I learned that Apollo 11 landed at half moon and that Apollo 8 circled the moon at new moon.
How important were the moon phases for the Apollo flights?
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The date of each landing was selected to achieve a favorable sun angle, with the sun low in the sky to the East of the landing site as described in this answer.
That means the phase of the moon as seen from Earth was dependent on the longitude of the landing site, but generally within 2 or 3 days of a waxing half moon.
Apollo 8 did not orbit during new moon, incidentally, but during the waxing crescent, two days before half moon. This was about the same phase as Apollo 11. Apollo 8 could thus observe the candidate landing sites in the Sea of Tranquility with similar lighting conditions as would be in effect during the landing attempt.
Good question. The moon phases were indeed important for the Apollo missions. The sun had to be in the right angle so the crew can see the lunar landscape. The sun had to be in an angle of 5° to 14°. The sun elevation angle changes about 0.5° per hour meaning the sun was at the right angle for 16 hours every 29.5 days. If the angle of the sunlight was greater than 16°, then it would create a "washout" effect where the light reflecting off the surface overflows the craters and any topographical features.
Here's a passage that answers your question from this source.
The time of a lunar landing was determined by the location of the lunar landing site and by the acceptable range of sun elevation angles. The range of these angles was from 5° to 14° and in a direction from east to west. Under these conditions, visible shadows of craters would aid the crew in recognizing topographical features. When the sun angle approached the descent angle, the mean value of which was 16°, visual resolution would be degraded by a “washout” phenomenon where backward reflectance was high enough to eliminate contrast. Sun angles above the flight path were not as desirable because shadows would not be readily visible unless the sun was significantly outside the descent plane. In addition, higher sun angles (greater than 18°) could be eliminated from consideration by planning the landing one day earlier where the lighting is at least 5°. Because lunar sunlight incidence changed about 0.5° per hour, the sun elevation angle restriction established a 16-hour period, which occurred every 29.5 days, when landing at a given site could be attempted. The number of Earth-launch opportunities for a given lunar month was equal to the number of candidate landing sites.