Lagrangian points

Ajaja

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Need help with calculation of Lagrangian points in cartesian coordinate system.
Input data:
Major body: positon <0,0>, speed <0,0> , mass = M1
Minor body: position <R,0>, speed <Vx,Vy> (elliptical orbit with eccentricity near 0, so Vx is not zero), mass=M2

How to calculate position and speed vectors of Lagrangian points (center of Lagrangian "area")?

---------- Post added 10-25-2009 at 12:41 PM ---------- Previous post was 10-24-2009 at 02:37 PM ----------

Nobody can help?
I just want to place here some objects in Orbiter and maybe to make simple celbody module addon for creating artificial satellites in these points.
For example: MJD 55129.0, Moon position relative to Earth: <403551905.78, 0,0> speed: <14.5409, 968.3182, 0> (data from Orbiter after few simple coordinate transformations).
Where is L1 point, and how to calculate speed vector for this point? Is there some equation for gravity, centrifugal, Coriolis forces?
 
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Calsir

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I did not reply any sooner because of the "elliptical" orbit. The circular case is pretty well known.

Starting from wikipedia, I found this calculator which may be useful: http://orbitsimulator.com/formulas/LagrangePointFinder.html

A treatise can be found here:
http://www.merlyn.demon.co.uk/gravity4.htm
and here
http://wmap.gsfc.nasa.gov/media/ContentMedia/lagrange.pdf . You will find ways to calculate the position of lagrange points.

These are all referenced by wikipedia: http://en.wikipedia.org/wiki/Lagrange_points

The aforementioned page states that the lagrange points move on orbits that are geometrically similar to the orbit of the secondary, with a matching period. While I could find no reference for this statement, the matching period argument certainly makes sense. About the similarity argument, the point is that the L-points will have the same geometry (equilateral triangles for L4-5, for instance) throughout the orbit of the secondary.

I have not performed any calculation, but I think that, to compute the velocity of any given L-point, you can roughly state it is equal to

Vlx = Vx * D/R
where Vlx is the velocity of the L-point, Vx is the velocity of the secondary, R the distance of the secondary, and D is the distance of the L-point. All the values should be relative to the centre of gravity, rather than the primary, but may work nonetheless. For Vly:
Vly = Vy * D/R
again.

I hope this makes sense.
 

Ajaja

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A treatise can be found here:
http://www.merlyn.demon.co.uk/gravity4.htm
and here
http://wmap.gsfc.nasa.gov/media/ContentMedia/lagrange.pdf . You will find ways to calculate the position of lagrange points.
Thanks. I found lagrange.pdf too but I didn't find first link. I'll readi it.

Vlx = Vx * D/R
where Vlx is the velocity of the L-point, Vx is the velocity of the secondary, R the distance of the secondary, and D is the distance of the L-point. All the values should be relative to the centre of gravity, rather than the primary, but may work nonetheless. For Vly:
Vly = Vy * D/R
again.

I tried it firstly. Got formula L1: (R*(1-(a/3)^(1/3)),0) from lagrange.pdf for position and ( Vx * D/R, Vy * D/R ) for speeed, put it into Orbiter, but ship is go away from L1 very fast. It's very approximate formula. So I'm looking for accurate equation to solve it numerically.
Maybe http://www.merlyn.demon.co.uk/gravity4.htm will help...
 

Calsir

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Try with L4 or L5, first. They should be stable for the Earth-Moon system. Of course, the influence of the sun can be quite strong, so I'd recommend to remove it :). You may want to try Sun-Earth L-points, but then the propagation could become an issue.

Dry humour aside, try to use the most precise propagator for Orbiter (the technical doc should help) and check that orbiter actually can use 3 sources of gravity: I fear that, since the L-points are, generally speaking, outside the moon SOI (or close by), only Earth and its primary (Sun) are considered, rather than the three of them. I cannot check, since I currently have no installation of orbiter (go ahead and burn me at the stake :p).

Also, if you can, try to plot the trajectory of the ship in the rotating frame to better understand its behaviour.

---------- Post added at 14:19 ---------- Previous post was at 13:40 ----------

I forgot: the velocities for L3, L4 and L5 are to be properly rotated in the reference frame.
 

Ajaja

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After few expirements with L1 in Sun-Jupiter system (without Saturn and other planets of course) i found:
1. D=R*(1-(a/3)^(1/3)) where a=M2/(M1+M2) from wiki is very inaccurate. Only for a<<1.
2. Solution of
M*L^5 - 3*M*L^4 + 3*M*L^3 - L^2 + 2*L - 1 = 0 (where M=M1/M2 and L - distance from M2 in units of R) equation from http://www.merlyn.demon.co.uk/gravity4.htm is good, but not the best.
3. Most accurate is solution of
u^2((1-s1)+3*u+3*u^2+u^3)=a*(s0+2*s0*u+(1+s0-s1)*u^2+2*u^3u+u^4) equation from lagrange.pdf (where a=M2/(M1+M2), u = L, (s0,s1)=(-1,1) for L1).
4. Seems like the speed of L1 is really ( Vx * D/R, Vy * D/R ).

