Why are Lagrange points so crucial to astronomy? What makes even simple gravity so difficult? How do you “put” something at a Lagrange point? I discuss these questions and more in today’s Ask a Spaceman!

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EPISODE TRANSCRIPTION (AUTO-GENERATED)

uh, general relativity is so hard, Einstein, this Einstein that 10 coupled nonlinear equations, black holes, gravitational waves When Yeah, we all talk about how hard gravity is. How how general relativity are equations to describe gravity are ferociously difficult to solve. They are are they are a real pain in the neck, folks. I'm serious. They're nasty. In fact, they're so nasty that there's only like, a dozen or so known full solutions. Everything else has to be done on the computer in order to make any progress. And and it's amazing that we're able to discover things like black holes in the first place. I mean, for for decades after Einstein developed GR. Basically, nobody used it because it was so dang hard. And yes, general relativity solutions for gravity are hard, but everyone acts like old school.

Newtonian physics is some kind of cake walk like Oh, yeah, yeah, yeah. Yo, you don't mess with Einstein and GR. That was a real feat of of human intelligence and a towering intellect and and but But Newton stuff. Yeah, whatever. You know, it's just easy. We teach it in high school. That's how easy it is. It's true we we teach Newtonian physics in high school and Einstein in physics in graduate school. It's It's different. And And, yes, Newton's Law of Gravity is much easier to write down. Uh, Einstein's equations for general relativity. There are 10 equations involved. They're all coupled. They're all connected to each other. You have to solve them all at the same time. Uh, they, they're they're a beast. And Newton's laws, it's It's just one equation. You know, The force of gravity is proportional to the masses of the two objects, uh, divided by the distance between them squared. And then there's this constant, the Newton's gravitational constant that floats around it like that's That's not that big of a deal.

That's a simple equation. But there's one thing I've learned in physics. It's that simple. Doesn't always mean easy. And the kind of Newtonian physics that you learn in high school that I learned in high school. Yeah, those are easy problems. II. I shoot a cannonball into the air and you and you calculate how long it will take to fall to the Earth. Yeah, you can do that. And for a really any kind of two body problem where you're looking at the gravitational interaction of two objects like the Earth and the moon or the earth in the sun, Uh, or a ball that I just tossed into the air in the rest of the earth. When I'm when I'm calculating the gravitational force of that, I can find what's called an exact solution. I can write it down with pencil and paper, a solution for that problem. And then once I have that solution, I can make any prediction I want. I can go as far into the future as I want. I know exactly how the moon will behave. I know exactly how the earth will behave.

I know exactly how that cannonball will behave I. I just know it because it's sitting right there in a in a nice little tidy equation that that high schooler could figure it out. But when we add a third object to the mix and this is when Omni Music should begin playing when we add a third object and we try to calculate the gravitational interaction between three bodies, this is called the Three Body Problem. When we try to make predictions, we try to write down a solution with pencil and paper that we can use to go out and do physics. Well, it gets a little tougher. There is no generic. Write it down once and for all. Solution for the three body problems. Seriously? None. It doesn't exist. There is no generic formula that allows you to plug in three different objects with three positions and three masses and just know how they will behave through their mutual gravitational interaction.

It's like if you've got a best friend, you know how you and your best friend are gonna behave. But then someone new shows up. When you go out to the movies now, it's all complicated and weird, and you don't know how to act. Well, that's the three body problem, this three body problem, this simple problem. It's easy to state. The problem was the gravitational interaction of three objects. It's simple to write, but it's almost impossible to solve. The simple problem made some of the giants the titans of astronomy, scratch their heads generation after generation. Newton himself touched on this a little in Principia Mathematica, just a tiny bit, and then he basically went well. I'll leave it as a homework exercise for the reader. You know someone else is gonna solve this way. Later on, in the late 18 hundreds, the three body problem would be used to accidentally discover chaos. That's how intense the three body problem is that it eventually led to an entire branch of study of physics and mathematics in the 20th century.

