# Why are leap seconds needed so often?

In Coordinated Universal Time (UTC), leap seconds are added to account for the slowing down of Earth's rotation. But the slowing down is said to be of the order of milliseconds in a century. Then why there were more than 25 leap seconds added to UTC in the last few decades alone?

• Regarding "the slowing down is said to be of the order of milliseconds in a century" - said by whom, and where? What precise quantity is that quote actually reporting? If it's a slowdown rate, a much more natural unit would be milliseconds per day per century - are you sure that that's not the case? – Emilio Pisanty Apr 22 '18 at 17:23
• @Mehrdad there's 5200 weeks in a century. A clock deceleration rate of 1 second/day/week means that after a century, the clock is losing 5200 seconds every day. A clock deceleration rate of 1 second/day/century means that after a century the clock is losing one second per day. It's really not that hard. – Emilio Pisanty Apr 23 '18 at 6:10
• @Mehrdad You seem to be completely confused about the difference between a velocity and an acceleration; they are completely different things and you go from the former to the latter by dividing a change by a time interval, i.e. by appending "per year" to the unit. I don't know why you think that addition "changes nothing" but that is dead wrong. – Emilio Pisanty Apr 23 '18 at 10:00
• @EmilioPisanty: No... I'm saying when people (read: normal people, not physics PhDs) talk about clocks running slow, they don't talk about it in terms of acceleration. They talk about the average deviation over a period of time. It's like saying "at my current acceleration I'd go an extra 1000 m in 1000 s".. that means over the next interval of 1000 seconds I'll gain an average of 1 meter every second. You can slice it and dice it however you want ("average 1 meter every second for the next second, or minute?") and get back the same thing ("any interval.. it's average"). – Mehrdad Apr 23 '18 at 11:19
• @TripeHound: And my point is that would be a wrong assumption. If you literally Google "earth rotation slow down rate" the first article is this Forbes article which says "slowing down by a few milliseconds per day"... and this is an article written by a Ph.D... not in physics. I don't know what articles you read, but every time I've seen a layman article on the topic, it's been in milliseconds-per-timespan like here, not acceleration. – Mehrdad Apr 23 '18 at 23:43

It's not the rate of change of the rotation speed that's important, it's the current rotation speed (in the rotating reference frame that stays facing the sun) not matching a 24h day.

Thus leap seconds (on average1) accumulate at a near-constant rate, because (as you point out) the average rate of change is low compared to the existing mismatch between actual day length and what our clocks say.

Remember that a leap second is an absolute offset added/subtracted, not a multiplier on the speed of our clocks that fixes the problem for the future until the speed drifts some more.

We're correcting the "error" in our time function by adding step offsets, not by changing the slope. The length of an SI second remains fixed, and the length of a day by our clocks remain fixed at 24 hours / 86400 SI seconds (with no leap second).

1. In practice the linear model doesn't work at all in the short-term: there's lots of year-to-year variation, and 1.5-2ms/day/century is only a long-term average. See @David Hammen's answer for a nice graph and more details. He commented:

Nine leap seconds were added in the first eight years after implementing the concept of leap seconds while only two were added over the 13 year span starting in 1999.

The chaotic short-term variation dominates over any period short enough to ignore the average slowdown.

## More details from the US Naval Observatory's Leap Second article

The SI second ($9 192 631 770$ cycles of the Cesium atom) was chosen to be $1 / 31 556 925.9747$ of the year 1900.

The Earth is constantly undergoing a deceleration caused by the braking action of the tides. Through the use of ancient observations of eclipses, it is possible to determine the deceleration of the Earth to be roughly 1.5-2 milliseconds per day per century.

Note the units of that measurement: it's ms per day per century, or $\Delta s / s / s$, like an acceleration, not a velocity. And definitely not 1.5 ms per century.

Purely coincidentally, a mean solar day is currently on average 2 ms longer than an SI day, so the current error-accumulation rate is 2 ms / day. It's been about 1 century since the defining epoch for the SI second. It takes less than 1000 days to need another leap second. . (There are various effects which make solar days differ in length, but on average they're longer than 24h and getting even longer.)

In another century from now (with constant deceleration of the Earth), we'll need to add leap seconds about twice as often as we do now, to maintain the cumulative difference UT1-UTC at less than 0.9 seconds.

