Can you see the flag on the Moon with a telescope?

The data about these measures is hard to find. But after some googling the best data I could find are in http://physics.ucsd.edu/~tmurphy/apollo/highlights.html . It it possible to make some estimations from the graphs.

There are three sets of three graphs. They are very similar, so let's see only the first one.

In the first graph there is a dark line in the center that shows that there are more photons coming at the expected time, than before or after that time. The signal is easy to see, but it is difficult to get any numerical values from this graph.

From the second graph it is possible to read some numerical values. Each bar shows the number of photons that arrive in a 100 ps interval. They have probably a Poisson distribution, but for simplicity let's approximate that by a Normal distribution.

Before the peak, there are approximately 10 +/- 10 photons coming every 100 ps. (These values are estimated from the graph with a lot of zoom, I can't be very sure about them. The values seem to be scattered between 0 an 20, so I assume that the average is ~10 and the standard deviation is ~10.) These are the photons that come from others sources. So if no mirror is present in the moon, or the laser dos not have enough power, then this is the expected value during all the experiment.

At the peak, there are approximately 400 photons in the same time interval. The difference between this value and the average of the first values is like 40 times the standard deviation. So it is almost impossible that they get these peaks by a random chance. For example, the probability to get a value that is bigger than 5 standard deviation is less than 1E-6, for 10 standard deviation it is less than 1E-23, and for 40 standard deviation it is almost almost 0. And they get not only one, but 8 intervals with more than 100 photons.

After the peak, they get more photons than before the peak for technical problems. I see something like 20 +/- 20. Whit these numbers, the chance of a 400 photons peak is bigger, but still almost 0, but I think that the correct way to do the estimations is using the first values.

The third graph is an auxiliary measure with an earth based mirror to get the dispersion caused by the system.

The analysis in the web page is more elaborated. They estimate the shape of the mirror and use the data from the third graph to explain the shape of the peak.

The gist of the story is: Use a large telescope to not only increase your collecting area, but focus your beam onto that surface: turn that measly 1 milliradian divergence laser into a 1 microradian divergence laser.

The laser retroreflectors are at least observable now, which is a pretty big distinction.

It is a fact that the NRO has been working on optical interferometers, and deployable reflectors, and probably has launched at least one.

1. if this impossibility is related to noise overwhelming the image signal (thereby causing it to be an unresolved blur)

2. and if we assume that this 3.5-mile diameter is for a naked eye looking through a telescope (i.e. we're talking about a single exposure)

3. isn't there some statistical process that could enable a reduction of the diameter of the lens, which would reconstruct a less-noisy image based on multiple exposures?

The article states "the Earth's atmosphere is never steady enough to allow resolution below about one arc second for most locations". But according to the European Southern Observatory one can do 1000x better:

"With AMBER on the Very Large Telescope Interferometer (VLTI), the astronomers were able to see details on the scale of one milli-arcsecond, corresponding to being able to distinguish, from the Earth, the headlights of a car on the Moon." -- http://amber.obs.ujf-grenoble.fr/spip.php?article154

Therefore a 6-foot-wide flag ("headlights of a car") would be visible as 1 or 2 pixels with the VLTI, which is made of four 8.2 meter reflectors.

I presume the actual flag left by the astronomers was somewhat smaller, so one would need a tad better telescope, but certainly not something with a "3.5 mile diameter".

Upvote me. HN readers need to be educated :-)

Asking if things discovered "a long time ago" have changed in light of modern development isn't un-called for. Especially given the vague statements.

>And, it would only be just visible as a small dot, it would not "look" like a flag at all.

Is similarly vague, and probably part of the reason for the question. We can build optical hardware with resolutions far in excess of our eyes per square unit. What we can see by looking through the scope directly has little to do with how we use more powerful modern telescopes, which appears to be how this is calculated. 20 years ago it would've been hard to predict high-density CCDs, how much more unlikely is it that something "A long time ago" took such things into account?

Lets ask another:

>The only method that could be used to (in theory) see something as small as the Flag on the Moon would be to use two optical telescopes set (for example) 1000 miles apart. This would easily provide the required resolution...

How does this help? Are they talking two Hubbles? Two 1.75 mile wide telescopes? There's zero explanation, despite such an absolute claim.

largest being built right now is 400m wide: http://en.wikipedia.org/wiki/Magdalena_Ridge_Observatory_Int...