From: alain.fournier@crulrg.ulaval.ca   
      
   Hop David wrote:   
      
   > I'm trying to model orbits with Lunar perturbations in Microsoft Excel   
   >   
   > My first step is just trying to get an ordinary ellipse without lunar   
   > peturbations. I use   
   >   
   > a = GMe/r0^2   
   > V1 = V0 + a*t   
   > r1 = r0 + V0t + .5*a*t^2   
      
   I'm not sure of what exactly you did here. The problem is at least   
   2 dimensional (with perturbations it's 3 dimensional), so those   
   equations are vectorial equations.   
      
      
   > Then the next row of cells uses the last row's r1 as r0, etc.   
   >   
   > So a path over (say) a ten second time increment gives me the endpoints   
   > of a small parabola fragment. The r1 at the end of a parabola fragment   
   > is slightly larger than the r1 end of a tiny ellipse fragment.   
   >   
   > Given this error I'd expected a slowing growing spiral rather than an   
   > ellipse and this is exactly what I get.   
   >   
   > There is no point in putting the moon's influence into each row of cells   
   > when my model of a simple two body is horribly inaccurate.   
   >   
   > I am hoping someone here will suggest better equations to put in the cells.   
      
   Try doing it with the Runge Kutta 4 method (or RK4 method). See any   
   textbook on numerical analysis or   
   http://en.wikipedia.org/wiki/Runge-Kutta_methods   
      
   To get an accurate model you will need to use a small step, this will   
   make your computations run slowly (specially so, if you do it with   
   Microsoft Excel). You can accelerate your computations by using the   
   exact equation of an orbit around a single mass point (elliptical   
   orbits) and then use RK4 to model the perturbations. So you would   
   compute a new elliptical orbit at each RK4 step, compute where the   
   orbit would bring you at the end of the step, then use RK4 to compute   
   how the perturbations will change that. Such a method will allow you   
   to use larger steps which makes computations run faster.   
      
   Alain Fournier   
      
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