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After Apollo 11 first landed on the Moon in 1969, there have been conspiracy theories that this never really happened and that it was all a hoax. In 2010 NASA released photos from its Lunar Reconnaissance Orbiter (LRO) mission of some landing sites which showed both a lunar lander and footprints (as dark trails).

The question is, how big (aperture) of a telescope would you theoretically need on earth to see the lunar module or even better, the dark trail of footprints? What about from orbit, without the atmosphere, would a smaller diameter scope work, how big would it need to be?

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For the record: the way this question is phrased it seems perfectly fine here. The key is that it reduces the motivating scenario (the moon landing conspiracy theories) to a concrete physical problem (required aperature of a telescope) and asks about that. If it asked "Has anyone seen these footprints?" then that would be a matter for Skeptics. –  David Z May 6 '12 at 5:10

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Google led me to this page which goes through the math to answer your question about the theoretical size of a telescope needed to resolve the lunar modules. It should come as no surprise that no, it can't be done with a small telescope: according to the math found at that link, you would need a telescope 100m in diameter to just about be able to see the LM on the moon (from earth). The largest telescope on earth is 10m in diameter.

He also posits that building such a telescope would be more expensive than going there and taking a picture yourself.

Another thing to consider is the atmospheric effects. If you've ever seen an amateur video of Jupiter made from pictures taken through the atmosphere, you've seen it bubble and deform as though you were observing it from underwater. The atmosphere is in fact a fluid and this would make observing details as small as footprints on the moon next to impossible.

Using an orbiting telescope would avoid this, but the situation's not much better: it would have to be of similar size to the ground telescope (a ~300km difference in altitude wouldn't do much for the resolution), and you would actually have to get the thing into orbit.

I think the LRO pictures are the best we're going to get for a while. (BTW, moon hoax believers love to say that the LRO pictures are also fake, since they, too, came from NASA (they didn't really). JAXA's KAGUYA spacecraft, while not photographing the footprints etc., did map the topography of the apollo landing sites and it matches perfectly with the pictures. If the landings were filmed in the desert, how did they get the scenery to match up exactly with what's on the moon at the "fake" landing sites, according to JAPAN'S space agency?)

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Just a small addition to your answer: Using ground based interferometry you could create the sizes needed to resolve a footprint. If only that footprint is bright enough to outshine everything around it. –  Tigran Khanzadyan Jun 1 '11 at 22:44
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Did someone say, "A telescope 100m in diameter"? en.wikipedia.org/wiki/Overwhelmingly_Large_Telescope –  Andrew Jun 15 '11 at 21:48
    
@Andrew that's a hilarious name for a telescope, and very appropriate as well :) too bad they're not building it –  Carson Myers Jun 16 '11 at 2:10
    
Yes, I've had an amateur/fan grade interest in the OWL Telescope ever since I first stumbled on the project. I was heartbroken when they downgraded it from pipedream to "Don't even go there." –  Andrew Jun 16 '11 at 10:58
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Some years ago, I read an article about a proposed "Heinlein Distributed Observing System" (named after Heinlein, I think, not his idea). Each individual telescope was to have a 1-kilometer mirror. They'd be placed in Earth's orbit around the Sun, using interferometry to give an effective aperture of 2 astronomical units, or about 300 million kilometers. If I recall correctly, it could theoretically read license plates at Alpha Centauri (or maybe it could just see individual people). I say we build it. –  Keith Thompson Jan 4 '12 at 4:30

It can't be done from the surface of the Earth. The atmosphere limits the best resolution you can achieve. Without adaptive optics that limit is 0.2 to 1.0 arc seconds depending on your location. Even at 0.2 arcsecond, you don't have the resolution as that corresponds to a linear size of about 375 meters. I don't have exact numbers but even if adaptive optics could get you an order of magnitude better, you'd still have a resolution of only about 40 meters, much larger than the lunar module or a foot print on the moon. So a ground based telescope is out.

If we go to space we remove the limiting effect of the atmosphere. The LRO camera had a resolution of one meter. If we want to shoot for that same resolution, then we just need to look to basic optics for the answer. Optical resolution is given by

R = lamda/diameter

where R is the angular resolution, lambda is the wavelength you are looking at, and diameter is the diameter of the telescope. At the lunar distance, 384,399 km, a one meter object has an angular size of 2.6 x 10^-9 radians. Assuming we are viewing in the middle of the visible spectrum at 500 nanometers, this means we need a telescope with a diamter of 192.2 meters.

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Keck AO can get about 0.05 arcsec, while Keck interferometer can get 0.005 arcsec. Of course, the real problem is that footprints do not have high contrast in the near IR. –  Jeremy Jun 1 '11 at 20:53
    
@dagorym +1 for the "back of the envelope" calculation. –  Alessandro Áleifur Jacopson Jul 1 '11 at 17:24
    
Hubble has imaged the Moon (because it has ultraviolet detectors) and its pictures at 0.1 arcsecond resolution still don't show anything smaller than a football field. –  Pete Jackson Jul 6 '11 at 10:57

One of the things the Apollo astronauts left on the moon was a corner cube mirror that is good for bouncing lasers off the moon. We do this regularly now, and it would be impossible without those mirrors, but I doubt this will convince the un-convincable. At the very least, one could argue that an unmanned spacecraft left them there. I guess the footprints would prove people, but again, that is thinking logically, which is probably not the way to go when dealing with deniers.

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Personally I think the kaguya maps of the landing sites is the most solid proof. They couldn't have known the terrain of their alleged landing site ahead of time, and even if they could, they couldn't rearrange the terrain of wherever they faked it to that magnitude without anyone noticing. Plus, the data came from Japan and so couldn't have been "faked" by the states to keep the coverup going. –  Carson Myers Jun 1 '11 at 21:31
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ALIENS LANDED ON THE MOON AND LEFT A MAGIC ARTIFACT THAT RETURNS LASER FIRE!!!!!!! –  Andrew Jun 17 '11 at 21:20
    
Wait, satire is frowned upon in this venue, isn't it. Sorry, the tinfoil hat brigade always brings it out in me. –  Andrew Jun 17 '11 at 21:22

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