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Lagrange points? (Read 9532 times)
EDG
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Lagrange points?
01/28/11 at 11:42:18
 
I guess this is more a science question, but if I can simulate it in GS then all the better Smiley
 
For the SF background I'm working on, I want to have 'jump points' located at the L4 and L5 points of the most massive objects in a planetary system. This has a few implications:
 
1) If there are no planets in the system, then it contains no jump points at all. This is OK, but I'm wondering what happens if you just have a system full of dust and rocks - would there be any L4/L5 points in there? If the only massive object of note was a small 10-100km diameter asteroid that had 'cleared out its orbit', would its lagrange points be stable at all? What would be the smallest object that can have stable L4/L5 points? I'm figuring that the jump points themselves are massless objects, but at really small scales I guess the mass of the spaceship itself might disrupt things!
 
2) If there were no planets, but multiple stars (e.g. a Binary or trinary system) then you wouldn't have L4/L5 points unless the mass ratio of the stars was below about 25:1 (so in most cases they'd be unstable you had a very massive star and a low mass star, such as a 10 solar mass star and a 0.4 solar mass star, which would be unlikely).  
 
3) In a solo or multiple system containing planets, the jump points would be found at the L4/L5 points of the most massive planet in the system. The jump points can exist in a system as long as stable L4/L5 points are located anywhere within it.  
 
How could I test this in GS? I'm not sure how one makes an object orbit an L4/L5 point, do I just create the lagrangian object with the same orbital parameters as the planet but with a mean anomaly of +60 or -60 degrees relative to the planet? (I guess it might drift a bit and go into an orbit around the L4/L5 point over time?). And if I put other planets in different orbits in the system, would they affect the stability of the L4/L5 points?
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Re: Lagrange points?
Reply #1 - 01/28/11 at 14:22:26
 
Quote from EDG on 01/28/11 at 11:42:18:
How could I test this in GS? I'm not sure how one makes an object orbit an L4/L5 point, do I just create the lagrangian object with the same orbital parameters as the planet but with a mean anomaly of +60 or -60 degrees relative to the planet?

 
I guess not, since if I do that I get two objects in the same orbit, with one 60 behind of the other, but if I switch it to a view centred on the planet then I just see the lagrange object orbiting the stationary planet (on an interior orbit relative to the sun). I suppose the problem is that I have no idea how to do those nice lagrange animations that others do here.
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Re: Lagrange points?
Reply #2 - 01/28/11 at 15:49:21
 
It should work if you create objects 60 +- from a massive body.  Create all three of the objects at the same time though.  Don't look at your planet's mean anomoly in the Orbital Elements interface and add or subtract 60 from that.  The problem is that mean anomoly is measured from the ascending node, and if your planet is orbiting in the ecliptic then there is no ascending node, and mean anomoly is arbitrairly reported.
 
I don't think there's any lower limit to the mass of the planet.  But the lower it gets, the more likely that other bodies will perturb your objects out of L4/5.  In a 2-body system (2 bodies with mass) and 2 more massless bodies, there should be no lower limit as long as you maintain a ratio greater than about 25:1
 
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Re: Lagrange points?
Reply #3 - 01/28/11 at 16:47:56
 
Quote from EDG on 01/28/11 at 14:22:26:
Quote from EDG on 01/28/11 at 11:42:18:
How could I test this in GS? I'm not sure how one makes an object orbit an L4/L5 point, do I just create the lagrangian object with the same orbital parameters as the planet but with a mean anomaly of +60 or -60 degrees relative to the planet?


I guess not, since if I do that I get two objects in the same orbit, with one 60 behind of the other, but if I switch it to a view centred on the planet then I just see the lagrange object orbiting the stationary planet (on an interior orbit relative to the sun). I suppose the problem is that I have no idea how to do those nice lagrange animations that others do here.

 
Actually that should work. First order of business first: make sure you are viewing in a rotating frame (in the View menu) with a period equal to the period of the planet.
 
If you have no other bodies in the system and the planet is in a near-circular orbit, it might take millenia or longer before your langrange object noticably starts to orbit the point. Add a few perturbers to the system or put the planet in a more elliptical orbit and this will start happening sooner. Or put the object at +/-58°. (In fact, L4 and L5 are stable for a surprisingly broad arc at the planet's orbital distance.)
 
