the Galileo probe flew by earth on its way to Jupiter and detected in our atmosphere both oxygen and methane, thereby "proving" the existence of life on earth. (actually science can never prove anything; it can only disprove things.) life, replenishing the methane (e.g., cows farting), is the simplest explanation for why the oxygen hasn't oxidized all the methane over geologic time scales. (life also replenishes the oxygen via photosynthesis. actually, photosynthesis, reducing carbon dioxide back to carbon, also does the key first step of producing methane.)
how easy is it to replicate this Galileo probe experiment on earth? detecting oxygen in the atmosphere is probably pretty easy. is spectroscopy the easiest? or, you could burn something then prove that the burnt product is the oxide of your input.
detecting and measuring the tiny amount of methane seems much more challenging. what is the best way? there are off-the-shelf methane-measurement devices. how do they work? (how might they fail if the atmosphere were completely different?)
an alien astronomer testing whether earth is suitable for life -- suitable for invasion and colonization -- would need to remotely detect the methane (and oxygen) in our atmosphere, probably through spectroscopy. consider limiting earthbound experiments detecting oxygen and methane only to remote-sensing techniques available to alien astronomers.
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