We solve this with computers because we can just crunch the numbers without having a solution on hand. We just say, Well, this is the gravitational force and then we're just going to advance time by a little bit, see where the new positions of the objects are and then recalculate the gravitational force, advance time a little bit, see where the new positions are and repeat that. But that's kind of cheating. I mean, not, not really, but like it's not giving us a solution. It's just crunching the numbers for us without without giving us an equation that we can write down on paper. And that's where we are. That's the reality of the three body problem of, of how difficult and nasty this problem really is. But to get to today's topic, uh, somewhere between Newton in the late 16 hundreds. And computers in the modern era is LaGrange. And yes, I never ignore a moment to savor some sweet, sweet, butchered French pronunciations on this show. And now, if we could combine a French pronunciation with a magnetic field reference Oh, that would that would be awesome.

Anyway, uh, today we're featuring a towering intellectual of 18th century France. Here we go. Joseph Louis LaGrange. Oh, wait a minute. Hold up. Um, it says here that, uh, he was Italian as in born and raised in Italy, and his actual name was not French at all. It was Italian. He just moved to France when he was, like, 50 years old, and then lived there for 20 years. And somehow we ended up with the French. Say, why did the French steal this dude's name? Why do we have to call him LaGrange? His I need to change characters to give him his his birth name. It was introducing Giuseppe Luigi. You know what, I? I kind of like LA grand better. It would be honestly, more fun to say LA grand points all the time like it's a chain of pizza restaurants. I mean, but honestly, if you want my accents to be better, then you need to contribute to Patreon. That's patreon dot com slash PM Sutter. Uh, if I reach my stretch goal my a, I will do accent training or something.

Anyway. La Grand was Italian but became a French naturalized citizen. Changed his name is even buried in the pan, which is like a big deal if you're French anyway, LaGrange or La Grand was a genius, a totally certified intellect, real big brain stuff here. Among his many accomplishments, he totally reformulated the mathematics of Newton into a form that's easier to do calculations with, and it is still more commonly used today. If if you open up a typical theoretical physics paper today, you won't find typically forces and acceleration, you know the language of Newton. Instead, you'll find something called Lagrangian in his honor. It's like the secret language that physicists used to talk amongst themselves and and definitely the topic for another episode. Anyway. He also tried to tackle the three body problem, and he largely failed. I mean, this is this is a dude who rewrote the laws of physics and still failed to solve the three body problem despite his genius.

But he did find some interesting bits. And those interesting bits are called LaGrange Points or, you know, lagrangian points if if you want to go to his original birthday. But I guess we don't because the French took it Anyway, I'm not bitter at all. What LaGrange did was to not look at the whole picture of three bodies. I mean and trying to come up with a generic solution. You know the two body problem When you're trying to find the gravitational interaction, try to predict how two objects will behave when they're interacting gravitationally. You can just take a general problem. It doesn't matter their sizes, it doesn't matter their masses, it doesn't matter their distances or their initial speeds. Uh, you can just write down a solution that that solves it all. You can't with the three body problem, LaGrange made some assumptions. He made one of the masses very, very, very, very, very tiny, as in so tiny it didn't matter so tiny that it the gravity of that object didn't really play a role. Like if you're looking at, uh, the earth and the moon and a speck of dust Yes.

Technically, the speck of dust is exerting a gravitational influence on the Earth and the moon, and technically it doesn't matter. So that was LaGrange's first big simplification. The other big simplification he made is that instead of trying to solve this all in the full three dimensional reality of just look at two dimensions, so like if the moon is orbiting around the Earth or the Earth is orbiting around the sun. He just looked at what happens in that two dimensional plane of the orbit and ignored any upy down motion. He made these two assumptions so that he could actually make progress. And he was. He was able to write down solutions of, If you have, say, an earth and a moon and a speck of dust. He was able to write down how that speck of dust would behave. This is huge. This is gigantic. The three body problem is insane.

It's one of the most difficult problems in all of physics and has been for centuries, and LaGrange was able to find a solution. It's it's narrow. There are caveats, but it's still a solution, and he was able to do it. And so he was able to write it down. This is before computers. This is before spreadsheets. This is before iPhones. I guess that's pretty obvious. He was able to write down a solution of how that speck of dust would behave, of how it would act of of what you could do with it. And in that solution in in this set of equations that he was able to develop, uh, with pen and paper, he found five points of interest. There are five interesting places. If you look at, say, the Earth sun system and examine what would happen to an imaginary speck of dust floating around the Earth and the sun. There were five places that that stuck out before I continue.

