The Physics Philes, lesson 74: Motion of Planets
In case you haven’t been keeping up, we’ve been learning about gravity for the past several weeks. Last week we saw how the universal law of gravitation tells us how satellites orbit a planet. Gravity is also a really important part of planetary motion. That’s what we’ll be talking about today.
When we talk about planetary motion, we have to discuss Kepler’s laws of planetary motion. Johannes Kepler deduced the nature of planetary orbits in the early 1600s. He figured out that three things were true of the orbits of planets:
- Each planet moves in an elliptical orbit with the sun at one focus of the ellipse.
- A line from the sun to a given planet sweeps out equal areas in equal times.
- The periods of the planets are proportional to the 3/2 powers of the major axis lengths of their orbits
For Kepler, these were just observations. He didn’t know why planets behave this way. But Isaac Newton was on the case. He figured out that these laws are the results of his laws of motion and universal gravitation.
Kepler’s First Law of Planetary Motion: Elliptical Orbits
We’re going to need one of my beautiful drawings:
OK, what we have here is an ellipse. There are a few parts of this we should define before we move on. The major axis is the largest dimension. In this picture, it’s basically the length of the ellipse along the x-axis. The semi-major axis the distance denoted with a blue line; it’s half the distance of the major axis. Elliptical orbits have two foci. These two foci are denoted as S and S’. If this were our solar system, the sun would be the yellow dot. It’s one focus. The red dot is the second focus, and there is nothing there.
There is another value we need to keep track of, and that eccentricity. It has nothing to do with your hipster beard or your pet capuchin monkey. It’s the distance of the foci from the center of the ellipse. The smaller the value, the more circular the ellipse. As you may guess, the eccentricity of a circle is zero. The planets of our solar system have fairly circular orbits. For example, the eccentricity of Earth is 0.017.
Since ellipses are not perfect circles with the sun at the center, it stands to reason that a planet will be closer to the sun at certain times in its orbit. Perihelion is when the planet is closest to the sun, and aphelion is when the planet is furthest away.
Newton showed that, for a body acted on with a force proportional to 1/r^2, the only possible closed orbit are ellipses. In addition, open orbits must be parabolas or hyperbolas. Unfortunately, I can’t go into the math of this. The textbook I’m learning from said – I’m not kidding – that the math is too difficult for this level of understanding. So I guess Kepler’s laws will have to stay observational for us right now.
Kepler’s Second Law of Planetary Motion: Areal Velocity
This law gets a little calculus-y, but I’ll do my best to make it understandable. We have a planet orbiting a sun. As the planet orbits, it will travel a certain number of radians. Over a small time interval, we can visualize the area swept out as a triangle between the planet at it’s initial position, its final position, ad the sun. Like this:
The area of this triangle is the rate at which that area is swept. It’s called the sector velocity. (It’s important to note, however, that this is only an approximation. The line between the initial and final position of the planet is actually slightly curved.)
This follows from Newton’s laws. We can express the area swept in this way:
This is just the equation for the area of a triangle, with r being the height and vsinΦ is the component of the velocity vector perpendicular to the radial line. In addition, rvsinΦ is 1/m times the angular momentum of the planet with respect to the sun:
Angular momentum is conserved!
We can also see why this has to be. The rate of change is equal to the torque of the gravitational force acting on the planet. The vector of the r vector from the sun to the planet and the F vector is directed from the planet to the sun. These vectors lie along the same line, so the vector product is zero. Angular momentum is conserved for any force that always acts along the line joining the particle to a fixed point. This is called the central force.
Kepler’s Third Law of Planetary Motion: Period of Orbit
This law should be relatively easy to wrap our heads around since we just talked about the orbits of satellites last week. We know all about circular orbits. Newton proved that the same principle holds for elliptical orbits, as well. All we have to do to find the period in an elliptical orbit is to replace the radius with the semi-major axis a and the Earth’s mass with the sun’s mass:
Notice that the period doesn’t depend on eccentricity. Elongated elliptical orbits with a semi-major axis a will have the same orbital period as something with a circular orbit with a radius of the same length. The object in the elliptical orbit will move at different speeds at different times in the orbit, while the object in a circular orbit will stay at a constant speed.
So that’s it for Kepler’s laws! Kinda cool, right? Next week we’ll probably talk about apparent weightlessness.
Featured image credit: Earth Observatory