OK, this may be arcane, but expression J.1 on page 343 of Astronomical Photometry (Henden and Kaitchuck 1990) has always troubled me. I was re-reading appendix J this morning and decided it was time to ask.
The cosθ factor in this expression has always bothered me. Assume the small patch of surface, ∆A, is small enough that it can be considered flat and every point on that surface is an omni directional radiator. Then as you move away from the normal to it, The apparent area of the patch decreases but the amount of light being emitted by the surface into every angle is constant as long as the solid angle under consideration remains above the plane of the surface (since light below the plane is absorbed by the star). So it seems to me that the apparent surface decreases but the intensity per apparent unit area increases so that the total power being radiated into the solid angle would remain constant rather than decreasing by the factor cosθ.
I realize that in reality this isn’t true because the light is emitted within a radial region within the star, and therefore, as you move away from the normal the average optical depth increases and therefore each radiating “point” it isn’t really an omni directional radiator. Is that really the effect for which cosθ corrects and decreasing apparent surface area is just a convenient way to visualize it?
Brad Walter, WBY