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Galilean Moon resonances (Read 11560 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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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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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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Re: Galilean Moon resonances
Reply #15 - 09/01/07 at 16:09:20
 
Smiley I get it
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Re: Galilean Moon resonances
Reply #16 - 09/01/07 at 21:18:27
 
Quote from Tony on 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?

 
Yes, Ganymede was the focus object, but I didn't set any Rotating Period at all.  
 
Now this is weird... I went to View -> Rotating Frame Adjustment, selected Ganymede as the focus and then a number came up in the Rotating Frame period box next to that (presumably derived from the mass of Jupiter and the orbital distance etc) and there was a check box that said "locked" which I ticked. I left "Clockwise" and "Rotating frame" selected too. When I press OK, I now see Io and Europa orbiting anti-clockwise around a stationary Jupiter, and Callisto orbiting clockwise around it, and Ganymede isn't moving.
 
If I select "Stationary" instead, the view suddenly changes to a "loopy" one with all the moons going in the same direction (anti-clockwise) around Ganymede (and Jupiter going round in a circle), but now Io has an interior three-looped orbit, and Europa has an interior orbit with a single loop (almost in the same place each orbit, but slightly misplaced on each cycle), and Callisto has a more complicated loopy external orbit that has four small loops spaced at 90 degrees around Ganymede (again, slightly displaced on each cycle).
 
(EDIT: I added a screenshot after Callisto (green) did one full cycle. Io is red and Europa is purple, Ganymede is the brown dot in the middle).  
 
Bwuh??  Huh
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Re: Galilean Moon resonances
Reply #17 - 09/02/07 at 01:29:55
 
Nice work Mal ! I got the same result .  
Thats why I added the charts of the periods above , indicating some deviations .  
As I was  not convinced about the exact nature of the resonance I googled and found the article ( almost with the speed of light  Wink)
 
http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1980CeMec..21..265 W&data_type=PDF_HIGH&whole_paper=YES&type=PRINTER&filetype=.pdf.  by W.Wiesel. The first sentences are :  
So the system might be at a "close resonance" .  
The article also mentions that the periods frequencies show two resonances with the Sun as Tony suggested .
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Re: Galilean Moon resonances
Reply #18 - 09/02/07 at 06:53:41
 
Wondering how stable a system in resonance is , I created the following system : Sun + 4 earths in a 1:2:3:4 resonance .  
The first Earth is at 1AU .  
Integrating for 8000 years I get the following result :  
Refering each SMA to its initial value it seems that the initial resonance is lost and that the system evolves to a near resonance dynamical state.  
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Re: Galilean Moon resonances
Reply #19 - 09/02/07 at 07:15:25
 
Running the same sim as above , but deleting the 1:3 resonating Erath , thus leaving the system in a 1:2:4 Resonance pattern gives the following evolution :  
Also in this case the ideal resonance vanishes with time
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Re: Galilean Moon resonances
Reply #20 - 09/02/07 at 07:45:42
 
The picture hereunder seems more comfortable : two planets in a 1:2 resonance gain their resonance again after some time but leave this state again aso ...
Seems as resonance causes a in each case a dynamically system .  
 
Just for fun : in the second picture the second planet was put in a polar orbit around the sun again in 1:2 resonance to the first . The aspect of the evolution is different but here also the system regains its resonance somewhile ( till Kozai arrives ?)
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Re: Galilean Moon resonances
Reply #21 - 09/02/07 at 08:39:55
 
What I don't get is what the number of loops in the orbits mean. Looking at the pictures on page 325 of the SSD book, I get the impression that the number of loops is usually the first number in the "X:Y" resonance (so a 3:4 resonance should have three loops). So why does Callisto have four loops if it's supposed to be a 7:3? Shouldn't it have seven?
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Re: Galilean Moon resonances
Reply #22 - 09/02/07 at 09:08:45
 
Yes , it is confusing sometimes . The look of the graph may have loops or lobes  which is function of the relative position of the planets and fi. also eccentricity .  
What counts is the number of revolutions a body makes .  
 
If you take Io as reference rotating frame and focus on Io you will see that :  
1. Europe makes 1 revolution when Jupiter makes 2 , so Io and Europe are in 1:2 resonance
2. Ganymede makes 1 revolution when Jupiter makes 4 , so Io and Ganymede are in 1:4 resonance.  
Counting is necessary .  
 
I also have a difficulty detecting the 7:3 resonance for G:C out of one frame and out of the loops.  
I do  it as following : in the rotating frame of Callisto count the orbits of Jupiter till Ganymede closes his loop , then go to the rotating frame with Ganymede and count the orbits of Jupiter till Callisto closes its loop . You will get the 7:3 .
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Re: Galilean Moon resonances
Reply #23 - 09/02/07 at 10:47:11
 
You image with Callisto's 4 loops does not appear to be in a rotating frame, and it is Ganymede-centered.
 
The images on page 325 are centered on the primary, in a rotating frame with a period that keeps one of the resonant particles stationary.
 
To create an analogy to page 325 with the Jovian system and show any moons that are resonant with Ganymede, you need a Jupiter-centered rotating frame set to Ganymede's period.  However, if you do this, you still won't see anything.  That is because the loops you see on page 325 are caused by the periapsis or apoapsis of the orbit.  Notice that the larger the eccentricity, the larger the loops.  In the case of Jupiter's moons, their eccentricities are very close to 0, which makes the loops disappear.  This is why I can't make an animated gif of the Galilean moon resonances comparable to the ones of Pluto / Neptune and Jupiter / Toutatis.
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Re: Galilean Moon resonances
Reply #24 - 09/02/07 at 12:18:16
 
Heres an animated gif using Mal's sim.  
The first frame is centered and related to Ganymede , the second to Callisto .  
Picture is taken after 60 days of simulation . Jupiter is shown in red .  
Couting the 'revolutions' of Jupiter in both frames gives the 7:3 ratio for Ganymede / Callisto
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Re: Galilean Moon resonances
Reply #25 - 09/02/07 at 12:27:29
 
hm, OK. I'll try it out here!
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Re: Galilean Moon resonances
Reply #26 - 09/03/07 at 09:16:31
 
Finally a simulation of three Earths in 1:2:3 resonance .  
As in the previous cases the system is dynamic and looses its initial exact resonance .  
In all the cases above with 3 or more planets the changes are smallest for the closest planet to the sun.  
This seems to be logical .
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Re: Galilean Moon resonances
Reply #27 - 09/04/07 at 00:30:41
 
Mal,
I prepared this answer to one of your posts a few days ago and I forgot to hit submit:
 
"Locked" computes Ganymede's period with each iteration in an attempt to keep it still, even if its period is dynamic.  I'm probably going to delete this feature as "Calibrate" is more useful.
 
The number that came up when you choose Ganymede from the Rotating Frame interface is Ganymede's period in seconds.  But since period is sometimes a dynamic number due to perturbations, Ganymede is still free to drift.  The Calibrate button should help you zero-in on the correct time.  Press "Calibrate".  The mouse changes to cross-hairs.  Then click Ganymede.  After it drifts a bit, click it again.  This tells it the drift.  The rotating period will adjust for the drift.
 
"Stationary" simply means not rotating frame.  You are viewing a Ganymedecentric universe if Ganymede is your focus object in the "Focus Object" interface.  This is also known as a Kepler's Pretzel.  If you try the same for Earth, zoomed out enough to see the rest of the solar system, you will see the universe the way the ancient civilazations perceived it.
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Re: Galilean Moon resonances
Reply #28 - 09/04/07 at 00:47:04
 
Cool. Thanks, I was wondering what all that other stuff was Smiley
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