I want to let you know that this show is brought to you by the wonderful folks at better help. That's better. Help dot com I. I know a lot of you listen to this show as a form of therapy A as a way of, of escaping the world and and just going among the stars on this wonderful journey. Uh, I am a big advocate for therapy. I personally see a therapist, and you would be surprised if you don't currently see a therapist how much they can really help you Just navigate a a difficult life. Just like you see a doctor to help you with physical conditions, you should see a therapist Better help dot com is a way to do that. That's convenient. It's affordable. Uh, these are licensed professional counselors that you can connect to online a range of expertise worldwide. It really is an invaluable resource. Uh, as a listener, you can get 10% off your first month by visiting our sponsor at better. Help dot com slash spaceman. You can join 1 million people who have taken charge of their mental health again.

That's better help. HE LP dot com slash spaceman. Now, to be fair, before I dig too much into LaGrange points Uh, previous to Joseph Louis LaGrange was Leonard Oiler. Leonard Oiler was a Swiss super genius and had cracked part of this problem 50 years earlier. Back in 17. 22 LaGrange, uh, wrote his paper in the 17 seventies, about a century after Newton. Uh, Leonard Oiler had already found three points of interest. Um Leonard Oiler Don't get me wrong. He is among super geniuses. A super genius. Like if you're a super genius, you wish you were Leonard Oiler. Uh, I could do a whole episode on Leonard Oiler in his contributions. Please feel free to ask. I would love to do an episode on him. Uh, but like, 99% of everything in physics and math is already named after Oiler, So we had to spread the love around a bit. So instead of Oiler Points or LaGrange Oiler points, they're just LaGrange points and and LaGrange, super genius that he was was able to take the work of Oiler and extend it.

So he took what Oiler accomplished, which was finding three points of interest in this limited case of the three body problem. And he found two more, which is pretty cool. These five points of interest called the LaGrange. Points are places of balance. There are places of equilibrium. There are places where everything is just right to give you some visual of one of these LaGrange points, let's say you're standing on the moon, you're hanging out. Just having a great time kicking around some regular playing golf. I don't know. You're on the moon and you take a rock and you hold it at eye level and you let go. You let go. What does that happen? It falls back to the moon. What if you reach as high as you can over your head and you hold that tiny little rock and you let go? What does it do? It It it falls back to the moon. Imagine you could stretch out your arm. Super, Super, Super Far like a million miles Far You could stretch out your arm so far that your your hand like grazed the edge of the earth's atmosphere like you're standing on the moon and you're stretching out your arm so far that that your hand is touching the earth's atmosphere and then you let go of the rock.

What will happen? It won't fall to the moon. It will fall to the earth. So there's a point somewhere between the Earth and the moon, where the gravity of the earth and the moon cancel each other out. You know if if you're on the moon side of that point and you let go of the rock and the rock will fall to the moon and if you're on the earth side of that point, the rock will fall to the earth. There's somewhere where it's just right, like the center of that teeter totter. With the earth on one side and the moon on the other, you can find that balance point. That balance point is the first LaGrange point. It's a point of interest. It's a point of equilibrium. It's a point where things cancel out. Now. It's not the only one. There are two more in a line with the earth in the moon, and I'm just using the Earth and the Moon as an example. Every pair of orbiting objects has its own set of LaGrange points, so there's five LaGrange points for the Earth moon, So system five LaGrange points for the Sun Earth system. Five LaGrange points for the Sun Jupiter system on and on and on.

Every you you take two objects in the solar system. There are gonna be five LaGrange points associated with them, but going back to the Earth and the moon. There's LaGrange, one, which is the point between the Earth and the moon, where the the gravity balances out. LaGrange 0.2 is on the far side of the moon in this earth moon system. Now, to think about this one, I want you to imagine you're out there. You're on the far side of the moon, OK? And you lift off a little bit and and you'd like to orbit the earth, you you would enjoy orbiting the earth, and that's what you're gonna do. If you're not attached to the moon, you're gonna to orbit the earth just like everybody else. Usually usually if you're orbiting the earth at a distance further than the moon, you would have a different orbital period than the moon because you're further away from the earth. So the earth's gravity out there is a little bit weaker, so you're gonna go a little slower.