Additionally, once the planet does start orbiting the lagrange point, it might take hundreds of sidereal years to complete a single orbit. So you might try increasing the graphics interval (first choice, until the frame rate looks too choppy) and stepsize (last resort if stepsize isn't doing it for you). Take a look at the attached sim. These objects take about 12 sidereal years to complete one orbit if the L4/L5 points.
 
Conflating the issue is the quasi-orbital motion of natural lagrange objects caused by the eccentricity of their orbits. This is seen in most Trojan asteroids. In the short term, this type of motion dominates, but over long periods of time this quasi-orbit itself orbits the Lagrange point in a motion usually called libration. (See the Neptune Trojans thread... the loops themselves (with a period of almost exactly one sidereal year) are quasi-orbits; the oscillating motion of those loops is the libration of the asteroid's mean position as it orbits L4/L5.)
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Re: Lagrange points?
Reply #4 - 01/28/11 at 17:18:56
 
Quote from Tony on 01/28/11 at 15:49:21:
It should work if you create objects 60 +- from a massive body.  Create all three of the objects at the same time though.  Don't look at your planet's mean anomoly in the Orbital Elements interface and add or subtract 60 from that.  The problem is that mean anomoly is measured from the ascending node, and if your planet is orbiting in the ecliptic then there is no ascending node, and mean anomoly is arbitrairly reported.

 
How do I create the objects though (especially if I have a circular, non-inclined orbit)? Would you be able to write a walkthrough for me please, because I'm not seeing how to create them all at the same time as you describe.
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Re: Lagrange points?
Reply #5 - 01/28/11 at 18:03:05
 
Screencaps. Create a new sim and make sure it is paused.
 
For the planet, set Longitude of Node, Argument of Periapsis (Periapsis = perifocus and is the term I prefer, since an ellipse technically has two foci), and Mean Anomaly all to zero. Make sure all random elements are also set to zero:
 

 
For the orbiter, same, only Mean Anomaly is 60:
 

 
That's pretty much it... although I do have to make a slight correction to what Tony said above: Mean anomaly is measured from the periapsis, not from the ascending node. So at a mean anomaly of 0, the object is at its closest approach to the central body; at mean anomaly of 180, it is at its farthest. The argument of periapsis is itself measured from the longitude of the ascending node. It took me a while to get that down, and until you do it can be pretty difficult to position objects exactly where you want them.
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Re: Lagrange points?
Reply #6 - 01/28/11 at 18:25:51
 
Quote from phoenixshade on 01/28/11 at 18:03:05:
... although I do have to make a slight correction to what Tony said above: Mean anomaly is measured from the periapsis, not from the ascending node...

 
Oops  embarrassed .  It still makes for an ambiguous situation when your orbit is circular.  So you need to create all your objects at the same time if you set ecc=0.
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Re: Lagrange points?
Reply #7 - 01/28/11 at 19:14:35
 
Just for fun, try the following:

  • File > New
     
  • Zoom out until the screen with is about 3-4 AU
     
  • Pause the simulation
     
  • Objects > Create Objects
     
  • Mass: 1 Jupiter mass, SMA: 1 AU, MA: 0, set all random stuff to 0%
     
  • Press Create
     
  • Objects > Create Objects
     
  • Number of objects 5
     
  • Press the SMA distribution until you see the words (Start, End)
     
  • Enter 0.98, 1.02
     
  • MA: 60
     
  • Press Create
     
  • If desired, repeat previous 6 steps, except MA: 300
     
  • View > Rotating Frame Adjustment.  Choose the object with 1 Jupiter mass.  Select rotating frame.
     
  • Preferences > Graphic Output 50
     
  • Save the simulation
     
  • Unpause and watch
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Re: Lagrange points?
Reply #8 - 01/28/11 at 19:37:58
 
Quote from phoenixshade on 01/28/11 at 18:03:05:
Screencaps. Create a new sim and make sure it is paused.

For the planet, set Longitude of Node, Argument of Periapsis (Periapsis = perifocus and is the term I prefer, since an ellipse technically has two foci), and Mean Anomaly all to zero. Make sure all random elements are also set to zero:

 
Thanks! Though actually I did exactly what you said here!  
 
Maybe I'm just not displaying it properly. It's quite likely that I'm not understanding how "Rotating Frame" works.  
 
Oh, I followed tony's instructions (though for an earthlike planet and a small mass)  and got a planet and a lagrange orbiter that were two dots sitting there 60 degrees apart around the sun, which is promising. I guess as phoenix says, not much is going to happen with this since there are no perturbers.  
 
