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Just as with Venus or Jupiter, we suppose a sphere illuminated at some intensity, in this case the standard intensity above Earth’s atmosphere, radiating light back and spreading it over a sphere of radius equal to our distance. So that light gets attenuated by a dimensionless factor, the square of the distance ratio (sphere radius / distance radius), which is just the half the visual angle (in radians) subtended in the sky: 1/2 1/2 degree = 0.0044 radians (at 360/2 degrees per radian). So we expect (1/0.0044)²= 50,000—fold attenuation, which is 12 magnitude factors of 100^1/5, and then further attenuated 15-fold = 3 magnitude factors more for losses in the dark rocks. With this assumption about the rocks, the Moon "should" look magnitude -27 +12 + 3 = -12. This is not far from observation of the full Moon under clear skies at night (-12.5). Notice, however, that with nothing to compare them to, Moon rocks don’t look dark from our distance. One could easily imagine the Moon to be made of chalk. "Optical illusion"? Here is a nice experiment to check, attributed to William Herschel. He noticed the full moon rising over Table Mountain near Capetown. Sunlight on the South African rocks contrasted brightly with the adjacent sunlight on Moon rocks. In both cases the sunlight had traversed about the same slant distance through Earth's atmosphere. Conclusion: Moon rocks are lots darker. You can try the same (and so can I) on 30 November using the wall of your (my) house in place of Table Mountain.

Notice also that we dealt only with the full moon here. Do you suppose a quarter-moon (half the visible disk illumined) sheds half as much light? It doesn’t. The intensity of moonlight falls off with surprising abruptness as the Moon’s phase advances At 90 degrees (quarter moon) the night is less than a tenth as bright as at full moon (0 degrees). There is a good "Discovery" exercise in this, too, if you like geometry and dare attempt a tricky integration over some trigonometric expressions. The result might help to refine our estimates last week regarding Venus.

Plotted from Allen, C.W., Astrophysical Quantitities, 3rd edition (1973), p. 143.