Can the US flags on the Moon be seen by terrestrial telescopes?

Question

Telescope manufacturer, Celestron, claims on their blog:

Can we directly see the flags on the Moon with an Earthbound telescope?

The answer is no. The largest of the American flags on the Moon is the one left by Apollo 17. It spans six feet when unfurled. A target that small at the quarter million-mile lunar distance is beyond the reach of any Earthbound telescope, even the most sophisticated professional observatories equipped with state-of-the-art optics.

To show the limitations of Earthly telescopes in observing ultra-small detail on the Moon, well-known astronomer Yuri Beletsky at the European Southern Observatory in Chile conducted an experiment using an eight-meter adaptive optics telescope. He tried to visually spot the 150-foot-long shadow of one of the Apollo landing stages still resting on the Moon. His attempt failed, showing that viewing the flags, targets at least ten times smaller, is impossible with current technology.

I am nearly sure some countries have the technology.

Is there still no technology to see the flags?

Answer 1

No.

Modern ground based telescopes can adapt to looking through Earth's atmosphere, but the problem is one of fundamental physics. The maximum theoretical angular resolution, the ability to distinguish small details, of an optical telescope is governed by the The Rayleigh criterion.

While the exact resolution and other details of modern spy satellites are classified, some idea of the trade-offs available can be made using simple physics. The formula for the highest possible resolution of an optical system with a circular aperture is given by the Rayleigh criterion:

resolvable size = (1.22 * wavelength * distance) / diameter of the mirror

Resolution decreases linearly with increasing wavelength of light or distance. It increases with the size of the lens. We can restructure the equation to find out how big our lens would have to be.

diameter = (1.22 * wavelength * distance) / resolvable size

A 6 foot (about 2 meter) flag on the Moon (ignoring that it would not be side-on to the Earth) has an angular diameter of about 0.0012 arcseconds. In visible light (400 - 700 nm), at the Moon's closest approach (350,000 km) would take a lens with a diameter of at least 85 meters.

There is no such telescope. The largest optical telescopes are only about 10-12m. The upcoming Extremely Large Telescope (ELT) will have a 39 m mirror and able to resolve 0.005 arcseconds. On the Moon that's about 7 m.

Very large radio telescopes, such as the recently collapsed Aracedo, or very large arrays exist, but these do not operate in the optical band. While it is possible to combine multiple telescopes into an interferometer to get higher resolutions, this is tricky for optical telescopes, and extra tricky for a close target which moves relatively quickly; it is generally used on very large, very distant point sources.

the interferometric technique is such that it is very efficient only for objects that are small enough that all their light is concentrated. For instance, an object with a relatively low surface brightness such as the moon cannot be observed, because its light is too diluted

For more detail about the capabilities of interferometers, ask on Astronomy.SE.

I am nearly sure some countries have the technology.

I'm guessing this is because we have very good spy satellites able to look down on the Earth; their abilities are not known and they tend to be much better funded than NASA, so it's possible they can resolve a large flag. Couldn't they be turned towards the Moon? The size of their mirrors is limited by the size of rocket fairings, only a few meters at most; even JWST, folded up, is only 6.5 meters. These operate in low Earth orbit just 500 km to 1000 km above the Earth. The Moon is 300-600 times further away. From our equation above we see that makes the job 300-600 times easier.

There's no reason for government to make a secret 85 meter ground-based optical telescope to spy on space. Nor would they likely be able to keep such a large and complex facility a secret. There's only so many places dark enough for such a sensitive telescope. Studies for a 100 meter Overwhelmingly Large Telescope (telescopes are getting so big, astronomers are running out of adjectives) found it infeasible with a price tag of over $20 billion, roughly the annual budget of NASA.

Answer 2

The Q wording about 'earthbound' isn't too clear if it includes those in orbit, but anyhow, the answer for those is 'no' too.

So can Hubble see the flagpole on the Moon? The answer is no, it cannot. The highest resolution that Hubble can achieve is about 0.03 arcseconds [per pixel] using its Advanced Camera for Surveys (ACS) array of cameras. The smallest object on the Moon that Hubble could observe is about the size of a football field.

Detailed ACS info here (repeats the 0.03 figure; FWTW, ESA gives a somewhat smaller number of 0.025 arcseconds per pixel for the High Resolution Channel of the ACS.) Note that the newer WCF3 installed in 2009 has slightly lower resolution (0.04 arcseconds / pixel) but a wider range of frequencies.

If that source is not enough, here's another NASA page saying:

The Hubble Space Telescope could just make out a crater 100 meters or yards across, the length of a soccer field.

The 4ft flags themselves are clearly out of the question. Possibly Hubble could discriminate the 150 foot shadow though, in the right conditions, but it would be like half a pixel at best, so it would require a fair bit of interpretation (probably from multiple images, using SAR-like techniques.)

N.B.: it's been rumored, but only partly correct that Hubble is related to the tech used in spy satellites, i.e. that it's a spy satellite pointed upwards, instead of downwards. The mirror diameter of 'leftover' US spy satellites is similar to Hubble's though, but some details differ, e.g. the focal length; the 'spy' sats consequently have "a much wider field of view". The WFIRST space telly that was the made with one of those leftover 'spy' mirrors has however many more sensors (and higher res) than Hubble's, i.e. an array -- 18 sensors of 16 megapixels each; but the ultimate resolution is the same, although WFIRST takes like 100 Hubble-like images simultaneously [also on a different spectrum]. WFIRST is slated to launch in 2027.

James Webb has a larger mirror, but because it operates at lower frequencies... that doesn't improve its resolution, relative to Hubble:

Despite its larger size, Webb delivers about the same resolution in near-infrared light as Hubble attains in visible light. A telescope’s resolution, the amount of discernable detail, is proportional to mirror size, and inversely proportional to the wavelength of the light observed. For Webb to resolve the same level of detail at longer wavelength infrared light that Hubble achieves at visible wavelengths, its mirror needs to be proportionally wider.

Apparently Webb has a focusing problem, in that it wasn't designed to focus on objects 'this close', but on distant galaxies. It also technically orbits the Sun, and not the Earth. And because of its position at L2, Webb is actually farther from the moon than Hubble. It's also not clear to me if the flag shadow would show in IR, the frequencies that Webb's sensor works at. It's also not possible to safely point Webb at nearby objects because that would at least temporarily 'blind' the ultra-sensitive sensors:

Because the telescope and instruments have to be kept cold, Webb’s protective sunshield is blocking the inner solar system from view. This means that the Sun, Earth, Moon, Mercury, and Venus, and of course sun-grazing comets and many known near-Earth objects cannot be observed.

(Fun factoid: "The sunshield gives Webb the equivalent sun protection factor, or SPF, of one million, NASA says.")

So, in theory, if you took the Webb mirror, a higher-res optical (not IR) sensor, and moved it back in earth orbit, it would be somewhat better at looking at the moon. But nobody is going to do that to see the shadow of a flag, on a few pixels, when you can send a spacecraft to orbit the moon and see it from there, like a spy satellites sees stuff on Earth (theoretically more clearly, because the moon lacks an atmosphere--but the LROC images from 2012 were not exactly impressive--the flag and its shadow were like 10 x 3 (blurry) pixels--LROC has 0.5 m/pixel best resolution, it seems.)