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). 2. (physics) What is the physical origin of the long period? -Veit
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
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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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
The reason we know that perpetual motion is impossible is that so many clever people have tried & failed. The reason we know that heavier than air flight is impossible is that so many clever people have tried & failed. Distinguishing these two cases is difficult. Rich ---- Quoting Veit Elser <ve10@cornell.edu>:
Isn't this obsession -- with looking for relationships where there almost certainly aren't any -- a waste of time?
On 2012-04-29, Veit Elser <ve10@cornell.edu> wrote:
Isn't this obsession -- with looking for relationships where there almost certainly aren't any -- a waste of time?
Why don't you ask John Horton Conway and Simon P. Norton? (Hint: Monstrous Moonshine :-) -- Robert Munafo -- mrob.com Follow me at: gplus.to/mrob - fb.com/mrob27 - twitter.com/mrob_27 - mrob27.wordpress.com - youtube.com/user/mrob143 - rilybot.blogspot.com
On 4/29/2012 12:36 PM, 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.
How are the transient approximate relationships between the orbits of random inconsequential chunks of space rock more meaningful than a permanent approximate relation between fundamental constants? But seriously, as far as wasting time goes, here's is a summary of my contributions to mathematics: - I worked on the OEIS. - I did a little work on Eric Weisstein's Treasure Trove of Mathematics. - I found a relationship between the Collatz conjecture and regular expressions. Jeff Shallit wrote a paper for me, "The 3x+1 Problem and Finite Automata", which gave me Erdos number 2. The proof in the paper is elementary to anyone versed in the subject, I think it ended up as a textbook problem. - I asked a question about prime divisors that sent John Conway and some associates to the blackboard for a few minutes. The result was the discovery of "Twin Peaks" (which see on MathWorld). - I found density waves in the Ulam sequence. Don Knuth actually contacted me about this. - John Conway was looking for a name for a polyhedron with holes in all its faces, and I came up with "holyhedron." This got me a mention in Pickover's "The Math Book", but it took true geniuses to actually construct one of these things. - I found the fifth taxicab number, which had in fact been published 3 years earlier. - I may have salvaged pi^4 + pi^5 ~= e^6 from the USENET dustbin. Doubtless it would have been rediscovered. - I made up an arguably clever graph for deciding if a number is divisible by 7, which is on Tanya's site. All in all, I think I've overachieved, but in all honesty, my greatest achievements barely rise above the level of a waste of time. Twenty minutes after I'm gone, I'll be about as noteworthy as any Sumerian bean farmer. So I don't worry much about wasting my time, given my talents, that's pretty much a foregone conclusion.
participants (6)
-
Adam P. Goucher -
David Wilson -
Henry Baker -
rcs@xmission.com -
Robert Munafo -
Veit Elser