New here: Hello all!

Read an article in the May issue of American Scientist re: low energy space travel using the Lagrange points (among other sources of momentum). It re-energized my curiosity about the Lagrange points.

The article talked about how the L1 and L2 points can be orbited by objects smaller than the primary and secondary gravity source. However, it left off the L4 and L5 points. The illustration showed how the smaller gravity source shapes the space near it within the larger gravity source's "well" to enable a much smaller object to orbit the L1 or L2 points. However, the article did not go on to discuss the L3, L4 or L5 points.

Can someone explain how the smaller object shapes space to create the L4 or L5 points? How does the L3 point happen at all? Perhaps you could point me to a good article on the subject?

Thanks in advance.

The Lagrange points are balance points between three forces: the gravity of the central mass, the gravity of orbiting mass, and the centrifugal "force" attendant on the rotating reference (since all these points rotate around the central mass with the same angular velocity as the orbiting mass). Centrifugal force ramps up steadily as we get further from the centre of rotation and is always directed outwards, whereas the gravity due to one of the masses decreases as the square of our distance from it, and is always directed towards the mass.
If we draw a line connecting the two masses and rotating with them, we can find three balance points between the three forces: these are L1, L2 and L3. There's one point between the two masses, at which the gravity of the central mass cancels the gravity of the orbiting mass + centrifugal force; one outside the orbiting mass, where the combined gravity of the central and orbiting masses are sufficient to balance centrifugal force; and one just outside the orbit of the orbiting mass but on the far side of the central mass, where the combined gravity of the orbiting and central masses is again sufficient to counter centrifugal force. (Since the gravitational effect of the orbiting mass is usually pretty small at that distance, this Lagrange point is usually no more than a tad further out than the orbiting mass's orbit.)
L4 and L5 are positioned at the apex of an equilateral triangle based on the central and orbiting mass. So they're equidistant from both masses, and sitting on the orbit of the orbiting mass. In this situation, the combined gravity of the two masses pulls directly towards the centre of gravity of the system, and is again sufficient to counter centrifugal force precisely. The interesting thing about L4 and L5 is that (within certain broad limits of the mass ratio of the two gravitating masses), they're stable to perturbation: stuff will lollop around them without needing to be steered. In contrast, L1, L2 and L3 are always unstable: once something starts to drift away from them, it needs some restoring force (like a station-keeping rocket engine) to push it back towards equilibrium.

Grant Hutchison

A picture is worth 1000 something or other....
L-points diagramed here with effective potentials drawn:
http://map.gsfc.nasa.gov/m_mm/ob_techorbit1.html

G^2

Was the article about Sun-Earth Lagrange points or the Earth-Moon Lagrange points? I suspect Sun-Earth because only L1 and L2 would be practical. The L1 and L2 points for the Sun-Earth system are near the Earth (a few hundred thousand km, I forget the precise distance), whereas the distance from Earth to L4 and L5 is about equal to the Earth-Sun distance--roughly 150 million km. And then L3 is on the other side of the Sun from us about 300 million km away. Thus only L1 and L2 are of any use in space travel. In fact, you'd probably have to use them if you wanted to get to any of the other Lagrange points.

Now there are also Lagrange points for the Earth-Moon system, and the Earth-Moon L3, L4 and L5 points are at about the same distance from the Earth as the Moon is. They might come in handy some day. The L1 and L2 points are close to the Moon and probably too risky to use. Hope this helps!

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CM - The article was mostly about the solar "superhighway" - using the unstable L1 or L2 points for low energy spacetravel. One of the things they talked about was a "hand-off" from the Sun-Earth L1 / L2 point to an Earth-Moon lagrange point. Interesting ideas which apparently are already in use.(Genisis mission, e.g.)

Grant H. brings an interesting point - he suggests (if I am reading correctly) the stable L4 and L5 points are created by equalibrium between the pull of the smaller body and the larger body on the third object. Yet the article supplied by GSquare suggests a curvature of space (unless that is merely a mathmatical curve). The Cornish article suggests that the L4 and L5 points are "hilltops" and when the third object begins to 'roll' downhill, the coriolis force then causes the third object to orbit the point.

