# 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?

• I rewrote your formulas using Mathjax instead of images of text. If you want to learn more about Mathjax, there is a very in-depth tutorial available at Mathematics Meta, namely the MathJax basic tutorial and quick reference. (I'm not sure I'd call it so basic, but that's Mathematics SE for you...) – user May 28 '17 at 19:07
• Thanks! (Sorry, I just now noticed this comment). I'll explore MathJax! I just did them in Word™ and screen-captured. – MasonChane May 29 '17 at 18:12
• MathJax can be a great aid for typesetting formulae, once you get used to its syntax (which can be somewhat awkward at times). It's basically the same thing as LaTeX, which is often used for typesetting scientific papers, but doesn't have all of the more advanced features. Just don't overuse it; it's terrific when needed, but remember that it comes at a cost in terms of rendering time. It's particularly troublesome in titles, where I recommend avoiding it unless it's absolutely required. If you can't figure it out, screenshots from whatever tool works for you are a decent alternative. – user May 29 '17 at 19:36

The periods of revolution of the Gallilean satellites of Jupiter correspond to n = 2, 3, 4, 5 in Dermott's formula:

$$\begin{array}{cc|c|c|} & & \text{(none)} & \text{Io} & \text{Europa} & \text{Ganymede} & \text{Callisto} & \text{(none)} \\ \hline \text{T} & \text{C} & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline 0.444 & 2.03 & 0.90 & 1.83 & 3.71 & 7.54 & 15.31 & 31.07 \\ \hline & & & 1.77 & 3.55 & 7.15 & 16.69 & \\ \hline \end{array}$$

This is an empirical formula, and there is no requirement to have a satellite for every n. In the same vein, the Titius-Bode law, an empirical formula which predicts the orbits of planets, has a gap at m = 3 (where the asteroid belt is).

• Ah ha! That's where I was going wrong--I was using the wrong values in the exponents, expecting that Io would be either 0 or 1, and also expecting an exact answer. Thanks!!!! – MasonChane May 28 '17 at 18:59
• I prettified your table. You're welcome. :-) – user May 28 '17 at 19:12
• @MichaelKjörling: Thank you. I was disappointed to see that Markdown tables don't work, but I had not thought to use LaTeX tables! This is most ingenious. – AlexP May 28 '17 at 19:18
• @AlexP It looks ugly once the table grows too wide, though. I think what's in this answer is about as wide as it can be and still look reasonable when rendered. Trust me, you do not want a $\LaTeX$ table extending into the margin. – user May 28 '17 at 19:20