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The search continues for other planetary systems like our own in the universe. But how do we see planets orbiting other stars (exoplanets, as they are called) that are light years away when we can hardly image stellar surfaces other than as tiny points of light and which are 100's to 1000's of times bigger in radius? The answer is that we don't actually "see" them, but rather we see the signature of them in other types of measurements.
One method of measurement is called astrometry. This method very carefully measures a star's position and watches for small side-to-side wiggles that an orbiting planet might cause in the star's position. Why the side to side wiggles? They arise because the star and the planet actually swing around their center of mass, in much the same way that two figure skaters on ice swing around their center of mass. Now imagine making one of the skaters a massive sumo wrestler. You might see the wrestler barely wiggle on the ice because the center of mass lies somewhere in the belly.
A second method involves a similar search for wiggles, but it uses spectroscopy to watch for the stellar spectrum to "wiggle" in wavelength. A stellar spectrum is a bit like a fingerprint. Stars generally emit light at all wavelengths (a rainbow), but depending upon the composition of the star, some very particular wavelengths will be "missing" from the spectrum and stars are classified according to which wavelengths are missing from the rainbow. If the star happens to be moving along our line of sight, these missing wavelengths will be shifted towards bluer wavelengths if the star moves towards us and redder wavelengths if the star moves away from us. The reason for this requires another story about special relativity, but that is for another day. If the star oscillates toward and away from us (think of the figure skaters again) as it will if it has planets in orbit about it, the missing wavelengths will alternately shift towards blue and then towards red as the star moves toward and then away from us in a repeating pattern.
A third method (and the last this piece will be discussing -- there are others) is called photometry and makes use of what astronomers call the light curve of the star. A light curve is simply a measure of how much light the star emits as a function of time. If a planet happens to move in between the star and us, the amount of light we see drops by a small amount. If this dimming is periodic, then we might say, "that's a planet!" However, we have to be careful because there are other physical process that might cause periodic dimming and brightening of a star.
Figure showing the Okayama 188-cm Reflector Telescope and an example of what measurements might look like for the photometry method. The researchers observed planet K2-3d, which is very similar to Earth, pass in front of its star. Image courtesy of National Astronomical Observatory of Japan here.