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Io and Callisto are satellites of Jupiter. Their periods are $1.77 d$ and $16.7 d$. What is the ratio of their distances from Jupiter?

I assume that I have to use Kepler's third law of planetary motion to find the ratio. I plugged the periods into the equation : $\dfrac{T_1^2}{r_1^3}=\dfrac{T_2^2}{r_2^3}$ and got the ratio of $89.02$. I am not sure of the units but is this the correct answer?

Kot
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  • So your ratio is $r_1:r_2=1:89.02$, is that right? After you squared each period duration and took the cube root of the resulting number? I get ${r_2\over r_1}\in [4,5]$. – abiessu Sep 16 '13 at 00:43
  • I forgot to take the cube root. So would the ratio be 4.47? – Kot Sep 16 '13 at 00:46
  • Correct. Technically, the ratio of the distances could be from the surface, but I expect the question is not referring to that. – abiessu Sep 16 '13 at 00:48
  • Would you happen to know the units? If any exist? Would it be 4.47 astronomical units? – Kot Sep 16 '13 at 00:49
  • In a ratio of equal units, all units are canceled. – abiessu Sep 16 '13 at 00:49
  • Oh that makes sense. Thanks for explaining this to me :) – Kot Sep 16 '13 at 00:50

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I think that you have to take the cube root of your answer. This is a frequent error, but one that can be easily corrected. However, you do not have to worry about the units, because a ratio of equal units will cancel, and will be dimensionless

cuabanana
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