How do they figure the distance between celestial bodies?

SDStaff Chronos,Straight Dope Science Advisory Board replies:

You probably think there’s a simple answer to this question, Mike. What’s frightening is that this is it. But we figure you’re old enough to take it.

There are a number of steps in the process, with the results of each step used to calibrate the next. To start with, we need to know the distances of things in the solar system. For this, we use something called Kepler’s Third Law. This states that for any object in the solar system, the orbital period P (in years) is related to the average radius of the orbit A by the formula P² = A³. The period can be determined easily by going out at night and watching the planet or whatever move. Plugging in the P gives us the radius A in astronomical units, or AUs. An AU is the average distance from the earth to the sun. To figure the length of an AU, we need to measure at least one distance. Usually, this is done by sending a radar pulse to either Mars or Venus, when it’s at closest approach to the Earth. Since we now know the difference |A_{Mars or Venus – AEarth|, and we already knew the ratio PMars or Venus}/P_Earth, the rest is child’s play.

OK, so now we’ve got the solar system licked. What about other stars? They’re too far away for radar to be any use. What we use here is a method called parallax (sometimes called trigonometric parallax). To see parallax on a small scale, hold one finger up at arm’s length in front of you, and look at it first with your left eye, and then with your right eye. Your finger will appear to change position relative to the background. Exactly how much depends on the distance b between your eyes (referred to as the baseline), and the distance D of your finger from your eyes. Specifically, for large distances, the formula is D = 2b/theta, where theta is the angle by which your finger shifts relative to the background, measured in radians (to get from degrees to radians, divide by 57.296). To use this method on the stars, we want the longest baseline we can easily get, which is the diameter of the Earth’s orbit. If you take a picture of an area of the sky one night, and then take another picture of the same area six months later, the nearby stars in that area of the sky will shift their positions slightly relative to the other stars. Parallaxes can be measured in this way for stars nearer than a few hundred lightyears; after that, the shift is too small to detect.

To get the distance of anything farther away than that, we usually use some sort of “standard candle” method. The apparent brightness of an object depends both on how bright it actually is, and on how far away it is. For example, if a hundred watt light bulb is 10 meters away, it’ll look exactly as bright as a 25-watt light bulb 5 meters away. A “standard candle” is an object of known brightness. If we then measure its apparent brightness, we can determine how far away it is. This can be used to determine any distance, as long as you can still see the object you’re using. For instance, we can use the distances from parallax to determine the average brightness of various colors of normal or “main sequence” stars, as they’re called. This gives us a standard candle good out to the various star clusters in our galaxy, a hundred thousand lightyears or so away.

Then, in some clusters, we see stars called Cepheid variables, whose brightness changes in a certain way with time. Their average brightness depends on their period, and if we see one in a cluster a known distance away, we can determine its brightness and thus determine exactly how the brightness depends on period, so we’ve got another standard candle. Since Cepheids are brighter than main sequence stars, we can use them at greater distances, allowing us to determine the distances of Cepheids in nearby galaxies, and thus the distances to those galaxies. This in turn allows us to determine the brightness of yet brighter standard candles, and so on. Currently, the largest distances are obtained from a certain type of exploding star called a supernova 1a. These are nearly ten billion times as bright as our sun, and can be seen nearly as far away as the edge of the visible universe. Oughta hold us for now.

SDStaff Chronos,Straight Dope Science Advisory Board

Send questions to Cecil via cecil@straightdope.com.

STAFF REPORTS ARE WRITTEN BY THE STRAIGHT DOPE SCIENCE ADVISORY BOARD, CECIL’S ONLINE AUXILIARY. THOUGH THE SDSAB DOES ITS BEST, THESE COLUMNS ARE EDITED BY ED ZOTTI, NOT CECIL, SO ACCURACYWISE YOU’D BETTER KEEP YOUR FINGERS CROSSED.