Hey Nick! Can you do a video about Lagrange Points?
Sure! That sounds fun.
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Hey Crazies.
Extreme gravity like you’d find around black holes gets a lot of press,
but you don’t have to take gravity to the extremes for it to do unexpected things.
Sometimes even Newton’s laws are enough.
Lagrange Points are one of those times.
They’re five special points around the orbit of a planet or moon.
Those Lagrange points orbit with the planet or moon,
so you can stick things in there like space probes and they’ll stay.
It’s pretty cool actually.
We take advantage of them all the time, at least the points around the Earth.
There’s a whole list on Wikipedia.
They love lists over there.
But there’s kind of a glaring question here.
Um, why do things stay there?
And, to answer that, we need to delve a little deeper into Newtonian gravity.
The simplest example would be the Earth.
If some tiny space rock wanders nearby, the Earth will exert a force on it,
which, in turn, lets us predict the path it will take.
That’s basically how Newtonian mechanics works.
An unbalanced force causes an interesting motion.
If there are no forces or all the forces are balanced, the path is just a boring straight line.
Ok, so, the single force from the Earth can only do one of three things.
It can speed the rock up, slow it down, or make it change direction,
or any combination of those that makes sense.
That’s what acceleration is.
There’s nothing unusual about that.
It’s typical of any unbalanced force.
That’s pretty much what Newton’s second law says.
To get the weird stuff like Lagrange points, we need two objects exerting a force simultaneously.
Let’s consider the Earth and Moon together.
This is what they look like to scale.
Any tiny rock that drifts into this space is going to have a force on it from both.
But, on the line connecting the Earth and Moon, the forces on the rock are in opposite directions.
See where we’re going with this?
Somewhere on this line, the forces should cancel each other.
Since the Moon is less massive than Earth, it should happen closer to the Moon.
Right about here.
Pretty cool, huh?
Except that’s not actually what we want.
At first glance, that looks like Lagrange Point number 1, but it’s not.
The forces on the rock might be balanced at the moment, but that won’t last very long
because the Earth and Moon are moving through space at different rates.
That’s highly unstable.
Lagrange points are always at least semi-stable.
So where did we make a mistake?
We don’t actually want the force to be zero.
Remember, these points move around as the Moon orbits.
Anything at those points will be traveling along curved paths.
Curved paths are accelerated paths
and acceleration means there should be an unbalanced force.
So we need to be a little more careful.
Rather than the two forces adding to zero in Newton’s second law,
we want them to add to give us curved motion instead.
This is the centripetal or center seeking acceleration.
Doing that puts the Lagrange point a bit farther from the Moon and a bit closer to the Earth.
This is Lagrange point number 1, or L1 for short.
It’s about 36,000 miles from the Moon
compared to the location from earlier which was only about 24,000 miles.
You could use the same technique to find L2 and L3, but this isn’t very elegant.
It requires us to already have some idea of where they are beforehand
and it isn’t very obvious why L4 and L5 even exist.
There’s a better way to do this, but we need to do two things.
First, we need a change in perspective.
While we know the Moon orbits the Earth, this would be a lot easier if it didn’t,
so let’s pretend like it doesn’t.
Can we just do that?
We do this kind of thing all the time.
It’s called a coordinate transformation.
Rather than having the Moon drag all the Lagrange points around,
let’s imagine our coordinates are going around in the opposite direction.
It’s like rotating our camera with the Moon’s orbit to make everything look stationary.
The second thing we need to do is stop talking about forces all together.
It’s going to be easier if we do this in terms of energy instead.
That gives us something like this.
We’ve got the gravitational energy from Earth, from the Moon,
and from the rotation we’re not seeing.
We should be able to map it across space and watch the Lagrange points just pop out.
Here it is!
Hmm, that isn’t so helpful.
Oh, I know! Let’s try shading instead of numbers.
Boom!
OK, that’s not so helpful either.
Maybe if we imagine the energy value as a height?
Oh.
I see what the problem is.
Lagrange points are actually really subtle, so we’re going to have to exaggerate things.
But, you know, I think we can still make that last graphic work.
Imagine you’re standing in the middle of a valley created by a circular mountain range.
The first thing you’d see is this flat ledge.
If you place a ball there, it’ll stay.
Any deviation from side-to-side and the ball will simply roll back to the ledge.
Any deviation forward or backward and the ball will roll away, down the cliff.
That’s exactly how L1 works in space.
Any deviation this direction and the space rock will return to L1.
Any deviation the other way and the rock will fall toward the Earth or Moon.
Mathematicians call that a saddle point and L2 and L3 work in a similar way.
In the mountain analogy, you’ll find L2 on the slightly higher ledge behind L1.
L3 is directly behind you, so you’ll need to pull a 180.
Again, a deviation side-to-side and the ball will simply roll back.
A deviation forward or backward and the ball will roll away.
For that reason, these three points are considered semi-stable.
L4 and L5 are a different story.
You’ll find those on either side at the highest peaks.
The mountains are really flat up there.
They’re very stable locations.
Wait wait, how can those be stable if the cliff goes down in all directions?
Yeah, well, that’s where our analogy breaks down.
That’s the problem with analogies.
They all break down somewhere.
In space, any motion of the rock will activate the Coriolis Effect.
That effect makes the path of the rock spiral back into L4 or L5.
In our analogy, it’s as if the motion of the ball changes the mountain,
which sounds a little strange, but rotating frames are weird like that.
Anyway, I’ve only been using the Moon’s orbit so I could draw things to scale.
The most useful Lagrange points are actually on the Earth’s orbit around the Sun.
If you want a space probe to monitor the Sun, you put it in L1, like SOHO.
If you want it to always point away from the Sun, you put it in L2, like Planck.
Unfortunately, those require course corrections every month or so.
L3 lasts a bit longer, but it’s not much use since it’s always blocked by the Sun.
It could never send us information.
L4 and L5 are so stable, they tend to collect space junk naturally.
They don’t even have to be related to the Earth.
All orbit has these points.
My favorites are L4 and L5 for Jupiter.
They’re filled to the brim with asteroids.
So, what are Lagrange Points?
They’re five special points around the orbit of a planet or moon.
They orbit with the planet or moon, so they’re locations of stability.
Objects can remain stationary with respect to the planet or moon.
Some of them are fully stable, while others are only semi-stable,
but the two least stable points are actually the most useful.
We regularly put space probes there for a consistent view of the universe.
So, what do you think about Lagrange points?
Cool? Not cool?
Let us know in the comments.
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The featured comment comes from Mikayla who said:
It doesn't matter if reality is discrete or not, because all numbers are just abstract ideas.
I like this approach because it means the answer doesn’t have to be open-ended.
Yes, pi exists, because all numbers exist in our minds and our minds exist.
I think.
Anyway, thanks for watching!