lew@ihuxr.UUCP (06/13/83)
There have been several inquiries about calculating planetary movements and many-body behavior. I not sure if this fills the bill, but you can get excellent results with a straight-forward application of "F=ma", combined with "F=m1*m2/r^2" (vector versions, of course). One simple trick which increases the accuracy tremendously is the following. Calculate the positions of the bodies half a time unit from the current time (ignoring acceleration), and use these positions to calculate the gravitational forces during that time unit. This comes much nearer to providing the correct accelerations than using the positions at the beginning of the time interval. When I was a TA at Lehigh in the introductory physics course, the students were given this method to calculate one period of a planet, given an initial position and velocity. I computed the orbital elements and compared them with the simulation and found a really fine agreement with a few hundred time units per orbit. I later used this method to estimate the magnitude of Jupiter's perturbation of Mars's orbit. My preliminary conclusion was that it was great enough that it should have defeated Kepler in his "war on Mars". Maybe Kepler had data points near nodes of the cyclic variation of the perturbative effects. Anyway, I've never resolved this question to my own satisfaction. Lew Mammel, Jr. ihuxr!lew
karn@eagle.UUCP (06/14/83)
This is a topic that has lately come near and dear to my heart (I'm doing the kick motor calculations for the amateur satellite Phase 3-B, due to be launched this Thursday morning). The method which Lew Mammel describes is known in the literature as the "Cowell Method", which is just the numerical integration of the second-order differential equations that describe an orbit. The beauty of this method is its simplicity and the ease with which it can include perturbing factors (other planets, air drag, earth oblateness, etc.) On the other hand, if the problem you're solving is a good approximation to two-body motion (i.e., one large body dominates the motion of your satellite) then you can integrate just the perturbing forces with respect to a reference two-body orbit, updating the reference orbit when you get too far away. This is Encke's method, and it allows larger step sizes (increasing program speed and reducing accumulated roundoff error) than Cowell's method. There are lots of methods for doing the numerical integration that these models require. Having no formal training in the subject, I'm only now becoming familiar with the Runge-Kutta method, which is apparently the simplest (but not the fastest or most accurate) algorithm available. It is, however, a refinement of the method which Lew describes, and is probably much more accurate. I'm learning to distrust anything a computer prints out with a decimal point wedged between digits... Phil Karn
norton@ihldt.UUCP (06/14/83)
So sorry at my ignorance, but what was Kepler's war on Mars? Mike Norton ihldt!norton