Leaping Seconds

When I was a kid my younger sister used to play basketball in a city league and my family would go watch. The game clock would reach the final seconds of each period and the kids in the stands would count down. 3! 2! 1! Zero!

The clocks were such that with less than one minute remaining, the clock would display tenths of a second. This resulted in a problem that bothered me but probably no one else ever. So as soon as (say) the 5 popped up the kids would shout “5!” when in fact there were 5.9 – or really six seconds left. This lead to the inevitable embarrassing moment when everyone would yell “zero!” and a second would elapse as the clock continued to count down from 0.9 before finally buzzing when time expired. Well, I was embarrassed for them anyway.

Today we’re going to have a global instance of a very similar thing. There’s going to be an extra second hanging around at midnight. It happens all at once though, so unless you live where midnight is actually at 00:00:00 UTC the problem won’t interfere with any Times Square style countdowns.

But there is an extra second, and it’s called a leap second. It’s mainly the moon’s fault. Tides rise and fall because the moon’s gravity pulls the water into a tidal bulge which the earth rotates under. This results in friction, which slows down the planet very gradually. Every year the rotation of the earth slows by about 17 microseconds. Over time this lag builds up and eventually the keepers of the various international standard clocks have to add a second in order to keep the clocks in synchronization with the spinning planet. It’s not a smooth slowing however, and so the corrections come irregularly. This year there’s one which will occur just after 5:59:50 p.m. Central Time today. The clocks will go (assuming your computer is clever enough to show it) …5:59:58, 5:59:59, 5:59:60, 6:00:00… with that funky 5:59:60 being the leap second.

The US Naval Observatory. Current home of Dick Cheney, future home of Joe Biden, and permanent home of the US Naval Observatory Master Clock.

As with the whole Pluto business, there’s some controversy about whether these leap seconds are a good idea. Because of the whole irregularity of the earth’s slowing, they can’t be predicted far in advance. That means embedded systems without easy access to internet time servers can’t account for the ones that haven’t happened yet. This can cause problems. On the other hand, there’s nothing requiring those systems to use UTC time. They could use GPS time, Unix time, or any of various other true delta-t standards. Conversion could happen later for non-critical user interface requirements. Me, I’d vote for keeping the leap seconds. I’d hate to eventually saddle our many-times great grandchildren with a clock that read midnight at sunrise.

Bonus Fermi problem! The earth has kinetic energy by virtue of its rotation, so if it’s slowing it’s losing energy at some average rate. What is this rate, in watts? Order-of-magnitude estimates are fine.

1. #1 Blaise Pascal
December 31, 2008

Fermi problem:

Roughly speaking, the change in rotational energy is IωΔω, where I is the moment of inertia, ω is the angular velocity, and Δω is the change in angular velocity.

ω is small, pi/43200 rad/s, I is big, about 100×10^36 kg m^2 for the Earth. It’s the Δω which is hard to estimate.

Wikipedia says the day gets longer by about 17us/century, so in 2000 the day should have been about 86400.000017 s long, so ω’= 2pi/86400.000017, and Δω is about 1.4×10^-15 rad/sec.

so ΔE is about 10^38 kg m^2 * 10^-5 /s * 10^-15/s = 10^18 J, over a century (about pi Gs), gives a power of about pi GW.

2. #2 Chris
December 31, 2008

“Wikipedia says the day gets longer by about 17us/century”
no, wikipedia and matt say the day gets longer by 1.7 ms/century

mass = 6 *10^24 kg
day =~ 9*10^4 s
year =~ 3 *10^7 s

I (sphere) = 2/5 * M * r^2 =~ 10^38 kg m^2
K (rotational) = 1/2 * I * w^2
delta K = (5 *10^37 kg m^2) * 4*pi^2 * ((day + .000017 s)^-2 – (day)^-2)

=~ (5 *10^37 kg m^2) * 40 * -2*10^-5/(day)^3 (rad/s)^2 =~ -4 *10^19 J

-4 *10^19 J / 3 *10^7 s =~ -10^12 W

3. #3 Blaise Pascal
December 31, 2008

Oops… Wikipedia says 17ms/century, not 17μs/century. That changes my answer by 3 orders of magnitude. Not 3 GW, but 3TW.