• It's worth noting that the error here is one or two milliseconds per DAY, not per century, so it takes less than 1000 days to be off by a second. – ShadSterling Apr 22 '18 at 3:51
• This is the answer that addresses OP's misunderstanding of "milliseconds in a century". – Jasper Apr 22 '18 at 11:07
• @Jasper: yeah, I thought the other answers were all failing to address the real misunderstanding, that's why I posted this :P – Peter Cordes Apr 22 '18 at 11:09
• @DawoodibnKareem part of the OP's misunderstanding seemed to be in thinking that leap seconds correct for changes in the length of a solar day due to the changing rate of rotation, prompting the question of why leap seconds appear to happen far more often than would seem necessary for that rate. OP is right about the rate of slowing, but wrong about how that relates to leap seconds. As this answer says, leap seconds correct for a different error, but this answer didn't mention the much higher rate of that error, without which it doesn't fully explain why leap seconds happen so much more often. – ShadSterling Apr 23 '18 at 0:05
• @ShadSterling: Updated with numbers because there's some interesting stuff to say. It was clear enough before, IMO: once you know that deceleration and current spin rate were independent, the confusion is resolved and you can obviously infer the current mismatch from the rate of leap second insertion. – Peter Cordes Apr 23 '18 at 6:37

It's a cumulative effect. Let's say that the mean solar day is about 1.5 milliseconds longer than the SI day of 86400 seconds. This difference accumulates every single day. After 1000 days, the total difference has become 1.5 seconds. After 18000 days, which is roughly 50 years, the total difference is 27 seconds. This is why 27 leap seconds have been inserted since 1972.

• This being a "cumulative effect" is a red herring though. I would take that out, or at least put it later in the answer, since it wouldn't explain the problem if the OP's statistic was correct. The real explanation is in your 3rd sentence, i.e. the OP got the statistic wrong. – Mehrdad Apr 23 '18 at 7:44
• No, the fact that this is a cumulative effect is not a red herring at all - it marks the difference between an acceleration and a velocity (which is just a "cumulative effect" of the acceleration), which is precisely the crux of the confusion in OP as posed. – Emilio Pisanty Apr 23 '18 at 10:02
• It depends very much what one chooses as the baseline. If, for example, the baseline rotation rate had been chosen as the average rotation rate in 1971, we would have to have had a number of negative leap seconds since 1972. That 1971 baseline would of course have broken the metric system. The baseline rotation rate instead is that of about 1820. Once a metric standard is chosen, the standards committees go out of their way to ensure that updates to the standard are consistent with the prior versions. – David Hammen Apr 24 '18 at 18:23

There are good answers here, but there remains a question — if atomic time has only been around for around 50 years, and the Earth's rotation is only slowing at a rate of about 0.5ms / day / century, how is it that we need a leap second every year or two already? Shouldn't we only be accumulating about 1 second of error per decade at this point?

And the answer to that involves a little history: the "atomic" SI second was chosen to maintain continuity with the second of ephemeris time, which was standardized in 1952. And although ephemeris time was based — rather impressively — on over 150 years of astronomical observations, the very length of those observations means that ET reflects the length of the mean solar day in about 1820. As such, the mean solar day was already longer than 86,400 seconds in the 1960s and 1970s by 1-3 ms per day. Which means that when the leap second was introduced in 1972, the error was already accumulating at a rate that required the regular introduction of leap seconds.

In fact, for some of the time since then, the Earth has been bucking the long-term trend and speeding up its rotation, causing the pace of leap second addition to slow down for a time, with none added at all between the end of 1998 and the end of 2005. But this is just a random fluctuation and in the long run, the trend will prevail and the error will grow at an increasing rate, requiring more frequent addition of leap seconds as the centuries roll on, unless some kind of calendar reform makes them unnecessary before then.

