By Don Koks, 2021.

Every few years, clocks around the world are paused for one second, either at the end of June or the end of December.
These pauses don't conform to a schedule; they are made only when needed, and confining them to June and December is just a
convention. Pausing a clock for a second is equivalent to inserting a second into its record of passing time. This
infrequently inserted second is called a *leap second*. We need leap seconds because our "international clock"
ticks just a little too quickly for our civil needs, and so must be stopped every once in a while. Because of the
disruption that stopping the international clock can cause, it's convenient to wait until that clock needs to be stopped for a
whole second, before doing so.

A century ago, the second was defined such that the length of the *mean solar day*—the average time between the
Sun's meridian transits—was 24 hours, or 24 × 60 × 60 seconds = 86,400 seconds. The use of "average"
here was problematic: how might this second be reproduced in the laboratory? With the advent of atomic clocks, the second
was redefined to be the length of time taken for a particular caesium isotope to emit 9,192,631,770 wavelengths of light when
undergoing a particular electronic transition. This modern second, based on the atom and so reproducible in the lab, is
the SI unit for *international atomic time*, "TAI".

When the second was redefined in this atomic way in 1967, researchers had to choose the number of wavelengths of light such
that 86,400 of the new SI seconds would fit exactly into one mean solar day. Given that any particular solar day probably
doesn't have the same length as a *mean* solar day, researchers cannot have been expected to get things exactly
right. They did very well, but fitting *exactly* 86,400 of the new SI seconds into one mean solar day was always
going to be a big ask, and they didn't quite get it right. If the SI second had instead been defined to be 9,192,631,997
wavelengths of caesium light, it would've been a much better fit to the mean solar day [1].
Additionally, Earth's spin rate slows such that the mean solar day lengthens by one or two milliseconds every century.

The result is that in recent decades, the length of the mean solar day has been approximately 86,400.001 SI seconds long. TAI clocks thus tick slightly fast compared to the Sun's motion in the sky. The consequences of this extra millisecond can be described by analysing the unrelated phenomenon of Earth spinning as it orbits the Sun.

Earth rotates approximately 365.25 times in one orbit around the Sun. For the purpose of this discussion, we'll
suppose that number to be exact. That is, 365.25 calendar days equate to one orbit. Our calendar year necessarily
comprises a whole number of these calendar days. Suppose that this whole number was set to be invariably 365. Then
such a calendar would finish its 365-day cycle a quarter of a day before Earth had completed one orbit. But our orbit
determines our seasons, and calendars were made to keep track of the seasons. So, this 365-day calendar would finish its
cycle a quarter day too soon to keep in step with the seasons. We could not address that problem after one calendar year,
since we could not very well start a new day at 6 a.m. on 1st January. But we *would* notice that four calendar
years (which require 4 × 365 days) finish a full day short of four orbits (which require 4 × 365.25 days), meaning
a full day short of what the date should be to mark the current season accurately.

For example, if the start of January marked a particular point in Earth's orbit, then at the end of 31st December four years later, another full day would have to pass before Earth returned to that point in its orbit marking the beginning of 1st January. In a sense, this 365-day calendar "ticks too quickly" for the needs of the seasons: the new calendar year begins too soon. So, every four years, we would have to pause such a calendar for one day, to delay—by one day—the next calendar year from beginning. That means, of course, adding a "leap day", traditionally at the end of February. When we do that, the calendar again lines up with the seasons to within one day.

Notice that the need for leap days does not imply that Earth is undergoing a long-term slowing in its journey around the
Sun. If Earth's orbital period *were* increasing, that certainly would impact our calendar. Nonetheless, the
need for leap days says nothing about any possible change in Earth's orbital period. Remember that—it'll be
important later.

In summary:

- 1 orbit = 365.25 calendar days.
- We want the orbit to align with the calendar, but subsequent orbits get out of synchronisation with the calendar.
- 4 orbits = (4 × 365 + 1) calendar days.
- So after 4 orbits, the calendar has gotten 1 day ahead of the orbit.
- Hence, after 4 orbits, pause the calendar for 1 day. That is, "insert a leap day" into the calendar.

The above discussion of 365.25 calendar days per orbit is analogous to a discussion of 86,400.001 SI seconds per mean solar day. In the same way that a 365-day calendar completes one cycle 0.25 days too soon to keep pace with the 365.25-day year of four seasons (one orbit, which is useful in our civil lives), a TAI clock counts out 86,400 SI seconds one millisecond too soon to keep pace with the 86,400.001 SI-second mean solar day (which is useful in our civil lives). Recall that we allow four 365-day years to elapse to accrue a 1-day difference of the 365-day calendar from the four-season year, and then we stop that calendar for one day, which in practice means inserting a leap day. Now consider that 1000 mean solar days require about 86,400,001 SI seconds. But this same time interval is required to mark out 86,400,000 "civil" seconds. Hence, TAI clocks get approximately one second ahead of the Sun every 1000 days (about three years). But civil life is in step with the Sun, and we require our clocks to reflect civil needs rather than atomic laws.

So, we might consider stopping TAI clocks for one second every three years to re-align their time with the mean solar
day. That is, in a way, what is done. But in practice, we choose never to stop TAI clocks; instead, the
precise-timing community has made a one-off copy of a TAI clock at some initial time. It has called that copy
a *coordinated universal time* clock ("UTC"), and has introduced the practice of stopping this UTC clock for one second
every three years approximately, to keep the UTC clock aligned with the mean solar day. There is no single official UTC
clock in the world; instead, UTC time is calculated from a set of clocks built for that purpose.

Stopping a UTC clock for a second is equivalent to inserting an extra second into its record of passing time without
stopping it. UTC incorporates these *leap seconds*, and is today's Greenwich mean time, our civil time.
Given the widely acknowledged difficulties that leap seconds give computer administrators, the idea of discarding leap seconds
for the foreseeable future is being considered in the timing community.