So how does this "rotating frame adjustment" thing work exactly? Is there any documentation describing it?
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Re: Lagrange points?
Reply #9 - 01/28/11 at 19:55:42
 
Quote from Tony on 01/28/11 at 19:14:35:
Just for fun, try the following:

 
I presume the lagrange objects should be very small/low mass, right?  
 
Interesting... they're spreading out!
 
Oh, ha! Now it looks like they're orbiting the lagrange point (in one of those long tadpole orbits)!
 
That's cute Smiley. Looks like I have something new to play with Wink
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Re: Lagrange points?
Reply #10 - 01/28/11 at 20:13:51
 
The rotating frame is great for looking at motions relative to a particular planet. (Case in point: the Lagrange objects we're talking about here.) What it does is rotates the camera. You can either specify the period of rotation (using the numeric field) or you can select any object in your sim and it will rotate the camera in sync with that object's orbit. Almost all orbits with resonances are more interesting this way. In fact, objects that otherwise seem mundane can have some very interesting properties that only become noticeable in a rotating frame.
 
Besides adding perturbers or Tony's example varying the SMA, try giving the planet a small eccentricity (around 0.05-0.08 is good) but put the lagrange objects in circular orbits, and view this in a rotating frame using the planet's period. Over time, the lagrange objects will start to develop eccentricity as well and start the looping motion characteristic of Trojan asteroids. I am pretty sure that the langrange object's eccentricity will end up very slowly oscillating over the range 0 – 2e (where e is the eccentricity of the planet).
 
Trust me, once you really start exploring the nuances of 1:1 resonances (and other resonances as well), you'll keep coming back to them. They are fascinating.
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Re: Lagrange points?
Reply #11 - 01/29/11 at 11:28:38
 
Quote from phoenixshade on 01/28/11 at 20:13:51:
The rotating frame is great for looking at motions relative to a particular planet. (Case in point: the Lagrange objects we're talking about here.) What it does is rotates the camera. You can either specify the period of rotation (using the numeric field) or you can select any object in your sim and it will rotate the camera in sync with that object's orbit. Almost all orbits with resonances are more interesting this way. In fact, objects that otherwise seem mundane can have some very interesting properties that only become noticeable in a rotating frame.

 
So what does changing the focus object do? If I focus on Jupiter in Tony's sim, that doesn't have it so that the camera is rotating in sync with the object's orbit, it centres the view on the planet instead, but it produces a very different view to the rotating one.  
 
Quote:
Besides adding perturbers or Tony's example varying the SMA, try giving the planet a small eccentricity (around 0.05-0.08 is good) but put the lagrange objects in circular orbits, and view this in a rotating frame using the planet's period. Over time, the lagrange objects will start to develop eccentricity as well and start the looping motion characteristic of Trojan asteroids. I am pretty sure that the langrange object's eccentricity will end up very slowly oscillating over the range 0 – 2e (where e is the eccentricity of the planet).

 
I'm wondering what happens to the lagrange points if the planet's orbit is eccentric (e.g. e=0.5 or more). Are they still stable?  
 
And is there a way to enter the L1/L2/L3 points too? It'd be nice if there was an option in the program to just place objects there to save us having to calculate the locations ourselves first.  
 
Quote:
Trust me, once you really start exploring the nuances of 1:1 resonances (and other resonances as well), you'll keep coming back to them. They are fascinating.

 
Oh definitely Smiley
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Re: Lagrange points?
Reply #12 - 01/29/11 at 16:20:54
 
Quote from EDG on 01/29/11 at 11:28:38:
So what does changing the focus object do? If I focus on Jupiter in Tony's sim, that doesn't have it so that the camera is rotating in sync with the object's orbit, it centres the view on the planet instead, but it produces a very different view to the rotating one.

 
The rotation period stays the same until you change it again through the View -> Rotatinge Frame Adjustment.
 
When you choose an object from the dropdown list, it sets the period equal to that object's period with respect to its REFERENCE OBJECT. This is independent of the Focus Object, which is always held in the center of the screen. Let's use a concrete example...
 
Say you have a simple Sun-Earth-Moon simulation. The sun is "Floating" (no reference object, its default state); the Earth has the Sun as its reference object, and the Moon has the Earth as its reference object.
 
Initially, the focus is the Sun, and you see the typical view of the sun in the center, with the Earth/Moon system forming a braided arc as it orbits the sun. Set the rotating frame to "Earth" and now the Sun and Earth appear nearly stationary, with the Moon orbiting the Earth.
 
