Well, it's been several days without any contribution from Steve. I'm sure (or at least I hope) he has a life outside the Internet, but it's suspicious that he stopped responding just when his feet were put to the fire. In all fairness I challenged him to "put up or shut up", but people like Steve rarely do either. They won't put up, but neither will they stop talking. So kudos to him at least on those grounds.

Lest the multitudes of readers following this gripping conversation think there is no more science to be had, let's solve the Dawes Limit problem that I challenged Steve to solve.

Dawes' limit is based on the phenomenon of diffraction -- the propensity, through optics, of an increasingly magnified object to devolve into a blob of light surrounded by diffraction rings. Diffraction is the result of the interaction of light with the surface of an optical element at the molecular level. As long as we have to build lenses and mirrors with molecules, we will have this problem.

But happily the larger we make our lenses and mirrors, the less diffraction hurts us. That's because the molecular "bumpiness" of a surface is proportionally finer in large elements. It's like building a model of the Sphinx out of Legos(tm). The Lego bricks are a fixed size -- as are molecules -- and of limited shapes -- mostly rectangular prisms. If your Lego sphinx is only two feet tall, its detail suffers from the limitations of the sizes and shapes of the blocks at that scale. If the Lego sphinx is 50 feet tall, the details are scaled up too and can be better represented with the bricks.

Diffraction limitation takes the form of a minimum angular resolution. That's the minimum angle separating two objects so that they can be seen as separate even through diffraction. If you hold a penny at arm's length, you notice that the penny may occlude (hide) a light switch from across the room, and perhaps even a truck from several blocks down the road, but not your computer screen just beyond your arm. So the question is not distance, but the angle formed by your eye and the two edges of the coin.

The moon, for example, has an angular dimension of about half a degree. That means if you draw a line between your eye and the left side of a full moon, and another from your eye to the right side, the angle between those lines will be half a degree. Coincidentally, the sun has the same angular diameter. It is much bigger than the moon, but it is also much farther away.

The minimum angular resolution for an optical instrument is given by the diameter of its principal aperture and the wavelength of light you wish to collect. The formula is

resolution = (1.22 * lambda) / diameter

Lambda is the wavelength of light. You put lambda and diameter in the same units. The 1.22 coefficient combines two factors: the particular contribution of Dawes, and the scaling factor that puts the answer in radian angle measurements. Dawes said that if the two diffraction blobs overlap by one radius, they can still be distinguished.

I stipulated that the KH-11 primary mirror can be no larger than 2.4 meters, because that's what can be put in the shuttle cargo bay, and we know that's what was used to launch them. I also specified 550 nanometers as the wavelength, because that's right in the middle of the human visible band.

550 nanometers (a nanometer is a billionth of a meter) is 5.5x10^-7 meter. So the angular resolution according to that formula is 2.79x10^-7 radian. We typically give the angular resolution of optical instruments in arcseconds. There are 60 arcseconds in an arc minute, and 60 arc minutes in a degree. Commensurately there are 206,265 arcseconds in a radian, so the answer comes out to 0.058 arcsecond.

We agreed that news type is 0.10 inch and that the Keyhole satellites orbit at an altitude of 300 (statute) miles, or 19,008,000 inches.

From basic trigonometry, tan (0.058 arcseconds) = 2.76^-7 ~= width/distance, and if the distance is 19,008,000 inches then the width corresponding to that angle at that range is about 5.25 inches. That respresents the *maximum* attainable resolution through those optics. (Other factors can make viewing worse, but no better.)

Or stated another way, features that are less than 5 inches apart cannot be distinguished by U.S. spy satellites. That includes words on a newspaper. The best you could hope for is to tell that someone is holding a newspaper.

So what does that mean for lunar distances? Recomputing with a distance of 240,000 miles (15,206,400,000 inches) yields 4,200 inches or about 350 feet.

Spy satellites can't see anything smaller than a sports stadium on the lunar surface from Earth orbit. There are other reasons why they can't be used to take pictures of the moon, but that should be sufficient for now.

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