Which brings two more questions: In Grant H.'s explanation, isn't the center of mass of the Sun-Earth system within the Sun(?), and if so, how is the mass of the Earth and its orbital velocity creating stable points both ahead and behind itself at one AU? If the pull is toward the center of mass, why do we have Lagrange points at all, and not just an object moving at solar orbital velocity on the same orbital plane as the Earth? (Are L4/L5 actually points at all?)

If Cornish is correct, and there is a change in the shape of space, can someone explain the effect of the coriolis force in a frictionless environment, and how that allows something to "orbit" a "hill" that isnt "there". (I'll stop there before I get more confused!)

I admit, I like Cornish's graphics, but Grant H.'s idea seems more logical.

I think L4 and L5 are not actually stable, the coriolis deflection keeps the small body from getting too far away but instead it meanders in a "horseshoe orbit" that alternately visits the L4 and L5 points, spending much more time near the LaGrange points than elsewhere (ergo the Trojan asteroids).

It is merely a mathematical curve, it is the effective potential. This has nothing to do with general relativity, which plays no important role in any of this, and there's no actual changes in the shape of space.

I'm not sure I understand the question, but Grant's description was quite accurate.

Ah, I think I see what you're asking. The LaGrange "points" are only points in the reference frame that orbits with the Earth, the "co-orbiting" frame. In the Sun frame, you are right, they go around just like the Earth does. This is true for all the LaGrange points. Note this is important for the article in question, because anything launched from the Earth picks up the Earth's 30 km/s of speed, and you would certainly not want to have to "stop" that!

"Stable" isn't quite the right word, I guess. What I meant was that, assuming the mass ratio of primary and secondary is below a critical threshold, things that start off comoving in the vicinity of L4 or L5 will assume a stable motion around that point, rather than wandering of into the void the way objects at L1, L2 or L3 would do.

Or a tadpole orbit, which is what the Trojans generally do: a broad "head" around the Lagrange piont, and a long "tail" extending back along the orbit towards L3. If they have enough energy they'll leak through L3 into the tail of the other tadpole, and end up doing the full horseshoe. I believe some of the Trojans asteroids can do this trick, and then settle around the opposite Trojan point, like defectors between the Greek and Trojan camps, though I don't know if this has been observed or merely theorized.

Grant Hutchison

I meant to ask ... was that an American comparative "quite" or a British superlative "quite"?

Grant Hutchiso

Quite so. (Seriously, I don't understand the difference-- can quite be comparative?)

[quote=grant hutchisonI believe some of the Trojans asteroids can do this trick, and then settle around the opposite Trojan point, like defectors between the Greek and Trojan camps, though I don't know if this has been observed or merely theorized.
[/QUOTE]
I don't know either. I don't think the timescale is suitable for actually observing it, I presume it would require many Jupiter orbits, perhaps many many! But the answer is probably known just from measuring the Trojans' energies.

It's the difference between "I'm quite sure" (I'm not entirely sure} and "I'm quite sure" {I'm certain). The second, superlative usage is supposedly more common in British English, with words like "rather" or "moderately" being deployed for the comparative role.
I think it depends a lot on where you were born and where you went to school, though.

Grant Hutchison

The center of mass (COM) of the Earth/Sun system is within the Sun, but in general the resultant pull of the two bodies is not towards the COM (just look at yourself--you're more attracted to the center of the earth than the COM). Since the earth rotates about that center of mass, the stable points would be points that also rotate about the center of mass.
I think that's what is meant by a stable point.

If you look at the graph of x^3-x^2, there is a hill at x=-1 and a valley at x=+1. An object at the top at x=-1 would be stationary, but unstable. Any push would send it far away from x=-1. An object at the bottom at x=+1 would also be stationary, but any push would send it rocking back and forth forever, around x=+1. That's stable though.

Well, the point itself isn't particularly stable. It's just the focus of a region which is quite tolerant to prturbation. In the real world, the chance of an object sitting stably precisely at the Trojan point for any period of time is negligible.
That's the shade of meaning I was trying to convey, or at least to stay consistent with.



