Celestial Mechanics on a Graphing Calculator

An analytic treatment of the Two-body problem is in Eric Weisstein's Treasure Trove of Physics. Marshal Hampton at the University of Washington has a page on Central configurations in the n-body problem with a lovely animation of a configuration studied by Lagrange. The Astronomy Workshop at the University of Maryland has a nice page on Orbital Simulations (but their Satellite Viewer crashed my Netscape). DIFFERENTIAL EQUATIONS AND OSCILLATIONS is a web resource by Rubin H. Landau of Oregon State University, with a description of Euler's Method and the Runge-Kutta Algorithm. Comparison of the efficiency of the algorithms is treated on G. Bard Ermentrout's Numerical Methods page at Carnegie Mellon (useful although some links are dead).

## 1. Newton's laws

Isaac Newton laid down the law in outer space. The vast majority of motion in space is governed by two laws, both first propounded by him (Principia, 1687).

1. The Law of Motion. A body in space travels in a constant direction at constant speed unless a force acts on it. And then the velocity vector changes (the body accelerates) according to the force. More specifically, if the body has mass m, and velocity v, and is acted upon by a force F, then
 dv F -- = --- dt m 
An informal way to understand this equation is to think that, if the body is traveling with velocity v at time t, then at an infinitesimally later time t + dt its velocity will be v + (F/m)dt. By that time its position will have changed by vdt. Acceleration in space. The body is traveling with velocity v at time t; at time t + dt its velocity will be v + (F/m)dt. By that time its position will have changed by vdt. At time t + dt the body feels the force vector corresponding to its new position, and the story recycles. (This image does not make complete sense because it is impossible to draw infinitesimal vectors. When the dts are sent back to the denominators everything has meaning, but the picture evaporates.)

2. The Law of Gravitation. Gravity attracts masses one to the other. In order to keep things simple we will not worry about the shape or composition of the bodies, and will consider only point masses. Suppose we have a mass m at position x and a mass m' at position x'. For future reference, their distance is d=|x'-x|. The force of gravitation pulls them towards their common center of mass at x0 = [mx+m'x']/(m + m'). Acting on m it has has direction x' - x (points from x to x'). Its magnitude is Gmm'/d2, G times the product of the two masses divided by the square of their distance. The constant G is universal and depends only on the units chosen.

To write the gravitational force acting on m as a vector, we take a unit vector pointing from x to x' : (x'-x)/d, and multiply it by the magnitude we have specified:

  Gmm'(x'-x) F = ----------- d3 

3. Putting it all together. If we have several bodies traveling through space, the gravitational force acting on each one is the vector sum of the attractive force it feels from each of the others. If we know the initial positions, masses and velocities, Newton's two laws completely determine the behavior of the system. What we would like to get is a set of explicit formulas allowing us to calculate where each body will be at any future instant in time. In general, such formulas do not exist. The differential equation assembled from Newton's two laws cannot be solved explicitly.

4. Numerical integration. How do we predict the motions of the satellites we send through the solar system? We go back to the infinitesimal scheme shown in the first image. If dt is replaced by a finite time increment Delta t, then the expressions vDelta t and (F/m)Delta t give approximate values for the change in position and velocity during the time interval Delta t. We can calculate the new, approximate, position and velocity and apply the same procedure to approximate the total change after 2Delta t, and we can do this over and over until we get to the time we are interested in.

Obviously, there is a lot of approximation going on; how can we tell if our predictions are accurate at all? As Delta t --> 0, the theory guarantees that the error will also go to zero, but will it do so fast enough for the computation to be carried out in a reasonable amount of time?

5. The restricted 2-body problem is a special case which can be solved explicitly. Suppose there are just two bodies, one of which has negligeable mass compared to the other. For example, a man-made satellite of mass perhaps 100 tons=105 kg and the Earth (mass=6x1024 kg); or the Earth and the Sun (mass=2x1030 kg). In this case, unless the smaller body has enough velocity to escape completely, it will travel forever in an elliptical orbit about the larger.

In this column we will investigate numerical integration in the restricted 2-body problem. The approximation algorithms are simple enough to be run on a graphing calculator. Readers are urged to experiment on their own with the programs. The relative slowness of calculators will make algorithm efficiency a crucial consideration. And our knowledge that after a year'' the smaller planet must return to exactly where it started will give us a simple test of algorithm reliability.

Tony Phillips
Stony Brook