Celestial Mechanics
Have you ever wondered how Copernicus, and Galileo and Isaac Newton and that Kepler guy figured out that the earth revolves around the sun, and not the other way around? When I go out at night and look up at the heavens, I have no idea what I’m looking at. Oh, I can find the North Star and the dippers as well as the next guy, and the moon is easy, but if you asked me to point out where any of the planets are, I’d be literally lost in space. It’s a shame too, because we’re supposed to be so much smarter than the ancients. But if I had to swear to it, I’d have to admit I can’t prove Copernicus was right, or disprove Ptolemy.
Well, not any more, because the other day I found a wonderful book on the internet that teaches all about celestial mechanics. You can download it for free in PDF format at Celestial Mechanics Link. It will take a minute or so to download depending on your connection speed, because it is about 45 MB in size, but it is well worth the read.
“An Introduction to Celestial Mechanics” by Forest Ray Moulton is not an easy book, and not a dumbed-down treatment of the subject. Written in 1902, it was intended for Senior college students and graduate students. But we’ve come a long way since then, haven’t we? I’m not completely through the book yet, so I haven’t gotten into the more advanced topics, such as perturbations but I’d say that the book requires an understanding of mathematics, especially the concepts of Calculus and Differential Equations, and a basic course in Physics for sure. For those who didn’t take those subjects in school, please read on because I think it’s important, even if you don’t read the book, to understand the process of how our present day understanding came about, and how we do the math today to track the planets, the moon, the asteroids and comets, and even other star systems.
We know from experiments that two bodies of mass attract each other with a force that is proportional to the product of their masses, and inversely proportional to the square of the distance between their centers. That means simply, the larger the masses, the stronger the force; and the farther apart the masses are, the weaker it gets. That’s only common sense. We word it in the more quantitative way, so we can take accurate measurements though. From these basic considerations, scientists formulate a number of equations of motion – 3 equations for each of the two masses; one is for the x direction, one for the y direction, and one for z – 6 equations. These equations are written in the form of 2nd order differential equations, because the force in each direction is found to be the second derivative of the momentum in that direction – and differentiation, in Calculus means finding the derivative. The derivative is simply the limiting value of a very small change in one quantity divided by a very small change in a second quantity. In this case we want to know how much the motion changes in each direction x, y, or z in a very small time, which we call the velocity, and we would denote each by the expression dx/dt, dy/dt, and dz/dt. If we take it one step further, we get the rate of change of the rate of change in each direction (acceleration) – our second order derivatives and multiplied by the mass, they are equal to the force. To solve these equations we need to go backwards and find x, y, and z, and to do this we integrate each equation twice. Integration is the inverse of operation of differentiation, which you also learn in Calculus.
We can simplify the problem immensely, by working with the coordinates of the center of mass of the two bodies, and in the process we find that the motions of one of the two bodies with respect to the other will always be in a plane passing through the center of the other.
Now that the we know the motion is planar, we can switch to a simpler coordinate system based in that plane. Instead of three coordinates, it now only takes two to locate the two bodies relative to one another, an angle and a distance. Solving these equations we end up with an equation for the distance, which by inspection we find to be that for a conic section, i.e. a parabola, hyperbola or ellipse with the origin at one focus. By comparing this equation to the standard mathematical equation for a conic section, we derive 6 constants or elements that determine the orbit. For example, one of these elements designated by the small letter “e” determines that the orbit is elliptical when e-squared is less than 1.
There’s somewhat more to it than that, more constants that we derive when we solve what is known as Kepler’s equation. Kepler’s equation allows us to find the exact position on the ellipse by measuring the time elapsed and the difference between the eccentric anomaly, an angle which can be solved for, and the angular distance it would have travelled at a uniform rate. These values designated E and M can be calculated as precisely as needed, by representing them in the form of an infinite series of terms, and just using the terms we need and discarding the rest.
That’s about the gist of it, except we do need to convert these values to our earth system of coordinates. When we have done that, one can then view the body in the sky by its right ascension or direction angle and its declination or angle of elevation. If the object is where we predict, we can assume our equations and our basic understanding of planetary motion is correct.
So if your kids ever ask you, how do you know the planets orbit around the sun, just tell them, “it’s really complicated and that’s why you need to study math,” and point them to this book.
Most of us don’t know why we study math, and what it is good for. I wish somebody had explained this to me, because I never could see how we had figured it out. Even after 3 years of college Physics, I knew about Newton’s laws and Kepler’s laws, but wondered why NASA used orbital elements to track the satellites and space shuttles.
Leave a Reply