Lagrange Points

John Baez

January 21, 2009


What are they?

What are "Lagrange points", also known as "libration points" or "L-points"?

These are all jargon for places where a light third body can sit "motionless" relative to two heavier bodies that are orbiting each other thanks to the force of gravity.

For example: imagine the Earth going around the Sun in a circular orbit. Then there are five Lagrange points where we can put a satellite. Three of them lie on a line through the Sun and the Earth. L1 is between the Sun and Earth, L2 is in the same direction as the Earth but a bit further out, and L3 is opposite the Earth, very slightly further out:



The other two Lagrange points are less obvious. L4 lies in the Earth's orbit approximately 60 degrees ahead of the Earth, while L5 lies in the Earth's orbit approximately 60 degrees behind the Earth.

By the way, both pictures on this website were created by Neil Cornish and other scientists who are studying the microwave background radiation with the help of a satellite called WMAP, located at the Lagrange point L2. You can click on these pictures to see more about this.

The Earth, Sun and Lagrange point L4 (or L5) almost form the vertices of an equilateral triangle, but not quite. It would be equilateral only in the limit where the Earth's mass is negligible compared to the Sun.

The really interesting thing is that the Lagrange points L4 and L5 are stable equilibria as long as the heavy body - the Sun in our example - is more than 24.96 times as massive as the intermediate-sized one. In other words, a small body put in orbit at the right speed at either of these points will tend to stay there! If you nudge it a little, it will drift back.

On the other hand, L1, L2, and L3 are unstable equilibria: the slightest nudge will make satellite at one of these points drift away. To keep it there, you'll need to equip it with thrusters that keep correcting its orbit now and then.

The stable Lagrange points are the most interesting for astronomy, because stuff tends to accumulate there. For example...

Trojan asteroids

So far a total of 1691 asteroids called "Trojans" have been found orbiting the stable Lagrange points of Jupiter's orbit around the sun. You can see them as blue dots in this animation:

The asteroid belt is the green stuff - ignore that. Look for Jupiter moving along the outermost circle, and see how the blue dots are clustered 60 degrees ahead and behind Jupiter. These are the Trojan asteroids!

The first of these was discovered by the German astronomer named Max Wolf in 1906, using a technique he pioneered: comparing two photographs taken on different night and seeing if any of the "stars" had moved. He discovered lots of asteroids this way! The first Trojan he discovered is called 588 Achilles, since it was the 588th asteroid to be discovered. It's 116 kilometers in diameter.

In general, the Trojan asteroids at L4 are named after Greek soldiers in the Trojan war, like Achilles, Nestor, Agamemnon and Odysseus. Those at L5 are named after guys on the Trojan side, like Patroclus, Priamus, and Paris.

I must thank Eric Vosselmans for informing me of the only two exceptions: Hektor, the Trojan prince who somehow wound up at L4, and Patroclus, the Greek companion of Achilles who accidentally strayed over to L5. But basically it's Greeks at L4, Trojans at L5:

For some mysterious reason there are more big asteroids at L4 than L5. Maybe this is why the Greeks won the war! But seriously: why are there more asteroids at L4? One of my correspondents has suggested a simple explanation: as Jupiter runs into asteroids, they tend to get stuck at L4 because it's in front of Jupiter. Could that be all there is to it? Maybe this could make a nice Ph.D. thesis in astrophysics.

Mars Trojans, Neptune Trojans, and Earth's strange companions

The asteroid 5261 Eureka is a "Mars Trojan", occupying the L5 point of Mars' orbit around the sun. It was discovered by David Levy in 1990. It's reddish in hue and it's deep within the stable Lagrangian zone of Mars, which suggests that it's been in this orbit for most of the history of the Solar system.

Some other Mars Trojans have been discovered since - at least four at L5 and one at L4. Hmm, this time there are more at L5! The plot thickens.

When I last checked, there were known known Saturn Trojans, but seven "Neptune Trojans", with unglamorous names 2001 QR322 and 2004 UP10. Asteroids in the outer solar system are called "Centaurs", but not too many are known. As of 2005, only about forty Centaurs are known that enter the region between Saturn and Uranus.

