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, `(a_x,a_y)`, of m imposed by the gravity force of `M` resulting in changes of `(v_x,v_y)`. Then increments of `(x,y)` are computed from `(v_x,v_y)`. The adjustable parameters are the initial distance, `r_1`, between `m` and `M`, the flight angle, `gamma_1`, of the initial velocity with respect to the vector `bbr_1`, the initial speed, `v_1`, 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, `nu_1`, of `m` with respect to the ellipse periapsis, `R_p`, is computed. From these results the very important rate of change of `nu`, `(dnu)/dt`, can be computed. It is the iteration of `nu` via`(dnu)/dt` that yields the position of `m` in its trajectory. Having obtained the initial nu1 and initial `(dnu)/dt` the values of initial `(x_1,y_1)` and initial `(v_(x1),v_(y1))` can be computed and these are all that are needed to begin iteration of the acceleration `(a_x,a_y)` to obtain successive values of `(v_x,v_y)` 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 `(x_M,y_M)`=(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.