Celestia Forums

Hi,

Does anyone have any definitions for the stable kidney-shaped halo orbits around the Earth-Moon L4 and L5 libration points.
From my research, they are about 1/2 million [edit:miles] long and have a period of about 89 days, but that's all I've found to date.
EDIT: Further research suggests these are probably difficult if not impossible to simulate accurately using keplerian elements, as they have more to do with "manifolds" and the Interplanetary Superhighway.
See this: http://www.freemars.org/l5/aboutl5.html
It's the green bits I'm interested in.

My current approach is to simulate these with a highly eccentric elliptical orbit around the L-point, which is not at all accurate.
Click on the picture for a demo: (~700kb)


If anyone's got any better ideas, I'd be glad to hear them.

I don't think that animation gets it right. The long axis on the white ovals should always align as close as possible with the Moon's orbit, and never go perpendicular to it. They're known as tadpole orbits. Tadpole orbits with enough energy to make it beyond the Moon's L1 point would be know as horseshoe orbits.

Celestia does not model gravity. So they are locked into whatever Keplerian orbit their starting conditions dictated.

Newtonian physics (this does not conflict with Kepler) numerically integrated would show these orbits properly evolving.

Since the Sun's gravitational tidal force through the Earth / Moon system is strong, and since the Moon's orbit around the Earth is elliptical, objects orbiting the L4 & L5 points in the Earth / Moon system are unstable. I've been able to simulate them (not in Celestia) for as long as 20 years, but they always escape.

I don't think that animation gets it right.

Nope, but I've just about given up on being able to model anything nearing reality in Celestia. (with regards to halo-orbits)
Of course the orbits I've depicted are impossible in reality as there's no massive object located at L4 or L5, but they do give periodic close passes to Earth and Moon.

Celestia does not model gravity. So they are locked into whatever Keplerian orbit their starting conditions dictated.

I've put in a feature request: http://celestiaproject.net/forum/viewtopic.php?p=67880#67880
(but I'm not holding my breath )

Newtonian physics (this does not conflict with Kepler) numerically integrated would show these orbits properly evolving.

But I don't think this can be done in Celestia. It's ellipses only! I might just have to settle for a circular orbit around L5.
Do you know if the orbit of Cruithne and it's relationship to Earth, is at all analogous to these halo orbits and their relationship to the libration points?

...objects orbiting the L4 & L5 points in the Earth / Moon system are unstable...

That's interesting, first time I've heard that. All the research I've read has suggested that the Coriolis force would keep an object in a stable orbit around L4 or L5 indefinitely. (assuming the right starting conditions)

Orbits around L4 and L5 points aren't really orbits at all. The object's only true orbit is around the parent body (Earth in your example). The "orbit" around L4 or L5 is simply a matter of perspective.

Look at the bottom 2 images on this page http://www.orbitsimulator.com/gravity/t ... frame.html
The right image shows the orbits of 2 objects, one around L4 and one around L5. But it only appears this way because the perspective frame is rotating. Without rotating it, you just have 3 objects orbiting the parent object in similar orbits, as shown in the bottom left image.

You call them halo orbits, but halo orbits are around L1, L2, and L3, and must be powered orbits where a spacecraft is capable of adjusting its own orbit since these L points are unstable.

The orbits you describe are tadpole orbits. In the bottom right image in my link, you can see that they sort of look like tadpoles.

As these objects approach the L3 point, they don't have enough energy to cross it and are turned back. If they did have enough energy to cross L3, they would continue to the other L point (4 or 5), circle around it, pass L3 again, circle around the other (4 or 5) point, and repeat this indefinately.

This is known as a horseshoe orbit. Here's a page I put together horseshoe orbits. http://www.orbitsimulator.com/gravity/a ... eshoe.html

Cruithne is very analogous to these orbits. Cruithne is in a horseshoe orbit, with a high eccentricity that gives it a kidney-bean shape. Its orbit also has high inclination, allowing it to "miss" one of the L4 or L5 points. Periodically, pertabutions from the other planets will alter Cruithne's orbit so it doesn't have enough energy to pass the L3 point. It will then be in a tadpole orbit until it is perturbed back into a horseshoe orbit.

Here's a link to my Cruithne page
http://www.orbitsimulator.com/gravity/a ... ithne.html

Thanks Tony,

I found your program very good for visualising these orbits, especially lagrange.gsim. (Is lagrange.gsim an accurate simulation of what the actual orbit of a lagrange captured object would be?) If that's the case, then I just need to work out how to convert your orbitsimulator parameters into Celestia's ElipticalOrbit definitions.

In laymans terms, the way I think of this orbit, in order to visualise it, is to think of an object initially at the L5 position in an orbit modelled on the Moon's orbit: It's orbit will be modified or perturbed by the Sun''s influence as the object orbits the Earth, such that when it is travelling towards the Sun it's orbital velocity relative to Earth will be increased by the Sun's influence, thereby moving it into a lower Earth orbit and ahead of the L-point; As it reaches the point between Earth and Sun it will be at it's closest approach to Earth, and the Sun's influence is reversed, such that it is now being slowed by the Sun, therefore it's orbit widens and slows again, and as it continues it's outward journey from the Sun, it is slowed further so that it now passes outside and behind the L-point. It will be at it's slowest and widest (apogee??) when it reaches the far side of the Earth from the Sun.
To me, this equates to an elliptical orbit modelled on the Moon's but with a much greater eccentricity. (This can definitely be modelled to some approximation within Celestia). It seems to me that this ellipse's axis must also advance as the Earth orbits the Sun, so that (perigee?? periapsis??) always occurs at the conjunction between Sun and Earth. (Not sure how to model that in Celestia)

All this means that the object orbits around the L-point in the opposite direction to it's orbit around the earth. ie. Clockwise rather than anti-clockwise. (This is confirmed by your simulation)

The thing I haven't grasped yet, is how to extend this conceptual model to many objects following each other around L5, but I think that would equate to varying their individual Epoch or MeanAnomaly/MeanLongitude's (in Celestia terminology), but I think that would have to include a corresponding variation in the axis alignment as well, to ensure each object reaches perigee at the appropriate time for it's epoch.

Am I thinking about all this in the right way?
Sorry if my thinking is a little simplistic, but I'm no planetary scientist.

PS. The other thing I haven't worked out, is why it's effective orbit around the L-point is 89 days as reported on some websites, but I'm thinking this may be to do with the fact that it's effective motion around the point, is retrograde to it's orbital motion.

I think I see the problem here, as the L4 and L5 points are being misinterpreted as orbiting around a point- Because they're not. Why?

Think for a moment- The Moon's orbit around the Earth is elliptical, not circular, but the L4/L5 points are always 60deg. either side of the Moon, providing that the relationship is seen along a circle- Which it isn't circular. Sooo... The L4/L5 points move in a constant relation to the Earth/Moon relation, and seems to "orbit" a common point- Which it doesn't.

Think of this invisible "circle" growing and shrinking depending on the Moon's distance- and angle- from the Earth.

Does that make any sense?

(Ah, Cruithne, a favourite subject of mine- One of several co-orbital "quasi-moon" NEOs... )

d.m.f.