This is the second time I've tried sending this message... The following question is the result of my on-going God-in-a-box computer, or GIAB simulations, using Visual Basic 2005 with .NET Framework. (This GIAB has decided to take a temporary water cooler break to smooze with fellow Vort deities.) I'm hoping another deity proficient in Calculus might be able to answer my current divine-inspired proclivities.
Using traditional computational algorithms over a fixed period of time it is easy to generate a 2D plot (algebraically) the orbital trajectory of a satellite around a fixed body. Beautiful circles, ellipses, and hyperbolic orbital paths can be plotted by exploiting the formula: [GravityForce] = [A pre-determined Constant value] / [Current-Radius]^2 Or, in a more simplified form: F = 1/R^2 By adding the current computed [GravityForce] as a vectored value to the current vector value "x" and "y" coordinates we determine the next position point that "x" and "y" will be by adding the combined vectored values to the current "x" and "y" positions. This results in the next "x" and "y" position and the next computed [GravityForce] value to be used in the next iteration. It's a simple technique. It works amazingly well with fast PCs we have at our fingertips today. The only problem with this simple algorithm is that it begins to break down if your differentiation samples become too big or coarse, particularly when approaching the center of gravity when sampling distances are their greatest. It turns into a predictability problem - ENTER the mysteries of CHAOS! If the computed vectored values between time periods become too big, there is the danger that the next computed trajectory will cause the satellite to veer off unpredictably, sometimes with so much additional acceleration that it leaves the orbit completely. The traditional way of getting around this problem is to take smaller and smaller samples in one's differentiation calculations. Unfortunately, this approach consumes additional computing time, threatening to slow the animation down to a halt. It is my understanding that the use of Calculus (a moment of silence please as we pay homage to Newton) gets around the thorny iterative differentiation problem. Is there a Vort deity-on-call that can point me to specific Calculus formulas that one would use to plot PRECISELY the predicted (x,y) positions and velocity on a 2D graph based on the specified time slice given? Do such devils in the guise of Calculus exist? Oh, Oh. Boss is approaching. Better get back to creation. Regards, Steven Vincent Johnson www.OrionWorks.com
