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Message started by frankuitaalst on 11/22/10 at 13:56:53

Title: Lagrange Points
Post by frankuitaalst on 11/22/10 at 13:56:53

Ever wondered about lagrange points ?
http://en.wikipedia.org/wiki/File:Lagrange_points2.svg gives a very good representation about this special points in space .
Especially the L4 and L5 points are of interest as they harbour the Trojans in the Sun-Jupiter system .
It's not easily to represent them in terms of gravitational potential  because the points in fact are at a maximum of the gravitational potential and the difference in gravitaional potential is hughe ( in fact infinite  as we go to the center of the sun ) . L4 and L5 represent a hill with small slope in potential while the Sun and Jupiter represent a hughe sink . So , very difficult to represent .  
The above picture adds the equipotential lines in order to represent the potential.

Title: Re: Lagrange Points
Post by frankuitaalst on 11/22/10 at 14:15:40

In fact the gravitational potential for a 1 kg mass ranges from -1.77e+11 to -2.66 e+8 Joules from the center of mass towards the L4 or L5 point .
( energy represented in a rotating frame with Jupiter )
This range is hard to represent by means of a colour plot .  One would hardly notice the hilltop.
A way to work around this hughe "hill" is to work iteratively  and start with the small region on the top.
Then gradually let increase the minimum value .
The animation hereunder starts at an intermediate potential level and gradually goes op the hill , towards the L4 and L5 region .

Title: Re: Lagrange Points
Post by frankuitaalst on 11/22/10 at 14:21:17

"Zooming" in further , i.e starting at an intermediate level close to the Lagrange hill and raising the potential in small steps gives the following picture :

The above animations were made with the following data : M1 = Msun ; M2 = 0.1 Msun ; Sma = SmaJup .

Title: Re: Lagrange Points
Post by Mal on 11/22/10 at 15:26:19

Cool! How are you making the animations?

And can you add other bodies in interior/exterior orbits to see if those mess up the stability of a planet's lagrange points?

Title: Re: Lagrange Points
Post by frankuitaalst on 11/22/10 at 15:37:02


Mal wrote:
Cool! How are you making the animations?
And can you add other bodies in interior/exterior orbits to see if those mess up the stability of a planet's lagrange points?

Euh .. with a lot of work and trial and error  :-[
I wrote a PBasic program to calculate the shapes , plotted tem and animated them /
If you are interested I can provide the program here

To second question : No , this isn't possible because the plots are in rotating frame . Other bodies , except from  those in coorbital motion cannot be represented because they move in such a frame

Title: Re: Lagrange Points
Post by rsanchez on 11/22/10 at 19:15:00

I'm interested in how you did it.  :)

Title: Re: Lagrange Points
Post by frankuitaalst on 11/23/10 at 10:44:57


rsanchez wrote:
I'm interested in how you did it.  :)

Here's the non compiled code , written in PowerBasic . It will generate some screenshots ( for k=1.....) representing the log of the effective potential .
Feel free to ask if you run in trouble .
If necessary I can make an executable in which the necessary parameters can be changed ...

Title: Re: Lagrange Points
Post by frankuitaalst on 11/23/10 at 10:59:59

The animation herunder shows the contours of effective potential .

It starts at a low potential . Gradually the potential rises till the hills at the L4 and L5 points are reached .
The animation halts at the L1, L2 and L3 points .
It is noticable that all the points on the same contour can be reached by a body , as the effective potential is constant along the contour line .
Given of course the body has zero velocity in this rotating frame.
Animated is the case : M2=0.1 M1 , where M1=Msun and sma = SmaJup .

Title: Lagrange Points in 3D
Post by frankuitaalst on 11/27/10 at 05:24:50

I couldn't resist going one step further  ;)
For those who have anaglyph 3D glasses :
The animation shows the gravitational potential , starting at the top at the L4 and L5 points and gradually decreasing towards the sink at the biggest mass.
It may take some time before eyes are adapted , but one schould clearly see the slow bulbs around L4 and L5 and the sinks towards the two masses.
Parameters were : M1=Msun ; M2=0.1M1 ; sma = 5AU .

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