixteen minutes fast or slow through the year — and it is the reason most of the world was content to stop.

How long until a minute

Once the leap seconds stop, the difference between the Earth’s rotation and UTC grows as the integral of the excess length of day, and the time for it to reach any given size depends on what the excess does next.

How long until the difference reaches a minute. The number of years after leap seconds stop until the Earth's rotation has fallen a full minute behind atomic time, against the excess of the day over 86,400 SI seconds at the moment they stop, with the tidal lengthening of 1.7 milliseconds a century added throughout. At the long-run average excess of about 1.5 ms the minute is reached after 77 years; at 0.5 ms after 113; at the present, slightly negative excess of about −0.3 ms, after 158, almost entirely because of the tides. A leap minute, if one were ever inserted, would arrive somewhere between a lifetime and three lifetimes after the decision — which is to say that the decision to stop the leap second is a decision to pass the problem to people not yet born, with a clear idea of its size and none of its timing.
Fig. 3 Years after leap seconds stop until the Earth’s rotation has fallen a full minute behind atomic time, against the excess length of day at the moment they stop, with the tidal lengthening of 1.7 milliseconds a century added throughout. At the long-run average excess of about 1.5 milliseconds the minute is reached after 77 years; at 0.5 after 113; at the present excess of about −0.3, after 158. A leap minute would arrive somewhere between one lifetime and three after the decision.

The spread in that figure is the whole problem. The tidal trend is known to a few per cent, and on its own it would carry the difference to a minute in about a century and a half from a start at zero excess. The decadal fluctuations are not known in advance at all; they have swung the excess by two to three milliseconds within twenty years, and a sustained swing of that size, integrated over a few decades, moves the time to a minute by decades either way. The drift’s size is set by physics everyone agrees on and its timing by a fluid core nobody can forecast.

How far the clock drifts from the Sun once leap seconds stop. The accumulated difference between the Earth's rotation and atomic clock time, in seconds, that builds up after leap seconds are abandoned, against years since then, for a day that now exceeds 86,400 SI seconds by −0.5 ms, 0.5 ms, 2.5 ms, each lengthening by 1.2 milliseconds a century as the tides brake the Earth. The drift is the integral of the excess: 4 s after 100 years, a minute after 212 for −0.5 ms; 40 s after 100 years, a minute after 129 for 0.5 ms; 113 s after 100 years, a minute after 58 for 2.5 ms. The present excess is close to zero — the day has recently been slightly shorter than 86,400 seconds — so the drift starts slowly or even backwards, and the tidal term then takes over and bends every curve upward. The time for the difference to reach a minute is set far more by the next century's core-driven fluctuations, which have swung the excess by two or three milliseconds within a few decades before, than by anything that can be predicted.
Fig. 4 The same drift with the net tidal trend reduced to 1.2 milliseconds a century — the lower end of what the combination of tides and glacial rebound allows — and a wider range of starting excesses, −0.5, 0.5 and 2.5 milliseconds. After a century the drift is 4, 40 and 113 seconds, and a minute is reached after 212, 129 and 58 years. The starting excess matters more than the trend over the first century, and the starting excess is a quantity that will have changed several times over before the century is out.

What has been decided is a threshold, not a schedule. The details remain to be negotiated — whether the tolerance becomes a minute, an hour, or is abandoned altogether, and whether any correction is ever inserted or the difference simply allowed to grow — and whichever is chosen, the next correction, if there is one, will be made by people who will know the Earth’s rotation for the intervening century and are not yet born.

How the Earth’s rotation is read

UT1 — the time kept by the Earth’s rotation — is not read off a sundial or a transit instrument any more, and the way it is measured is part of why the decision to stop leap seconds could be taken calmly.

The Earth’s orientation in space is measured against the most distant objects there are. Radio telescopes separated by thousands of kilometres observe the same quasars simultaneously, and the difference in the arrival time of each wavefront at the different telescopes — measured to a few picoseconds — gives the direction of the baseline between them relative to the quasars. Because the telescopes are fixed to the Earth, a sequence of such measurements gives the Earth’s orientation against a frame defined by hundreds of quasars billions of light-years away, which do not move on the sky by any measurable amount. From that orientation come three numbers: two for the position of the pole on the Earth’s surface, which wanders in a circle of a few metres with a period of about fourteen months, and one for the rotation angle about it, which is UT1.

The rotation angle is known to about ten microseconds of time from a day’s observations, and it is published by the International Earth Rotation and Reference Systems Service together with forecasts for the following months. The difference UT1 − UTC is therefore a measured, published number, known far more precisely than any leap-second rule needs, and anyone who needs the Earth’s orientation already reads it from the bulletin. The leap second was never how astronomers kept track of the Earth; it was how everyone else was protected from having to.