Why so few? Well, most of these have been discovered since 2000, so we may just be starting to find them. But, researchers at the University of California at Los Angeles and elsewhere have done some computer simulations to study this question. They found that niches for stable orbits are rare in the outer solar system. Amusingly, they named the hypothetical Saturn Trojans "Bruins". Why? Because that's what UCLA's football team is called - and they wanted to get back at astronomers at the University of Southern California, whose football team is called the Trojans!

Unfortunately, the Trojans are way ahead of the Bruins.

Are there any "Earth Trojans" - asteroids at the Lagrange points of the Earth's orbit around the Sun? Astronomers have searched for them, and gotten some tantalizing possible glimpses, but there's no definitive evidence. This website is the place to go for more information - it even has some movies showing what Earth Trojans would do if they existed:

Though no Earth Trojans are known, our planet does have some amazing companions. The best known is the asteroid 3753 Cruithne. Cruithne is not at a Lagrange point. Instead, it moves in a very complicated spiralling horseshoe-shaped orbit relative to the Earth! For a beautiful discussion of this by the discoverers of this phenomenon, complete with animated movies, see:

An even more complicated companion asteroid of the Earth is called 2002 AA29. It oscillates between being a quasi-satellite of the Earth and having a horseshoe orbit!

Extra moons, alien spaceships, space colonies and dust clouds

What about the Lagrange points of the Moon's orbit around the Earth? Is anything lurking there? If the Earth had any smaller moons besides the one we all know, this is where they'd probably be!

In fact, people have been looking a long time for extra moons orbiting the Earth. Some people even claimed to find them! The history is pretty interesting. It began when Frederic Petit claimed three observers had seen a small second moon... but it picked up steam when Jules Verne read about this and put it in his novel From the Earth to the Moon:

""Is that possible?", exclaimed Michel Ardan, "the earth has two moons?"

"Yes, my friend, it has two moons, although it is usually believed to have only one. But this second moon is so small and its velocity is so great that the inhabitants of Earth cannot see it. It was by noticing disturbances that a French astronomer, Monsieur Petit, could determine the existence of this second moon and calculated its orbit. According to him a complete revolution around the Earth takes three hours and twenty minutes.... "

This got amateur German astronomers searching for a little moon they called "Kleinchen"... and in 1898, one Dr. Georg Waltemath claimed to have found a whole system  of small moons! In 1918, an astrologer even added an extra moon to his horoscopes: the dark moon "Lilith". For more of this amusing history, you've gotta read this:

However, the most recent and scientific search for objects at the Earth-Moon Lagrange points seems to be this:

Believe it or not, Freitas claims to have found an alien spacecraft at L4!

Just kidding... this isn't one of those crackpot websites you've probably seen. Actually, his search proved that no solid object the size of Skylab or larger is parked at L4 or L5.

But that doesn't mean these places are completely boring. In fact, there are dust clouds at L4 and L5. They're about four times as big as the Moon. They're very wispy and insubstantial, but they've been photographed a couple of times. In 1990 the Polish astronomer Winiarski found that they are a few degrees in apparent diameter, that they drift up to ten degrees away from the Lagrange points, and that they're somewhat redder than the usual "zodiacal light" - the light reflecting off dust in the solar system.

In 1974, a physics professor at Princeton wrote a paper that advocated putting space colonies at one of the Earth's Lagrange points, namely L5:

The idea caught on and an organization formed to advocate doing this: the L5 Society! They merged with the National Space Society in 1987, but you can still read their old newsletters:

But so far, it seems there's just dust at the Earth-Moon Lagrange points. So, what about the Lagrange points of moons of other planets?

Well, Saturn has an icy moon called Dione, 560 kilometers in diameter - and 60 degrees ahead of Dione, right at the Lagrange point L4, there is a tiny moon called Helene! Here's a picture of Helene:

There's also a tiny moon called Polydeuces trailing behind Dione at the L5 point: Even better, Saturn has another moon called Tethys, quite similar to Dione, 530 kilometers in diameter, which has two Lagrange moons. There's a little one called Telesto at the L4 point, 60 degrees ahead of Tethys, and another little one called Calypso, 60 degrees behind Tethys!

Does anyone know other natural occupants of Lagrange points? You'd think that some of the big moons of Jupiter would have some of their own Trojans. Jupiter has plenty of moons: Metis, Adrastea, Amalthea and Thebe, Io, Europa, Ganymede, Callisto, Leda, Himalia, Lysithea, Elara, Ananke, Carme, Pasiphae and Sinope, and eleven more discovered in 2000 and named in 2004. But I don't know any of these that are sitting at Lagrange points!

