## Thursday, August 30, 2012

### Q: Why is the SI second defined as 9,192,631,770 periods of the transition between two states of cesium-133?

In 1895, the astronomer Simon Newcomb published his "Tables of the Sun," based on observations of the sun's position from 1750 to 1892. (http://en.wikipedia.org/wiki/Newcomb%27s_Tables_of_the_Sun) These calculations turned out to be reliable enough that astronomers continued to use them until the 1980s.

Before 1956, the second was defined as the mean solar second, or in other words, 1/86,400 of the time the earth takes to spin around on its own axis and see the sun again each day. But because the moon's gravity and the tides are slowing down the earth's spin, this is not a stable quantity.

From 1950-1956, the international authorities agreed to redefine the second to be the "ephemeris second," based on the speed of the earth's orbit around the sun in 1900, as predicted in Newcomb's tables. The earth's orbit around the sun is not slowing down, at least not on anything like the effect on the earth's spin around its own axis. (In practice the ephemeris second is measured by looking at the moon's orbit around the earth and taking pictures of what stars the moon is near.)

Because Newcomb's tables cover observations from 1750 to 1892, the "ephemeris second" corresponds to the mean solar second at the middle of this period, or about 1820. (http://tycho.usno.navy.mil/leapsec.html)

Meanwhile, from 1952 to 1958, astronomers from the U.S. Navy and the British National Physical Laboratory measured the frequency of cesium oscillations in terms of the ephemeris second. (http://www.leapsecond.com/history/1958-PhysRev-v1-n3-Markowitz-Hall-Essen-Parry.pdf) Cesium is even more stable than the orbit of the earth around the sun.

There are a few ways to do the calculations that they show in the paper (having to do with exactly what period they observed over and whether they corrected for some subtleties re: the moon's orbit), giving results between 9,192,631,761 and 9,192,631,780. The average was 9,192,631,770.

In 1967, this became the official definition of the SI second, replacing the ephemeris second. But the reason the number is what it is is because Newcomb analyzed observations from 1750 to 1892, and the middle of that period is 1820, and that's how fast the earth was spinning on its axis in 1820.

## Saturday, August 25, 2012

### Q: If one day is not exactly 24 hours and is in fact 23 hours, 56 minutes, shouldn't the error add up, and shouldn't we see 12 a.m. becoming noon?

You're right that a "sidereal" day is about 23 hours, 56 minutes, 4 seconds. But this is not a day in the everyday sense.

A sidereal day is how long it takes the earth (on average) to make one rotation relative to the faraway stars and other galaxies in the sky.

If you find a star that is directly above you at midnight one night, the same star will be directly above you again at 11:56:04 p.m. the next evening.

Similarly, if you were sitting on the star Proxima Centauri looking through a powerful telescope at earth, you would see Toledo, Ohio, go by every 23 hours, 56 minutes, and 4 seconds.

However, we don't keep time by the faraway stars -- we measure time by a much closer star, the sun! And we are actually in orbit around the sun, orbiting in the same direction that the earth is spinning on its own axis. From our perspective, the sun goes a little slower in the sky because we are also orbiting around it.

How fast are we orbiting around the sun? We make one full orbit every year, or roughly 366.25 sidereal days.

So after a year, the faraway stars will have done 366.25 rotations around the earth, but the sun will only have done 365.25 rotations. We "lose" a sunset because of the complete orbit. (The extra quarter day is why we need a leap year every four years.)

So there are 365.25 "mean solar days" in 366.25 "sidereal" days. How long is a "mean solar day"? Let's do the math: One sidereal day is 23 hours, 56 minutes, 4 seconds, or 86164 seconds. Multiply this by 366.25 sidereal days in a year, and you get 31557565 seconds. Divide by 365.25 solar days, and we get that a solar day is.... 86,400 seconds. That's 24 hours exactly!

It's this "mean solar day" (24 hours) that is the normal definition of day.

If you want to do the math more exactly, a sidereal day is 86164.09054 seconds, and a tropical year is 366.242198781 sidereal days. That works out very closely.

(P.S. Unfortunately, the earth's spin has been slowing down because the moon is sucking away the earth's energy. Every time the high tide of the Atlantic Ocean slams into the east coast of North America, the earth slows its spin a little bit. The definition of the second is based on the speed the earth was spinning back in 1820, and we have slowed down since then. As a result, we occasionally have to add in a "leap" second to the world's clocks. See http://online.wsj.com/article_email/SB112258962467199210-lMyQjAxMTEyMjIyNTUyODU5Wj.html?mod=wsj_valetleft_email)