What is the equation for finding the mass of Jupiter based on Callisto's orbit?

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Discussion Overview

The discussion revolves around deriving the equation for calculating the mass of Jupiter based on the orbital characteristics of its moon Callisto. Participants explore the application of Kepler's laws and Newton's law of gravity in this context, addressing the specific equation provided in a physics problem.

Discussion Character

  • Homework-related, Technical explanation, Exploratory

Main Points Raised

  • One participant seeks clarification on the equation 4π²(R)³ / Gt² used to find Jupiter's mass, expressing confusion over its derivation.
  • Another participant suggests that the equation is based on Kepler's third law, implying a connection to Newton's explanations.
  • A different participant provides a derivation using Newton's law of gravity, showing how to relate centripetal force to gravitational force, leading to the mass of Jupiter.
  • Some participants note the importance of the gravitational constant (G) in the equation, indicating its role in determining Jupiter's mass from the gravitational effects on Callisto.
  • One participant expresses a sense of familiarity with deriving the relationship but acknowledges overlooking simpler arguments presented by others.
  • Another participant encourages not to overcomplicate the problem, suggesting a more straightforward approach.

Areas of Agreement / Disagreement

Participants generally agree on the use of Kepler's laws and Newton's law of gravity to derive the mass of Jupiter, but there are variations in the approaches and interpretations of the equation. No consensus is reached on a single method or explanation.

Contextual Notes

Some assumptions regarding the derivation of the equation and the definitions of variables are not explicitly stated, which may affect the clarity of the discussion. The discussion also reflects varying levels of comfort with the mathematical derivations involved.

DocZ1219
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Hello, i am a student taking a physics course online since my school doesn't offer it. I have come across one problem that i nor any science teacher I have access to can help me with. If you would please explain it to me then i would be more than grateful. Thank you

The problem reads...
"One of the moons of Jupiter is Callisto. It has a mean distance of 1.883 x 10^6 kilometers from Jupiter and has a period of 16.7 days. What is the mass of Jupiter?"

Can anyone please explain why the professor set up the equation as 4pie^2(R)^3 \ Gt^2 ?

Thank you
 
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Welcome to PF!
He's using Kepler's 3.law, if I'm not mistaken.
(LONG derivation..)
 
not so hard actually;
consider Newton's law of gravity, F = GmM/r^2. This law serves as our centripetal force:
mv^2/r = GmM/r^2. Simplify this and use v = r*(omega)=r*2*pie/T and you'll get rid of m and get the exact equation for M as you have.
 
As arildno said, that equation is practically just Kepler's third law, as explained by Newton (i.e. - the equation explains why Kepler's third law is true). The equation's rearranged to find the gravitational constant for Jupiter with one small variation.

If you divide the universal gravitational constant (G) out of Jupiter's gravitational constant, you'll get the mass of Jupiter. That's why the G is in the denominator of your equation.
 
niehls said:
not so hard actually;
consider Newton's law of gravity, F = GmM/r^2. This law serves as our centripetal force:
mv^2/r = GmM/r^2. Simplify this and use v = r*(omega)=r*2*pie/T and you'll get rid of m and get the exact equation for M as you have.
BLAARGH!
I'm so used to derive this to gain the relation in terms of the semi-major axis that such elegant arguments as yours are overlooked..:redface:
 
don't make it too hard on yourself :)
 
alright... thank you all very much. I guess i just got a little overwhelmed when he substituted all those equations in. Thanks again! :biggrin:
 

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