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With the naked eye, would it be easier to see the moon when it's close to the sun, because it might reflect some more light (or not, I'm not sure) or farther from the sun because the sun shines brighter, same with how we can't see the stars?

Also, would it be easier to see it a few minutes after the sun sets and the moon is on the verge of setting, maybe at the +5° on the azimuthal grid?

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The further the moon is from the sun the easier it is to see. There are two reasons. Firstly, when it is close to the sun it gets lost in the glare from the sun. The sky close to the sun is very bright, and there isn't enough contrast to see the moon against the bright sky. Secondly, as the moon gets further from the sun, more of the illuminated side of the moon is visible and so the moon becomes brighter. Brighter moon → easier to see.

When I say "close to the sun" remember this only means "appears close in the sky" the actual distance to the sun only changes a little over the year.

The moon is easier to see against a darker sky, so if the sun has set, it is more likely to be visible. There was a new moon at 2:30 GMT on 12th April 2021 and the moon will become visible in the next few days (which is astrologically significant to Muslims)

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