The Lagrangian Points
By Paul Cooper
Newton’s first law of motion says in effect that an object which is stationary will continue to be stationary and an object which is moving will continue to move in a straight line at a constant speed unless it is pushed by a force. This law is quite important. It explains, for example, why it is distinctly difficult to fire a bullet around a corner; why chairs cannot levitate or roam about the room at will; why if you turn too tight a corner on a bike you fall off and why the Moon revolves around the Earth. The first three of these are well attested by experience, but the fourth perhaps requires some explanation.
Imagine the Moon alone in the universe, moving along quite happily. Newton’s first law tells us that it will move in a straight line at a constant speed. However, we know that the Moon goes around the Earth in a circle (more or less), not in a straight line and furthermore, there must be some force acting on the Moon in order to persuade it not to continue in a straight line, as by Newton’s law it ought.
By now you can probably guess what I am getting at. This force is provided by the Earth’s gravitational pull. The Moon’s circular orbit is the result of the Moon trying to behave properly and move in a straight line and the tug of the Earth trying to make it deviate from the straight and narrow.
In addition to this, you can readily imagine that the harder the Earth pulls the tighter the Moon's path about it will be and the faster the Moon will go. There is therefore some relationship between the strength of the Earth’s gravity and the time it takes the Moon to go around it.
Of course, it is not just the earth that is pulling on the moon. It is a mutual affair in which the Moon is also pulling on the Earth, and the strength with which they pull upon one another depends on their masses and the distance between them. The actual result of doing the necessary calculations is the following formula:-
Instead of screaming, shouting or running away, take another look. It is not as frightening as it first appears and in any case it can’t bite. The letter "T" stands for the time the Moon takes to go around the Earth. The Greek letter p (pronounced ‘pie’) represents a number, approximately 3.1416. ‘M’ is the mass of the Earth, ‘m’ the mass of the Moon and ‘G’ another number, about 0.000 000 000 066 7, which tells us how intrinsically strong gravity is (not very, judging by the size of the number).‘R’ is the distance between the Earth and the Moon and R3 just means R x R x R. Once you realise what the various terms mean, the equation loses some of its power to scare. Now we shall look it boldly in the eye, put some numbers in and defy it to give us the wrong answer. The relevant figures are:
R = 384, 000, 000 metres
M = 5, 970, 000, 000, 000, 000, 000, 000, 000 kilograms
m = 73, 400, 000, 000, 000, 000, 000, 000 kilograms
Do not be put off by the big numbers. Putting them in the formula gives:-
This large number of seconds translates into 27 days and 3 hours, that is indeed the time it takes the Moon to go around the Earth. The formula actually applies to any two bodies in orbit around each other, not just the Earth and Moon.
Notice that in the previous sentence I wrote ‘around each other’. Because if you think about it, not only does the Earth pull the Moon about but the Moon, having it’s own gravity, retaliates. The situation is more complicated than I made it out to be. How, then, are we to make any sense of it, if both bodies are moving?
The answer lies in coathanger wire. Imagine that you can somehow get hold of a massive piece of coathanger wire (from an equally massive coathanger) a quarter of a million miles long, very thick and very strong. Stick the Moon onto one end and jab the other end into the Earth (North America might be a good place). You are now the proud owner of a thing that looks like a huge and peculiarly asymmetrical dumbbell.
Now, being one object, it behaves like one object. In particular it possesses a centre of mass (marked ‘B’ on the diagram) and in some ways it behaves as if it is a small but very heavy ball located at this point. To get back to Newton’s laws of motion, if the whole unwieldy arrangement is pushed, the point B will travel in a straight line and if it is stationary the point B will be stationary. The important thing is that B will never rotate. No matter what the Earth and Moon may do the point B effectively remains stationary.
This, then, gives us a way of describing what the Earth and Moon are doing. They are both orbiting around their centre of mass, which remains stationary.
We are finally in a position to explain what the Lagrangian points are and, more importantly, why they exist. Let us introduce a third body into the system, but a body so small that it does not affect the motion of the Earth or Moon. Suppose, for want of something better, that the third body is an asteroid. We then have the following situation:-
The two dotted lines represent the gravitational forces the asteroid experiences, towards the Earth and towards the Moon. As you can see, they do not generally point in the same direction. Without going into the matter in detail, you can probably also see that the effect of these two forces will be to make the asteroid move along the direction of the arrow. It is as if the asteroid is attracted towards the point marked ‘A’ which lies somewhere along the line joining the Earth and the Moon.
Imagine that the asteroid is now placed in such a position that the result of the two forces, from the Earth and the Moon, is to make the asteroid move towards B, the centre of mass of the Earth and Moon.
Now, this situation is exactly the same, as far as the asteroid is concerned, as if a large mass was put at B. Consequently, just as the Earth and Moon orbit about one another, the asteroid and the fictional body at B will try to do the same. However, earlier I introduced the get out clause that the asteroid is very small. Consequently, to all intents and purposes, the asteroid will orbit about B.
Imagine now, in one final leap of faith, that the force pulling the asteroid towards B is just sufficient to make the asteroid orbit about B in the same time that the Earth and Moon go around each other. Then the whole triangle, Earth, Moon, asteroid, will be rotating around B at the same rate, and the asteroid will not change its position relative to the Earth and the Moon. Such a stable position is known as a Lagrangian point, after Joseph Louis Lagrange, that well known French mathematician who discovered that they should exist.
Whether or not you believe it, there are in fact five points around the Earth and Moon at which the conditions for a Lagrangian point are satisfied:
This diagram should (hopefully) make it clear how L4 and L5 come to exist. L1, L2 and L3 are simply points on the line joining the Earth and Moon at which their combined pull allows a potential asteroid to orbit around B, just as before.
I’m afraid that I can’t say a great deal more (and you probably think that I’ve said quite enough already). To explain exactly how the forces conspire to act in just the right direction and with just the right strength requires a bit of mathematics (about three pages of it) and I have been specifically warned against this, on pain of being shouted at.
Footnote: In case you were wondering, I have marked the centre of mass as "B" because this is the initial letter of the technical word for it, the Barycentre. This comes from Greek ‘baros’ meaning ‘heavy’ and English ‘centre’ meaning ‘centre’.