Oleg Zabluda's blog
Friday, July 27, 2012

Moon during the day is something that still surprises some people.

Moon during the day is something that still surprises some people. But it's a regular occurrence, which is obvious if you think about it for a second.

On the photo below, shot yesterday before sunset, the Moon is in the first quarter, i.e. you can see exactly half of it illuminated. If the Sun was infinitely far away, Sun-Earth-Moon angle (elongation) would be exactly 90 deg, but in reality it's less then 90 deg, and measuring this angle is the same as measuring the Earth-Sun distance in units of Earth-Moon distance.

Aristarchus did it in ~300BC. He measured the angle to be 87°, thus the distance is tan(87°)=19. He knew Moon diameter was 1/3 the Earth diameter, and that Sun and Moon have equal angular sizes, from which it immediately follows that Sun's diameter was 19/3~=6 times greater the Earth's, and volume was ~300 times greater. This was the first time people had serious reason to believe that the Earth, being much smaller, moved around the Sun, which is much larger.

The exact time when the Moon is exactly half-illuminated, is notoriously hard to tell with a naked eye. Aristarchus knew it, and his numbers were lower limits. The correct elongation is 89.86°, tan (89.86°)=400, i.e. Sun is 400x farther then the Moon, Sun's diameter is 400/3.7=109x larger then Earth's, volume is 1.3 million times larger, but mass is only 0.3M times larger.

There are two reasons people didn't believe in heliocentric system at the time. One was that they did not see yearly stellar parallax. They knew that this means the stars could simply be very far away, but they could calculate just how far away (~2000 AU), but saw no point in such a universe. Nowadays we would call it Hierarchy Problem. First person, who was fine with it was Copernicus. I am quite surprised he didn't propose that stars are just like Sun, only much farther, since he was the first who knew the distance to them well enough. Numerology would rock!

Second reason was the thinking that if Earth moves around the Sun, how come stuff does not come flying. Again, the first person who was fine with it was Copernicus, with explanation like "Earth's is a natural motion, and everything on it moves naturally".

Note also that in the picture, the Moon is not illuminated along the straight line  from the Sun, but ~45° to the upper-right. That's because, as seen from Earth, distant straight lines are great circles, and projection of Sun-Moon one coincides with ecliptic.

Как хорошо известно, Луна выходит днём чтобы вести лунных зайчиков в гости к солнечным зайчикам: