Dermott's Law and Major Moons

I'm trying to work with Dermott's Law to develop a generalized "formula" for assigning major moons to gas- and ice-giant planets, but it doesn't seem to work.

If I use the values specified for Jupiter: $T_0=0.444$ and $C=2.03$, and I assume that Ganymede would be considered the third major moon of Jupiter, the orbital period I get out of the equation is:

$$\begin{gathered} T(3)=0.444\times {2.03}^3 \\ T(3)=0.444\times 8.365 \\ T(3)=3.714 \end{gathered}$$

... which is just over half the correct value of 7.155 days.

If I use the known orbital period for Ganymede (7.155) days, and determine the value for $n$, I get 3.930:

$$\begin{gathered} T(n)=T_0{C}^n \\ 7.155=0.444\times {2.03}^n \\ \frac{7.155}{0.444}={2.03}^n \\ 16.115={2.03}^n \\ {\log }_{10}16.115=n\times {\log }_{10}2.03 \\ n=\frac{{\log }_{10}16.115}{{\log }_{10}2.03} \\ n=\frac{1.028}{0.307} \\ n=3.930 \end{gathered}$$

... which is not even an integer, let alone the 3.0 I was expecting.

I find a similar problem with Io, Europa, and Callisto, which come out as:

$n_{\text{Io}}=1.956$

$n_{\text{Europa}}=2.941$

$n_{\text{Callisto}}=5.128$

... where I would expect the values to be 1, 2, and 4 (or maybe 0, 1, and 3).

Has anybody else worked with this? Can you tell me what I'm not understanding about it?

posted almost 8 years ago

MasonChane‭

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