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Leap Day

Published on Feb 29, 2008 at 11:51 am. 1 Comment.
Filed under calendars.

I’ve written about the calendar a number of times before, including one posting about years, and even a post on why the New Year starts January 1. But, since this is the first leap year since I started Astroprof’s Page, it seemed natural that I blog about Leap Day. However, I looked at some of the blogs that I read a lot and found that Professor Astronomy and the Bad Astronomer had already written about leap year. So, you should check out what they have to say, too.

I teach about time and calendars in my astronomy classes. Historically, astronomers were the ones who were in charge of measuring the passage of time. That continued until the days of atomic clocks. Now, atomic physicists keep track of time. But, talking about time and the passage of time is still a lot of fun for me. I have a whole lecture dedicated to calendars. There’s really a lot more to all of that than most people realize. I don’t have time to go into all of it here, though.

You should read the other people’s postings on leap year, but I’ll still summarize the key points here. The problem with the calendar is that we want the years to match the seasons. We want the year to start shortly after the winter solstice. We want the vernal equinox to be near March 21. We want the summer solstice to come in late June. But that won’t happen with a calendar of 365 days. You see, the Earth doesn’t orbit the Sun in an exact number of days. The Earth’s sidereal year, the time it takes to make one orbit, is 365 days, 6 hours, 9 minutes, and almost 10 seconds. But, as Earth goes around the Sun, it is also slowly shifting the direction of its tilt and the exact path around the Sun that it takes. These effects cause the seasons to repeat, not every sidereal year, but slightly more often. The seasons repeat every 365 days, 5 hours, 48 minutes, and just over 45 seconds. We call this the tropical year, and it is the average time between one vernal equinox and another. That is almost 6 hours more than 365 days. So, if the calendar ran for only 365 days every year, then after four years, the calendar would have started a day early. Four years of 365 days is 1460 days. But, four cycles of 365 days and 6 hours is 1461 days.

So, the calendar would be ahead of the seasons by a day. Well, that’s only a day, right? What’s so bad about that? Well, after a century, it would be off by 25 days — nearly a month! And, after 500 years, it would be off by over 4 months. But, a year of 366 days isn’t any better. In fact, it is worse. Because four years of 366 days is 1464 days. That still doesn’t match up with 1461 days. In fact, it is now three days off, instead of one! After just 40 years, the calendar would be out of sync with the seasons by a month!

So, that is why do leap year. We go for 365 days for three years, and then 366 days for one year. The extra day is stuck onto the end of February for historical reasons (in the ancient Roman calendar, they always seemed to mess around with February). So, this cycle of four years is 1461 days, matching the seasons.

But, it isn’t a perfect match. Remember, the seasons repeat every 365 days, 5 hours, 48 minutes, and a fraction over 45 seconds. That is just over 11 minutes shorter than 365 days 6 hours. What this means is that after four years, instead of being off by a full day, the calendar is actually off by only 0.969 days. Well, that’s pretty close to a day. So, having a 366 day year every four years is pretty close. Unfortunately, it isn’t exact. So after 100 years, the calendar is again off by almost a day, but in the other direction. There would be one day too many. So, the way that the calendar works is to leave off a leap year every century. But, even that isn’t perfect. Now, there is still one day too few every 400 years. So, instead, we leave off a leap year three times every 400 years. The last time was in 1900. The next time will be in 2100, and then in 2200, and again in 2300. The rule is that we leave off a leap year every centennial year unless it is even divisible by 400 (which is why we had a leap year in 2000).

But, most of my students haven’t heard all of this. Whenever I am teaching, I try to elicit comments from my students. I ask, “Why such-and-such ?” I sometimes get all kinds of interesting answers. Leap Year is always one that seems to get them. Apparently in the schools around here they don’t teach the kids about why we have leap year, only that we do. Or else, they teach it in elementary school and people forget it. At any rate, one common misconception that I find all the time is that people think that leap year has something to do with daylight saving time. Of course, they don’t have anything to do with one another. All we are really doing in Daylight Saving Time is just setting the clock differently. We are simply calling it later so that everyone get up earlier. The calendar does not in any way depend on how we set the clock. Now, since so many people don’t understand leap year and daylight saving time, that is actually a pretty ingenious connection. After all, they hear all the time on the news that daylight saving time adds an hour of daylight. Most people are smart enough to know that you don’t really change the rotation or orbit of the Earth when you mess with the clock (though clearly not everyone, as evidenced by this). So, they figure that leap year must somehow fix that. That extra hour of daylight has to be compensated for somehow by adding a day to the calendar. Of course, that has nothing to do with it. But, just for fun, lets think about how such a system might work.

If Daylight Saving Time really did add hours to the day, then to fix things, you’d need to take days away from the year, not add them. So, perhaps instead of adding a day every four years, we are really taking days away from three years out of four. In that case, leap year would be the true length of the year. But, does the math work? To look at that, consider this year. Daylight Saving Time begins March 9 and ends November 2. That is 238 days. An extra hour per day would be 238 hours, which corresponds to nearly 10 days. So clearly this doesn’t work out to any sort of leap year scheme.

Another thing that students often come up with is that they associate leap year with the difference between sidereal and synodic days. A sidereal day is the actual rotational period of the Earth. It is 23 hours, 56 minutes, 4.09 seconds. That is just about 4 minutes short of the 24 hour clock day. The clock day is supposed to be in sync with the synodic day. The synodic day is the time between when the Sun is on the meridian (the imaginary line running from north to south across the sky through the zenith) until it is on the meridian again. But, that varies over the course of the year due to Earth’s elliptical orbit around the Sun. So, we take the average over the year. But, there is about a 4 minute difference between the sidereal and synodic days. This is why stars appear to rise and set 4 minutes earlier each day. Students quickly pick up on the fact that this nearly 4 minute difference over the course of a year adds up to nearly one day. So, they quickly jump to the conclusion that this is the source of the leap year. But, it adds up to a day every year, not every four years. So, that can’t explain a leap year every four years, either. Interestingly, though, the discrepancy adds up not to one 24 hour day, but rather it adds up to one sidereal day every sidereal year. If you think about it, that makes sense. The whole reason that the sidereal and synodic days don’t match is that the Earth is moving around the Sun. So, each day, the Earth has moved a bit in its orbit, meaning that it has to turn a bit more to line up with the Sun the same way that it had the day before. Thus, one complete orbit requires one complete turn, just to stay in sync.

-Astroprof