Was the command/service module visible from the lunar surface?

Question

While on the moon, either outside or inside the lunar module, could the astronauts see the command/service module in orbit?

Answer 1

I'll calculate the visibility of a diffuse 50% gray sphere with a 6 meter diameter. Averaging over all Sun-object-observer configurations that might be similar to a 4 x 11 meter shiny cylinder since both will scatter light over more than a hemisphere.

Equations are from this answer and from sources linked therein:

$$M_{Abs} = 5 \left(\log_{10}(1329) -\frac{1}{2}\log_{10}(\text{albedo}) -\log_{10}(D_{km})\right).$$

$$ m = M_{Abs} + 5 \log_{10}\left(\frac{d_{SR} \ d_{RE}}{1 \ \text{AU}^2 O(1)}\right), $$

First lets calculate the absolute magnitude of our "asteroid". With an average albedo of 0.5 and a diameter of 6 meters the first equation gives an absolute magnitude of +27.5 which means that if it was 1 AU from the Sun and 1 AU from you, that would be its apparent magnitude.

Then let's calculate the apparent magnitude with it about 150 km away, which is one 1E-06 AU. The second equation gives an apparent magnitude of -2.5 under these assumptions. That's Venus bright!

There will of course be phase angle corrections (the $O(1)$ (order unity) terms) and variations with orientation of the craft, but even if it were 10 times dimmer it would still be magnitude 0!

Was the command/service module visible from the lunar surface?

Yes, it should have been possible to see without much trouble given the following caveats: if it passed within say 400 or 500 km away, as long as it was substantially above the horizon, there was nothing blocking it, and the astronauts knew were to look and weren't using their gold reflective visors to protect their eyes, which they probably were.