And so what would happen is if you detach from the moon on the far side of the moon and you're just hanging out, uh, the moon will race ahead of you. You'll see the moon sailing away. It's going ahead in its orbit. You're further from the earth, so the gravity of the earth is weaker, and so your own orbit is gonna be a little bit slower. But if you're right next to the moon. Well, then you have to account for the gravity of the moon itself. Like if you're, uh, literally standing on the moon, then the gravity of the moon is dragging you along with its orbit. You're attached to it. And if you're too far away from the moon, well, the moon is just like you don't You don't even feel the moon anymore, and it's just gonna race ahead of you. But there is a point where naturally the moon is gonna wanna pull ahead of you because it's going faster than you. But the moon's gravity itself is pulling you along. And there is a very, very specific point where you don't have to do any extra work where even though you are further away from the Earth than the moon is and you should have a slower orbit, the gravity of the moon is keeping you going.

It's pulling you along with it. And that is the second LaGrange point that is L two again, a place of balance where you want to be drifting away from the earth. You want to be going slower, but the gravity of the moon is pulling you along And so everything balances. LaGrange 0.3 is the exact same scenario. But instead of the far side of the moon, you're at the far side of the earth. Now LaGrange 0.4 and LaGrange 0.5. These are a little bit weirder. And and this is why you know Oiler himself. He found those three that are in the line The L one, L two and L three. LaGrange found the 4th and 5th points, and the 4th and 5th point sit at a corner of a triangle where the legs of the triangle imagine you're floating out, you know, around the earth hanging out and you can see the earth and you can see the moon. There is a specific point, a place where the gravity of the earth is proportional to the gravity of the moon.

And it's where the distances to the moon and the earth are the same. And so you can imagine this point. You can draw a triangle from the earth to the moon to you. And when this triangle is an isosceles triangle, you know, 60 degrees every angle, that is the fourth LaGrange point. And then on the other side, that's the fifth LaGrange point. The way I like to think of these LaGrange points is when you're here. When you're in this spot sitting at this triangle, you are drawn to the earth but the moon holds you back like you wanna fall a little bit to the earth. But the moon's like No, no, no, no, no, no, no. Come back here. Come back here just a little bit. And then you're drawn to the moon. But the earth is like No, no, no, no, no. Come back, Come back, Come back. So there's like a tug of war here happening altogether. These five LaGrange points are places of balance. They are places where the gravitational forces, the the centrifugal forces balance each other out.

L one, L two and L three. The three LaGrange points they're sitting in the line. Even though they're places of balance, they are unstable. So if I were to place a speck of dust at that first LaGrange point, which for me personally is the easiest one to understand between the earth and the moon, the moon is pulling on it equally from the earth. And I were to let go if I were to just breathe on it like a little. Well, then it will end up tumbling towards the earth or the moon. It's a balance point, but an unstable balance point, like you can put a pencil on its tip. And, yeah, it's balanced, but any little nudge, it will send it crashing. So L One and L two and L three are all unstable points of equilibrium. But L four and L five are actually stable. Uh, there is one caveat Here. Uh, the the mass ratio between the two objects has to be greater than 1 to 25. Uh, just the way the math works out in order to make these stable. But L four and L five are stable.

That means if I put a speck of dust to L four L five at the corners of these triangles, following in the orbits, and I nudge them a little bit, I breathe on them a little bit. They'll tend to go back to L four or L five. They're stable. And the reason here of all the things, is the Coriolis effect. You know, the Coriolis effect, Like like we live on a rotating earth and The speed of rotation at the equator is different than the speed of rotation at the pole. And this causes some twisty motion. This is how we get things like hurricanes. Uh, at L four and L five. Remember, everything's in motion. Everything's in orbit. Nothing is still in the solar system. You're you're moving around. And if you're at the L four or L five point in the Earth Moon system, you are in orbit around the earth. And so if you move, you are rotating. You are orbiting. If you move, you will experience a chore effect. It's a real thing. And at L four and L five, the Coriolis effect ends up having a direction that pushes you or nudges you back to L four and L five.

