Basic Orbital Dynamics

This explanation was written up by Caleb Pool, who very kindly gave me a bunch of awesome feedback on my Kinematics book, and then wrote this cool explanation of Orbital Dynamics that I thought you might find interesting/useful/entertaining.

Basics of Orbital Dynamics

When we look outside, we see ground that is pretty flat.  Trees go straight up, buildings stand up, and the ground is perpendicular to those.

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Even buildings look like they stand up straight.

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Water looks very flat when we stand next to it.

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If one side of a pond is higher than the other, the water flows until it is level again.

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Gravity pulls the water equally until it looks flat from a close perspective.  But what about when we start zooming out?

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The water still looks flat to us.  We zoom out some more.

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And some more:

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And some more:

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And a lot more:

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It seems the earth is round.  But we knew that.  How can all the water stay on without falling off?  Gravity pulls objects (and water) toward the center of the earth.

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Although, from our perspective, things fall down and water looks flat, the bigger story is that things fall toward the earth’s center and water curves around the earth.

 

The earth spins about an axis passing through the poles. This rotation causes a force that is largest around the equator, and it makes the earth’s water and land bulge slightly:

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This bulge makes the earth’s radius through its equator about 3,963 miles (6,378 km), when its radius through the poles is 3,950 miles (6,356 km).  A bulge that causes a 13-mile difference in radius between the poles and equator seems like a lot, and it is, but that bulge is not very big compared to the earth’s overall size.   That bulge plays an important role in many orbits.  The basics of an orbit work even if we consider the earth to just be a sphere for now.

 

Suppose you find an extremely high mountain that reaches above the earth’s atmosphere and you hike to the top, carrying an extremely powerful cannon.

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You fire the cannon.

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But that’s not far enough.  You increase the power and fire again and again:

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You keep increasing the power on this magic cannon and firing away from your mountaintop.  The cannonballs fly so far that, as they fall toward the earth, the earth starts curving away underneath them.

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You aren’t satisfied with just shooting cannonballs so far away they land on the opposite side of the earth, so you continue to increase the power:

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And look what just happened!  The cannonball went all the way around the world!  If you step out of the way, you get to watch it zoom by really fast (remember, you had to fire it fast enough to go that far!).  What happens now?  The cannonball doesn’t just stop dead in its tracks after it goes around once; it keeps going around.  And around.

It turns out that you have just made a satellite and launched it into orbit.  Until it gets slowed down by the thin atmospheric gas at the edge of space and burns up, your cannonball will continue to fly through space, following the same path as it circles the world.  Your special cannonball now joins the moon (earth’s only natural satellite) and a host of about 40,000 man-made objects as it endlessly circles in orbit.

 

As gravity pulls the cannonball toward the earth, it falls, but the earth’s surface curves away just as fast, so it never gets any closer to the earth as it goes around.

 

Now that you are satisfied with your powerful cannon, you look at other orbiting objects to see how they behave.

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Objects in orbit that are farther away from the earth go around it more slowly.  This is primarily because earth’s gravity is weaker the farther away from the earth you go.

 

Example:  Which is farther away from the earth?  The moon, which goes around once a month, or a GPS satellite, which goes around twice each day?

 

Answer:  the moon is farther away.

 

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