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:

$$ T(3) = 0.444 \times 2.03^3 \\ \space \\ T(3) = 0.444 \times 8.365 \\ \space \\ T(3) = 3.714 $$

... 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:

$$ T(n) = T_0C^n \\ \space \\ 7.155 = 0.444 \times 2.03^n \\ \space \\ \frac{7.155}{0.444} = 2.03^n \\ \space \\ 16.115 = 2.03^n \\ \space \\ \log_{10}{16.115} = n \times \log_{10}{2.03} \\ \space \\ n = \frac{\log_{10}{16.115}}{\log_{10}{2.03}} \\ \space \\ n = \frac{1.028}{0.307} \\ \space \\ n = 3.930 $$

... 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 7 years ago

MasonChane‭

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