Like, if you grabbed the asteroid 16 Psyche and hammered it out into a disc of 1 mm thick iron foil and curved it into a telescope mirror with 2.4x the radius of the Sun, could you resolve details on the surface of an exoplanet? At what resolution?
Could you make the mirror arbitrarily bigger and continue to get better resolution?
What are the formulas for calculating this and what are the limits, if any?
If you make a big enough mirror could you see individual houses on Proxima b? Could aliens with big telescopes have videos of the formation of the moon in their libraries?
Best Answer
Some theoretical answers were provided, but here's a practical answer from an astronomer's point of view.
(First off, the resolving power is given by diameter, not surface area. So we will talk about diameter here.)
For visible light, the practical resolving power of a 100mm diameter mirror is 1 arcsec. A 200mm mirror: 0.5 arcsec. And so on. This is the rule used by astronomers.
A mirror "with 20x the surface area of the Sun" (apparent area, I assume) would have approx 4.5x the diameter of the Sun, which is 6.3 x 10^6 km. Such a mirror would have a resolving power of 1.6 x 10^-11 arcsec.
At 10 light years distance, such a mirror would resolve details as small as 7 meters.
Please note this discussion is entirely theoretical, as there are no known technologies to manufacture such a big mirror with the required precision for visible light astronomy optics - which means the surface error cannot be bigger than 100 nanometers - in fact, for a good mirror, the acceptable error is 4x ... 5x smaller. There's no way to maintain such precision across millions of kilometers of reflective surface.
Currently, the biggest monolithic mirror is 6m in diameter and it never performed very well. The biggest well-performing monolith mirror is 5m. The biggest segmented mirrors are 10m in diameter, with a 40m project having had its initial funding approved very recently.