• This isn't fully accurate - ephemeris time is based on the year length using 1900 as the baseline epoch, not 1820. – Emilio Pisanty Apr 23 '18 at 8:06
• @EmilioPisanty - It is fully accurate. It's just missing a piece or two. Ephemeris time was defined in 1948 based on Newcomb's late 19th century model of the Earth's orbit, which in turn was based on measurements of the solar day over a span of 140 years, centered on 1820. – David Hammen Apr 23 '18 at 20:49
• @DavidHammen Is there a refernce to that analysis, and how they extrapolated to 1900 (apparently it is day 0.5 of January 1900, so really it's mid day of the last day of 1899, so they get a full night's observing without fiddling with the clocks IIRC) – Philip Oakley Apr 24 '18 at 7:45
• @Philip-Oakley References: An influential analysis towards ephemeris seconds was Spencer Jones' (adsabs.harvard.edu/abs/1939MNRAS..99..541S), see also many other references in (en.wikipedia.org/wiki/Ephemeris_time). The ephemeris second was only proposed originally for scientific use, and would have left mean solar time in use for civil life that did not need such precision. This caveat was later ignored by standards bodies, which partly explains the perversity of adopting a standard second that has always been a bit too short. – terry-s Apr 24 '18 at 9:03
• @Philip-Oakley Starting the astronomical day at noon was a very old tradition, the official almanacs only converted to the civil-midnight day-start from 1925. – terry-s Apr 24 '18 at 9:05

Why are leap seconds needed so often?

TL;DR: We need leap seconds somewhat frequently because there's a two hundred year old bias (a non-zero offset) in the definition of a second being 1/86400th of a day.

A one thousand word (aka one picture) explanation:

Source: Public domain Wikipedia Commons page Deviation of day length from SI day.

The grey and green curves in the above plot show the variations in the length of day (the length of one mean solar day less 86400 seconds) from 1 January 1962 to 31 December 2016, in units of milliseconds (left vertical axis). The grey curve shows smoothed daily values while the green curve shows a 365 day running average. The red curve shows the area under the curve, with the zero point set at 1 January 1972 (when the leap second concept was introduced). The red dots show when each leap second was introduced. The red curve and red dots are in units of seconds, the right vertical axis.

Note that the length of day exhibits a lot of variability. This short term variability is a result of exchanges of angular momentum between the five differentially rotating parts of the Earth: The atmosphere, the oceans, the crust and mantle, the outer core, and the inner core. What you can't see in this short 55 year span is that length of day also exhibits long term trends. These long term trends are due in part to changes in the Earth's inertia tensor (the Earth is still rebounding from the end of the last ice age) and in part to a secular transfer of angular momentum from the Earth to the Moon.

The transfer of angular momentum from the Earth to the Moon means that a day is now longer than it was in the distant past. While the rate at which the Earth transfers angular momentum to the Moon is very small, this inexorably builds up over time. The day is considerably longer now than it was 4.5 billion years ago (a day is conjectured to have been about four to six hours long shortly after the Moon first formed), and is a good deal longer than it was 2.5 billion years ago (the first reliable observations based on tidal rhythmites).

The day is now also a tiny bit longer than it was a couple of centuries ago. That couple of centuries is key to answering the question "Why are leap seconds needed so often?" Our concept of a day comprising 24 hours, or 86400 seconds, is based on how long a day was a couple of centuries ago. The long term trend makes the Earth rotate a tiny bit slower now than it did back then. This results in a bias (a non-zero offset) in the curve. The bias in the green curve results in the red curve, the area under the green curve, exhibiting a secular growth. Leap seconds are added when the red curve gains a second, more or less.

Detail: The practice is to add or subtract a leap second on June 30 or December 31 when the absolute difference between UT1 and UTC exceeds 0.6 seconds. Leap seconds have always been positive. There has never been a need for a negative leap second due to the ~200 year old bias in the definition of a second.

Earths rotational speed is effected by multiple factors both local to the planet (such as weather) and external (such as solar system gravitational perturbations).

Your questions inherent assumption that only 1 factor influences it and only at 1 constant rate is incorrect.

The speed actually increases and decreases in mathematically chaotic and therefore difficult to predict ways. The IERS attempt to predict changes up to 6 months in advance with the goal of keeping the delta between the highly accurate "atomic time" and civil UTC time below 0.9 SI seconds by adding or removing leap seconds to UTC at the most mathematically and politically appropriate time of year.

