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Explaining Orbits with Square Planets


originally posted: December 19, 2007
last updated: November 11, 2013

Recently, while reading an old space adventure novel (Heinlein's The Rolling Stones) to my 8-year-old son, I paused to give a quick explanation of the basic physics of space travel to him. For part of this, I dug up an old explanation of why satellites don't fall down that I had invented long ago when I was a college student. I've always thought that this was much easier to understand than the explanations I remember from high school physics, and my son seemed to get it (even faster than I could lay it out), so I thought I'd do a little write up here. I've recently heard the process of being in an orbit described as "falling over the horizon" which is completely apt, but probably completely mysterious to the novice. This discussion should clear that up.

Preliminaries

To start with, let's review a few basic facts of physics.

Suppose you are drifting around in the depths of space, far, far away from the gravitational attraction of any star or planet. You throw a ball. On the earth, if we throw a ball it starts slowing down the moment it leaves our hand, and it's path curves downward until it hits the ground. But in deep space, without the air to slow the ball down or gravity to change it's path, it would just keep moving in a straight line, at the same speed we threw it, forever. (This is Newton's first law of motion.)

Now, let's move to the surface of a planet. One without air. Now you throw your ball straight up. Because of the force of gravity, the ball will start slowing down the moment it leaves your hand. How quickly it slows down depends on how strong the gravity on the planet is. On the earth it slows down at a rate of 9.8 meters per second per second (roughly speaking, that means that every second it's speed is decreases by 9.8 meters per second). Eventually it will slow down so much that it stops, and begins to fall back down. On the way down, it speeds up at the same rate that it slowed down on the way up (9.8 meters per second per second on the Earth). So, when it comes back to your waiting hand, it will have regained all the speed it lost, and, if your hand is at the same height, the ball will hit it at the same speed you originally threw it. If you throw the ball upward at 1 mph, it lands in your hand at 1 mph. If you throw the ball upward at 100 mph, it lands in your hand at 100 mph.

For our next experiment, we'll get a bit fancier. Instead of throwing the ball straight up in the air, we'll throw it upward at an angle of 45 degrees. That is half way between straight up (90 degrees) and horizontal (0 degrees). What happens then?

This is easy to understand if we think of the ball's motion as it leaves our hand as having two parts. It has some speed upward, and some speed horizontally. If we are throwing upward at 45 degrees, then the vertical speed starts out exactly the same as the horizontal speed. We can then consider what happens to the vertical speed and horizontal speed separately.

The horizontal speed isn't effected by gravity, because gravity is vertical. So, by Newton's first law, the ball will keep moving horizontally at whatever speed I threw it, just like in our first experiment.

The vertical speed will behave just like our second experiment, where we threw the ball straight up. It will move upward, slowing all the time, then start moving downward, and when it returns to the level of your hand, it will have the same velocity downward that it had upward to start with. In other words, it will be moving downward at 45 degrees. The path it follows is actually a curve called a parabola, as shown in the picture below. The grey arrows show the direction of gravity, and the green curve shows the flight path of the ball.

If you throw the ball harder, it will go further, partly because it has more time in the air because it's greater vertical speed will carry it higher before starting to fall, and partly because it's horizontal speed is faster. But no matter how hard or softly you throw it, as long as you throw it upward at a 45 degree angle, it always lands at a 45 degree angle. In fact, the same thing is true of other angle. If you throw it upward at a 60 degree angle it lands at a 60 degree angle. If you throw it upward at a 15 degree angle it lands at a 15 degree angle. And no matter how fast you throw it, or what angle you throw it at, the speed it lands with is the same as the speed it was launched with. That's all we need to know to start talking about orbits.

Square Planets

We're going to start by talking about orbits around square planets. These are completely imaginary objects that we are inventing for our temporary convience. Our square planet will look like this:
So the square planet has four sides that people can walk around on. The direction of gravity is indicated by the gray arrows. It is always perpendicular to the side of the planet. This is handy because once we start throwing things around on the square planet, things will work just like they did in the example above, so long as we stay on just one side of the planet.

But that's not what we are going to do. Instead we're going to stand as close as we can get to one corner of the planet, and then throw our ball upward at 45 degrees, throwing it just hard enough so that it will just skim past the next corner of the planet, as shown in the picture below.

