Find the location knowing the resonance using Kepler's third law

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The discussion focuses on determining the locations of celestial bodies in Mean Motion Resonance (MMR) with Jupiter, specifically the 4:1 and 3:1 resonances, using Kepler's Third Law. The user, Tom, seeks clarification on how to apply the law, particularly whether to use 1/4 of Jupiter's orbital period (Pj) for the 4:1 resonance to find the semi-major axis (a). The conversation emphasizes the importance of understanding the relationship between time and distance in orbital mechanics, particularly how time squared relates to length cubed in the context of Kepler's law.

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Kovac
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TL;DR Summary: .

I need to find the location of following bodies MMR with Jupiter: 4:1, 3:1, with the help of Keplers third law.Keplers third law:
1697974988128.png
, where P is the orbital period in Earth years, a= semi major axis in AU.
For Jupiter: Pj =
1697975054421.png
years.

Now my question is, to find the location of 4:1, should I simply take 1/4 * Pj as the new P? (Since 4 orbits are made with each Jupiter orbit)
And then use the formula again with
1697974988128.png
to find the position for a? Meaning I need to solve for a with the new P?
 
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I'd think a bit about units/dimensions first! How can a time squared equal a length cubed?
 
Could you explain in words what this phrase means?
Kovac said:
MMR with Jupiter: 4:1, 3:1,
Such as: to what variables do the 4:1 and 3:1 apply, and in which direction?

Cheers,
Tom
 

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