From: alain.fournier@crulrg.ulaval.ca   
      
   Hop David wrote:   
   >   
      
   >   
   > This sounds like a sensible approach. The hard part is "compute a new   
   > elliptical orbit at each RK4 step," Given the position and velocity   
   > vectors, this would be easy to do at peri and apogee when the vectors   
   > are perpendicular. But how to do this when position and velocity vectors   
   > aren't perpendicular isn't immediately obvious to me. Maybe getting the   
   > angular momentum from the cross product would be a good place to start...   
   >   
   > Thank you for your help.   
      
      
   You have position and velocity so you can compute the energy:   
     -G Me/R + 0.5 v^2, where G is the gravitational constant   
   Me is the mass of Earth, R is the distance to the center of   
   the Earth and v is the speed. One of Kepler's laws states   
   that the area of the orbit that is swept in a given amount   
   or time is constant, i.e. the dot product of the speed vector   
   and the position vector (vector from center of Earth to position   
   on orbit) is constant. So you know that at apogee you will   
   have the same energy and the same speed.position dot product.   
   At apogee the speed.position dot product is just the speed   
   times distance from Earth center. So you have two equations   
   and two unknowns (speed and height) at apogee. The equations   
   are quadratic so you will find two solutions. If you do   
   the same for perigee you find the exact same two equations   
   and two unknowns. So one solution is for the apogee and the   
   other for the perigee. With apogee and perigee you are   
   all set to have the equation of the orbit.   
      
   I hope this helps. If not I can elaborate a little more.   
      
   Alain Fournier   
      
   --- SoupGate-Win32 v1.05   
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