[throwaway_yy2Di]

    The planet is about 0.05 AU from Proxima Centauri, meaning
    we need an angular resolution of about 1.9e-7 radians to
    even distinguish it from its host star. Is that realistic?

Much more than that; that's the angle for HALF-maximum brightness, but since the star's many orders of magnitude brighter than the planet, you'd need a much larger reduction than 1/2. Unfortunately, the diffraction-limited pattern [0] has fat tails -- it's not Gaussian, the brightness is slow to drop off away from the center (polynomially slow? [1]). I understand you'd need >100 times the FWHM angle in practice, on the order of 1" for JWST for instance [2]

This is why coronagraphs will be so useful.

[0] https://en.wikipedia.org/wiki/Airy_disk#Mathematical_details

[1] a log-log graph shows the envelope is close to inverse-cubic (x^-3)

[2] http://nexsci.caltech.edu/workshop/2016/NIRCam_Planets_and_B...

[Thorondor]

Technically, 1.22λ/D is the angle for the first dark circular ring of the Airy disc (first zero of the relevant first-order Bessel function [0]). But you are still right that the host star needs to be blocked out in some way to produce a useful image. I think NASA is working on some ways to do this, see [1].

[0] https://oeis.org/A245461

[1] https://exoplanets.nasa.gov/resources/1015/

[throwaway_yy2Di]

You're exactly right; good catch.

[throwaway_yy2Di]

Here's a better answer on /r/askscience:

    Can we directly image the planet from earth?

    1. "The planet/star contrast is 10^-7 " This basically means
    for every 10,000,000 photons from the star, we would measure
    ~ one from the planet.
    
    2. "Current instrumentation using adaptive optics and
    coronography on 10 m class telescopes (like Sphere on VLT or
    Gemini Planetary Imager) aims at achieving a contrast of
    10^-6 to 10^-7 at an angular resolution of 100-200 mas"
    
    3. "The planet has a separation of 38 mas".
    
    4. Therefore with the best planet imagers we cannot
    currently directly image the planet. Our best hope is the
    E-ELT which should have first light in 2024.

https://www.reddit.com/r/askscience/comments/4zdkra/askscien...