This leaves artificial satellites of the man-made sort.

Satellites at unstable Sun-Earth Lagrange points

Despite the desires of the L5 society, I don't know of man-made satellites at the Lagrange points of the Earth-Moon system. However, people are already putting artificial satellites at the Lagrange points of the Earth's orbit around the Sun! L1 is home of SOHO, the Solar and Heliospheric Observatory. This allows SOHO a nice clear view of the Sun. In April 2001, NASA put a satellite called WMAP into the Lagrange point L2. WMAP stands for Wilkinson Microwave Anisotropy Probe: it measured bumps in the cosmic microwave background and discovered amazing things about the amount of dark matter and dark energy in the universe.

Amusingly, a blob of dust trailing the Earth showed up nicely in measurements of the cosmic infrared background radiation by a precessor of WMAP, the famous satellite COBE:

This blob has a radius of about a tenth that of the earth's orbit. So far, attempts to find similar dust clouds at the Lagrange points of Mars and Jupiter have failed:

Perhaps the asteroids there screw things up?

The interplanetary superhighway

In fact, the Lagrange points are useful for much more than just parking  a spaceship - they also play a crucial role in a clever system for travelling around the Solar System while using almost no fuel! It's called the "interplanetary superhighway", or IPS for short. The unstable Lagrange points L1 and L2 are the important ones here, since you can easily slip onto them and then back off. They're like stepping-stones in a game of interplanetary hopscotch!

In fact, they've already used the IPS for a number of space missions, including the Genesis mission - the one that collected samples of solar wind, then came back to Earth... and then crashed into the desert in Utah at 320 kilometers per hour because of a faulty switch.

I hadn't heard about the interplanetary superhighway until Nathan Urban explained it to me. Here's what he said:

I've just recently been reading about the "Interplanetary Superhighway" (IPS). Have you heard of it? If not: It turns out that the L1 and L2 points act as "gateways" for very low-energy transfer orbits - better than Hohmann transfers. The IPS works by transferring between unstable and stable manifolds winding off and on unstable orbits like at L1 and L2.

The recent Genesis probe, the one that crashed, used such an orbit. It went out to the Earth-Sun L1 point, made 5 orbits on some crazy Lissajous curve to collect the solar wind, then returned back to Earth, passing by the L2 point to delay reentry until daylight in Utah for the planned helicopter capture - all without deterministic trajectory changes! They did have to use engines to correct the orbit, since they didn't get it exactly right and there were other requirements of the spacecraft, but in principle you could come up with a wacky orbit like this that would have completed the entire Genesis mission and returned the probe to Utah without any course corrections past the initial boost.

In fact, you can design an orbit that goes around the Sun some number of times, the Earth-Sun L1 point some number of times, the Earth some number of times, the L2 point some number of times, and the region outside the Earth's orbit some number of times - for any  list of numbers you want!

Before being fully understood as such, the IPS was first used in the ISEE3 mission in 1978. Later in 1991, it was used to rescue a Japanese satellite called Hiten stranded in the wrong orbit without enough fuel to reach the proper orbit by traditional means like Hohmann transferring.

As you mention on your page, it's now popular to put satellites (like Genesis, SOHO, WMAP) etc. in "halo orbits" at Sun-Earth Lagrange points. Well, it turns out that using an IPS trajectory, you can go from the Earth-Moon L1 point to the Sun-Earth L1 point or back with a course correction of only 50 meters/second!! This has led to a proposal for a "Lunar L1 Gateway", a manned space station at the lunar L1 point that can be used to service missions at the solar L1 point - just bring them back to the gateway station for servicing, and send them back to their halo orbits afterwards, with almost no fuel.

They've also used the IPS to design a Jovian "Petit Grand Tour" mission to visit four of the Jovian moons on a screwy trajectory that has it oribiting each moon a certain number of times and going on to the next, using little or no fuel.

It has been shown that Comet Shoemaker-Levy 9, the one that crashed into Jupiter, was following one of these chaotic IPS orbits through a Lagrange point. Apparently there is some evidence that the dinosaur-killer asteroid followed one to Earth.

I found the IPS to be an interesting application of chaos theory... you heard a lot about it in the '80s and early '90s, it was very faddish, then you didn't hear much about it anymore... this is one application not just for pure science (e.g., stability of the solar system), but applied science (transportation within the solar system).