The chain of offsets, rebuilt

The leap seconds are one link in a chain of offsets between the time scales astronomy uses, and stopping them changes one link and leaves the others.

TT − UT1 = 69.364 s at 2020, from three unrelated causes. The four time scales an astronomical calculation passes through, at 2020, in order of their offset from TAI and not to scale — UTC and UT1 differ by 0.18 s out of a 69 s span, which on any proportional axis puts their two rows inside one pixel of each other. Every offset is printed beside its own row. Three different KINDS of number are stacked here and telling them apart is the whole point. TT − TAI = 32.184 s is a definition, chosen once so that Terrestrial Time continued the ephemeris time it replaced; it will never change. TAI − UTC = 37 s is a count, of leap seconds inserted one at a time by a committee, and it changes by exactly one whenever they decide. UT1 − UTC = −0.18 s is a measurement of where the Earth actually is, published weekly and never allowed past 0.9 s, which is what the leap seconds are for. Add them and ΔT = TT − UT1 = 69.364 s, the same quantity the ΔT curve reaches 69.4 s for at this epoch — and every one of the 5.7 hours of it at the far end of that curve accumulated one leap second's worth at a time.
Fig. 5 The four time scales an astronomical calculation passes through, at 2020, in order of their offset from TAI and not to scale. TT − TAI = 32.184 seconds is a definition and will never change. TAI − UTC = 37 seconds is a count of leap seconds, which stops changing when they stop. UT1 − UTC, a measurement of where the Earth actually is, has been held within 0.9 seconds by the leap seconds. After 2035 the second offset freezes and the third grows without limit, and every calculation that needs the Earth’s orientation — a telescope’s pointing, a satellite’s ground track — will read UT1 − UTC from a bulletin instead of assuming it is small.

For astronomy the change is mostly bookkeeping. Every precise calculation already converts from UTC to UT1 using the measured difference, which the International Earth Rotation and Reference Systems Service publishes weekly from very-long-baseline radio observations of quasars; the difference simply stops being guaranteed small. Software that assumed it was below a second — that took UTC as a good enough approximation to the Earth’s orientation — will point telescopes wrongly by 15 arcseconds per second of accumulated difference, which is the width of a star’s image in poor seeing after a single second and a degree after four minutes. The assumption was never correct to better than a second, and after 2035 it will fail visibly within a few decades.

The deeper offset is unaffected. Terrestrial Time and the relativistic time scales that ephemerides are computed in never had leap seconds, and the difference between them and the Earth’s rotation, ΔT, will go on growing as it has for three thousand years.

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.
Fig. 6 ΔT = TT − UT1 from 700 BC to 2020, on a logarithmic scale, with a parabola fitted to the older entries. It is quadratic because the Earth is not merely slow but slowing, and its coefficient is a length-of-day rate that agrees with the tidal figure computed from angular momentum. The leap seconds were the last seventy years of this curve cut into one-second steps; stopping them does not change the curve at all.

The clock that telescopes point by

The one kind of timekeeping that has always depended on the Earth’s rotation rather than on atomic seconds is the one astronomers use at the telescope.

Sidereal time measures the Earth’s rotation against the stars rather than the Sun, and it runs four minutes a day faster than solar time because the Earth’s orbital motion carries the Sun eastward among the stars by about a degree a day. The local sidereal time at a telescope is the right ascension currently on its meridian, which is what a telescope needs in order to point: a star’s hour angle is the sidereal time minus its right ascension. And sidereal time is computed from UT1, not from UTC — it is a statement about the Earth’s orientation, and the leap seconds were only a device for keeping UTC close enough to UT1 that the difference could be ignored for rough work.

After 2035 the difference will stop being negligible within a few decades. An error of one second in the time used to compute the sidereal time points a telescope fifteen arcseconds from its target at the celestial equator, which is several times the field of view of a spectrograph’s slit; an error of a minute points it a quarter of a degree away. Every observatory control system already applies UT1 − UTC, so the change will pass unnoticed there. The systems at risk are the ones written on the assumption that UTC tracks the Earth — small-telescope software, satellite-tracking programs, antennas for amateur and educational use — and the risk grows slowly enough that it will be discovered by someone whose pointing has drifted rather than announced by anyone.

A calendar decision, again

Deciding to let a clock drift from the thing it was meant to track, and to defer the correction to a later generation, has been done before, with the calendar.

The Julian calendar’s year of 365.25 days is eleven minutes longer than the tropical year, and by the sixteenth century the accumulated error had moved the spring equinox ten days away from the date the Church used to compute Easter. The Gregorian reform of 1582 corrected it by removing ten days and dropping three leap days every four centuries, and the year that is not a whole number of days has been approximated well enough since that the calendar drifts by a day only in about three thousand years. The reform was adopted over three and a half centuries, country by country.

