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@SteveLinton's nicely written and sourced answer about using strong gravitational lensing by the Sun or even Jupiter as a kind of telescope to resolve the surfaces of exoplanets is really interesting, and the link cited there is indeed quite readable.

For the bare minimum resolution I considered of 1E-10 to make a exo-Jupiter 10 pixels wide at 7 light years, or even the 1E-12 he mentioned to do roughly the same at 1000 light years, what would a telescope like that be like?

Is it a maneuverable JWST that is so far from the Sun that it just scans back and forth building up an image of whatever is behind the Sun (or Jupiter) or a rigid 10x10 array of JWSTs?

Would each telescope just be a light collector for that pixel, or would this be more like a "light field camera" where the angular information from the telescope's focal plane at each telescope position could be used to improve the spatial resolution of the whole thing by "computational de-blurring"?

Would each telescope need a coronagraph to block the light from the Sun (or Jupiter) while collecting the light from the far dimmer Einstein ring-like structure surrounding it?

The reason I ask that is that gravitational lenses are barely "lenses" from the point of view of telescopy. The deflection of a ray by a "normal" or thin lens increases linearly proportional to distance from the axis, whereas for a point gravitational source the deflection is inversely proportional. In order to image an extended source like a planet or star's disk rather than just just collect the light of the Einstein ring from an unresolved star, you have to play games with geometry.

For more on light fields, see the question and great answer to Is a “Light Field” useful in mathematics, or just in marketing?

I'm not interested in just opinions, but instead answers that are built from reasonable, cited sources, or unsourced but derived from solid physics and math principles.

enter image description here

Source

A remote light source passing behind a gravitational lens. There is a large point mass in the center acting as a lens. The aqua circle is how we would see the light source if there was no lens, while the white spots/circle is the light source as seen through the lens. If the light source is collinear with the earth and lens, the image is an "Einstein ring". When the source is off this line we see a double image. As it moves far away, one of the images gets fainter while the other one is almost not affected by the lens any more (thus coinciding with cyan circle).


Stanford plenoptic camera array used to research light fields

Source

Stanford plenoptic camera array used to research light fields

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    $\begingroup$ this video mentions a "string of pearls" or "train of spacecrafts" doing a one way trip along the focal line $\endgroup$
    – user19132
    Jul 25, 2023 at 12:16
  • $\begingroup$ @jkztd thanks! That's a wonderful video overview. $\endgroup$
    – uhoh
    Jul 25, 2023 at 14:26

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Here are some papers that describe what a gravitational lens telescope would look like.

"FOCAL Mission Concept" http://kiss.caltech.edu/workshops/ism/presentations/FOCAL%20Mission%20Concept%20JOHNSON.pdf

"Mission to the Gravitational Focus of the Sun: A Critical Analysis" https://arxiv.org/abs/1604.06351

Note that Jupiter is a much more difficult option for a gravitational lens -- the sun's lens starts at 550AU away, while Jupiter's starts at 6100AU.

I will update this answer with more information from the links as I have time later today.

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    $\begingroup$ Thanks, the first link is quite thin, really just "we are going to think about this". The second is the exact same "readable" ArXiv paper I mention in the first paragraph that (at)SteveLinton uses in his answer. Hopefully you can add some more here, but you are at least on the right track. Welcome to Space! $\endgroup$
    – uhoh
    Nov 5, 2018 at 18:53
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The minimum focal distance (where light rays grazing a spherically symmetric object first meet) is given by the formula $z=r^2/2r_g,$ where $r_g$ is the Schwarzschild-radius of the object and $r$ is its actual radius. More generally, light rays with impact parameter $b$ meet at $z = b^2/2r_g.$

Plug in the Sun and you get about 550 AU. Plug in Jupiter and you get ~5,800 AU.

So no, using Jupiter as a gravitational lens is not feasible. Even for the Sun, we need observing spacecraft at a distance of at least 650 AU (we don't want the light to appear too close to the Sun's disk, because we need to block out the Sun to observe the faint Einstein ring) and that's more than 4 times the distance traveled by Voyager 1 so far, in nearly half a century.

Thinking of the Solar Gravitational Lens as a projector, it projects an image of an exoplanet that is many kilometers wide. An observing spacecraft in that image region, looking back at the Sun, would see the Sun, surrounded by a faint Einstein ring (light from the target) on the much brighter solar corona background. The intensity of this Einstein ring represents the signal at that "pixel" in the image plane. So yes, the observing spacecraft will act as a single-pixel detector.

Such a spacecraft would require at least a 1-meter telescope in order to resolve the Einstein ring from the Sun, and to be able to block out the Sun using a coronagraph.

Putting aside the challenges of navigation, removing the corona background and other issues, in principle a single spacecraft could then "scan" the image plane, moving back and forth, collecting data from each pixel. Of course if we had hundreds of spacecraft, they could form a grid and take an instant "snapshot" of the image plane. That would make the job much easier. However, I don't think such a mission is realistic.

In actuality, we will still need multiple spacecraft, because simply navigating to the right spot is a challenge, stationkeeping even more so. Having one spacecraft as an inertial reference might allow us to precisely track the positions of the other spacecraft with respect to the (moving) image, and multiple spacecraft would also help us measure and remove background, such as the solar corona, leaving us only with the residual stochastic (shot) noise.

For reference, here is our paper that provides a reasonable description of how such an SGL mission might take place: https://arxiv.org/abs/2207.03005. It also has a fairly complete bibliography detailing our work on modeling the problem, though we have also published a couple of papers since.

And yes, the lenses are "barely lenses" with significant spherical aberration and thus plenty of blurring. Averaging the PSF in the geometric optics limit over the aperture of the observing telescope's lens, we get a reasonable expression using elliptic integrals, which reduces to $\propto\rho^{-1}$ behavior (with $\rho$ being the distance from the optical axis in the image plane) for larger $\rho$. The blur can be removed by deconvolving the image, but this can substantially decrease the SNR; so depending on the desired resolution and the noise in the data, a blurred but recognizable image might be preferable over a pile of noise.

enter image description here

For what it's worth, here's a simulated result using the Earth as a stand-in, from one of our most recent papers (https://arxiv.org/abs/2306.07832), detailing a simulation that also takes into account the moving nature of the target. The image evolves as we collect initially fewer than a hundred pixels' worth of data, all the way up to around 6,000.

enter image description here

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