How many pixels could an image of Proxima b taken by James Webb have? I know it's very difficult for the James Webb to image Proxima b without a coronograph (I have been told by NASA scientists that they don't know yet whether they will be able to do so), but I wonder how it is possible to mathemetically calculate 1. Whether it could theoretically image the planet (even if it takes very long exposures), and 2. How many pixels the image would have.
I assume it might as little as just 1 pixel, but I would like to know how to calculate it in order to get an aproximation of the number of pixels.
 A: TL;DR:
Any planet around Proxima Centauri (or any other exoplanet around any other star) will be unresolved. That means it is so small and far away, that it would appear as a point source of light to all intents and purposes. The pixels in the high angular resolution coronagraphic imaging are about 0.03 arcseconds and this is comparable with the angular resolution of the telescope. This means that it is possible that most of the light of the planet would fall in 1 pixel unless the telescope is "dithered" to record a sequence of pictures with small offsets in order to properly locate point sources. However, an image of Proxima Cen b cannot be obtained with JWST because Proxima Cen (the star) is too close to it ($\leq 0.04$ arcsec) and too bright, and the coronagraph stops have a radius of $\geq 0.4$ arcsec.
Details:
The main high angular resolution coronagraphic imaging instrument on JWST is NIRCam. It operates between wavelengths of 0.6 and 5 microns and at the short wavelength end has pixels that map to 0.032 arcseconds on the sky. These pixels are actually too big to sample the native optical angular resolution, limited by the size of JWST, which is $\sim 1.22 \times \lambda /6.5{\rm m}$. This is 0.023 arcseconds at 0.6 microns but in practice it is a bit worse because the telescope isn't perfect.
That means an image of a point source at those wavelengths could in principle fall in one pixel, but more likely it will be spread over about $2\times 2$ pixels, because the image of a point source does not appear as a "point" on the detector, but is smeared out to some extent. The angular resolution of 0.023 arcsec is more akin to a full-width half maximum for the image at that wavelength.
To image an exoplanet however, you probably would not work at the shortest wavelengths because the contrast between the brightness of the star and the planet would be increased at longer wavelengths. At longer wavelengths, the angular resolution of the telescope becomes worse, but you reach a sweet spot at about 2 microns where the angular resolution of the telescope is equivalent to 2 pixels on the detector. At this wavelength the image of a point source will always have a diameter of at least $2\times 2$ pixels, but this is still an "unresolved image" - no detail is discernable. What it does give you the ability to do, which isn't possible with a single, "undersampled" picture at shorter wavelengths (although could be with done with multiple images and a clever dithering procedure), is to accurately say, to a small fraction of a pixel if the data are good enough, where the "photocentre" of the image is (i.e. where the image is brightest).
In principle then, you could get some indirect indication of the surface features of an unresolved source (a star or a planet) by seeing whether the photocentre moves (as something rotates) or whether it changes with wavelength.
In practice, for Proxima Cen b, this isn't viable. The star and the planet are themselves separated by (at maximum elongation) by $\leq 0.04$ arcseconds (the orbital inclination is not known, so it could be smaller than this). Whilst this is just about resolvable with JWST + NIRCam at the shortest wavelengths, this would only be possible if the objects had similar brightness (e.g. the image of the star+exoplanet would be broader than that of a point source). But the exoplanet, which if it is an Earth-like or even a sub-Neptune-sized planet, has a brightness that is largely determined by light reflected from the star. Given that the planet is likely to be $\sim 10$ times smaller than the star, have an albedo $<1$, and be separated from the star by about 100 stellar radii - then the reflected brightness will be many orders of magnitude fainter than the star.
The coronagraphic imaging is meant to help out with these cases. If you can block the light from the star, then it becomes more feasible to detect the faint light from the exoplanet. However, the coronagraph cannot work miracles. If the exoplanet is only separated from the star by about the angular resolution of the instrument then there will still be lots of starlight that "spills over" and swamps the faint light expected from the exoplanet. For this reason the smallest coronagraphic stops on the JWST have a radius of 0.4 arcseconds and so you couldn't effectively obscure Proxima Cen and have Proxima Cen b visible.
I think you are more likely to see JWST images of (unresolved) Jupiter-sized planets separated by at least an arcsecond from their parent stars, but possibly some Neptune-sized things at $\geq 0.5$ arcseconds from cool M-dwarfs. Proxima Cen b is out of reach for imaging.