In summary:

- 1 mean solar day = 86,400.001 SI seconds.
- We want the mean solar day to align with a UTC clock's display, but subsequent days get out of synchronisation with the UTC clock's display.
- 1000 mean solar days = (1000 × 86,400 + 1) SI seconds.
- So after 1000 mean solar days, the UTC clock's display has gotten 1 second ahead of the mean solar day.
- Hence, after 1000 mean solar days, pause the UTC clock for 1 second. That is, "insert a leap second" into the clock's record of time. In practice, this isn't done exactly every 1000 days; it depends on small changes happening to Earth's spin rate.

Recall the point above that the need for leap *days* does not imply that the time needed for one orbit of Earth
around the Sun is increasing. The analogous statement is true for leap *seconds*: their existence does not imply
that Earth's spin rate is slowing. Earth's spin rate *is* slowing; but that fact cannot be inferred from the
current need for leap seconds. Leap seconds tell us only that our UTC clocks tick too quickly to match the mean solar
day. UTC clocks are no different to your wristwatch: if your watch gets a little ahead every day, you cannot infer that
Earth's spin is slowing. You cannot infer anything; but you might make the infinitely more probable guess that your watch
ticks faster than it should. And so you correct it occasionally. Leap seconds are the same idea: they are just a
correction to UTC clocks because those clocks tick too fast.

Nonetheless, it is widely believed—incorrectly—that the ongoing need for leap seconds is due to Earth's spin rate slowing. You will even find this written in the US Naval Observatory's Circular number 179, which describes the International Astronomical Union's utterances on time, with the words "the tidal deceleration of the Earth's rotation, which causes [the civil time] to lag increasingly behind [atomic time]". This lag (which is the ever-increasing lag of UTC behind TAI described above) is caused by the UTC clock ticking too quickly for civil purposes; but whether that too-quick ticking is caused by a slowing of Earth's rotation is another question entirely.

Earth's spin slows at a rate of one or two milliseconds per century (and is thought to be somewhat erratic, and even
speeding up from time to time). This long-term slowing of rotation has presumably contributed to the 1-millisecond
mismatch mentioned above, but is not solely responsible for the mismatch. The definition of the SI second was based on
day-length data that was already outdated at the time of that definition; consequently, more than 86,400 SI seconds equalled a
mean solar day from the very start. But given the difficulties of measuring the length of a mean solar day—and
even *defining* such a day—we should not expect the numbers to have been perfect from the start. Hence,
whatever the reasons might be, the SI second is slightly too short for 86,400 of them to equal the current mean solar day.

It follows that if Earth's rotation were to stop slowing today, we would still need to insert leap seconds into UTC
indefinitely, for the straightforward reason that a mean solar day (a civil day) is currently longer than 86,400 SI
seconds. UTC clocks simply tick too quickly for civil needs, and so, as with any fast-ticking clock, they must be paused
from time to time. The leap second accomplishes just that. Even if Earth's rotation rate began
to *increase*, leap seconds would still be required until the 1-millisecond mismatch vanished. So, a need for leap
seconds does not imply that Earth's spin is slowing, even though that spin *is* slowing.

Leap seconds are tied to Earth's current angular *velocity*, not its current angular *acceleration*,
notwithstanding the fact that the angular acceleration certainly affects the angular velocity. If a leap second were to
be required at ever shorter intervals, we would conclude that the mean solar day was lengthening, and it would certainly follow
that Earth's spin was slowing.

Perhaps we should "fix the mechanism" of our fast-ticking UTC clock by redefining the SI second slightly. That's possible, but in practice it might cause more problems or confusion than leap seconds currently cause.

Compare the tick rates of two super-accurate clocks, one on Earth's equator and the other at one of its poles. In the
inertial frame in which Earth spins, the equator clock is moving, and that motion exerts a *slowing* on its tick rate
compared to the pole clock. But the equator clock happens to be in a slightly weaker gravity field—a consequence of
Earth's oblate shape—and this weaker gravity exerts a *quickening* on its tick rate compared to the pole
clock. Remarkably, at least to a high approximation, these two effects cancel, and time *everywhere* on Earth's
geoid (approximately mean sea level) proceeds at the same rate. This can be verified from the Schwarzschild metric in
general relativity, and can be argued on general grounds without appealing to a metric. (Whether it is *exactly*
true is a knotty problem in relativity, and is not known.) This fact is used to define TAI. Modern precise timers
create plots of the length of the day that they interpret as showing irregularities in Earth's spin rate from day to day.
Currently this is achieved by "very long baseline interferometry" measurements of astronomical radio sources, made by radio
dishes widely spaced on Earth. The measurements made by these dishes are correlated to deduce the geometry of the dishes
relative to the radio sources, and Earth-rotation specialists use this data to infer knowledge of Earth's spin rate as a
function of time.

That said, were Earth's spin rate to change, then for the TAI that governs the measurements of these dishes to remain well defined, the dishes would have to continue either to remain on the geoid, or to have their readings interrelatable via the concept of a geoid. If Earth's spin rate changes on a time scale of one day, then it's not clear whether Earth's actual rocky surface (which dictates where the dishes are) is able to adjust continuously to conform to that geoid. In fact, Earth's geoid does appear to be continually changing on the order of at least tens of centimetres. So, it's not clear whether the timing measurements that say Earth's spin rate is changing have come from a set of radio dishes whose clocks are ticking at the same rate. In that case, all bets are off that the numbers make any sense at all. Precise timing is not a straightforward subject!

[1] Daniel Kleppner, *Time Too Good to be True*, Physics Today, March 2006, page 11.