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The question Why is this video showing radio waves transmitted from a radio telescope? and this answer to it got me thinking. If atmospheric seeing at visible wavelengths is the result of refractive index inhomogeneity, would it also be a similar problem for mm to cm wavelengths? From a quick search, the index of refraction of air at STP is about 1.0003 (visible) and 1.0002 (radio).
If it is not, is there a way to understand quantitatively why it is not a problem?
In fact, the techniques of adaptive optics are already being used in radio astronomy. They are implicit in the basic imaging algorithms (e.g., CLEAN) used to produce maps from radio interferometers. In those cases, they are usually being used to correct for the artificial structure introduced by the way the interferometer samples the sky, rather than for structure imposed by the intervening material. But at low frequencies (1 GHz and below, certainly) they are also used to correct for the artificial structure imposed on the incoming radio wavefronts as they pass through the ionosphere. Current large low-frequency instruments (such as the LWA and LOFAR) rely heavily on these methods.
The purpose of adaptive optics is to reach or approach the diffraction limit of the system, which is the maximum resolution achievable due to the wave nature of electromagnetic radiation. The formula for the diffraction limit (in radians) is approximately $\lambda / D$. For a 30-meter radio telescope observing the 21-centimeter line, this works out to 0.007 radians, or about 24 arcminutes. This is much larger than the sub-arcsecond diffraction limit of an optical telescope; no matter what you do with your telescope you can't do better than this, so seeing is simply not a factor for single-dish radio astronomy.
