Why is NASA unwilling to aim the Hubble telescope at the alleged Apollo lunar landing sites and show us once and for all? Quoting: Thor's Hamster
Simply because the resolution of the HST is not sufficient see anything that small.
The theoretical resolution of a telescope is calculated using the formula
R = 11.6 / D
where R is the the angular size of the object in arc-seconds and D is the diameter of the mirror in centimeters. The HST mirror is 2.4 meters (240 cm), so we can calculate that its theoretical resolution is 11.6 / 240 = 0.05 arc-seconds. (For comparison, the diameter of the moon as viewed from the Earth is half a degree, about 1800 arc-seconds
The is a slight problem with this however. Due to factors involving interference patterns and the wavelength range of visible light, the smallest resolvable object is about twice the theoretical resolution. This is given using something called Nyquist's Theorem; you can read about it here.
[link to searchcio-midmarket.techtarget.com
So effectively, the HST's resolution is about 0.1 of an arc-second. But the biggest things the astronauts left on the moon are the lunar descent stages. They are 4 meters across, and since they are about 400,000,000 meters away, their angular diameter is only 0.002 arc-seconds.
If you don't believe me, you can calculate this for yourself. The formula is...
d / D x 206265 = R
where d is the actual size of an object in meters, D distance to that object in meters, and R is the resulting angular size in arc-seconds
This makes the descent stages 50 times too small to be detected by the HST. They would have to be a lot bigger to be seen at all. In fact, if you do the math (set Hubble’s resolution to 0.1 arc-seconds and the distance to 400,000 kilometers) you see that Hubble’s resolution on the Moon is about 200 meters! In other words, even a football stadium on the Moon would look like a dot to Hubble.
This fact surprises a lot of people (it certainly surprised me until I understood THE SCIENCE). They’re used to seeing the detail in HST images, galaxies and wisps of gas in beautiful nebulae. But those objects are far, far larger than the Moon. The HST's resolution is 0.1 arc-seconds no matter how far away an object is. Those wisps of gas appear to be finely resolved, but they’re billions of kilometers across.
Using a bigger telescope won’t help much. You’d need a mirror 50 times bigger than Hubble’s to see the descent stages at all, and we don’t have a 100 meter telescope handy.
So, when you understand the science involved, you understand why the scientists will not point HST at the moon. It would be an utter waste of valuable telescope time.