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September 12th, 2019, 11:55 AM   #1
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Planet alignment problem

Me and a group of students used lowest common multiple to work out how often all the planets would be lined up as about 84 million years.

The problem with this approach is we're only taking whole orbits into account; when all the planets are in the same position as t=0.

Any ideas for improving this approach to find alignments in between, say after 38002.7 earth years etc.

I guess we need a "to within" figure, so let's say to within a degree?

Your thoughts Maths world?
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September 12th, 2019, 12:44 PM   #2
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The simplest thing you could do is take today's snapshot of the elliptical orbits of the planets run those forward in time until they align.

There's not going to be any closed form solution for alignment.

That's a bit simplistic as the planets themselves do affect the orbits of other planets thus changing the orbits slightly as time progresses. That might not have much affect over 100 or 1000 yrs or so but over 85 million years it just might.

To be more accurate you'd just have to run a simulation of the 10 body system starting with a known set of orbital parameters at a known time. It's a formidable simulation but it's doable, at least for reasonably "small" amounts of time.
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