I've stumbled around trying to figure this out, but it just isn't my area and I haven't gotten far. This is for writing about an idea of a field of mirrors on the moon, on the terminator when the moon is at first quarter, that are aimed so the light is reflected towards the Earth. I am trying to determine how big that field of mirrors would need to be in order for it to be easily visible to the naked eye to a person on Earth within the area where it would be visible.
On the moon the sun has an angular dimension of 0.5o, and the Earth 2o . Taken from Wikipedia's entry on Sunlight, it has an intensity at mid-day of 1050 W/m2, 93 lumens per watt, and an illuminance of 98000 lux.
Based on the angular dimensions of the two bodies, the reflection of the sun on the surface of the Earth would be a circle with a radius one quarter that of the Earth, right? That means a circle with an area very close to 8 000 000 km2. After this point I get confused.
Each square meter of the mirror is reflecting the whole area of the sun over that 8 000 000 km2. So each square meter is distributing 98 000 lux over that area?
From here I found a reference that the formula to convert lux to apparent magnitude is $m=-2.5 log I - 14.2$ , where $I$ is lux, which is lumens per m2. So if I want to determine the size of mirror field so the magnitude is 0, then I need to set it up so $-2.5 log I$ is 14.2, making $I$ about 0.75.
If I am doing this right, I get a result that a mirror field with an area of 61 km2 would be necessary. Is that right?