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Resonance patterns (Read 7797 times)
frankuitaalst
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Resonance patterns
10/12/07 at 07:54:30
 
Planets or asteroids orbiting in a resonance to another body make curious patterns if seen in a rotating frame .
The patterns vary as the resonance number varies , but also with varying eccentricity .  
It is not always directly clear out of a picture which resonance occurs if one looks at an orbit .  
In order to get some feeling with the patterns I created some frames in which one parameter has been changed .  
The first picture shows a simple 1:1 resonance .  
The following frames show the patterns with increasing eccentricity , starting with e=0.1 , to e=0.9 in steps of 0.1.
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« Last Edit: 10/12/07 at 15:07:23 by frankuitaalst »  

1to1Resonance.gif
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frankuitaalst
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Re: Resonance patterns
Reply #1 - 10/12/07 at 07:55:46
 
The following frame shows the 1:2 resonances , starting with e=0 to e=0.9.
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1to2Resonance.gif
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frankuitaalst
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Re: Resonance patterns
Reply #2 - 10/12/07 at 08:20:46
 
1 to 3 resonances give the followiong pictures , starting with e=0...to e=0.9
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1to3Resonance.gif
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frankuitaalst
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Re: Resonance patterns
Reply #3 - 10/12/07 at 08:40:08
 
The 2 to 3 resonance gives even more funnier pictures .  
e=0.....0.9
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2to3Resonance.gif
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frankuitaalst
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Re: Resonance patterns
Reply #4 - 10/12/07 at 12:34:55
 
The 3 to 4 resonances give an even more complicated pattern .  
e=0....0.9
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« Last Edit: 10/12/07 at 13:46:09 by frankuitaalst »  

3to4Resonance.gif
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frankuitaalst
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Re: Resonance patterns
Reply #5 - 10/12/07 at 12:37:15
 
In case the body orbits at the inside of the reference we have the same resonance structure as above , but the pattern is totally different .  
Heres the 4 to 3 resonance .  
e=0...0.9.  
Somewhere between e=0.1 and e=0.2 the orbit is "squared".
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« Last Edit: 10/12/07 at 14:13:48 by frankuitaalst »  

4to3Resonances.gif
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frankuitaalst
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Re: Resonance patterns
Reply #6 - 10/12/07 at 14:16:06
 
3 to 5 Resonances give the following ...
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3to5Resonance.gif
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frankuitaalst
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Re: Resonance patterns
Reply #7 - 10/12/07 at 14:16:53
 
while at 5 to 3 it looks like this ...
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5to3Resonances.gif
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frankuitaalst
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Re: Resonance patterns
Reply #8 - 10/12/07 at 15:22:17
 
An innerplanet in a 3:2 resonance gives the following picture
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3to2Resonance.gif
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frankuitaalst
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Re: Resonance patterns
Reply #9 - 10/12/07 at 15:23:26
 
Whereas a planet in 2:1 resonance evolves like this
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2to1Resonance.gif
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Tony
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Re: Resonance patterns
Reply #10 - 10/12/07 at 17:19:20
 
There's a link on the main page describing these types of resonant patterns: http://www.orbitsimulator.com/gravity/articles/page325.html
You've probably figured out by now that the number of loops represents one of the numbers in the resonance.
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Re: Resonance patterns
Reply #11 - 10/12/07 at 18:05:25
 
I love seeing this stuff, just reminds me how amazingly cool nature can be Smiley
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Tony
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Re: Resonance patterns
Reply #12 - 10/12/07 at 18:16:47
 
I agree.  One of these days I'm going to start a thread only for screenshots of cool looking stuff.  Only brief explanations required.
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Re: Resonance patterns
Reply #13 - 10/13/07 at 00:01:53
 
Just to check because i am not sure if this is right. To create a 1:n resonance the outer object must describe an orbit with n times the orbital period. To get this the increase in sma major axis must be such that a(n) =a1*n(2/3)? From keplers (1st or 2nd)law; square of period is proportional to cube of sma.
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« Last Edit: 10/13/07 at 03:13:38 by shellandtube »  

The shortest distance between two points is a straight line, or is it?
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frankuitaalst
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Re: Resonance patterns
Reply #14 - 10/13/07 at 00:43:13
 
Quote from Tony on 10/12/07 at 17:19:20:
There's a link on the main page describing these types of resonant patterns: http://www.orbitsimulator.com/gravity/articles/page325.html
You've probably figured out by now that the number of loops represents one of the numbers in the resonance.

I wasn't aware of this . Thanks !
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frankuitaalst
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Re: Resonance patterns
Reply #15 - 10/13/07 at 03:00:52
 
"a(n) =a1*n(3/2). "
This is correct indeed . Thats how i calculated the SMA and put it in the simulations .
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Tony
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Re: Resonance patterns
Reply #16 - 10/13/07 at 10:57:19
 
There's a bunch of javascript calculators on this page:
http://orbitsimulator.com/formulas/
 
The 4th one down will compute your semi-major axis directly, rather than as a fraction or mutliple of another object's orbit.
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frankuitaalst
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Re: Resonance patterns
Reply #17 - 10/13/07 at 12:34:29
 
Is  this link to the formulas also choosable from the main menu ?
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Tony
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Re: Resonance patterns
Reply #18 - 10/13/07 at 19:44:30
 
Quote from frankuitaalst on 10/13/07 at 12:34:29:
Is  this link to the formulas also choosable from the main menu ?

No, it's only available on-line.  They're written in a different programming language, javascript, so the only way I could include them is to make them open in a browser window.
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frankuitaalst
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Re: Resonance patterns
Reply #19 - 10/19/07 at 10:06:32
 
As Tony mentions the number of loops represents one of the number of the resonance .  
Here is a 3:1 resonance with increasing eccentricity 0....0.9.
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3to1Resonance.gif
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