- Timo Budarz
The Solar System
Planets orbit the sun due to the sun's gravitational pull on them. While it's true that planets pull on one another, those influences are much smaller. We will ignore those interplanetary interactions in our discussions. Since the sun is so much more massive than any of the orbiting planets (it is almost half a million times more massive than earth), we will assume that the sun is stationary in spite of planets tugging on it. While this is a common assumption for such calculations, it should be noted that the wobble of the sun due to planetary gravitational influences is one of the primary methods of detecting extrasolar planets - or those orbiting stars besides our own sun. The general problem of orbital motion in which a planet's trajectory is guided by the pull of the sun, or some other central mass, leads to an elliptical orbit. However, since a circle is an ellipse with zero eccentricity, circular orbits are one type of orbit. We will focus on circular orbits because the math is much easier than the case of the elliptical orbit, and we can still learn some general principles from analyzing our results. Consider a mass like our moon orbiting the earth. We know from our early chapter on fundamental forces, that there is a gravitational force that pulls the moon toward the earth and likewise pulls the earth toward the moon. This means that both objects undergo acceleration according to Newton's second law. While this is true, the mass of the moon is around 1% of earth's mass, which means that the earth's acceleration is only 1% of the moon's. Given this fact, it is common to assume that the earth in such situations is fixed in place, as if its acceleration is zero. We imagine the earth at the origin of the coordinate system with the moon moving around the origin in a circular path. Just keep in mind that this is not quite true, and yet these calculations will give us rather accurate results. As mentioned above, for planets orbiting the sun, we will assume a fixed sun as well.
Moon's Orbit Around Earth
The force drawing the moon toward the assumed-fixed earth has a magnitude of where the 'r' in the denominator is measured from the center of mass of the moon to the center of mass of the earth. Recall that the universal gravitation constant is Assuming that the gravitational force by the earth on the moon plays the role of a centripetal force acting on the moon, we can write: In looking at this result there are a few things that we can learn.
- The first is that the mass of the moon doesn't affect its orbital period. This is true of orbital motion in general. We could in principle replace the moon with a baseball that is at the same distance and moving with the same velocity and it would track through space just the same. What we'd lose, however is the earth's tides since the gravitational force from the baseball wouldn't pull nearly the same on earth's oceans.
- There is a direct connection between the size of an orbit (its radius in this case) and the time it takes the orbiting body to go around, or the period. These two are linked by the laws of nature. This relationship was first noted by Johannes Kepler's third law in 1619.
- The only way to change the ratio on the right side of the equation is to have control over the gravitational constant (not an option), the value of pi (also not an option) or to change the central mass. For planets this central mass is the sun. For the moon, it is earth.
Kepler's Laws of Planetary Motion
The first known proposal that the sun is the center of the solar system was in 270BC by a Greek named Aristarchus of Samos. Nearly 1600 years elapsed until Polish astronomer Nicolaus Copernicus proposed the same - allegedly by reading the ancients and not by observations of the heavens. There were heavy debates going on regarding the structure and dynamics of the solar system through the 1500s and 1600s. Johannes Kepler was a German mathematician born in 1571. He was hired by Tyco Brahe (a Danish astronomer and nobleman) to analyze some of the best astronomical data in existence at the time. There is a detailed account of the interesting history at this Wikipedia site: Kepler History. Ultimately over a decade of analysis led Kepler to propose three laws of planetary motion. The first two were published in 1609, and the third law in 1619. The laws were: 1. Planetary orbits are elliptical with the sun residing at one focus of the ellipse. 2. The planetary orbits sweep out equal areas in equal time. See this graphic: Graphic. We will see in a future chapter that this amounts to conservation of angular momentum for orbiting systems. 3. Planetary orbital periods are related to the semi-major axis of their orbital trajectory by This means that any planet orbiting our sun should have the same ratio as any other planet. This is not strictly true, however, since the inner planets always help the outer ones maintain their orbits. In this sense outer planets get pulled inward by a larger mass than the inner ones. Kepler came up with these laws a few decades before Newton. It was convincing work, and therefore Newton's burden, when he came up with a law of universal gravitation, was to prove that his law of gravitation could reproduce these Kepler's laws. To test this, Newton needed to invent the mathematics of calculus (and differential equations), which he did. It was really this work that prompted Newton to publish his treatise on his laws of motion after two decades during which he just kept them to himself! We will not prove Kepler's laws using calculus and differential equations, since it's difficult to do. Rather we will assume planetary orbits are circular and starting there see if we can come up with the second and third laws. Since a circle is an ellipse of zero eccentricity, this is a fine thing to do. It should also be noted that the eccentricities of the planetary orbits are quite small. From mercury to Neptune they are 0.2, 0.01, 0.02, 0.1, 0.05, 0.06, 0.05, 0.01. They are all very nearly circular! If we skip over the second law and focus rather on the third law for a moment, you will see that our result from the moon's orbit already proves this. Our expression with the sun's mass instead of the earth's is: We use the mass of the sun since that mass represents the central object around which the planets are orbiting. Since all the planets orbit the same central sun, the left side of the expression is the same for every planet. That means the right side must be the same as well. Regarding the second law for circular orbits, it is trivial. In a circular orbit there is only a centripetal force of gravity. That means the speed of the orbiting planet remains constant while it makes a steady turn around the sun. Since the orbit is circular, the distance to the sun is constant. So area swept out can be written as the radius of the orbit times the arc length, or The rate of change of this area (or the rate at which it sweeps) if we assume radius is constant, is just Since both r and v are constants, the planet sweeps out constant area in time.