This link has the calculation:
From the link:
Yes, the flag is still on the moon, but you can't see it using a telescope. I found some statistics on the size of lunar equipment in a Press Kit for the Apollo 16 mission. The flag is 125 cm (4 feet) long, and you would need an optical wavelength telescope around 200 meters (~650 feet) in diameter to see it. The largest optical wavelength telescope that we have now is the Keck Telscope in Hawaii which is 10 meters in diameter. The Hubble Space Telescope is only 2.4 meters in diameter - much too small!
Resolving the larger lunar rover (which has a length of 3.1 meters) would still require a telescope 75 meters in diameter.
Even barely resolving the lunar lander base, which is 9.5 meters across (including landing gear), would require a telescope about 25 meters across. And in reality you would want a couple (or a few) resolution elements across the object so that it's possible to identify it. (Otherwise it'll look like a one pixel detection, not an image, and I don't think people would be convinced by a couple pixels!)
In addition, with a ground based telescope, you have to deal with distortion by the atmosphere as well, so you'll probably want something considerably larger than 25 meters if you want a good, believable, image of the lander. We don't have anything that big built yet!
So there's really no way to image equipment left behind by the astronauts with current telescope technology.
More details for the mathematically inclined: How did I calculate this stuff? Well, here's the procedure. Let's take the case of Hubble and find out what the smallest thing it can see on the surface of the Moon is.
Resolution (in radians) = (wavelength)/(telescope diameter) or R= w/D. This is a formula from optics.
So for Hubble we know that the telescope diameter is 2.4 meters (it's not very big - it had to fit into the Shuttle.) Also, we know that visible wavelength light is in the range 400-700 nanometers. I'll use 600 nm, because it's somewhere in the middle and I've used it before for this calculation.
If you use all units of meters and do R= (600e-9)/(2.4) = 2.5e-7. Well, that gives us the resolution of Hubble in radians which isn't too intuitive, but we can convert to meters on the surface of the Moon.
To find the spatial extent that 2.5e-7 radians is at the distance of the moon, set up a triangle between Earth and the Moon, where R is the angle in radians that we calculated, x is the side opposite angle R which corrosponds to the object on the moon, and the adjacent side is the Earth-Moon distance. Then you have Tangent(R)=x/(distance Moon).
The distance to the moon is 384,400 km. So converting to meters again and plugging in R and d_moon will give you a size in meters of the smallest size thing HST can see.
- When you do this you get 96.1 meters (315 feet). The astronauts didn't leave anything this big! If you look at this HST image of the Moon you can see that they say "Hubble can resolve features as small as 280 feet across." I think they used 500 nm as their wavelength instead of 600 nm, but it's the same order of magnitude as what we got here.
So there's no way HST can see anything humans left behind. HST can do a good job of studying large-scale geology, like craters, which is what the images were of. People and their stuff are just really small on a planetary scale!