The atmophere refracts the starlight passing through it. The effect is similar to the refraction of light by a plane piece of glass. In this way, for each thin slice of the atmosphere, the refraction is goverened approximately by Snell's law: n1Sin(theta1) = n2Sin(theta2).Because the atmosphere varies in density and hence refractive index, you must calculate an effective index of refraction by multiplying together the effect of all these thin slices. A good derivation is given at http://ganymede.nmsu.edu/~holtz/a535/ay535notes/node6.html

The effect is to make stars appear higher in the sky than they really are, and let's call this difference the refraction angle. The refraction angle is approximately given by r = (1-n2)tan(alpha)where alpha is the observed zenith angle. Because the refraction angle is not linearly dependent on the zenith angle, two stars separated by some fixed difference in altitude will appear to *converge* as they mutually set, or to *diverge* as they mutually rise. This has consequence in the accuracy of using guide stars to guide an exposre that are not at the center of the image being captured. In fact, for wide field exposures at relatively low altitudes no amount of guiding can correct for the differential rate of refraction over the course of an exposure *within* the image.

Using the calculated refraction as a function of zenith angle, I have produced some graphs that show how much a putative guide star and an object diverge in the course of an exposure of a given length depending on inititial zenith angle and separation. The change in zenith angle is taken to be change in RA due to the exposure multiplied by sin(lattitude) which is taken as 40 degrees. Note that this is the maximun change in zenith angle that could be observed located along the celestial equator. Note also that the change in zenith angle due to refraction and the calculated differential refraction will in general result in a shift in both declination and RA. The number shown for differential refraction are just the change in zenith displacement between the guide star and object.