Moon Terminator Spoiler
Moon Terminator SpoilerBelow is a complete solution to the Moon terminator question given on
and referenced on InterestingQuestionsForInquiringMinds.
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The explanation below was given by AndyMorris and later expanded.
Imagine standing on the North Pole at the moment of the equinox, when the sun crosses the Earth’s Equator ( http://www.astrosociety.org/education/publications/tnl/14/14.html). Gravitational 'up' points in the same direction as ecliptic north.
Suppose the Moon is at its quarter, with the Earth-Moon-Sun angle being 90 degrees. Then the lunar terminator runs from ecliptic north to ecliptic south, and so the terminator will appear to run vertically.
If you walk (briskly) towards the Earth's equator along the sunrise (or sunset) line, towards the moon, the moon will appear to rise in the sky until it is above your head. During all this time the terminator will continue to appear vertical.
If on the other hand you walk towards the equator at 90 degrees from the sunrise line, the moon will stay above the horizon, but will appear to twist. When you arrive at the equator, the moon will have twisted by 90 degrees and the terminator will be horizontal. If you then walk along the equator towards the moon, it will rise in the sky until it is above your head.
Now do all the above without assuming that it's the equinox. Instead of the Equator, simply use the GreatCircle? defined by the plane through the centres of the Earth, Moon and Sun. Instead of the North Pole, use the appropriate point relative to that plane. This allows us to take into account the inclination of the Earth's axis, the inclination of the Moon's orbit, etc.
So, to summarize,
Anon - do you honestly think your quibbles add to either your reputation or the quality of this page?
Yes. For example, the tropics are defined solely in terms of the sun and the Earth, whereas you need to define a region which relates to the moon. That region therefore needs to be defined properly. -- Anon
The evidence suggests, then, that you are incapable of adding such refinements.
Not at all. I didn't want to complete the answer in relation to the horizontal terminator, just to point out that the current answer wasn't adequate. -- Anon
So you don't want to improve it, you just want to point out that the existing answer is wrong. You claim you could provide a better one, but you choose to withhold your wisdom from us.
The correct answer for this type of question tends to be less interesting than the process which leads to its discovery. I wanted to find out how others would reach a correct answer. -- Anon
[Fine. Since from my point of view you, Anon, are among the "others", I would now like to see the process by which you, Anon, reach a correct answer.]
I haven't yet worked out a full answer. At present, I'm not sufficiently interested in the problem to do so. -- Anon
[You've already demonstrated considerable interest in the problem, but I suspect you are presently unable to solve it. Your behaviour suggests that you are frustrated at being unable to solve it, and nit-pick at Colin in order to relieve your frustration. We have already seen this demonstrated on one of Colin's other problems -- the twin 1kg weights and horizontal spring-balance -- in which you nit-picked at what was an obvious hypothetical scenario before producing an initial incorrect answer. This behaviour reminds me of some of my high-functioning AspergersSyndrome students, who sometimes react to a difficult problem by criticising the problem or even the individual who has set the problem.]
It's basically simple geometry and a few facts about the motion of the moon and the Earth.
[In other words, you haven't solved it yet.]
Neither has Andy, though he's nearly there.
Your answer regarding a horizontal terminator suffers from the problem that you don't identify precisely where on Earth the sun can be overhead at midday (or underfoot at midnight). "In the tropics" is too vague.
You are hereby invited to give a complete, correct and enlightening answer.
Consider the limiting case of the solution for the vertical terminator, where the moon is directly overhead. The observer moves an inch in the appropriate direction so that he can now claim the moon is no longer directly overhead and the terminator is horizontal. The appearance of the terminator has changed too little for the eye to detect.
The limiting case requires that the observer be in the tropics, or close to the tropics, to allow for the inclination of the Moon's orbit.
That's irrelevant; for the vertical terminator, the time is sunrise or sunset, not midday or midnight. Does it take 6 hours to move an inch?
So finally you get to it. Your point is that the horizontal terminator can be seen at any time the Moon is visible and is at the quarter. Places from which it can be seen are those that lie on the intersection between the Earth's surface and the E-M-S plane. Such places are necessarily in, or very close to (allowing for orbital inclinations, etc.), the tropics.
One can't give a satisfactory answer if the question is imprecise as to what knowledge may be assumed, what is to be given, and in how much detail. I still hold that "close to the tropics" is a pretty vague way of specifying the location. How close, and where are the tropics anyway? How is "when the moon is at the quarter" different from what you give in the question? For the type of question you're giving, the correct answers are of less interest than the methods used to obtain them. On certain occasions in the past when I've given an acceptable answer (to another question), you've promptly removed it as a spoiler.
This seems to assume that the moon's orbit of the Earth lies within the plane of the Earth's orbit of the sun. If that were precisely true, there would be frequent lunar eclipses, which isn't the case.
No, the last paragraph makes it clear that you can do all this relative to the plane through the centres of all three bodies.
The sun and moving Earth determine a moving circle on which the moon must lie in order that the Earth-Moon-Sun angle is 90 degrees. An arbitrary lunar orbit could miss that circle entirely.
Your first sentence is wrong and the second is irrelevant.
The Sun-Moon-Earth angle varies continuously and is large during New Moon and small during Full Moon. Common sense tells us that at some point it must go through 90 degrees, but the Intermediate Value Theorem (quoted in every good calculus book and proven in some) says the same thing.
The centres of the three bodies, unless colinear, uniquely determine a plane. Consider the two points on the surface of the Earth defined by the perpendicular to that plane through the centre of the Earth. Define a plane through those points and through the centres of the Earth and Moon. That plane more-or-less defines the terminator when the S-M-E angle is 90 degrees. If you like you can be more precise and allow that the terminator is actually slightly further back from the Sun than that line. That simply gives a marginally different plane. The plane cuts the Earth in a GreatCircle? (because it goes through the Earth's centre), and from any point on it from which the Moon is visible, the terminator will appear as a vertical straight line.
Effectively the Sun-Moon-Earth plane defines an "equator" and the points on the perpendicular line define the "North Pole" and "South Pole".
My first sentence was correct (or nearly so, since the moon's distance from the Earth is not constant), since it referred to an arbitrary lunar orbit. The actual orbit is not arbitrary, and your argument uses additional facts about the actual orbit to show that the angle will sometimes (for an instant) be 90 degrees.
The questions were as shown below.
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