### Out on the range

Posted on March 10th, 2014 by george.
Categories: Uncategorized.

Got the following question from my friend Sean and thought I’d repost my answer here for those of you looking for a relatively simple explanation of Lagrange points.

Question:

Hypothetically, lets say there are two bodies of equal mass. We’ll assume they’re identical in every respect (size, density, etc). We’ll also assume that the distance between them is always constant. It seems reasonable that exactly half way in-between them their gravitational pull would be negated by the opposite mass. Therefore, if an object were to fly precisely in-between them at that point, it would not be affected by either’s gravitation pull. To the best of your knowledge is this presumption true? And if so, is there a term to describe such a zone of gravitational equilibrium? Or would an object be caught in between?

Your intuition is correct. Such a spot in space is known as a libration point. Where it gets interesting is when you mix in orbital mechanics. God doesn’t reach down with two giant hands and hold two spheres motionless in space. The laws of gravitation tend to put everything into motion, always accelerating unless balanced by another force. On Earth, our bodies want desperately to accelerate toward the center, but the surface of the Earth provides a force that exactly balances that gravitational pull and holds us in one place. The only way two massive bodies can maintain a constant distance between each other is to orbit one another in a shared circular orbit (as opposed to an elliptical orbit, where the distance would be constantly changing). This is the simplest case, called a two-body system. Each object wants to accelerate directly toward the other, but each also has a large amount of momentum directed ninety degrees away from the other body. The centripetal (outward) acceleration due to that momentum exactly balances the gravitational pull between the two objects. So as the sphere moves forward, instead of traveling in a straight line (which it would do if the other body were not there) it gets pulled a little toward its companion. Simultaneously, the companion traveling in the opposite direction gets pulled a little toward the original sphere. Add each increment of forward and lateral motion as you march forward in time step by step, and eventually the two bodies trace a path about a common central point called a barycenter. Here’s how two identical bodies can travel in non-circular orbits around a common barycenter: This is what it looks like when the masses are different. The barycenter of the Earth-Sun system is inside the Sun itself because the Sun is so much more massive. It starts to look like the maps of the solar system you’re accustomed to seeing. Even though the planet is small, you can see that it still causes its central star to wobble. This is one of the methods we use to find planets around other stars. When a planet is large and close to its sun, we can see the wobble with our current telescopes.

In the special case of symmetric bodies in a circular orbit, the barycenter (center of mass) is a libration point. If you placed a third body there, it would not accelerate toward either of the original two spheres. This is not because it is unaffected by the gravitational pull of either primary body, but because those two pulls balance equally.

Now here’s where it gets fun. The gravitational forces and the centripetal acceleration don’t just balance between the two bodies. When you invoke Kepler’s equations of motion and Newton’s law of gravitation and solve for the points in space where the gravitational pull due to both bodies matches the centripetal force needed to orbit in lockstep with them, you arrive at FIVE libration points. Euler discovered the first three that exist in a straight line traced between the two main bodies, but Lagrange discovered four and five, so they named the libration points of a three-body system after him. The one in the center of two identical bodies is the easiest to conceive of, since equal forces pull in opposite directions. This is called the L1 point. As soon as you consider a system with bodies of different masses, however, the barycenter and the L1 point part ways; the former moves toward the heavier body, and the latter marches closer to the lighter one. But there are also points on the outside of the two bodies where all the forces balance.

Let’s switch to a Sun/Earth model so it’s easier to understand. The further you orbit from a central body (the Sun in this case), the slower your orbital speed (but the higher your orbital potential energy). If you aligned a Sun-orbiting satellite with Earth at a much greater distance from the Sun and let it go, the Earth would start outpacing it on the way around the Sun. For example, Mars does not co-rotate with Earth. It follows Kepler’s laws and orbits with a period of about 1.9 Earth years. This is not to say the Earth does not exert gravitational forces on Mars. It most definitely does. The Sun, all the planets, every asteroid and all objects in the solar system (and, to an unmeasurable degree, the universe) feel the Earth’s gravitational pull. The reason the Sun dominates this neighborhood is because it is the most massive. Remember, too, that the Sun is in orbit around the center of the Milky Way galaxy. The reason we don’t orbit the galaxy directly is because we are so close to the Sun relative to the galactic center. Long story short the Earth-generated forces felt on Mars are not as large as the pull of the Sun, and Earth is so far from Mars (even at closest approach) that the overall effect on Mars is tiny over the age of the solar system.

Now if you move the Sun-orbiting satellite close enough to the Earth, its gravitational pull eventually gets strong enough that it drags the satellite along behind it, even though the satellite is still orbiting the Sun in a higher orbit, and should be orbiting more slowly than the Earth. This is L2. For reference, it is outside of the Moon’s orbital distance. Move the satellite too close to Earth, though, and it will orbit Earth instead of the Sun. L3 is a little harder to grasp because it’s all the way on the other side of the Sun from the Earth. But it’s still in a higher orbit than Earth, so it should be going slower. Once again, however, Earth pulls the satellite along and causes it to orbit at Earth speed in its higher orbit where it should have a longer period and be outpaced by Earth. This is in spite of the fact that Earth is two astronomical units (186 million miles) away and the force due to Earth’s gravity traces a line through the Sun itself! If the Earth were the only planet orbiting the Sun, L3 would be a stable libration point, meaning if the satellite drifted the forces would conspire to bring it back (as long as you ignore solar radiation pressure, the fact that the Earth is not a perfect sphere and that the Sun is squashed, among other perturbing effects). Maintaining your position near a Lagrange point (or in any desired orbit) is called “station keeping”. But remember how the Earth is on the other side of the Sun from L3? And remember that Venus exists? In real life the Earth-Sun L3 point is unstable, because the gravitational pull of Venus is much larger than Earth’s pull on a fictional satellite at L3, no matter where Venus is in its faster-moving orbit. Venus will quickly destabilize a satellite at L3.

L4 and L5 are wacky and hard to conceive, which is why Euler missed them and Lagrange got all the points named after him when he discovered four and five. These two are at the point where the distance between the satellite and the Sun and the satellite and Earth are equal to the Earth-Sun distance, forming an equilateral triangle. The forces once again balance and the satellite is held in a higher orbit while still rotating around the Sun at Earth speed. The best way I know to think about L4 and L5 is that the presence of the Earth and its small gravitational pull causes the satellite to orbit the Earth-Sun barycenter, not the center of the Sun. It’s not as easy to understand as the case where the satellite is “pulled along” as in L2 and L3, but if you stare at this diagram long enough, you’ll see that the presence of the Earth is effectively shortening the radial distance between the satellite and the center. As we know from Kepler, shorter radial distance equals faster orbital speed, so the satellite orbits as fast as Earth even though it’s further from the Sun. This diagram shows the Earth and the Moon, but the physics is the same. Here are all the Lagrange points together. Haw haw haw haw. Bonus: you can orbit a Lagrange point in what’s called a “halo orbit”. There’s nothing there for you to revolve around, of course, but your momentum and the gravitational forces of the other two bodies balance perfectly to conspire to make it seem as if there is. NASA’s multi-billion dollar James Webb Space Telescope plans to use a halo orbit around the Earth-Sun L2. Bonus bonus: JWST looks like an imperial star destroyer.  