Three Planet Galaxy and Stock Market Chaos

[BMAD]

A galaxy consists of three planets, each of them moving along a straight line with its own constant speed. If the centers of all three planets happen to lie on a straight line (some kind of eclipse) the inhabitants of each planet go nuts (they cannot see their two neighbor planets all at once), start talking about the end of the world, and the stock market crashes. Show that there will be no more than two such market crashes on each of these planets.

[dgreening]

This is not getting a lot of attention, so I am going to jump in.

I think there must be something I am missing.

If the paths are no coplanar, then I don't think there are positions that will support 2 alignments.

Let's simplify the problem, by making the paths of the 3 planets coplanar.

If the 3 paths are parallel, then I think there can be only 1 point in time/ space that they will line up -- after that, they will never be in alignment again.

If we imagine that the paths of the 3 planets intersect and that the planets are travelling on paths as follow [i can't figure out how to post a picture that I created, so I am going to revert to compass points for ease of illustration]:

• "Green" planet is travelling due east [90 degrees] towards the point of intersection
• "Red" planet is travelling due ESE [120 degrees] towards the point of intersection
• "Black" planet is travelling ENE [60 degrees] towards the point of intersection

If they line up before the intersection, there is a possibility that they will line up again after the intersection. After that, they will continue to diverge.

This second alignment requires just the right mix of relative speeds [probably Green is the middle speed while one of the others is faster and one is slower..

Not really a proof, but I think that is the answer. [sorry about having to revert to compass points].

[bonanova]

I agree with dgreening's suggestion by example of only two alignments - one before and one after a point of closest approach. But if you were to visualize those two alignments, I think their two connecting lines could be skewed, not requiring them to be coplanar. Take any two lines, in fact. Color their end points red and green and their midpoints blue. The same-color points define the paths of the respective planets. Again there are only two cases. They are either aligned forever (one long stock market crash) or only at the two lines.

But two questions arise.

[1] Does the middle planet have a stock crash? It eclipses the end planets from each other but itself is not eclipsed. Or does having the end planets appear on opposite horizons satisfy the OP condition "cannot see their two neighbor planets all at once"? I think the latter, finding hard to imagine each of the three planets being an "end" planet twice.

[2] If we call the planets A B and C, imagine the lines AB and BC. In the most general case they will never be collinear. Imagine that they are collinear at some moment in time. With arbitrary planet velocities the lines will never again be collinear. But imagine that they collinear at some later moment as well. I have a nagging thought that with constant velocity, straight-line planet paths, the lines will forever be collinear. Their joining line would trace a ribbon-like shape in space.

If that conjecture were proved in the plane I think it would also be true in 3D.

[bonanova]

My previous post speculated that if aligned planets travel at constant speed on straight paths they will stay aligned. The figure below shows this is not so, and a moment's though confirms they would have had to coincide, not just align.

Three planets colored red blue and green travel left to right from one state of alignment to another. Lines connecting the red and green planets as they travel is shown as well. The blue planet is seen to begin and end on that line but not to travel with it.

The paths may be coplanar but they need not be.

Extension of the figure to the left and right shows the blue planet diverges from the red-green planet line on both sides, never to join it again.

If the colors are reordered, BRG on the left and GRB on the right and we sketch in the connecting lines for the red and green planets at intermediate points we see a situation where the middle planet becomes an end planet. Here it's even more clear that alignment is not maintained.