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Would we be able to detect (via emitted radiation, or its gravity) a rogue planet between us and Proxima Centauri? How big would that planet need to be?

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Not today, but it should be possible to illuminate this planet by a powerful EMW blast or giant laser powered projector. –  Asphir Dom Aug 29 '13 at 19:52
    
How far can it be from the sun? Can it be only 10% of the distance to Alpha Centauri? 1%? –  Ben Crowell Aug 30 '13 at 3:21
    
@BenCrowell - you can treat it like a parameter. Looking for JWST's sensitivity data to produce a couple of size vs. range graphs for a better answer. –  Deer Hunter Aug 30 '13 at 7:20

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Transit-based methods and astrometry won't help here (a rogue (not rouge) planet is highly unlikely to move across the line-of-sight between us and neighboring stars more than once), but microlensing events like that (provided there's another star behind the planet and Proxima) are possible.

Of course, if you don't make repeated observations, you cannot be sure how far the planet is, what its proper velocity vector is, and whether you are detecting a regular planet or a rogue one.

In principle, Earth-sized planets are currently detectable via microlensing.

When James Webb Space Telescope is launched, it may be able to spot and possibly track a rogue planet with a still liquid core, heated above the background temperature.

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