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If there was a theoretically perfect mirror the size of say, our solar system, somewhere out in space that had a focal point of literally earth. What would that look like to a space or earth telescope? Could it be distinguished from other parts of space? Could you identify earth in it? (A la By putting a mirror in space, would we be able to see into the past?)

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    $\begingroup$ This is a great question, but I’m voting to close this question because I think it more properly belongs on the World Building or Space Exploration Stack Exchange sites. $\endgroup$ – Connor Garcia Dec 14 '20 at 5:40
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    $\begingroup$ @ConnorGarcia we should only close questions when they are off-topic where they are asked, not for "I think it's better asked elsewhere" by itself. Some questions are on-topic in multiple sites and some are off-topic everywhere. There isn't always a 1:1 correspondence between questions and their sites. The OP decides where to ask. We can make suggestions, and we vote to close if we are pretty sure the question is off-topic locally. $\endgroup$ – uhoh Dec 14 '20 at 9:42
  • $\begingroup$ If the mirror had a focal point of Earth, that would stand out as it would increase the light reflected to Earth. If it didn't have the focal point of Earth and just reflected stars, I think it still might be visible as mirrors don't reflect all wavelengths of light. With a good enough look, it should be recognizable as a mirror, I would think. $\endgroup$ – userLTK Dec 14 '20 at 10:06
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If there was a theoretically perfect mirror the size of say, our solar system, somewhere out in space...

Great thought experiment so far!

...that had a focal point of literally earth...

Rats! I was hoping for a flat mirror so the answer would have been slightly simpler.

It is true that concave mirrors are "magnifying mirrors" and if we stay closer to the mirror than twice its focal point (see below) we can see a magnified image. I don't believe that it provides better optical resolution than a normal mirror, but since the construction of our eyes is fixed (we can't change the optical system nor "pixel density") we use the mirror to "blow up" the image in the same way that a photographic enlarger blows up the image on a negative without improving the resolution of the image.

https://commons.wikimedia.org/wiki/File:Magnifying_cosmetic_mirror.jpg source

In optics image information is encoded in the wavefront at any point whether in focus at that point or not. If we know the mirror's diameter and distance from Earth, we can apply the principle of diffraction in a simple way no matter how complicated the rest of the optical system.

For a hard-edge circular mirror (top-hat shaped apodization) we know that the Airy disk is the right principle to apply, and a simple definition of resolution gives us

$$\theta \approx \text{1.22} \frac{\lambda}{d}$$

for the angular resolution. Let's use 500 nm green light for $\lambda$ and twice Neptune's orbital radius (60 AU) for $d$. That gives us $4 \times 10^{-18}$ radians.

If the mirror were at the distance of Proxima Centauri or only $4 \times 10^{+16}$ meters, then the impact on resolving a wavefront from the perfect mirror based only on diffraction of a circular aperture would be of the order of centimeters!

The focal length of a concave mirror is the parallel-to-point distance, so we would really need the mirror to be at roughly the point-to-point distance of twice the focal length.

If the concave mirror brought our image back to us, say on a sheet of paper, it would be incredibly dim i.e. pretty much no photons except from the Sun itself. But if we ignore that then we would see ourselves blurred by about 1 centimeter.

If the image was kilometers in front of us then we could focus a telescope on that image in space and re-image it at the entrance pupil of an eyepiece or on to a sensor.

If the telescope were 1 meter in diameter, we could see roughly a 1 meter wide swath of Earth.

Of course this doesn't work so simply because the mirror would have to be pointed such that the location of Earth 8.5 years ago would be imaged where we were today.

Conclusions

Yes this is kinda possible from a Gedankenexperiment point of view; a 60 AU mirror out at Proxima Centauri with a focal length of twice that distance could produce an image of Earth nearby Earth (light time considerations in force) and we could look at that location in space with a large diameter telescope and see a patch of Earth 8.5 years ago roughly vignetted by the size of the telescope's aperture.

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