I say "nearly stationary" because, while the frame rotates at a fixed rate, the earth's speed in its orbit varies due to eccentricity. So it forms a small elliptoid, slightly flattened on the side facing the sun, that completes its circuit once a year. The greater the eccentricity, the larger this elliptoid will be. If there's some second object in the middle of that elliptoid whose gravity isn't really strong enough to have captured the first object, it's in a quasi-orbit, but I digress...
 
If you don't want to see that elliptoid, change the focus to the Earth. Things will still look fine, because relative to one another both the earth and sun rotate each other at the same rate. So you're free to zoom in on the earth and the sun will stay nearly stationary, exhibiting the same elliptoid motion. (Right now you can't remove it altogether; I think that might be the purpose of the currently-unsupported Locked checkbox, which as far as I can tell does nothing for now.)
 
If you want the Earth-Moon system to remain stationary, you'd have to go to View -> Rotating Frame Adjustment... and select "Moon" from the dropdown. With either the Earth or the Moon as the Focus Object, things will look fine, with the other object showing the same small elliptoids due to eccentricity. Zoom out to include the sun in the view and things look a bit strange, with the sun apparently orbiting a stationary Earth/Moon once a month... so you probably wouldn't want this view at those scales. Choose the sun as the focus and it looks even weirder, with the sun stationary and the Earth/moon appearing to revolve once a month, side-by-side with each other. So again, not the view you'd want.
 
Now, what would you USE each of these views for?
 
Rotation Period = Earth; Focus = Sun: Particularly good for looking at objects in resonant orbits with the earth, such as many of the near-earth asteroids. Also good for looking at Lagrange Points L3, L4, and L5.
 
Rotation Period = Earth; Focus = Earth: Good for exploring the near-Earth Lagrange points, L1 and L2. Also a good for viewing the effects of solar perturbation on the orbit of the Moon.
 
Rotation Period = Moon; Focus = Earth: Good for looking at objects at Lunar lagrange points; Lunar perturbations of high-orbiting satellites (or Low-Energy Transfer spacecraft, such as Japan's Hiten lunar mission of 1991), Earth-Moon Lagrange points.
 
Rotation Period = Moon; Focus = Moon: Limited only because there aren't a lot of bodies orbiting the moon. In my Apollo simulation, I use this view for operations such as the lunar orbit insertion and trans-earth injection burns, since these are dependent on the position of the Earth.
 
Quote:
Quote:
Besides adding perturbers or Tony's example varying the SMA, try giving the planet a small eccentricity (around 0.05-0.08 is good) but put the lagrange objects in circular orbits, and view this in a rotating frame using the planet's period. Over time, the lagrange objects will start to develop eccentricity as well and start the looping motion characteristic of Trojan asteroids. I am pretty sure that the langrange object's eccentricity will end up very slowly oscillating over the range 0 – 2e (where e is the eccentricity of the planet).


I'm wondering what happens to the lagrange points if the planet's orbit is eccentric (e.g. e=0.5 or more). Are they still stable?

 
Run the simulation I suggested above and see. I ran it to check my assumption about oscillating eccentricity. I was correct; the lagrange orbit oscillates in eccentricity but remains stable. (Good thing, or else 5,000+ trojan asteroids would be wrong.)
 
There is not a CIRCULAR orbit at L4/L5 that is stable, but if you match the eccentricity of the planet and rotate the argument of periapsis by 60°, you'll get a stable — albeit elliptical — orbit. Although it will always stay 60° ahead of or behind the planet, the LINEAR DISTANCE between them will change (and will always equal the distance between the planet and the sun). I have attached a gsim file illustrating this. It takes a long time, but the L4 object as actually more stable than the apparently motionless L5 object, which will eventually leave its station and start librating all over the place.
 
Quote:
And is there a way to enter the L1/L2/L3 points too? It'd be nice if there was an option in the program to just place objects there to save us having to calculate the locations ourselves first.

 
These points are chaotically unstable, and the errors within the integrator have a deterministic influence on the outcomes. Nevertheless I've written a Macro that generates gsims from Excel spreadsheets and calculates things like Lagrange points. I'll release it here after debugging.
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Re: Lagrange points?
Reply #13 - 01/29/11 at 16:50:02
 
On the page http://orbitsimulator.com/formulas is a Lagrange Point calculator.  It's assuming a perfect 3-body system.  It'll be interesting to see ps' Excel sheet.
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