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Lagrangian Points (Read 18190 times)
frankuitaalst
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Lagrangian Points
06/19/08 at 13:40:40
 
On the Bautforum there's actually a thread going on about the stability of the Lagrangian points of the Earth and Moon .  
http://www.bautforum.com/questions-answers/74989-orbit-question-3.html
Theres a reference to the Neil Cornish paper which calculates the Lagrangian points  
http://map.gsfc.nasa.gov/ContentMedia/lagrange.pdf
 
With the formulas in this paper one can calculate the position and velocities of the bodies residing in the L1,L2,L3 points  : ( referenced to the mass center of Earth/Moon )  
"Earth", -4665880.79,  -12.455,  
"Moon", 379334119.21, 1012.617
"MoonL1", 321375133.17, 857.90
"MoonL2", 443782166.58, 1184.659
"MoonL3", -385944079.29,  -1030.262
 
Using the following data ( as Tony did )  
For Earth: 5.9736E+24 kg
For Moon: 7.34764E+22
G: 6.6725985E-11
R =384000000 m  
 
All of the points are known to be unstable. But how "unstable" ?  
Tony is correct as he says the L3 point is rather stable . (L3 being the point opposite to the moon ) .  
To find out the degree of instability the above data were input in the Picard Integrator and ran for 1 year . Screenshots were made every  1/100 year.  
One can see how all the test masses leave their original position after a short time .  
L3 keeps its position for a rather long time . L1 seems to become a moon of our moon . Smiley
I'll try to post a rotating frame also which is far more expressive ...  
( hint : click on the shortcut to open a separate frame )  
 
 
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L123PointsMoonEarthCentered.gif
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frankuitaalst
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Re: Lagrangian Points
Reply #1 - 06/19/08 at 13:52:26
 
The same simulation as above but in a rotating frame to the Moon and centered to Earth shows the stability of the L1...L3 points .  
Moon is a simple white dot at the right of the screen .  
L1 starts to move away after +/- 17 days , L2 after +/- 25 days , while L3 ( at the left ) keeps quit for about 204 days .  
L1 is captured as a moon of our moon . L2 goes into a wide orbit around Earth ,while L3 becomes a second moon .  
I don't know how sensitive these motions are for small deviations from the initial settings .
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« Last Edit: 06/19/08 at 23:00:09 by frankuitaalst »  

L123PointsMoonEarthRotating.gif
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Tony
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Re: Lagrangian Points
Reply #2 - 06/20/08 at 16:19:37
 
I believe the formulas in Cornish's paper are only approximations that don't yield good results when the mass of the secondary is not insignificant compared to the primary
 
I made a javascript calculator to compute the L points:
http://orbitsimulator.com/formulas/LagrangePointFinder.html
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frankuitaalst
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Re: Lagrangian Points
Reply #3 - 06/21/08 at 00:24:20
 
Very nice and quick calculator Tony  Smiley. What algorithm is used ?  
The first input value mentions semi major axis . Is this half the distance between the two bodies ?
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frankuitaalst
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Lagrangian Points L1 and L2 , mass ratio
Reply #4 - 06/21/08 at 13:29:08
 
In order to get some feeling with the position of the L1 and L2 points of a two body system I made the following animation ...
X-axis shows at zero the mass center . X=1 is the position of the secondary .  
The Y-axis represents the value of uČ(....) = .... in the Cornish paper referenced above .  
The L1 point is located at the intersection of the blue line with 0 ; the L2 point is positioned at the intersection of the pink line .  
The parameter M2/(M1+M2) is increased in steps of 0.02 . So the animation starts at zero mass of the secondary .  
Animation stops at the point where the two masses are equal .  
The positon of the L2 point tends to be around 1.2 R , whereas L1 approches to the center of mass when M2=M1 . This is logical .  
However L2 seems to be frozen at 1.2 R in this case , which I didn't expect . So , even in this case there seems to be a L2 point ?  
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AnimL1L2Points.gif
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Re: Lagrangian Points
Reply #5 - 06/21/08 at 13:37:20
 
I like the animated graph  Smiley
 
Semi-major axis is the full distance.  For example, to find the Earth/Sun Lagrange points, enter 1 AU for semi-major axis.
Since it's a javascript, you can View > Source and see the code.  It works by taking a guess at the Lagrange point distance and comparing the acceleration from gravity to the centrifgual acceleration.  It then continuously refines its guess until that number is below a certain threshold set by the maximum number of digits javascript variables can handle.
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frankuitaalst
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Re: Lagrangian Points
Reply #6 - 06/21/08 at 13:53:20
 
Quote from Tony on 06/21/08 at 13:37:20:
I like the animated graph  Smiley

Semi-major axis is the full distance.  For example, to find the Earth/Sun Lagrange points, enter 1 AU for semi-major axis.
Since it's a javascript, you can View > Source and see the code.  It works by taking a guess at the Lagrange point distance and comparing the acceleration from gravity to the centrifgual acceleration.  It then continuously refines its guess until that number is below a certain threshold set by the maximum number of digits javascript variables can handle.

I like this graph too ... Tells a lot more then a formula .
Any idea what the L2 at M2=M1 might be at x/R =1.2 , ie beyond the secondary ?  
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Re: Lagrangian Points
Reply #7 - 06/21/08 at 19:14:00
 
Very nice... both the calculator and the animated graphs... Good times.
 
I'm not getting any value for Velocity with respect to primary. Is this working yet?
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Re: Lagrangian Points
Reply #8 - 06/21/08 at 19:23:05
 
Quote from Dan on 06/21/08 at 19:14:00:
...I'm not getting any value for Velocity with respect to primary. Is this working yet?

 
To the best of my knowledge, every field works.  Velocity remains blank until you fill in a value for secondary mass.  But after all 3 inputs are filled, you should have values in all the outputs.  So if you're still not getting a value let me know.  It may be a browser issue.  What browser are you using?  I've only tested it against IE and Firefox.
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frankuitaalst
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Lagrangian Points : L2
Reply #9 - 06/22/08 at 02:55:57
 
The applet works fine over here ...no problem .  
I simulated the case of 2 equal masses ( both Earth sized) , separated by 384.000 km .  
The L2 point (yellow) remains at is position for a couple of revolutions but finally smashes into Earth .  
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L2equalMasses.gif
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frankuitaalst
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Lagrangian Points
Reply #10 - 06/24/08 at 10:12:13
 
Applying the Cornish equation in reference above to varying mass ratios of the secondary to the primary body I got the following animation .  
The graph shows the position of the Lagrangian points to the primary and secondary ( blue dots ) for increasing mass ratios M2/M1.  
It's interesting to see that the L4/L5 points don't change at all , L1 and L2 change a lot , while L3 is not so much affected by the varying mass ratio.
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« Last Edit: 06/24/08 at 22:55:08 by frankuitaalst »  

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