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Chapter 1: Astrophysics

How Far is a Parsec?

Dear Dr. SETI:
How far is a parsec?

The Doctor Responds:
Short, sweet, and to the point, Anonymous.

Usually, when someone declines to sign a name to a query, it is because he or she is afraid to show ignorance. In this case, however, your question reveals a good deal of knowledge, so you should not have been hesitant to sign your name. Specifically, the wording of your question shows that you know a parsec is a unit of distance. This puts you at the head of the class -- and way ahead of those scriptwriters in Hollywood who seem to think that parsec is a unit of time

The above is, of course, a direct reference to everybody's favorite science fiction action-adventure movie, the 1977 classic "Star Wars." In illuminating the speed of his beloved ship, pilot Han Solo boasts that the Millennium Falcon completed the Kessel run in under twelve parsecs. Any space pilot worth his wings would have known better. Star Wars fans have ever since tried to rationalize the error, by claiming that the shortest distance between two points in hyperspace is not a straight line at all, and thus by reducing the distance to 'under twelve parsecs', Solo was proving the abilities of his ship, and his piloting skills. But, I digress.

The short answer, Anonymous, is that one parsec equals about 3.26 light years. But you probably wanted to know how it was derived. (If you didn't, you wouldn't still be reading this, would you?) OK, so here's the long answer.

Parsec stands for parallax-second. Let's define the parallax part first, and then deal with the second (which I'll admit sounds like a unit of time, thus confusing Han Solo, but really isn't. Bear with me, please.)

It's pretty hard to directly measure interstellar distances from Earth. I mean, who are you going to get to hold the other end of the tape measure? So, we measure them indirectly. One useful technique involves the concept of parallax. But before we go there, we have to talk about the orbit of the Earth.

Try to picture the Earth orbiting the Sun. Let's assume we go around in a perfectly circular orbit. (OK, I know Kepler said it's really an elipse with the Sun at one focus, but humor me here...) The radius of the Earth's orbit (or, more properly, the distance from the center of mass of the Earth to the center of mass of the Sun) is a distance which we call one Astronomical Unit, or AU. Remember that it takes us one year to orbit the Sun. So, if you picture the position of the Earth right now, where will it be in exactly six months? Why, on the exact opposite side of the Sun, of course. So the Earth has moved to a new location in six months' time. And how far is that location from where we started? Two orbital radii, or one orbital diameter, or two AU away.

In case you were wondering, one AU is about 150 million kilometers, so the opposite sides of the Earth's orbit are about 300 million kilometers apart.

Now, we have established that in a six month period, the position of the Earth shifts two AU. Because of the Earth's motion around the Sun, if we look at a distant star right now, and then sight the exact same star six months later (from our new vantage point of two AU away), then the location of that star will appear to have moved. Actually, it is we who have moved, but being geocentric creatures, we perceive stellar motion. We call this apparent motion the star's parallax. The angular change in apparent position of that distant star is called parallax angle.

We could choose to measure the parallax angle in radians, or degrees, or arc-minutes, or arc-seconds, or any other unit of angular measure. Let's agree to measure parallax angle in seconds of arc (one arc-second is 1/3600 of a degree, or about five millionths of a radian -- in other words, a small angle!) Notice that I'm saying seconds of arc, not seconds of time.

Parallax angle can be used as a rough measure of distance, because the closer to Earth the object being viewed, the greater the observed parallax (and conversely, the further the object is from Earth, the less parallax will be observed). You can prove this to yourself by holding your thumb up at arm's length, and looking at it alternately with your right eye and your left eye. See, your thumb seems to move! Now, bring your thumb closer to your face, and repeat the experiment. Your thumb seems to move even more.

Similarly, as the Earth goes around the Sun, a very nearby object would have a very large parallax angle -- perhaps measured in degrees. Conversely, a very distant one -- say, another galaxy -- might exhibit a tiny parallax angle -- maybe on the order of a millionth of a degree, or less. (This is, in fact, exactly how we know the relative distances to different objects in interstellar space.) Well, it stands to reason that, at some intermediate distance, there could be an object which exhibits exactly one arc-second of parallax over a six month period. We call this unique distance one parallax-second, or parsec.

Finally, since one light year equals about 9.46 million million kilometers, and we said a parsec is 3.26 light years, simple multiplication tells us that one parsec is a distance of almost 31 million million kilometers. A long distance, to be sure!

But don't forget that our nearest neighboring stars are about 4.2 light years away. And that's greater than a parsec (actually, it's nearly 1.3 parsecs). So how many stars are there within one parsec of Earth?

Only one: the Sun.

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