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Horseshoe Orbits Initial Conditions (Read 3755 times)
rsanchez
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Horseshoe Orbits Initial Conditions
11/10/10 at 23:38:13
 
I want to run a simulation of horseshoe orbits, and I'm trying to follow the initial conditions presented in this paper:
 
http://adsabs.harvard.edu/abs/1981A%26A...103..288T
 
I just wanted to know if anyone knew how Taylor derived the parameters in numerous tables throughout the paper. I've read it many times but the math is a little over my head. Basically, I'm guessing that in the x column, 1.002 for example represents 1.002 times the separation between Jupiter and the Sun in a given coordinate system. I guess Taylor intended for all the quantities to be dimensionless, which is why I can't figure out how he arrived to the y-dot velocity values he gives in the tables, and how to translate those to length/time units I can use in a simulation. I just want to see if having the initial position and velocity would produce the same orbital patterns that are plotted in the paper.
 
If anyone can help out, I would appreciate it very much!
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frankuitaalst
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Re: Horseshoe Orbits Initial Conditions
Reply #1 - 11/11/10 at 01:23:39
 
Thanks for the link rsanchez . Welcome here !
As far as I understand Taylor uses dimensionless coordinates centered on the mass-center , where he has set GM equal 1.  
Sun and Jupiter are here at -mu , 1-mu respectively on the x-axis
You might try the following :
 
1. scaling the x-axis :  
While the Sma of SunJupiter is 778547200000 m
Xsun: =778547200000 * (-mu)
XJup := 778547200000 *(1-mu)  
XAst : = 778547200000 * X given  
 
mu is given and is equal to  Mj/(Ms+Mj) ; Mt=Ms+Mj
 
2. Scaling the velocities  
Vysun= -sqrt( GMt / abs(Xsun))
VyJup = sqrt (GMt/ abs(XJup))
Vyast = - Vgiven * sqrt(GMt/ abs(Xast))
 
I haven't tried this myself. I hope this works .  
Please note these positions and velocities are relative to the mass center . If the sun is the center of your simulation you should convert the above values to the sun-coordinates .  
 
Edit : when working with the above formulae I saw some flaws . Will try to correct them
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« Last Edit: 11/11/10 at 03:03:46 by frankuitaalst »  
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frankuitaalst
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Re: Horseshoe Orbits Initial Conditions
Reply #2 - 11/11/10 at 05:12:44
 
I think there's an easier way to convert .  
 
1. For the positions :  
The scaling factor is : 778547200000 ( ie = sma.Jup ) , while in the paper the sma = 1
Xsun = - mu *  SmaJup = -mu * 778547200000 =-743350581.5
XJup = (1-mu) * SmaJup = (1-mu) * 778547200000 = 7.77804E+11
Xast = Xgiven * 778547200000 = -8.26825E+11
 
2. For the velocities :  
 
The velocity of Jup is given by : v Jup = sqrt ( G*Mt * (1-mu)^2 / sma ) = 13051.22627
The velocity of Sun is given by : vSun = - vJup * Mjup/Msun = - 12.46120548
 
The scaling factor for the velocity is = 13063.69939.  
This value is obtained by using the above formula once with GMt=1 and sma =1 ; and the second time with the real values of GMt and Sma.  
So the velocity of the asteroid is : V ast = V given * 13063.69939 = 1417.54202
Please note the above values are referenced to the center of mass , which has zero velocity .  
 
Edit : when plugging in the above values I see I don't get the desired orbits . Maybe the velocity parameter ( 0.10851)  is described in a rotating frame , as a would have expected a value close to 1 . I think I must check this out  Sad
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frankuitaalst
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Re: Horseshoe Orbits Initial Conditions
Reply #3 - 11/11/10 at 06:36:23
 
Finally got it and tested it  Smiley.  
The value of ydot was indeed expressed in rotating frame.  
Here's a spreadsheet for conversion of the given values .  
Plug in the values for x and y dot in the yellow marked input fields .  
The values to use are at the bottom of the sheet .  
 