Nathan also sent me a lot of cool links to more information on the interplanetary superhighway. You can find these at the end of this page.

The mathematics of Lagrange points

Finally, before I quit, I want to say a bit about the fascinating mathematics of Lagrange points.

In 1772, Joseph-Louis Lagrange shared a prize with Euler for work on the orbit of the Moon. Euler had discovered L1, L2, and L3, but Lagrange did a more complete analysis of the problem including L4 and L5 in his Essai sur le problème des trois corps. You can read this paper if you go here, choose "Lagrange" as the author, select the 6th volume of "Oeuvres / Joseph Louis de Lagrange. 6 / publ. par les soins de J.-A. Serret" and go to page 229.

For a modern proof that there are exactly five Lagrange points, formulas for where they are located, and a proof that L4 and L5 are stable while the other three are not, try this:

I can sketch the argument here, but it gets pretty tricky when it comes to the stability of L4 and L5.

Let's use the Sun-Earth system as an example. If we work in a coordinate system that rotates around the Sun along with the Earth, we see that a light third body would be affected by four forces:

Luckily, the Coriolis force is zero when the smaller third body is at rest in this rotating coordinate system - this simplifies the analysis! The gravitational force pulls this body towards the Sun and the Earth. The centrifugal force repels it from the Sun, and gets stronger further away.

Thus, to find the Lagrange points we need to see where the gravitational force and centrifugal force cancel out. One can work out where this happens:



The lines here are "potential lines" - lines of a potential whose gradient is the gravitational force plus the centrifugal force. The Lagrange points are "critical points" of this potential - places where the force is zero. An object exactly at one of these points will remain there. But if you nudge it slightly, it will tend to move in the direction of the arrows. It will either move back toward the Lagrange point, as shown by the red arrows, or further away, as shown by the blue ones.

In some cases this is actually easy to see without any calculations. For example, suppose we have a body perched at L1, between the Sun and the Earth. If it moves any closer to the Earth it will be more attracted to the Earth, because the Earth's pull will be stronger and the Sun's weaker. Similarly, if it moves any closer to the Sun it'll be more attracted to the Sun. So, it's like a ball sitting on a hilltop: the slightest nudge in either direction and it will roll off! That's why see blue arrows pointing to left and right at L1.

Similarly, suppose we have a body at L2. If it moves closer to the Earth it will be more attracted to the Earth, because both the Earth's and Sun's pull will be stronger - this time they're pulling in the same direction. Conversely, if it moves away from the Earth it will be less attracted to the Earth. So again, we see blue arrows pointing to the left and right at L2. L3 is similar.

It's a little harder to intuitively explain the red arrows at L1, L2 and L3, but it's not all that hard for L1 and L2. Think about it.

The really tricky cases are L4 and L5. It's hard to see why these are places where the gravitational force plus centrifugal force add to zero. And it's even harder to see why these are more stable locations than L1, L2, or L3.

In fact, these Lagrange points look even worse than L1, L2 or L3, since they have no red arrows: no directions you can nudge an object where it will tend to move directly back.

But in these cases, the Coriolis force also plays a crucial role! If the object at L4 or L5 starts moving either towards or away from the Sun, the Coriolis force will push it to one side, sending it into orbit around the Lagrange point! And it turns out this effect manages to stabilize the equilibria at L4 and L5.

This argument should not be enough to convince you - and indeed, I don't see an easy way to explain this stuff to a skeptic using nothing but plain English. Neil Cornish's mathematical proof uses calculus and matrices! It's not that complicated... but it's a great example of how gravity has tricks up its sleeves that can only be revealed using math.


References

Here are Nathan's references on the interplanetary superhighway: Here are some other online references for further study: Bryan Erickson pointed me to the following papers about computer simulations of asteroids in the outer Solar system - including the hypothetical "Bruins", or asteroids at the Lagrange points of Saturn. Note that some of the coauthors are from UCLA:

Here are some other references, which unfortunately are not available for free online. It's sad that even with the rise of the free online archive, astronomers are still burying their papers in journals that can only be accessed for a fee. But, you'll see from the abstracts that people are looking for Earth Trojans and even Trojan planets in other solar systems!


© 2009 John Baez
baez@math.removethis.ucr.andthis.edu

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