Well, you’re absolutely right that radar is an ideal way of measuring distances to objects; with radar you bounce a radio or microwave pulse off of the other object (a planet, for instance) and wait for it to come back. The Arecibo Observatory in Puerto Rico is one of the best observatories for doing this kind of work, and it’s not limited to planets, though objects that are large and nearby are easiest. Asteroids and comets are also good targets for radar observations, and the radar allows us to not only get great distances but a general idea of the shape of the object.
But radar has limited usefulness once the object you’re interested in gets too far away, and when we need to get a distance from an object in the outer regions of our solar system, or for the nearest stars, we have to find another option. That option is parallax, which is also pretty straightforward as astronomy distance measurements go, because it’s mostly just geometry.
We’re well familiar with parallax as a phenomenon, even if we’ve never had the name to apply to it. Parallax is simply that objects which are close to you will appear to move, relative to some distant object, if you move. It’s why, when you take a picture out of a moving car’s window, the scenery along the roadside will be blurred out, whereas the distant scenery is captured without any motions. The roadside has a large parallax effect relative to the background. You can do this at home, too – close one eye and hold a finger out at arm’s length. Get your finger to cover up some object on your wall – a light switch or something. Now close that eye and open the other one. Your finger will appear to jump sideways away from the object it was covering. That’s parallax.
With a little math you can figure out how far away that apparently moving object is. All you need to know is how far apart your vantage points were (in the finger example, the distance between your eyes), and how far the object (your finger) appeared to move. With that information, you can work out how far away the object must have been in order for the angles to work out. If the object were closer, it would appear to have moved more. If it were farther, it would move less.
The other thing you can change is how far apart your viewing positions are. The farther apart, the more obvious the effect. For astronomical distances, we can make use of this by measuring the positions of stars when our planet is at opposite ends of our orbit around the Sun. Six months apart gives us viewing positions which are 186 million miles apart, instead of the few inches between your eyes. That allows you to see even the tiniest changes in a star’s position, relative to even more distant stars.
However, once you get beyond a few hundred parsecs, this kind of measurement gets really hard to do, and even with the best telescopes, parallax is only measurable out to about 1000 parsecs. Considering that we’re sitting 8,000 parsecs from the center of our galaxy, that doesn’t get very far. We’re going to need another method to get even farther away.
This is where Cepheid Variables come in. Cepheids are an interesting class of star which change their brightness over time in a predictable, repeating pattern. And, very usefully for distance measurements, that repeating pattern changes depending on how intrinsically bright the star is, a discovery made by Henrietta Swan Leavitt in 1902. We can therefore use the speed of the Cepheid’s pulse to tell if it’s faint in our skies because it’s intrinsically dim, or because it’s faint because it’s far away.
We know how bright the Cepheid should be, because of its pulse, so any fainter means it’s farther away. We know how brightness fades with distance – twice as far away means eight times as faint. The mismatch between how bright the Cepheid appears in the sky, and how bright it should be gives us this distance. This method works well throughout our galaxy and out to the nearest galaxies beyond us. To go even farther out in the universe, we need an even brighter tracer – supernovae.
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