Happy New Tropical (or Anomalistic or Sidereal) Earth Orbital Time Period!

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Jan 1, 2014
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I decided a while ago to make this post an annual BA tradition on Jan. 1, which is a more polite way of saying it's a repost and I'm sometimes a tad on the lazy side. I've punched it up here and there over the previous version, added a number (can you find it?), and switched up the images, just to make it fresher. Enjoy.

Yay! It’s a new year!

But what does that mean, exactly?

The year, of course, is the time it takes for the Earth to go around the Sun, right? Well, not exactly. It depends on what you mean by “year” and how you measure it. This takes a wee bit of explaining, so while the antacid is dissolving in your stomach to remedy last night’s excesses, sit back and let me tell you the tale of the year.

Round and Round She Goes

Let’s take a look at the Earth from a distance. From our imaginary point in space, we look down and see the Earth and the Sun. The Earth is moving, orbiting the Sun. Of course it is, you think to yourself. But how do you measure that? For something to be moving, it has to be moving relative to something else. What can we use as a yardstick against which to measure the Earth’s motion?

Well, we might notice as we float in space that we are surrounded by billions of pretty stars. We can use them! So we mark the position of the Earth and Sun using the stars as benchmarks, and then watch and wait. Some time later, the Earth has moved in a big circle and is back to where it started in reference to those stars. That’s called a “sidereal year” (sidus is the Latin word for star). How long did that take?

Let’s say we used a stopwatch to measure the elapsed time. You'll find that it took the Earth 31,558,149 seconds (some people like to approximate that as pi x 10 million = 31,415,926 seconds, which is an easy way to be pretty dang close—better than a half a percent accuracy). But how many days is that?

Well, that’s a second complication. A “day” is how long it takes the Earth to rotate once, but we’re back to that measurement problem again. But hey, we used the stars once, let’s do it again! You stand on the Earth and define a day as the time it takes for a star to go from directly overhead to directly overhead again: a sidereal day. That takes 23 hours 56 minutes 4 seconds = 86,164 seconds. But wait a second (a sidereal second?)—shouldn’t that be exactly equal to 24 hours? What happened to those 3 minutes and 56 seconds?

I was afraid you’d ask that—but this turns out to be important.

It’s because the 24-hour day is based on the motion of the Sun in the sky, and not the stars. During the course of that almost-but-not-quite 24 hours, the Earth was busily orbiting the Sun, so it moved a little bit of the way around its orbit (about a degree). If you measure the time it takes the Sun to go around the sky once—a solar day—that takes 24 hours, or 86,400 seconds. It’s longer than a sidereal day because the Earth has moved a bit around the Sun during that day, and it takes a few extra minutes for the Earth to spin a little bit more to “catch up” to the Sun’s position in the sky.

A diagram from Nick Strobel’s fine site Astronomy Notes (shown here; click to embiggen) helps explain this. See how the Earth has to spin a little bit longer to get the Sun in the same part of the sky? That extra 3 minutes and 56 seconds is the difference between a solar and sidereal day.

OK, so we have a year of 31,558,149 seconds. If we divide that by 86,164 seconds/day we get 366.256 days per year.

Wait, that doesn’t sound right. You’ve always read it’s 365.25 days per year, right? But that first number, 366.256, is a year in sidereal days. In solar days, you divide the seconds in a year by 86,400 to get 365.256 days.

Phew! That number sounds right. But really, both numbers are right. It just depends on what unit you use. It’s like saying something is 1 inch long, and it’s also 2.54 centimeters long. Both are correct.

Having said all that, I have to admit that the 365.25 number is not really correct. It’s a cheat. That’s really using a mean or average solar day. The Sun is not a point source, it’s a disk, so you have to measure a solar day using the center of the Sun, correcting for the differences in Earth’s motion as it orbits the Sun (because it’s not really a circle, it’s an ellipse) and and and. In the end, the solar day is really just an average version of the day, because the actual length of the day changes every, um, day.

The Sun Rose by Any Other Name

Confused yet? Yeah, me too. It’s hard to keep all this straight. But back to the year: That year we measured was a sidereal year. It turns out that’s not the only way to measure a year.

You could, for example, measure it from the exact moment of the vernal equinox—a specific time of the year when the Sun crosses directly over the Earth’s equator in March—in one year to the vernal equinox in the next. That’s called a tropical year (which is 31,556,941 seconds long). But why the heck would you want to use that? Ah, because of an interesting problem! Here’s a hint:

The Earth precesses! That means as it spins, it wobbles very slightly, like a top does as it slows down. The Earth’s wobble means the direction the Earth’s axis points in the sky changes over time. It makes a big circle, taking over 20,000 years to complete one wobble. Right now, the Earth’s axis points pretty close to the star Polaris, but in a few hundred years it’ll be noticeably off from Polaris.