The file in annex is an excel sheet and should be renamed : change .txt with .xls
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rsanchez
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Re: Horseshoe Orbits Initial Conditions
Reply #4 - 11/11/10 at 09:57:18
 
Thanks for that spreadsheet!
 
Out of curiosity, were you able to get the orbits to look like the plotted orbits beyond one orbit? For me, the orbits in Figures 4a-d were only stable for one orbit. Actually, pretty much the only stable orbits I got that looked like the orbit plots are the ones in Figures 3a-c.
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frankuitaalst
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Re: Horseshoe Orbits Initial Conditions
Reply #5 - 11/11/10 at 10:29:08
 
yes , I did some simulations , in fact the 3a...d and noticed all last only for one period . Then the orbits seem to switch to another shape.  
 I guess the asts come to close to Jupiter and can't maintain their orbit . I don't know why exactly this is .  
Maybe the initial values are to close to the stability limit . Maybe the mu value of the paper ( which isn't exact after the four significant decimals )  may play a role . I don't know exactly .
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Re: Horseshoe Orbits Initial Conditions
Reply #6 - 11/11/10 at 11:02:44
 
Welcome, Rsanchez!
 
If you're using the Beta version ( http://www.orbitsimulator.com/cgi-bin/yabb/YaBB.pl?num=1176774875 ), here are instructions for making 20 smooth horseshoe orbits around Jupiter.  I'm just letting Gravity Simulator create them, rather than using the data in the article.  They quickly evolve away from smooth.
 
File > New
 
Pause the simulation
 
Objects > Create Objects
* name: Jupiter
* SMA: 5.1 AU
* mass: 1 Jupiter mass
* Mean Anomaly : 180
* Press "Set All to 0%" to prevent it from generating random values
* Press Create
 
Objects > Create Objects
* Number of objects: 20
* Color: something different than Jupiter
* SMA, Press the distribution button until you you see "Start / End", then type 5, 5.2 in the SMA box (you can experiment with different ranges centered around 5.1.  The closer they are to 5.1, the longer they'll remain smooth.)
* mass : 0
* Mean Anomaly: 0
* Press "Set All to 0%"
* Press Create
 
View > Rotating Frame Adjustment
* Choose Jupiter from the dropdown list
* Click the "Rotating Frame" and "Clockwise" options
* Press OK
 
Zoom out the screen to about 20 AU
 
File > Save As...
* give it a name
 
Unpause and watch.  You can increase the time step to 4096.
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frankuitaalst
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Re: Horseshoe Orbits Initial Conditions
Reply #7 - 11/11/10 at 11:26:26
 
Thanks for the hint Tony . This method should work indeed .  
Having the "create data file " on one is then able to define a certain obtaoned shape with an initial sma value ..!
 
Concerning the data given in the paper I've noticed that the outcome is very sensitive to the mu value , which is different ( value of 1965 ) from now .  
Further the G value should be checked out ( is the Gvalue the Gvalue in the simulator ? )  
In the file in annex I've changed the mu to the mu in the paper . The results are better now when simulating .
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rsanchez
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Re: Horseshoe Orbits Initial Conditions
Reply #8 - 11/11/10 at 18:56:44
 
Thanks for the tips. I just thought it would've been nice if I could reproduce the results of that paper.
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Re: Horseshoe Orbits Initial Conditions
Reply #9 - 11/12/10 at 06:15:25
 
Quote from rsanchez on 11/11/10 at 09:57:18:
Thanks for that spreadsheet!

Out of curiosity, were you able to get the orbits to look like the plotted orbits beyond one orbit? For me, the orbits in Figures 4a-d were only stable for one orbit. Actually, pretty much the only stable orbits I got that looked like the orbit plots are the ones in Figures 3a-c.

 
Yes stability is a problem for some of them . If I read the paper correctly the abs(s) should be smaller than 1 , which isn't the case for the given initila parameters .
I made an animation for the orbits in Fig a...f . Result is hereunder . The period is set to 350 years  
When running for 35000 years only the 3b and 3f are stable.
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JupiterHorseshoes1to6.gif
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