---------- Post added at 11:27 PM ---------- Previous post was at 08:29 PM ----------

I checked stability of Earth-Moon L1 in Orbiter. It's only 1 km drift from L1 after 4 days on 10000x !!! All planets, all "Realism" and "Perturbations" are ON. I think it's good result :) Magic number (u from equation 3 above) for Earth-Moon system is 0.15093423356147271228 ;)
Scenario with DG accurate in L1:
Code:
BEGIN_DESC

END_DESC

BEGIN_ENVIRONMENT
  System Sol
  Date MJD 55130.0000001157
END_ENVIRONMENT

BEGIN_FOCUS
  Ship GL-01
END_FOCUS

BEGIN_CAMERA
  TARGET GL-01
  MODE Cockpit
  FOV 70.00
END_CAMERA

BEGIN_HUD
  TYPE Orbit
END_HUD

BEGIN_MFD Left
  TYPE Orbit
  PROJ Frame
  FRAME Ecliptic
  REF Moon
END_MFD

BEGIN_MFD Right
  TYPE Orbit
  PROJ Frame
  FRAME Ecliptic
  REF Earth
  TARGET MOON
END_MFD

BEGIN_SHIPS
GL-01:DeltaGlider
  STATUS Orbiting Earth
  RPOS 183071346.42321 3922551.79904 -290225040.08469
  RVEL 690.74023 71.72056 437.11283                  
  AROT -0.00 -0.00 0.00
  PRPLEVEL 0:0.962562 1:0.993001
  NAVFREQ 0 0 0 0
  XPDR 0
  AAP 0:0 0:0 0:0
END
END_SHIPS
 

Calsir

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4. Seems like the speed of L1 is really ( Vx * D/R, Vy * D/R ).


It makes sense if the statement from wikipedia about the similar orbit is correct, since it keeps the relative geometry similar.

I checked stability of Earth-Moon L1 in Orbiter. It's only 1 km drift from L1 after 4 days on 10000x !!! All planets, all "Realism" and "Perturbations" are ON. I think it's good result :)

Check that 10000x time compression does not introduce the orbit stabilisation algorithm. I do not know if it actually has any adverse effect, but you may want to check it with the newtonian propagators. Anyhow, it is a good result :cheers:.

---------- Post added at 16:42 ---------- Previous post was at 13:38 ----------

I let your scenario run on the latest orbiter beta with no gui for half a month, 14 days (3 minutes :) ).


here is the status of the boat:

Code:
BEGIN_DESC

END_DESC

BEGIN_ENVIRONMENT
  System Sol
  Date MJD 55144.0718098514
END_ENVIRONMENT

BEGIN_FOCUS
  Ship GL-01
END_FOCUS

BEGIN_CAMERA
  TARGET GL-01
  MODE Cockpit
  FOV 70.00
END_CAMERA

BEGIN_MFD Left
  TYPE Orbit
  PROJ Frame
  FRAME Ecliptic
  REF Earth
  TARGET Moon
END_MFD

BEGIN_PANEL
END_PANEL

BEGIN_SHIPS
GL-01:DeltaGlider
  STATUS Orbiting Sun
  RPOS 101526056297.46 -10678872.80 107977679654.99
  RVEL -22724.874 -73.962 19586.439
  AROT -0.00 -0.00 0.00
  PRPLEVEL 0:0.962562 1:0.993001
  NAVFREQ 0 0 0 0
  XPDR 0
  AAP 0:0 0:0 0:0
END
END_SHIPS

At first glance, it seems that the ship drifted a little too much, but I may be wrong. Also notice that its primary is the Sun.
 

Ajaja

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At first glance, it seems that the ship drifted a little too much, but I may be wrong.
Yes. It's far away from L1 (>30M). Seems like you did not change ErrorLimit for Moon from 1e-5 to 1e-8 in Moon.cfg ;)
In my tests with latest beta I have ~1.2M shift from L1 ater 14 days (3.8days-1k, 7.3 - 10k, 10.6-100k, 13.8-1000k).
I use Orbiter_ng.exe without graphics module for testing (FPS ~12000, so step with 10000x <1s). Fresh Orbiter install (orbiter091023beta) without addons, only ErrorLimits for all planets and Sun changed to 1e-8.

Also notice that its primary is the Sun.
I created scenario by hands. Orbiter in this point saves position as STATUS Orbiting Sun, not Earth.

Scenario for L2:
Code:
BEGIN_DESC

END_DESC

BEGIN_ENVIRONMENT
  System Sol
  Date MJD 55130.000001916262
END_ENVIRONMENT

BEGIN_FOCUS
  Ship GL-01
END_FOCUS

BEGIN_CAMERA
  TARGET GL-01
  MODE Cockpit
  FOV 70.00
END_CAMERA

BEGIN_HUD
  TYPE Orbit
  REF AUTO
END_HUD

BEGIN_MFD Left
  TYPE Orbit
  PROJ Frame
  FRAME Ecliptic
  REF Moon
END_MFD

BEGIN_MFD Right
  TYPE Orbit
  PROJ Frame
  FRAME Ecliptic
  REF Earth
  TARGET Moon
END_MFD

BEGIN_SHIPS
GL-01:DeltaGlider
  STATUS Orbiting Earth
  RPOS 251802434.6442 5395220.7267 -399184870.2655
  RVEL 950.066353 98.646794 601.219588            
  AROT 0.00 -0.00 -0.00
  PRPLEVEL 0:1.000000 1:1.000000
  NAVFREQ 0 0 0 0
  XPDR 0
  AAP 0:0 0:0 0:0
END
END_SHIPS
 
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River Crab

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For future reference, this addon should be useful for scenario building:
[nomedia="http://www.orbithangar.com/searchid.php?ID=3644"]Lagrange MFD 0.7[/nomedia]
 

Ajaja

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For future reference, this addon should be useful for scenario building:
Lagrange MFD 0.7

If you want to place an object into Lagrange point, then accuracy is very impotent. Some points are very unstable. I didn't find how to set correct position and velocity for a vessel using Lagrange MFD 0.7, so I wrote another MFD: [nomedia="http://www.orbithangar.com/searchid.php?ID=4582"]Lagrangian points MFD v0.2[/nomedia]
 
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