So if you're at L four, you're hanging out, you're doing things. You see the earth over here, you see the moon over there, and they're and they stay exactly where they are because you're in sync, you're in orbit with everything and you move a little. You're going to feel a Coriolis effect that's going to nudge you right back to where you started. So L four and L five. The 4th and 5th LaGrange points are stable. Equilibrium and naturally objects tend to collect there. It's true, uh, the the biggest example is with the Jupiter system. The sun Jupiter system. Sun and Jupiter have five LaGrange points. Three of them are in a line, and the 4th and 5th are at these 60 degree angles in the orbit. So the these are points in the orbit of Jupiter that lead and follow the planet itself. And since they're stable, things can collect there. If just a random rock finds itself there, it's it's gonna feel a force that keeps it there. And these are what we call the Trojan asteroids.

There are clumps groups of asteroids that lead and follow Jupiter by 60 degrees because they're hanging out at the LaGrange points, those points of stable equilibrium. The sun Earth system has its own set of five points. And yeah, we're not as massive as Jupiter, so we have a little bit of hard, harder time collecting stuff. But yeah, there are. There are a couple of tiny rocks leading and following the earth in our orbit around the sun because they're collecting it. L four and L five. The other three LaGrange points even though they are unstable, The the fact that they are points of equilibrium makes them handy because, yeah, they're unstable, but not very unstable. You know, they're kind of stable. So if you put something there, it's easier because all the forces do balance there. You only have to do a little bit of extra station keeping to keep yourself there. You just you know, you'll, like, slowly drift away from L two example and and and then you just give yourself a little squirt of gas and and you go back to L two.

You know, it's not hard to keep yourself there, you know, it's not like, right up against the Earth or right up against the moon. Yeah, you know, you're pretty chill there. And so we like to put satellites there. We we tend not to put stuff at LaGrange 0.3 of the Sun Earth system because that's on the opposite side of the sun. Kind of hard to get to, but, uh, L one L one is about 1 1/100 of the distance from the earth to the sun. So you just go a little bit closer to the sun. Boom. You're at L one for the Sun Earth system, and that's useful for solar observations. That's why we put, uh, Soho, the solar observatory there because it's it can just hang out, not have to work very hard, unstable, but not very much so. And so it doesn't need to expend a lot of fuel to keep its station. And the sun is always just right there on one side, and then the earth is on the on its back side. So you can put your cameras on one side and your communications gear on the other, and you're done. We also use L two a lot. This is of the sun earth system.

This is on the far side of the earth, farther away from the sun. It's about three times farther away than the moon is. Uh, we like to put observatories there. The James Webb Space Telescope is there. It's a great place for observations because it keeps the sun in one spot. So your your solar panels are always gonna be, uh, nice and juicy. If you need to chill yourself out, which the James Webb does, you can just put your so solar shield in one spot and keep yourself nice and chilled. The sun's not gonna move around in all sorts of crazy, random directions. It's always gonna be in the same spot in the sky for you. And, uh, the earth is always gonna be in the same spot on the sky for you. So again, you can put your communications gear there and and you're good. And so that's why we put James Webb also, uh, like W map is there plank is there hanging out? Uh, you know, we don't use those anymore, but, you know, we're collecting our own little, uh, collection of satellites there at L two because it's a handy spot. And we learned about these LaGrange points through the math of LaGrange through the solution to this essentially unsolvable problem.

The three body problem. We learned about these places of equilibrium through his work. So hundreds of years ago, we were able to discover these LaGrange points centuries before putting them to use. And that's just how awesome LaGrange is or la Grande is. You know, whichever one you want. And as for the LaGrange points, whether they're stable or unstable. You know, no matter what, they're a pretty great place to hang out. Thank you to Paul S on email at Murray Wilson on Twitter and Brandon B on email for the questions that led to today's episode and thank you to the top patreon contributors. That's patreon dot com slash PM Sutter. We got Justin G, Chris Barbeque, Duncan M, Corey D, Justin Z, Nate H and E Aaron Scott M Rob H, LOL Justin Lewis M Paul G, John W, Aaron J, Jennifer M, Gilbert M, Tom B, Joshua and Kurt M Those are the top contributors, and I appreciate all the contributions. Please send me some more questions. Ask us spaceman at gmail dot com. Hashtag ask us Spaceman. Uh, go to the website website.

Ask us spaceman dot com to check out all the old episodes, Buy a book, How to Die in Space, Your Place in the universe and catch me on social media and I will see you next time for more complete knowledge of time and space

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