## A line drawn perpendicular to a line through the tips of the horns of the crescent moon doesn't point to the Sun! Why not? (Advanced)

Hi. Could you help with the following problem when the Sun and the Moon appear to both be visible at the same time:

If you look at the Moon at such times, imagine a line connecting its cresent points, then bisect that line and run a perpendicular line from it towards the Sun. Now, since the Sun is the source of the light reflected by the Moon, you would expect that perpendicular to point directly at the Sun. However, it does not. In fact, it misses the Sun by some considerable arc. We have looked on the 'Net for explainations for this observation and we are very skepictal of the few we have found. Some even involve invoking "great circles" as if the light were travelling in an arc from Sun to Moon!

Any help greatly appreciated!

I believe the the solution to the problem is this: the Earth is far enough from the Sun that, to good approximation, the Sun's rays are parallel when they reach us.

In our typical earthly experience, rays from most sources of illumination are divergent. Thus, if you and your friend are standing near a street lamp, if you each draw a line through your shadows, both lines will point back toward the streetlamp, and the lines will diverge noticably as they move away from the lamp. Your shadows show the direction of the rays of light from the lamp, and the light rays are clearly divergent.

Now, if you and your friend stand further away from the lamp, and again draw lines through your shadows, you'll probably note that they are still divergent, but not to such a strong degree. In fact, the further you get from the lamp, the closer to parallel your shadows will be, indicating that the rays from the lamp are become more parallel the further away from the light source you are.

Another way to think about this is that if you and your friend each point at the lamp, your arms will be more divergent the closer you are to the lamp, and more parallel the further you are away. If you and your friend stand some distance apart, far away from the lamp, and point to it, it may seem to you that your friend isn't really pointing at the lamp, which is directly analagous to this problem! You may have noticed this problem if you ever tried to point out a star or another object in the sky to a friend--in order for him to see what you're pointing at, he almost has to stand right behind you and look over your shoulder.

The Earth and Moon are a good distance away from the Sun. Thus, if you drew lines through the shadows they created, they'd be almost exactly parallel. Thus, the line perpendicular to the Moon's terminator shouldn't point toward the Sun. It should point parallel to a line drawn from you (or, equivalently, the Earth) to the Sun.

Next time you see the Moon, check this out. Keep in mind that there is also some confusion created by our human brains trying to percieve straight lines imbedded in the seemingly dome-shaped sky (the great circle arguments you reference) so it's easiest to keep things straight when the Moon is a crescent, and thus relatively close to the Sun in the sky.