The Physics Philes, lesson 73: In Orbit
Are you guys ready for MOAR GRAVITY KNOWLEDGE? What a silly question. Of course you are. Let’s get to it.
So far we’ve been discussing how gravity causes object of some mass to basically fall toward each other. It’s a mutual force that causes bodies to tug at each other. When one body is way more massive than the other, the smaller body will fall toward the larger. But what about objects that orbit other objects? How does that work?
Let’s first think about projectile motion. Think about standing on the roof of your house with a ball. Say you push that ball off the roof in a straight line. The harder you push the ball off the roof, the further from the building it will land. If you push that ball hard enough, that ball would go all the way around the Earth. This is what happens with satellites when they are in orbit.
There are a couple of different types of orbits: open and closed. In open orbits, the projectile keeps going and going away from it’s starting place. In closed orbits – which is the kind of orbit we’ll be discussing today – the orbits are ellipses (or a circle, which is just a special kind of ellipse) or fragments of ellipses.
Since circular orbits are the simplest, let’s talk about those. The only force acting is the gravitational force between the satellite and the Earth, and this force is always in the direction of the Earth. Because we’re talking about a uniform circular orbit, we know that the speed of the satellite is constant. The satellite in orbit is not falling toward the Earth. It’s falling around the Earth.
Pretty weird, right?
OK, so we know the speed is constant, but how do we know what speed that should be? If you’ve been keeping up with these posts over the past year and a half you already know almost everything you need to figure this out: Newton’s second law of motion and Newton’s universal law of gravitation.
For uniform circular motion, the magnitude of the acceleration is:
with r being the radius measured from the center of the Earth. Acceleration is always directed toward the center of the orbit. Since gravity is the only force acting on the satellite, the net force on the satellite of mass m is given by the universal law of gravitation:
The force of gravity is in the same direction as the acceleration. We can set this equation equal to Newton’s second law of motion (F=ma) while substituting “a” with v^2/r, like this:
Now all we have to do is solve for v:
Notice that how fast the satellite needs to go depends on how high we want the satellite to be, not the mass of the satellite. In fact, the mass of the satellite doesn’t figure into this equation at all.
This relationship is true for anything in orbit around another thing, even people. Let’s think about an astronaut in the International Space Station. Not only is the ISS a satellite, so are the astronauts and cosmonauts on board. And like other satellites, the only significant force acting on them is the gravitational force between the people and the Earth. Nothing is pushing on the astronauts; they are in a state of apparent weightlessness. (True weightlessness only occurs if you are infinitely far from any other mass. In that case no gravity is acting on you at all.) So the astronauts float.
But wait! We can know more about the orbit of a satellite! We can also figure out the period.
We know that speed is distance divided by time. In a circular orbit, the distance traveled is the circumference of that circle, or 2Πr. So to find the velocity of a satellite we solve:
with T being the period, or time it takes to make one complete revolution. We can set the equation equal to T and replace v with the velocity equation we found earlier. Like so:
Does that make sense? We inverted the velocity equation because it’s in the denominator. (Don’t worry if you didn’t know you could do that. I had forgotten you could do that until embarrassingly recently.) Anyway…BLAMO! Now you can figure out how long your satellite will take to orbit the Earth.
There is one lasts satellite-related topic we should discuss before we adjourn for the week, and that is the satellite’s mechanical energy.
Remember that the mechanical energy E is equal to the sum of kinetic energy K and the potential energy U. We know that kinetic energy is 1/2mv^2 and we know from last week that gravitational potential energy is -Gm_Em/r. If we replace v with the velocity equation we found earlier and simplify, we get:
Aaaaaand voila! The total mechanical energy in a circular orbit is negative and equal to half of the potential energy. If the orbit is low, the mechanical energy will decrease due to air resistance from the outskirts of the atmosphere. As the air drags the satellite, the orbit will decrease until it can’t sustain it anymore and comes crashing down to Earth.
The math of orbiting satellites seems like it should be really complicated, but it’s actually pretty straight forward. I’ll admit I was a little surprised. But we can’t rest on our laurels! Next week we start looking at Kepler’s laws.
Featured image credit: Wikipedia