Experiment 2

On 18th November last year I reported on a lunar experiment and pointed up the need for further research into the direction in which the horns of the moon were pointed. A topic being given further interest in that it had been a challenge issued by the naval uncle, perhaps 15 years ago, and so something which I have thought about off and on since. But we now had lift-off and a second experiment was conducted during January and the first part of February this year, so spanning an entire lunar cycle.

This second experiment took the form of observing the moon, its shape and orientation at various times of day and from various places in and round central and south west London. Shortly into the experiment I decided that it was not possible to solve the problem by cerebration alone, and that while I could cerebrate a diagram, I would get far better results if I were to commit the diagram to paper. A trick which certainly worked but which perhaps only worked because the problem was reducible to two dimensions. Maybe professional mathematicians with geometrical problems in three dimensions need to resort to the 3D representations of the sort that can be produced by engineering drawing packages and in four dimensions are stuck with cerebration, which perhaps highlights their need for a big cerebrum. I made do with Powerpoint, which, I might say, in its 2013 incarnation seems to have taken a big step forward.

Continuing the line taken in November, we assume that for present purposes we can neglect various complications. We assume that the earth and the sun are stationary. That the distances between the three bodies are large relative to their sizes. That there is a single plane with contains the three bodies and the orbit of the moon around the earth (with atmospheric distortions meaning that you would not think that this was the case by looking). That we are observing the moon from a point on the equator.

We have thus taken a position intermediate between that of the ancients and moderns. The moon and the sun are vaguely comparable heavenly objects, both a long way away. The sun and the earth might both be stationary, but we do allow that the earth rotates on its axis (perpendicular to the plane of the diagram), anti-clockwise on the diagram, and that the moon rotates around the earth, orbits the earth if you will, clockwise on the diagram.

Other points on the diagram are that the lines ab and AE are parallel and both perpendicular to the line CG. The line DH is perpendicular to the line BF. All lines pass through the centre of bodies which they touch.

The first eureka was the realisation that the moon was not something that happened at night. The sun and the moon both moved around in the sky because the earth was rotating at the rate of one rotation per day. The sun rises and then sets some 12 hours later to rise again after a further 12 hours; the moon does the same sort of thing. When there is no moon, the lunar day is the same as the solar day. When there is a full moon they are 12 hours out of phase. At other times they are just out of phase. And, if one is standing on the moon side of the earth, to the right of the line a-b, one can see the moon and if one is standing on the dark side, one can't.

But exactly what you see does depend on where the moon is in relation to the sun. The sun illuminates the half of the moon marked by the points D-F-H, whereas what you see is the half marked by the points A-C-E, of which rather less than half, the segment marked by the points D-E is illuminated.

So the moon as a whole is always seen as a circular disc while the illuminated portion is something more complicated. In the state of the moon illustrated, the right hand side of the illuminated portion is right hand side of the great circle through A and E, that is to say the circle with A and E diametrically opposite which is perpendicular to the plane of the diagram. While the left hand side is the left hand side of the great circle through D and H, viewed from the side and thus as an ellipse. So the horns, rather blunt in this case, point from right to left, away from the sun.

The moon then rotates clockwise around the earth, moving down in the diagram, and after it has crossed the line of no moon, the horns flip round, and point from left to right. What has not changed is that they are still pointing away from the sun. So the answer to the avuncular challenge is that the horns of a waning moon point left and the horns of a waxing moon point right.

And so, it is now time to consult the mullahs regarding the moons which appear on the flags of many Islamic nations.

PS: during this post I have been impressed by how difficult it is to communicate something which now seems clear to me and which, in the scheme of things, is relatively simple. To the point that I am not very satisfied with what I have done here and will have another go in due course. It would take me a long time to write a textbook.