## How Long Is A Year?

How long is a year? 365 1/4 days, right? Close. How could we figure this out? Think about that. It's not particularly obvious.

OK, let's say we have a telescope, and a watch. The watch shows us that the sun is overhead every 24 hours. Not exactly. But, over a period of a year (or several years), it averages out to almost exactly 24 hours. 24 hours is not how long it takes the earth to rotate once, however. 24 hours is how long it is from noon to noon. But the earth moves a ways around the sun in that 24 hours. Actually, the earth has turned more than one rotation in 24 hours (see the diagram).

To see how long it actually takes the earth to rotate once, we have to look at the stars, which is why we brought along our telescope. We point our telescope at a bright star. As the earth turns, the star seems to move out of our field of view. The next night, about 23 hours and 56 minutes after our first observation, the star is again in the center of our field of view. The earth has rotated once in 23 hr. 56 min. That's 4 min. short of a 24 hour day.

Next time, our telescope is pointing at the star 8 min. short of two days, and then 12 min., and then 16 min. After a year, the earth has rotated a whole day less than we would think by counting days by the sun.

From this we can get a fairly simple ratio:

`y = (1 day - 4 min) / 4 min`

Here, y is the length of a year. Remember that the 1 day, above, is a 24 hour day, not the 23 hr. 56 min "day" with respect to the stars. Solving for y, we find that one year is equal to 359 days. Is that right? I thought it was 365 days. Well, our 4 min. is not very accurate. We need a more accurate measurement of the earth's rotation period, by looking through our telescope.

If we use very high power, and make sure our telescope doesn't move, we find that the rotation period is 23 hr. 56 min. 4.091 sec. (I got that from the World Almanac). Using this value,

`y = (1 day - 235.909 sec) / 235.909 sec`

we now get y=365.2429 days. That sounds about right. Our last digit (9) is not accurate, at all. We only started with 6-digit accuracy. So 365.243 is as accurate as we can be with that data.

So, you don't have to wait around for a whole year, to measure the length of a year. Two observations, on consecutive nights, do the trick.

We use a 365 day year, with a leap year every 4 years. That's 365.25 days per year. But, three of every four century-years is not a leap year. By doing that we are subtracting off 3 days every 400 years, or .0075 days every year. That gives us a year of 365.2425 days. That may, or may not, be more accurate than our calculated value of 365.243 days.

By the way, the year 2000 is a leap year, while 1700, 1800, and 1900 were not.