Isn't this obsession -- with looking for relationships where there almost certainly aren't any -- a waste of time? Only Adam Goucher took up my challenge of discovering a relationship that was actually meaningful. The three orbital periods TI=1.769137786 TE=3.551181041 TG=7.15455296 when turned into frequencies fI=1/TI=0.565247098 fE=1/TE=0.281596457 fG=1/TG=0.139771137 can be "explained" by just two numbers: fI=4f1-f2 fE=2f1-f2 fG=f1-f2 f1=0.14182532 f2=0.00205418 The 4:2:1 pattern is the "Laplace resonance" and the tiny correction, or large period 1/f2=486.8 days, is probably related to the periapsis precession you get from a slightly non-spherical gravitational field (an oblate source such as Jupiter). Adam's continued fraction method is also how I first approached the problem. Veit On Apr 12, 2012, at 6:22 AM, Adam P. Goucher wrote:
A puzzle I gave my mechanics class:
The average orbital periods of the Jovian moons Io, Europa and Ganymede are:
TI=1.769137786 TE=3.551181041 TG=7.15455296
These are taken from Wikipedia; the time unit is days.
1. (math) On the basis of just these numbers, infer the existence of a much longer period (on the order of hundreds of days).
That's a nice puzzle. The first thing I did was to enter TE/TI, TG/TE and TG/TI into a continued fraction calculator, to produce best rational approximations. The first few convergents for TE/TI are:
2/1 275/137 * 3027/1058 3302/1645
And those for TG/TE are:
2/1 137/68 * 3153/1565 3920/1633
Finally, the convergents for TG/TI are:
4/1 89/22 93/23 275/68 *
The asterisked convergents suggest a ratio of:
TI : TE : TG = (1/275) : (1/137) : (1/68)
This means there is a large period of 275 TI = 137 TE = 68 TG. For each of the orbital periods you have provided, this results in the following approximations to the large period:
275 TI = 486.51289115 days 137 TE = 486.511802617 days 68 TG = 486.50960128 days
= approx 486.51 days.
(Technically, a better method than using continued fractions is the LLL lattice reduction algorithm. However, since the data are so precise, the continued fraction method was adequate.)
Sincerely,
Adam P. Goucher
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