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I know it's very difficult for the James Webb Space Telescope to image the exoplanet Proxima b without an external 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 for the number of pixels.

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    $\begingroup$ Its worth noting that the JWST will have a similar angular resolution to Hubble, but in a different spectrum and with greater sensitivity. $\endgroup$
    – James K
    Feb 6, 2022 at 9:34
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    $\begingroup$ @JamesK The Webb's angular resolution will be worse than that of the Hubble. $\endgroup$ Feb 6, 2022 at 10:49
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    $\begingroup$ Seeing even the coarsest of details on an exoplanet is well beyond the capabilities of any human-made optical or infrared telescope, ground-based or space-based, now or in the foreseeable future. The Hubble has an optical resolution of about 0.05 arcseconds. The Webb, about 0.1 arcseconds. At a distance of 4.2 light years, Proxima Centauri b is about 86 microarcseconds across. $\endgroup$ Feb 6, 2022 at 10:59
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    $\begingroup$ @DavidHammen JWST’s resolution will be better than HST’s in the wavelength range where they overlap (0.6-1.6 microns). $\endgroup$ Feb 6, 2022 at 12:36
  • $\begingroup$ Maybe it had not been a bad idea to have some ultraviolet devices on the JSWT, too. So at least the close exoplanets could have been systematically mapped, giving irreplaceable information about their stats and distribution. $\endgroup$
    – peterh
    Feb 7, 2022 at 15:58

2 Answers 2

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The JWST will have an angular resolution of about 0.1 arcseconds, which is similar to Hubble.

The star, Proxima, has an angular size of 0.001 arcseconds, and the planet Proxima b (a super-earth) would be an order of magnitude smaller than that.

You could fit 10000 Proxima b planets into one pixel of a JWST image.

Indeed at 7.5 million km from the star, both the planet and the star would fit into a single pixel, as they have an angular separation of 0.04 arcseconds.

To illustrate here are the star "C" and the planet "b" simulated in a single 0.1 arcsecond pixel (the brown square).

enter image description here

And, zooming in to resolve as disks, here are the star and planet. You can see that resolving details on exoplanets is well beyond current technology.

enter image description here

It is possible that the JWST can do spectrographic analysis of the Proxima system, which could reveal properties of the planet. One of the main science goals is to learn about the physical and chemical properties of planetary systems, and investigate the potential for the origins of life in those systems.

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    $\begingroup$ “The JWST will have an angular resolution of about 0.1 arcsecond” — this is not, in general, true; it is true only for wavelengths of ~ 3 microns. It will be worse for longer wavelengths and better for shorter wavelengths. $\endgroup$ Feb 6, 2022 at 12:40
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    $\begingroup$ That is right. Consider this a ballpark figure, to illustrate the gulf between the resolution of the current generation of space telescopes and the size of exoplanets. $\endgroup$
    – James K
    Feb 6, 2022 at 12:46
  • $\begingroup$ @JamesK thanks for your answer. Would the small light dot have any color? $\endgroup$
    – James
    Feb 7, 2022 at 0:38
  • $\begingroup$ *even if that color is extremely light $\endgroup$
    – James
    Feb 7, 2022 at 0:44
  • $\begingroup$ Yes, that's what a mean by spectrographic analysis. This takes the light from the star and planets and splits them up by wavelength (ie colour). Of course the "light" that this telescope sees is all infrared, and so invisible to us. $\endgroup$
    – James K
    Feb 7, 2022 at 7:17
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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. An image of Proxima Cen b cannot be obtained with JWST in any case 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.

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.

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  • $\begingroup$ Note that with subpixel dithering of multiple exposures, it should be possible to recover Nyquist sampling for shorter wavelengths. (Not that this would help for Proxima Cen.) $\endgroup$ Feb 6, 2022 at 13:18
  • $\begingroup$ @PeterErwin True. I was careful to say "a single 'undersampled' picture". $\endgroup$
    – ProfRob
    Feb 6, 2022 at 15:23
  • $\begingroup$ @ProfRob thanks for your answer. Wouldn't be possible to move the telescope a bit to the side, so proxima centauri isn't in the midde of the coronograph and it doesn't cover proxima b? What I mean is having the coronograph covering most of proxima centauri surroundings, but less on the side where proxima b is. I assume less light would be blocked, but perhaps enough to image the planet? $\endgroup$
    – James
    Feb 7, 2022 at 0:39
  • $\begingroup$ The point is that if you don't have the star in the centre of the coronagraph then more light spills around the edge. In the case of Proxima Cen b it is impossible to separate it from the image of its star. It is too close. $\endgroup$
    – ProfRob
    Feb 7, 2022 at 7:29

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