Remember too, that our seasons depend on the Earth’s tilt. Because of this slow wobble, the tropical year (from season to season) does not precisely match the sidereal year (using stars). The tropical year is a wee bit shorter, by 21 minutes or so. If we didn’t account for this, then every year the seasons would come 21 minutes earlier. Eventually we’ll have winter in August, and summer in December! That’s fine if you’re in Australia, but in the Northern Hemisphere this would cause panic, rioting, people leaving comments in all caps, and so on.

So how do you account for this difference and not let the time of the seasons wander all over the calendar? Easy: You adopt the tropical year as your standard year. Done! You have to pick some way to measure a year, so why not the one that keeps the seasons more or less where they are now? This means that the apparent times of the rising and setting of stars changes over time, but really, astronomers are the only ones who care about that, and, not to self-aggrandize too much, they’re a smart bunch. They know how to compensate.

OK, so where were we? Oh yeah—our standard year (also called a Gregorian year) is the tropical year, and it’s made up of 365.25 mean solar days (most of the time, actually), each of which is 86,400 seconds long, pretty much just as you’ve always been taught. And this way, the vernal equinox always happens on or around March 21 every year.

But there are other “years,” too. The Earth orbits the Sun in an ellipse, remember. When it’s closest to the Sun we call that perihelion (the farthest point is called aphelion). If you measure the year from perihelion to perihelion (called an anomalistic year, an old term used to describe the shape of an orbit) you get yet a different number! That’s because the orientation of the Earth’s orbital ellipse changes due to the tugs of gravity from the other planets, taking about 100,000 years for the ellipse to rotate once relative to the stars. Also, it’s not a smooth effect, since the positions of the planets change, sometimes tugging on us harder, sometimes not as hard. The average length of the anomalistic year is 31,558,432 seconds, or 365.26 days. What is that in sidereal days, you may ask? The answer is: I don’t really care. Do the math yourself.

Let’s see, what else? Well, there’s a pile of years based on the Moon, too, and the Sun’s position relative to it. There are ideal years, using pure math with simplified inputs (like a massless planet with no other planets in the solar system prodding it). There’s also the Julian year, which is a defined year of 365.25 days (those would be the 86,400 seconds-long solar days). Astronomers actually use this because it makes it easier to calculate the times between two events separated by many years. I used them in my Ph.D. research because I was watching an object fade away over several years, and it made life a lot easier.

Where to Start?

One more thing. We have all these different years and decided to adopt the tropical year for our calendars, which is all well and good. But here’s an issue: Where do we start it?

After all, the Earth’s orbit is an ellipse with no start or finish. It just keeps on keeping on. But there are some points in the orbit that are special, and we could use them. For example, as I mentioned above, we could use perihelion, when the Earth is closest to the Sun, or the vernal equinox. Those are actual physical events that have a well-defined meaning and time.

The problem, though, is that the calendar year doesn’t line up with them well. The date of perihelion changes year to year due to several factors (including, of all things, the Moon, and the fact that we have to add a leap day every four years). In 2013 perihelion was on Jan. 2, but in 2014 it’s on Jan. 4. Same thing with the equinox: It can range from March 20 to March 21. That makes using orbital markers a tough standard.

Various countries used different dates for the beginning of the year. Some had already used Jan. 1 by the time the Gregorian (tropical) calendar was first decreed in 1582, but it took time for others to move to that date. England didn’t until 1752 when it passed the Calendar Act. Not surprisingly, there was a lot of religious influence on when to start the new year; for a long time a lot of countries used March 25 as the start of the new year, calling it Lady Day, based on the assumed date when the archangel Gabriel told Mary she would be the mother of God. Given that a lot of ancient Christian holidays are actually based on older, Pagan holiday dates, and the fact that this was on March 25—very close to the equinox—makes this date at the very least suspicious.

Still, in the end, the date to start the new year is an arbitrary choice, and Jan. 1 is as good a day as any. And as a happy side effect it does help establish the Knuckle Rule.

Resolving the New Year

So there you go. As usual, astronomers have taken a simple concept like “years” and turned it into a horrifying nightmare of nerdery and math. But really, it’s not like we made all this stuff up. The fault literally lies in the stars and not ourselves.

Now if you’re still curious about all this even after reading my lengthy oratory, and you want to know more about some of these less well-known years, then check out Wikipedia. It has lots of info, but curiously I found it rather incomplete. I may submit something to them as an update (like how many seconds are in each kind of year; they list only how many days, which is useful but could be better).

I have to add one more bit of geekiness. While originally researching all this, I learned a new word! It’s nychthemeron, which is the complete cycle of day and night. You and I, in general, would call this a “day.” Personally, if someone dropped that word into casual conversation, I’d challenge them to a duel with orreries at dawn.

Incidentally, after all this talk of durations and lengths, you might be curious to know just when the Earth reaches perihelion, or when the exact moment of the vernal equinox occurs. If you do, check out the U.S. Naval Observatory website. It has tons of gory details about this stuff.

Hmmmm, is there anything else to say here? (Counting on fingers.) Years, days, seconds, yeah, got those. (Mumbling.) Nychthemeron, yeah, Gregorian, tropical, anomalistic … oh wait! I know something I forgot to say:

Happy New Year!