I disagree, slightly. I don't think that mathematics has exhausted the possibilities of different types of approximations using readily computed expressions. Perhaps some obscure numerical relationship will point the way to new methods of approximation?? At 09:36 AM 4/29/2012, Veit Elser wrote:
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