Grant Hutchison

Thanks for the answers and input thus far! - More Q's:

I think I am starting to see how the coriolis effect works to establish equilibrium at L4 / L5 - but the idea of a "horseshoe orbit" around both points is throwing me a bit... I can picture the orbit, but not what keeps it stable (i.e. if the third object falls off from L5 and is passed by L3, how does L4 pick it back up and give it enough inertia to speed past L3 again on its way back to L5?)

Here's another question: L3 is supposed to be opposite from and slightly farther out from the COM of the system than the smaller mass - i.e. in the case of the Sun/Earth system, slightly farther than 1 AU and on the far side of the Sun. Presumably, this is because the equilibrium is established through countering the centripetal force of the combined gravity of the Sun and the Earth by the greater centrifugal force on an object at that distance in the rotational frame of reference.

If this is the case, is the earth's gravity merely passing through the sun and adding its effects to the total gravitational pull at that point? (meaning there is no such thing as a "gravitational shadow", and that an object falling toward the COM of the system falls faster if it is travelling in regions where it is aligned with the two large masses, (until it gets to a point where the gravitational attraction of either mass exceeds the pull of the other)?

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hhEb09'1 wrote - The center of mass (COM) of the Earth/Sun system is within the Sun, but in general the resultant pull of the two bodies is not towards the COM (just look at yourself--you're more attracted to the center of the earth than the COM). Since the earth rotates about that center of mass, the stable points would be points that also rotate about the center of mass. (havn't figured the quote function yet)

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hhEb09'1 - isn't that just because I am so close to the earth that its pull overwhelms that of the sun? -- I mean, if we imagine a solar system of just the Sun and Earth (no other massive objects) - wouldn't a distant third object entering the system be drawn toward the COM unless and until it entered a region where the gravity of either single body overwhelmed the general attraction of the whole system?

The reason I ask this is because I presume that the Earth/Sun Lagrange points aren't specifically tied to the Earth itself, but are actually measured from the Sun and the COM of the Earth/Moon system - and that Jupiter's Trojan points are actually derived from the COM of Jupiter and its moons, not merely the COM of Jupiter itself. Is this correct?

I think you're maybe imagining that L4 has to turn the object around to send it back to L3. That's true only in our rotating reference frame. In a non-rotating frame, all we'd see is an object that alternatively moves around the sun a little faster and then a little slower than Jupiter, so it catches up as far as one Lagrangian, but begins to slow as it passes it, so that it eventually lags behind Jupiter. Then it lags for many years while it passes through L3, and approaches the other Lagrangian, which speeds it up again. The velocity changes are low and the time scale large.

There's no such thing as a gravitational shadow. An object will fall under the force exerted by whatever combination of masses exists around it, but closer masses will exert a force larger according to the inverse square of their distance: this means that in general an object won't fall towards the centre of mass of the system; it requires unusual geometry for that to happen, such as the alignment of masses you mention.

Grant Hutchison

I understand what you were trying to convey, but I think that it is at odds with what we actually mean when we talk about stable points. In fact, I'm pretty sure that if one were able to set an object at one of those points, with the appropriate conditions, it would stay there for longer than a negligible period of time.
But the Lagrange points are not that distant. They're actually at the balance points where the effect of the Earth and the effect of the Sun add up so that the result does point to the COM with the appropriate magnitude--otherwise, the points would not be lagrange points.

I do believe that all Lagrange points are unstable in the sense that if you put an object right at one of them, it will "roll off" right away. The issue about L4 and L5 is that the coriolis effect quickly deflects such an object, similarly to the way air is deflected to circulate around a high pressure center. The high pressure center is not a stable point, yet the feature persists. I don't know how that works at L1 and L2, but apparently not as well, because people always talk about needing thrusters to maintain that orbit.

The L4 and L5 points are truly stable.That appears to be the analogy expressed at Gsquare's link, the lagrange point analysis by Neil Cornish. He talks about L4 and L5 being at the top of potential hills, but he is talking about his effective potential. That's an effect of the choice of coordinate system, which, in the language of Cornish's approach, ends up being a potential hilltop, but subject to coriolis forces that keep a particle from leaving the hilltop.