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Old April 25 2013, 08:02 PM   #6
Location: Kentucky
Re: Planet around Alpha Centauri - lets talk about it

Well, let's have some fun with a calculator and see what it would take to image a planet at that distance.

The distance to Alpha Centauri is 4.367 light years and a light year is 9.460528e12 kilometers, so the distance is 4.131e13 km. The resolution of a telescope (separating two stars by eye) is res = 4.56 arc seconds / diameter in inches. Trigonometry says the sin of one arc-second is 4.848e-6 times the distance, so an arc-second resolution can distinguish two objects that are 200 million kilometers apart at the distance of alpha centauri.

The Hubble has a resolution of about 0.05 arc seconds, so it can resolve to about 10 million kilometers at that distance. (The Earth is 12,742 km in diameter. Dawe's limit (the 4.46/d" formula) for telescope resolution says the Hubble's 94.5 inch diameter aperture should resolve to about 0.0483 arc seconds.)

Supposing we wanted a telescope that could distinguish between two planets and Earth-diameter apart at the distance of Alpha Centauri, we'd need a telescope that could resolve down to 12,742 km or 127.42 km, which by trigonometry would be 1.767e-8 arc-seconds and require an aperature of 4,000 miles. To resolve an Earth-size planet into an image size of 120x120 useful pixels would take a 480,000-mile diameter telescope, which if centered on the Earth would span out to the orbit of the moon. We couldn't build a single mirror that vast, but all we need are dozens or hundreds of small mirrors in orbit whose angle can be controlled so their surface is aligned to within an 1/8th wavelength of light. We could probably do that in the not so distant future (relative to launching anything to Alpha Centauri).

Scaling such a set of satellites up isn't all that hard, so I'd imagine that we could be producing useful images of nearby planets from our own solar system long before we could construct a ship to visit them.
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