The leap-second decision is the opposite choice with the same structure. The calendar reformers chose to keep dates tied to the seasons at the cost of a discontinuity; the metrologists have chosen to keep the time scale continuous at the cost of letting it drift from the Sun. Both decisions trade a correction that is visible and disruptive for one that is slow and invisible, and the drift chosen now will accumulate, like the Julian calendar’s, into something a future generation must decide what to do about. The leap second was an attempt to correct the clock continuously, in small steps; stopping it returns to the older habit of correcting rarely, in large ones, or not at all.

What the figures assume

The excess length of day is drawn as a straight line in time. Its real history is a tidal trend with decadal fluctuations of several milliseconds, and the fluctuations are the reason the forecast is uncertain; drawing three straight lines from three starting values brackets the possibilities without modelling them.

The net tidal trend is taken as 1.7 milliseconds a century, the tidal braking of about 2.3 less the glacial-rebound acceleration of about 0.6. The balance depends on the rate of rebound and on the modern redistribution of water from ice sheets to oceans, and the second of those is changing in a way the historical rate does not include.

The leap-second rates by decade are simple counts, and a count per decade is a coarse measurement of an average day; the published length-of-day record, measured daily from space geodesy, shows the same trend with far more detail, including the seasonal and tidal variations of a millisecond that the counts average out.

And the time to a minute assumes no correction at all. The rule eventually adopted may allow a larger tolerance, a scheduled leap minute, or something else; the figures show only how quickly the Earth would bring the question back.

Still open: whether civil time keeps any tie to the Earth

The resolution of 2022 decided that leap seconds will stop and left open what, if anything, will replace them. One possibility is a larger step at long intervals — a leap minute every century or so, announced far in advance. Another is to abandon the tie altogether, and let UTC become a renamed atomic time scale that drifts from the Sun indefinitely, as the calendar would have drifted from the seasons without the Gregorian reform. A third is to leave the choice to the future. Each has a different consequence for navigation, astronomy and law, and each depends on a rate of drift that the core of the Earth will set and no one can predict. The rule that has governed civil time for half a century was an assumption that the Earth’s rotation is worth tracking to a second; what replaces it will say, in effect, how much the rotation of the planet still matters to the time its inhabitants keep.

Where this goes next

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. How steady each kind of clock is, by how long it is averaged. The Allan deviation — the typical fractional difference in frequency between successive averages of length τ — against τ, for representative noise models of caesium beam, hydrogen maser, caesium fountain, optical lattice clock. Each falls as τ^−½ while random frequency noise dominates, then levels off at a floor set by slower, correlated noise, and a clock with a systematic drift eventually rises again. A caesium beam is still improving at the longest averaging drawn, reaching 10⁻¹⁴; a hydrogen maser reaches its best, 7.3·10⁻¹⁶, after about 27 hours; a caesium fountain is still improving at the longest averaging drawn, reaching 1.2·10⁻¹⁶; an optical lattice clock is still improving at the longest averaging drawn, reaching 10⁻¹⁸. No clock is best at every averaging time — the maser is steadiest over hours and wanders over weeks, the fountain is noisier over minutes and more accurate over months — and a time scale is built by using each where it is best. The curves are typical shapes, not the specification of any particular instrument. A time scale that is finished after it is used The world's reference time is not kept by any clock. It is an average of about four hundred and fifty, weighted by how steady each has been, published a month late, and then recomputed a year later against the handful of clocks that define the second — so the best available time for any moment is known only long after the moment has passed. The best clocks are now so good that averaging them requires knowing each one's height to a centimetre. Five rules, and how fast each one leaves the seasons behind. The accumulated difference between a calendar and the seasons, for five leap-year rules, over 4000 years. Each is a straight line whose slope is the rule's fraction minus the tropical year's 0.2421897, and nothing else about a calendar matters to this plot — not which months are long, not where the year starts, not what anything is called. The Julian quarter is out by 11.25 minutes a year, which is one day every 128 years and is why ten days had to be removed in 1582. The Gregorian rule takes 3223 years to lose a day and the Persian 4264, in the other direction — the older rule is the better one, at a denominator of 33 against 400. The 128-year rule is level on this scale: 454545 years to a day, which is longer than any calendar has been kept — and it is not a separate invention but the Julian rule with its own error subtracted, since 1/4 − 1/128 = 31/128 exactly and the Julian drift is exactly one day per 128 years. What the figure cannot show is that the tropical year is itself shortening, by about half a second a century, so the slopes drawn here are the ones at the present epoch and every line is very slightly curved. A year that is not a whole number of days The tropical year is 365.24219 days, so every calendar is a fraction chosen to approximate 0.24219. The continued fraction says which fractions are best, and the one in use is not among them.

The objects this essay names

Each one links to every other essay that touches it.

Coordinated Universal TimeCore–mantle couplingΔTInternational Atomic TimeLeap secondLength of dayTidal frictionUniversal Time