One way is a spectrograph. It essentially breaks light down (like one of those glass pyramids). By checking what sorts of wavelengths are most commonly being reflected and absorbed by something you’re pointing at (as well as a bunch of other things, like temperature) you can work out what materials it is made from.
The key concept to understand is that you can use the interactions of light with a substance (especially a gas) to identify what that substance consists of. This is because when atoms absorb a photon, it can cause one of the electrons in the atom to jump up to a higher energy level. Eventually that electron drops back down to the lower level, and when it does so, it emits a photon of the exact same energy. This is called an emission spectra, and you can use an instrument called a spectrograph to split the incoming light into different wavelengths, and you’ll see spectrum lines. Through a lot of work, scientists determined what lines form for each element, much like a fingerprint.
This is how we discovered helium. We detected it from solar light.
Now, if you have light passing through the atmosphere of a planet, you end up with an absorption spectrum, where instead of lines of light, you get black lines. This is how JWST is analyzing the planets of transiting exoplanets (but we do have to filter out the spectrum caused by the star itself). While the planet is passing in front of the star, some of the light gets absorbed. But in much the same fashion, we can use the “fingerprints” we’ve taken of the elements to detect what elements are present in the exoplanet’s atmosphere.
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