In our preliminary discussion, we figured out exactly what will happen in this case. As the ball passes that next corner, we know that it's speed will be the same as the speed it had when we threw it, except that it will be going 45 degrees downward instead of 45 degrees upward. But then what happens? Well, the guy who threw the ball can't see what happens, because he's gone and thrown his ball over the horizon, but we can turn our picture 90 degrees to the left and see what happens.
OK, looking at the second side of the planet, we see that all the sudden the ball is flying upward at 45 degrees instead of downward at 45 degrees. Of course, the ball hasn't turned a corner or anything, it's just that the direction of "down" changed when we passed over the corner of the planet. Since down isn't the same direction anymore, we aren't going down any more.

What happens next? Well, the ball is again moving at the same speed it was when we first threw it, and it is again headed 45 degrees upward, so the same thing will happen as happened when we threw it in the first place. It will fly up in a parabola, and come down at a 45 degree angle, just missing the next corner of the planet:

Well, plainly the same thing is going to happen on the next two sides of the planet, the ball is going to come flying up behind the guy who threw it, and, if he has the good sense to duck, it'll go right by him, on the very same path he originally threw it on, 45 degrees upward, and just fast enough to clear the next corner of the planet. If nothing stops the ball, it will keep going forever. We've successfully placed the ball into orbit around our square planet.
Of course, if there were air on our square planet, the air would slow the ball down. Then it wouldn't be able to keep up the speed it needs to keep falling over the horizon. That's why all the satellites orbiting the earth are way up high, above most of the atmosphere, but on an airless planet, there is no reason a satellite couldn't be orbiting so low that you have to duck when it goes by. (Actually, thin wisps of atmosphere do reach high enough above the earth so that they gradually slow down satellites. That's why Skylab eventually fell down.)

More Sides to the Story

But, of course, planets aren't really square. They're more circular.

We can approach the question of circular planets by degrees though. Let's start with a hexagonal planet, a planet with six sides. Hexagons are nearer to being circular than squares. We'll make the same assumptions about gravity over each side being perpendicular to the side.

For the orbit trick to work right on a hexagonal planet, we need to throw our ball at a shallower angle than before. We know the ball will be going downward at the same angle that we launched it when it crosses the first corner. When the ball crosses that corner, we want the motion to change from the downward angle, back to the upward angle we started with. On the square planet, the direction of gravity changed by 90 degrees when we crossed from one side of the planet to the next. That was enough to change 45 degrees downward to 45 degrees upward. So 45 degrees was the magic number on the square planet. On a hexagonal planet, the direction of gravity only changes 60 degrees with each side. So on a hexagonal planet we'll launch the ball 30 degrees upward, just fast enough to miss the next corner. Crossing the corner will change the direction of "down" by 60 degrees, so our 30 degrees downward becomes 30 degrees upward and the whole orbit thing works out.

Obviously we can do the same thing on an n-sided planet. On an n-sided planet, the direction of gravity will shift by 360/n degrees at each corner, so we'll throw the ball upward at half that angle, 180/n degrees.

If we make n into a really, really big number, then we'll have a planet that is essentially circular. We'll be launching our ball upward at essentially zero degrees. In other words, we'll be throwing the ball straight horizontally, just fast enough that the curve downward due to gravity's pull, exactly matches the constant change in direction of gravity, and the ball travels around our perfectly circular planet in a perfectly circular orbit.

Once long ago I did the math on this. First I worked out the length of each side of an n-sided planet with a radius of r (that's simple trigonometry). Then I wrote down the standard formula for how strong the gravity of a planet with mass m is at a distance of r from it's center. Then I worked out the formula for how fast you'd have to throw a ball upward at a 180/n degree angle in a gravity field of that strength, if you wanted it to travel the length of a side. Combine all these formulas, take the limit as n gets extremely large, and, sure enough, you end up with the correct formula for how fast you need to throw a ball to put it into a circular orbit in the real world. I'll leave reconstructing that calculation as an exercise for the enterprising reader however, and thus save myself the bother of having to type lots of formulas in HTML. I assure you, drawing the pictures was enough of a headache for me.

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