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Telescope technology is advancing. Logically there should be a point where clear images of the landing sites are visible.

I am guessing a 1km telescope would be plenty. If we currently have 100m telescopes and had 30m a few decades ago will we get to 1km by 2050-2100?

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    $\begingroup$ There were several moon landings. How did you come up with the 1km size? We do not currently have 100m telescopes. The largest optical telescopes currently are 12m in size, with 30m-50m under construction and expected first lights in 2025. Noone can say if telescopes will grow in size, that requires funding and engineering breakthroughs. $\endgroup$ – AtmosphericPrisonEscape Nov 11 '19 at 19:20
  • $\begingroup$ If my figuring is right, the Overwhelmingly Large Telescope, if it had been built, could resolve to 16 meters. So not that, unless a differently-colored pixel counts. $\endgroup$ – Greg Nov 11 '19 at 20:34
  • $\begingroup$ Which moon landing? Pick your favorite one, I suppose. Does it change the answer? $\endgroup$ – Greg Nov 11 '19 at 20:35
  • $\begingroup$ Apollo obviously $\endgroup$ – user33880 Nov 11 '19 at 21:11
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    $\begingroup$ These questions are related, but I'm not sure though if answers there also answer your question. Picture of equipment left on the Moon? and also Were the Apollo lunar activities observed from Earth? $\endgroup$ – uhoh Nov 11 '19 at 23:31
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Using this page you can calculate how big a telescope must be to view any object at any distance:

http://win98.altervista.org/telescopio.html

Page also lists data for some known telescopes and some famous site, for example Hubble telescope and Apollo 11 landing ste:

  • Hubble-Moon resolution (400000 km, 2400 mm telescope): 93 m
  • Apollo Lunar Excursion Module (LEM) diameter leg-to-leg: 9.4m

To view Lunr Module as big as a pixel from Earth, you'd need a 21338.16 mm aperture telescope (21 meters). But it would be just 1 pixel; to distinguish details you'd need at least a 0.1 meters resolution (0.0001 km), which gives 2133816.27 mm (2133 meters).

Formulas:

  • ResolutionArcSec = 116/Diam_mm
  • ResolutionArcSec = 4.51/Diam_in
  • TargetMinimumDiam_km = DistanceKm * Math.tan(ResolutionRad)
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In theory, a VLBI (very long baseline interferometric) set of telescopes can provide the resolution you need. Basically, placing a few telescopes of reasonable aperture far apart from each other and relaying their images to a common collection system (the interferometer) will give you excellent high-spatial-frequency images. You will not get much of the low-spatial-frequency data due to the overall sparcity of the imaging system.

However, maintaining phase relationships on the order of a couple hundred nanometers over a 2-km spacing is really really hard. VLBI is much easier for radiotelescope systems, of which there are a number in operation at this time.

The longest current optical baselines are only 40 to 80 meters and the longest one currently under construction is the Magdalena Ridge Optical Interferometer which

will have ten 1.4 m (55 in) telescopes located on three 340 m (1,120 ft) arms. Each arm will have nine stations where the telescopes can be positioned, and one telescope can be positioned at the center.

This is sufficient to have of the order of 1 meter resolution at the Moon's distance, but being optimized for star-like sources it's not clear if it will be able to image extended objects like the surface of the Moon.

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