4. #4 Nomen Nescio
December 31, 2008

a lot of Unix systems are effectively on GPS time anyway, thanks to a lot of public time servers being fed their time synch from a GPS receiver. better coordination between UTC and GPS might be a useful goal, in fact.

5. #5 Abby Normal
December 31, 2008

I’m just glad to know I’m not the only one who’s been embarrassed/annoyed by the second after zero at sporting events.

6. #6 Peter
December 31, 2008

As we go forward, will become necessary to add more and more leap seconds every year, until in about 6000 yrs it will be 1 every day! Clearly this becomes a pain at some point keeping track of these adjustments. One option would be to define a standardized time system that is no longer continually adjusted to keep it aligned with the Earth’s rotation at all, then use a simple algorithim to calculate a “common” time for practical uses. Depending on the precision required, such an algorithim would simply be an offset calculable from the date (within a second) or it could also include a correction for the time of day as well (fraction of a second). Real precise time needs would simply refer to the standard time system and not common time at all.
We would still use a 24 “hour” clock matched to the Earth’s rotation therefore common time “seconds” would no longer be quite the same as standard seconds (.00005 longer in 6000 years) but there would not be an extra one every night either!

7. #7 Tom
December 31, 2008

a lot of Unix systems are effectively on GPS time anyway, thanks to a lot of public time servers being fed their time synch from a GPS receiver. better coordination between UTC and GPS might be a useful goal, in fact.

Got you covered. The USNO provides the time to GPS. GPS time doesn’t include leap seconds, though — that offset is added in separately.

And Matt forgot to add “and where Tom works” to his caption. 🙂

8. #8 Tualha
January 1, 2009

It’s actually mostly her fault.

9. #9 Tom
January 1, 2009

It’s actually mostly her fault.

But when she stops spinning, the effect ceases.

10. #10 Bob Sykes
January 1, 2009

All these problems would go away if we would just use sundials.

11. #11 a lurker
January 1, 2009

Trivia: We would still need leap seconds even if the Earth’s rotation stopped slowing down. The second is traditionally 1/60*1/60*1/24th of a solar day (obviously). The second was redefined in terms of an atomic process in a way to approximate what it was way back when. Since then the solar day has gotten longer. Even if the Earth’s rotation stops slowing down, the second will still be shorter than 1/60*1/60*1/24th of the current solar day. Thus leap seconds will be needed even if the slowing down of rotation stops. (And also thus the young-earth argument that leap seconds prove the Earth is young also fails.)

Dick Cheney’s “home” has the United States master clock? I imagine a really bad science fiction story can be made from this: The VP orders that starting January 19, every second last an solar day thus preventing noon of January 20 and thus staying in power. (Cheney is smart enough to know that it would not work and values his hide and wealth enough to stop him even if he is not smart enough.)

12. #12 rob
January 2, 2009

K=1/2Iw^2
I=2/5MR^2
w=2pi/T

so, K=(4/5)*(pi^2*MR^2)/(T^2)

power is dK/dt= – (8/5)*(pi^2)*(MR^2)/(T^3) * dT/dt

M=6e24 kg, R=6e6 m, T=8.6e4 s, dT/dt=17e-6 / 3.2e7

so power is about -10^12 watts.

13. #13 Riesz Fischer
January 5, 2009

I guess if the Earth’s rotation is slowing then the Moon’s revolution must be speeding up, to conserve angular momentum.

14. #14 Tom
January 6, 2009

I guess if the Earth’s rotation is slowing then the Moon’s revolution must be speeding up, to conserve angular momentum.

The moon’s orbital distance is increasing at ~ 4cm/year