My recent talking about central forces got me going. There’s interesting stuff about what central forces allow things to orbit one another, and what forces allow for closed orbits. And I feel like trying out a bit of real mathematics, the kind that physics majors do as undergraduates, around here. I should get something for the student loans I’m still paying off and I’ll accept “showing off on my meager little blog here” as something.
Central forces are, uh, forces. Pairs of particles attract each other. The strength of the attraction depends on how far apart they are. The direction of the attraction is exactly towards the other in the pair. So it works like gravity or electric attraction. It might follow a different rule, although I know I’m going to casually refer to things as “gravity” or “gravitational” because that’s just too familiar a reference. I’m formally talking about a problem in classical mechanics, but the ideas and approaches come from orbital mechanics. The language of orbital mechanics comes along with it.
And it is too possible that the force would point some other way. Electric charges in a magnetic field feel a force perpendicular to the magnet. And we can represent vortices, things that swirl around the way cyclones do, as particles pushing each other in perpendicular directions. We’re not going to deal with those.
The easiest kind of orbit to find is a circular one, made by a single pair of particles. I so want to describe that, but if I do, I’m just going to make things more confusing. It’s an orbit that’s a circle. And we’re sticking to a single pair of particles because it turns out it’s easy to describe the central-force movement of two particles. And it’s kind of impossible to describe the central-force movement three particles. So, let’s stick to two.
When we start thinking about what we need to describe the system it’s easy to despair. We need the x, y, and z coordinates for two particles. Plus there’s the mass of both particles. Plus there’s some gravitational constant, however strong the force itself is. That’s at least nine things to keep track of.
We don’t need all that. Physics helps us. Ever hear of the Conservation of Angular Momentum? It’s that thing that makes an ice skater twirling around speed up by pulling in his arms and slow down by reaching them out again. In an argument I’m not dealing with here, the Conservation of Angular Momentum tells us the two particles are going to keep to a single plane. They can move together or apart, but they’ll trace out paths in a two-dimensional slice of space. We can, without loss of generality, suppose it to be the horizontal plane. That is, that the z-coordinate for both planets starts as zero and stays there. So we’re down to seven things to keep track of.
We can simplify some other stuff. For example, suppose we have one really big mass and one really small one: a sun and a planet, or a planet and a satellite. The sun isn’t going to move very much; the planet hasn’t got enough gravity to matter. We can pretend the sun doesn’t move. We’ll make a little error, but it’ll be small enough we don’t have to care. So we’re down to five things to keep track of.
And we’ll do better. The strength of the attractive force isn’t going to change because we don’t need a universe that complicated. The mass of the sun and the planet? Well, that could change, if we wanted to work out how rockets behave. We don’t. So their masses are not going to change. So that’s three things whose value we might not have, but which aren’t going to change. We’ll give those numbers labels that will be letters, but there’s nothing to keep track of. They don’t change. We only have to worry about the x- and y-coordinates of the planet.
But we don’t even have to do that, not really. The force between the sun and the planet depends on how far apart they are. This almost begs us to use polar coordinates instead of Cartesian coordinates. In polar coordinates we identify a point by two things. First is how far it is from the origin. Second is what angle the line from the origin to that point makes with some reference line. And if we’re looking for a circular orbit, then we don’t care what the angle is. It’s going to start at some arbitrary value and increase (or decrease) steadily in time. We don’t have to keep track of it. The only thing that changes that we have to keep track of is the distance between the sun and the planet. Since this is a distance, we naturally call this ‘r’. Well, it’s the radius of the circle traced out by the planet. That’s why it makes sense.
So we have one thing that changes, r. And we have a couple things whose value we don’t know, but which aren’t going to change during the problem. This is getting manageable. (Later on, when we’ll want to allow for elliptic or other funny-shaped orbits, we’ll need an angle θ. But by then we’ll be so comfortable with one variable we’ll be looking to get the thrill of the challenge back.)
When I pick this up again I mean to introduce all the kinds of central forces that we might possibly look at. And then how right away we can see there’s no such thing as an orbit. Should be fun.