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Right now, our resolution + light gathering power are still far too low to take direct images of exoplanets, so we're limited to subtracting the planet spectra from the parent star spectra when the planet undergoes a transit (and this isn't going to be possible for decades, according to Jim Kasting's latest book). So in this case, it seems that light gathering power is more important.

So, Is angular resolution more important when we want to measure the spectra of an Earth-like Exo?

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To study the spectra of Earth-like planets in transit across their stars, we'll need an observatory in space so that we won't be affected by spectral lines from the Earth's atmosphere. Unfortunately, we won't then be able to use large apertures on Earth to smooth down the noise. We'll also want to go to wavelengths of several microns in the infrared to get deep absorption lines, and also get to a weaker portion of the spectrum of the host star. Then it's simply a matter of integrating for a long time during transits and subtracting the spectrum of the star during transit from the spectrum when there is no transit. This will give the absorption spectrum due to the ring of atmosphere visible around the disk of the planet. But transits are few and far between so it takes a long time to build up integration time.

One way around needing an observatory in space may be to choose a star with a high radial velocity so that a planet's spectral lines will be red or blue-shifted relative to absorption from the Earth's atmosphere. Then the largest ground-based observatory could be used, or even a consortium of all the largest telescopes be employed at the same time to get the best possible signal.

Using interferometric techniques in space to localize on only a portion of the star being transited, would one day offer a good way to improve the signal-to-noise ratio.

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Using coronagraphy we have actually imaged exo planets, though they are little more than blobs at the moment. Atmospheric spectra have been taken but the people working in the field are well aware that the science is only just beginning and lacks both quantity and quality of observation.

There are several pupil masking schemes which act to increase the contrast ratio of planet to star that look excellent in theory but none on research equipment as far as I know.

As Pete says space born interferometry is the way to go, both to avoid atmospheric absorption in the Near to Far infrared and to access longer baselines.

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