Gravity Simulator http://www.orbitsimulator.com/cgi-bin/yabb/YaBB.pl General >> Discussion >> Lagrangian Points http://www.orbitsimulator.com/cgi-bin/yabb/YaBB.pl?num=1213908048 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.htmlTheres a reference to the Neil Cornish paper which calculates the Lagrangian points http://map.gsfc.nasa.gov/ContentMedia/lagrange.pdfWith 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,  -1030.262Using the following data ( as Tony did ) For Earth: 5.9736E+24 kgFor Moon: 7.34764E+22G: 6.6725985E-11R =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 primaryI 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.