OK, what exactly are these, and how do you go about finding them?
I was always under the impression that the L points were stable-orbit-type places where earth’s gravity balanced the sun’s gravity. However, I was reading an article about the new Cosmic Background Explorer-type satellite, and it’s supposedly at L2, which is something like 1.5 million km further away from the sun than Earth, and on the Earth-sun line.
So, what gives? Am I just entirely wrong with what I think the Lagrange points are?
::humbly awaiting the assistance of those more knowledgeable than myself::
There are five of them in any two-body system. Using Earth/Moon for our example, two are in the Moon’s orbit, 60 degrees before and behind the Moon. One is at the point where Earth and Moon gravity are equal, on a line between the Earth and Moon. One is at the same point on the far side of the Moon (think this one through – the Moon’s gravity attenuates faster than the Earth’s does, given the difference in masses, so there’s a point beyond the Moon where they equal out again). The final one is on the opposite side of Earth from the Moon, nearer than the Moon’s orbit, again on the line through their centers of gravity, and I confess I have no idea why this one exists whatsoever. Any astrophysics buff care to explain it?
Sure, the earth and moon’s gravity are equal at some point beyond the moon. But, on the far side of the moon, both bodies are pulling in the same direction. Doesn’t seem too stable to me, but maybe I’m just missing something.
Anyway, I’ll go to bed now and let my poor little brain percolate on this, and hopefully I’ll awake enlightened. Or, at least mildly brighter.
I am sure I have read somewhere that the L4 and L5 points (the ones angled ahead and behind the Moon’s orbit) are the only truly stable ones. All are theoretically stable, but an object deviating slightly from the L5 and L4 positions tend to return to that position, while an object deviating slightly from L1, L2 or L3 will drift further and further off course at a steadily increasing speed.
Lagrange points aren’t quite the points at which gravitational forces from the two planets cancel out (there’s only one of those) - it’s where the gravitational attractions hold a small body in an orbit of constant relative location (the 2 planets are not stationary; they’re spinning as a unit).
In the earth-sun system, L1 lies between the two; L2 is on the far side of the sun; L3 is “behind” the earth; and L4 & L5 are in the earth’s orbital path, 60 degrees ahead of/behind it. 1,2 and 3 are unstable, apparently, while L4 & L5 are stable “so long as the mass ratio between the two large masses exceeds 24.96.” I the sun-earth system, it most certainly does.
The linked page gives a general description of the hows & whys, but it shies away from a full-blown derivation. They have a hyperlink to a derivation, but it didn’t work when I tried it. I’m not an astrophysicist, so I don’t know much more than what I read on that site.
The above are correct. The five Lagrange points are as described and take into account the gravitational attraction of the two large masses and the orbital motion of the third mass. The L1-L3 points are considered stable, but not very.
Think of it this way; when an object is at the L1-L3 points, the gravitational attraction of the two larger masses is in the same line. If the object drifts off to one side or the other then the gravitational attraction would tend to pull the object back into the line. (Think of a string stretched between two points, then pull the string to one side.) Unfortunately, the points are only stable if the object drifts off at a right angle to the line. (Hmm… that may only be true for L2. I think there are other specific angles for L1 and L3.) At any rate, if the object drifts in a different direction the forces are no longer balanced and it continues to drift off.
L4 and L5 are fairly stable. Objects which drift off from them tend to drift back. In fact, there are small clusters of asteroids in Jupiter’s L4 and L5 points called the Trojan Asteroids. Because of this, the L4 and L5 points are sometimes called the Trojan points.
If you go into the co-rotating frame (where the Earth and Moon are stationary) the Lagrange points are where gravity and centrifugal forces balance. If you plunk an object down (again, still thinking in the corotating frame of reference) at or near the Lagrange points, it’ll stay stationary, or oscillate around the Lagrange point. (What’s really wacky is the “horseshoe orbit” where the object actually wanders between L4 and L5. . . see Janus and Epimetheus, moons of Saturn, for a real-world example!)
Remember, centrifugal forces are “fictional” forces. In this case they arise because the Earth and Moon aren’t stationary, they’re really orbiting their center of mass. If you think about the problem in the inertial (not corotating) frame, the Lagrange points are just where gravity and the motions of the Earth and Moon consipire to let the object orbit the Earth with the same orbital period as the Moon, as brad_d said.
I’ve actually spent some time at work trying to calculate these things, and keep making stupid little mistakes. Maybe I’ll try sometime this weekend…I seem to be much smarter at home than I am at work.
For a good fictional treatment, try “Sucker Bait” by Asimov - he posits a habitable planet in a binary solar system, by placing the planet in one of the Lagrange points.
Um, I’m not an astronomer or anything, but as I recall, Janus and Epimetheus don’t wander between Lagrange points. The unusual aspect of their orbits is that they are only about 50 km apart, 90000 km above Saturn. As they approach each other, they exchange momentum, and switch places.
I do remember reading about a near-Earth minor planet/asteroid that does have a horseshoe shaped orbit. The web site even had a cute little animation to illustrate the weird orbit, but I’m damned if I can find it now.
Does anyone else think that they’re going to have a hard time getting this out of their heads?
By the way, the horshoe orbit of that asteroid isn’t any different, in principle, from the orbits of Janus and Epimetheus. The Earth has to gain or lose a little bit of momentum at each interaction, but since our hunk of rock is so much bigger, the chance is negligible. The horseshoe shape only comes about because we’re looking at it from the Earth’s frame of reference: Observers on Janus could equally say that Epimetheus has a horseshoe-shaped orbit (except that Janians probably don’t know what horses are).
Yes, they do switch places (i.e. which one is closer and which is farther from Saturn)–that’s a characteristic of the horseshoe orbit. You might have seen diagram of the orbits that didn’t look too much like horseshoes, but that’s because it’s not in the coorbital frame, which is pretty tough to explain, though dynamicists love it.
Here’s a cite: Yoder, et al., Astronomical Journal, vol. 98, Nov. 1989, p. 1875-1889. Sorry, can’t find anything online but vague references in asteroid 3753 Cruithne pages.
Right you are… dug up a fair number of references to Epimetheus and Janus sharing a horseshoe orbit. The source that I remembered (probably “The Nine Planets”) didn’t mention that particular aspect of their orbits (strangely). A Google search of “Epimetheus horsehoe” turned up numerous references to 3753 Cruithne, as well.
Oh give me a locus, where the gravitons focus…
And the three body problem is solved.
Where the microwaves play, down at three degrees K…
And the cold virus never evolved.
chorus
Home, home on LaGrange…
Where the space debris always collects.
We posess, so it seems, two of man’s greatest dreams…
Solar power, and zero-G sex.
You won’t need no oil, nor a Tokomak coil…
Solar stations provide Earth with juice.
Power beams are sublime, so nobody will mind…
If we cook the occasional goose.
chorus
I’ve been feeling quite blue, since some crystals I grew…
Grew too big, to fit through the door.
But from slices I sold, Hewlett-Packard, I’m told…
Made a chip that was seven foot four.
chorus
I’m sick of this place, it’s just McDonalds in space…
And living up here is a bore.
Tell the girls not to cry, when I kiss them goodbye…
'Cause I’m moving next week to L4!