• What particular problem, if any, would there be with selecting one of two or three lengths of "civil second", which differ by about 0.05ppm, on January 1 of each year? The only things that would need to care about the varying length of a civil second would be programs that need to convert between civil time and atomic time, or those that keep civil time and have their own time base which is accurate to better than 0.05ppm. Having the rate at which time flows change by 0.05ppm would seem much less disruptive than having some "five-second" intervals be a whole second longer. – supercat Apr 23 '18 at 22:31
• @supercat thats an interesting question. I recall google used a similar scheme recently to handle leap seconds internally. I suggest posting it as a seperate question though. I'm not sure which would be the correct stackexchange though. – John McNamara Apr 24 '18 at 7:04
• @supercat: The nightmare scenario is that some jurisdictions adopt your idea and some don't, so you need to keep this info in zoneinfo files and crossing timezones changes the time by +-1h (or whatever) and retroactively do a leap second and undo the accumulation or something? I think and hope that's too obviously horrible for anyone to consider. 0.05 ppm is lost in the imprecision of most real clocks, so computers would typically just use NTP exactly as before with some multiplier on the local clock rate. But stuff with precise clocks could maybe have their assumptions violated. – Peter Cordes Apr 24 '18 at 8:55
• @PeterCordes: I suppose that perhaps a slight improvement to my rule would be to say that 180 days starting on Jan 1 and July 1 would be adjusted (or not) by 1/16,552,000 and the remaing days wouldn't. That would allow both times to precisely coincide on Jan 1. and July 1 of every year. Having some references use leap seconds and some not wouldn't cause problems if systems that expect leap seconds consistently use references that use it. Systems that don't use leap seconds are going to have unavoidable deviations from leap-second time no matter what they do, but... – supercat Apr 24 '18 at 14:53
• ...they could have less deviation if they were synchronized to a clock that smeared leap seconds over 180 days (I think Google's leap smear spreads the leap second over one day, but smearing it over 180 would seem even better). – supercat Apr 24 '18 at 14:54

A good comparison for this is the US national debt vs. the US national deficit. The US debt is currently about 21 trillion dollars, but the US national deficit for 2017 was less than 700 billion dollars. The leap seconds are like the debt, they keep adding up. The change in the length of day is like change of the deficit, which seems small by comparison to the debt.

• More (or also) importantly, the change in the deficit is small compared with the deficit. – Peter Cordes Apr 22 '18 at 11:12
• So... the deficit is a yearly increase in debt and the national debt is a single growing total? You might want to reevaluate how immediately familiar those terms are to a physics (i.e. non-financial, and also generally non-US, English-as-a-second-language) audience, and maybe provide more precise definitions of those terms as appropriate. – Emilio Pisanty Apr 22 '18 at 19:45

Leap seconds are not added to account for the slowing of the Earth's revolution. They are added to account for the fact that the Earth's rotation and its revolution (about the sun) are not perfectly sychronized. Adding an extra day every four years (except for certain unusual circumstances) helps but still does not correct perfectly for the mismatch between rotation and revolution. That is why leap seconds are employed.

Edit: @JBently is correct (see comment below). I did conflate the two different "leap" corrections. Time-keeping is complicated and old memories are sometimes misleading. Jim Garrison also has a point since the day length varies slightly due to weather and geologic factors. Hence the leap second additions are unpredictable.

• The leap second is only weakly related to the orbital period of the earth, it is strongly related to the rotation period of the earth. The leap second is added because the duration one second (cesium hyperfine transitions) is unrelated to the duration of one day (1 rotation). Cesium atoms have no awareness of whether or not the earth has completed one rotation, so cesium atoms cannot say when today becomes tomorrow on the calendar. – Steve Allen Apr 21 '18 at 17:56
• It’s even more complex. A “day” (noon to noon) varies over the year by up to ~15 minutes. Google “equation of time”. – Jim Garrison Apr 21 '18 at 19:57
• -1 for being incorrect, but to to clarify where the error has come from, this answer is conflating leap seconds with the fact that the algorithm for calculating leap days occasionally deviates from the usual "year divisible by 4" rule in order to make the corrections that this answer describes but incorrectly attributes to leap seconds. – JBentley Apr 21 '18 at 21:55
• I downvoted. This is entirely wrong. Leap seconds are there because the mean solar day is slightly longer than 86400 seconds. It has nothing to do with the length of a year. – Dawood ibn Kareem Apr 22 '18 at 4:10
• You can edit your answer to make it correct, instead of just editing to add a section that says it's wrong. (Or delete it). Your edit is mostly just a comment-reply, which should still be a comment. – Peter Cordes Apr 22 '18 at 20:34

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