Paul - It still seems to me you can get the antenna theorem without invoking the reciprocity theorem twice. And that first invocation is somewhat indirect. Also, you assume that the antenna boresight directions are aligned, so your result appears to have less generality than it really has.

Let's see if I can defend these claims:

From a receiving standpoint, the concept of effective area is straight forward - the equivalent broadside barn door area whose area times times an incident plane wave flux equals the power available at the antenna terminals. An antenna used for receiving has its greatest effective area in the boresight direction and smaller effective areas in other directions. But anyhow, A_eff is

A_eff(theta, phi), and the normalized effective area, A_eff(theta,phi)/A_eff(0,0) is, by definition, the receiving pattern. Of course A_eff accounts for losses; if the antenna is lossy, A_eff is less.

A less straight forward way to characterize a receiving antenna is by its antenna solid angle, Sigma_a , which is related to its beam size. Beam size is some interest even if we are receiving a signal from a distant point source: we want to know how critical the pointing is and also how well the antenna supresses signals from directions other than the pointing direction (the antenna's performance as a spatial filter). Since the antennas directional characteristics are entirely specified by A_eff(theta,phi), beam solid angle must be a function of A_eff(theta,phi). The definition chosen for beam solid angle is

how far downwe may want to know how, but radio astronomers often look at extended sources, and need to know how well.

Likewise, there is no problem with the