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

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To find the location of celestial bodies in a 4:1 or 3:1 mean motion resonance with Jupiter using Kepler's third law, the orbital period (P) of the resonant body should be calculated as a fraction of Jupiter's orbital period. For a 4:1 resonance, P would be set to 1/4 of Jupiter's period, and the semi-major axis (a) can then be determined using the formula a^3 = P^2. The discussion also raises a question about the dimensional analysis of the formula, specifically how time squared relates to length cubed. Understanding the variables involved in these resonances and their directional implications is crucial for accurate calculations. The conversation emphasizes the importance of careful unit consideration when applying Kepler's laws.
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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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