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Galilean Moon resonances (Read 11542 times)
Tony
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Galilean Moon resonances
08/30/07 at 00:05:48
 
Because of their nearly-round orbits, it's difficult to make an animated GIF showing the resonances in the Galilean moons similar to the animations showing the Neptune/Pluto or Toutatis/Jupiter resonances.  But its easy to do using a graph of periods, similar to the one on the Janus/Epimethus page in the Simulations link.  In this illustration, the graphs from top to bottom represent Io, Europa, Ganymede, and Callisto.  The x-axis spans 11+ years, ~1 Jovian year.  The y-axis is period, but its pointless to put values since this is a mosaic of graphs.  It's simply meant to highlight how the periodicity in the inner 3 moons line up with each other, demonstrating that lock each other into their 1:2:3 resonance.  Callisto is not in resonance with the other 3, and the graph shows this.  Its changing period seems to have no periodic relationship to the inner 3 Galilean moons.

I wonder if the Sun itself might be part of this resonance, since at first glance it appears that there are exactly 2 cycles per Jovian year.  I'll run a longer simulation to confirm or shoot down this theory.
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Re: Galilean Moon resonances
Reply #1 - 08/30/07 at 01:03:02
 
I'm not convinced about Callisto... it looked to me that it was very close to a 7:3 mean motion resonance with Ganymede when I was looking at the numbers for my thesis... can gravsim verify or deny that at all?
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Tony
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Re: Galilean Moon resonances
Reply #2 - 08/30/07 at 01:36:05
 
You're right, it is very close to 7:3  Shocked.  I never realized that.  I'll look into a more convincing way to demonstrate whether it is or isn't.  But the graphs don't seem to show any corallation between its dynamic period and those of the inner 3.  I believe I've heard (but I don't have any references now) that Callisto is not part of the resonance, but that its orbit is expanding and ultimately it will join the resonance in 1:2 with Ganymede, setting up a 1:2:4:8 for the Galilean moons.  
 
"Very close" is completely different from "exactly" in a time-averaged sense, as "exactly" (or mean motion resonance) implies that some mechanism is restoring the resonance when the moons try to chaotically drift out of resonance, while "very close" allows one of the moons to pick up an extra orbit over long periods of time (i.e. 701:300).
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Re: Galilean Moon resonances
Reply #3 - 08/30/07 at 07:26:47
 
Hrm.  
 
Maybe you should take a look at my PhD thesis (especially Chapter 4, from page 82 onwards). I never did publish anything about this beyond my thesis, but maybe I should try to get something out there if this really is new...  you can download it by rightclicking this link.
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Re: Galilean Moon resonances
Reply #4 - 08/30/07 at 20:09:30
 
I get an interesting result when I graph the periods of two moons divided by each other (i.e. Period of Europa / Period of Ganymede.  The ratio does not fluxuate around 0.5 like I would expect.  Instead, it fluxuates around 0.4966.  This is weird.  Any idea why?  A resonance should fluxuate around an exact interger ratio.  At 0.4966, sooner or later, one of the moons is going to pick up an extra orbit.  I'm going to run another few sims to try to demystify these moons.
 
Re: Callisto, Googling for Galilean Resonances shows many references that talk only about resonances between the first 3 moons.  But your 7:3 observation should not be ignored.
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Re: Galilean Moon resonances
Reply #5 - 08/31/07 at 08:26:01
 
One question : the periods of the moons were they calculated with respect to the sun or to Jupiter ?  
I think the only difference is the zero or isnt it ?  
Concerning your question : how long was the sim run ? The longer it was run the more an initial value is averaged and can go to the exact ratio it should have .
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« Last Edit: 09/01/07 at 00:35:30 by frankuitaalst »  
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Re: Galilean Moon resonances
Reply #6 - 09/01/07 at 00:46:40
 
Here's a gsim file I made of Jupiter and the four galileans using the numbers on page 236 of my thesis.  
 
I tried this with a timestep of 2 and then went to View -> Rotating Frame with a focus on Ganymede and got a very odd loopy pattern that didn't quite seem to repeat itself (the next loops were always a bit out of step with the previous ones).... I had Io at mean anomaly 0 and Europa and Ganymede at 180, and Callisto at a random one.  
 
I thought if they were in a 4:2:1 resonance then I'd see a four-lobed orbit for Io and a two-lobed orbit for Europa if I centred a rotating frame on Ganymede?
 
Anyway, try the attachment and see what it looks like... Smiley
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Re: Galilean Moon resonances
Reply #7 - 09/01/07 at 02:23:21
 
The problem with visualizing the resonance, and seeing lobes is that the orbits are very close to circular.  Was Ganymede the focus object? (middle of the screen), and was the Rotating period set to Ganymede's orbital period?
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Re: Galilean Moon resonances
Reply #8 - 09/01/07 at 06:17:10
 
I get more or less the same anomaly ( should we call it the IEGC anomaly  Wink ? ).  
Graph was generated with 6 bodies ( sun , Jupiter  and the four Gallileans ) .  
Data were taken from the JPL database , integrated for exactly 2 Jovian revolutions .  
The periods were calculated referenced to the Ganymede period and further related to the "exact" values of 1:2 , 1:4 and 7:3 .  
It seems that neither of the values is correct .  
Io / Ganymede deviates by almost 1% of the value 1:4 .  
The best fit is the fit Callisto / Ganymede !
Do you get the same results?  
 
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frankuitaalst
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Re: Galilean Moon resonances
Reply #9 - 09/01/07 at 06:50:09
 
Even referring to the orbit of Callisto doesn't give exact results ... Sad
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Re: Galilean Moon resonances
Reply #10 - 09/01/07 at 06:55:54
 
I'm wondering with proximity to jupiter and jupiter's mass if relativity may be having an influence...Just a thought
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Re: Galilean Moon resonances
Reply #11 - 09/01/07 at 13:23:38
 
The proximity of Jupiter is surely a point , but the velocities of the moons are in the order of 30 km/s, far beyond the speed of light , so this effect may be neglected . I wonder if the sattelites are really in exact resonance  . Or do tidal forces have to be taken into account ?
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Re: Galilean Moon resonances
Reply #12 - 09/01/07 at 14:26:35
 
Quote from frankuitaalst on 09/01/07 at 13:23:38:
...but the velocities of the moons are in the order of 30 km/s, far beyond the speed of light...

You sure about that?  Wink
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Re: Galilean Moon resonances
Reply #13 - 09/01/07 at 14:46:04
 
The velocity of Io in m/s is according JPL , at 01/02/07:  
2.410342345960552E+04, -1.705496983601301E+04, -4.895335786823480E+02
so about 20-30 km/s .
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Re: Galilean Moon resonances
Reply #14 - 09/01/07 at 15:09:58
 
and the speed of light is?... Shocked
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