As I recalled it on the night Neil Armstrong walked on the moon, the moon was a waxing crescent, about half way between new and first quarter. I remember going outside and looking at it and thinking that people are there for the first time. I was a senior in college at what is today California State University, East Bay. The following year I started grad school in geology at Stanford.
Today I checked my recollection with a planetarium program, Celestia, and confirmed it. The Moon was in Virgo, below the Ecliptic; so the descending node was precessed to somewhere around Cancer. I don't know when there was a solar eclipse in 1969, but it might have been in June or July of that year.
I speculate that NASA chose the phase at Mare Tranquillitatis, with heating of the space suits or the LEM in mind. The sun had just risen at that phase angel, and so the lunar surface has not warmed too much.
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