Course of an Approaching Body, m, Influenced by gravity force of a Much More Massive Body, M This animation computes the trajectory of m using two different iteration means. The first iteration is just a polar integration of Kepler's equations of motion where the angle, nu, with respect to M is iterated and from that the distance, r, between m and M is computed from the equation for an ellipse. The second iteration is a simple integration of the acceleration, (ax,ay), of m imposed by the gravity force of M resulting in changes of (vx,vy). Then increments of (x,y) are computed from (vx,vy). The adjustable parameters are the initial distance, r1, between m and M, the flight angle, gamma1, of the initial velocity with respect to the vector r1, the initial speed, v1, of m with respect to M as well as the gravitation force coefficient of M, GM. From these parameters the ellipse semi-major axis, a, and eccentricity, e, are computed. In addition, the initial angle, nu1, of m with respect to the ellipse periapsis, Rp, is computed. From these results the very important rate of change of nu, nuDot, can be computed. It is the iteration of nu via nuDot that yields the position of m in its trajectory. Having obtained the initial nu1 and initial nuDot1 the values of initial (x1,y1) and initial (vx1,vy1) can be computed and these are all that are needed to begin iteration of the acceleration (ax,ay) to obtain successive values of (vx,vy) and (x,y) in the Cartesian iteration process. The animation keeps a running printout of the speeds of both iterations and these usually agree to a precision of 3 digits. For simplicity M is assumed to be fixed at position (xM,yM)=(0,0) and, because M/m is essentially infinite, M remains at (0,0). Some choices of initial values will result in an eccentricity, e, greater than 1. The trajectories for these will be hyperbolic rather than elliptical. Also these initial values will result in strange results for the Kepler elliptical parameters a, b, and Period. However, the resulting (r,nu) and (x,y) iteration trajectories still agree. In order to provide better acceleration double integration accuracy to obtain (x,y) of m, the learner has access to the integration timestep. For the equations in their most concise form the learner should click the third link.
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r1 gamma1 v1 GM timeStep Single Step