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Message started by frankuitaalst on 06/19/08 at 13:40:40

Title: Lagrangian Points
Post by frankuitaalst on 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 . :)
I'll try to post a rotating frame also which is far more expressive ...
( hint : click on the shortcut to open a separate frame )



Title: Re: Lagrangian Points
Post by frankuitaalst on 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 .

Title: Re: Lagrangian Points
Post by Tony on 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

Title: Re: Lagrangian Points
Post by frankuitaalst on 06/21/08 at 00:24:20

Very nice and quick calculator Tony  :). What algorithm is used ?
The first input value mentions semi major axis . Is this half the distance between the two bodies ?

Title: Lagrangian Points L1 and L2 , mass ratio
Post by frankuitaalst on 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 ?

Title: Re: Lagrangian Points
Post by Tony on 06/21/08 at 13:37:20

I like the animated graph  :)

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.

Title: Re: Lagrangian Points
Post by frankuitaalst on 06/21/08 at 13:53:20


Tony wrote:
I like the animated graph  :)

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 ?  

Title: Re: Lagrangian Points
Post by Dan on 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?

Title: Re: Lagrangian Points
Post by Tony on 06/21/08 at 19:23:05


Dan wrote:
...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.

Title: Lagrangian Points : L2
Post by frankuitaalst on 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 .

Title: Lagrangian Points
Post by frankuitaalst on 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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