Question

These are parametric equations for a Keplerianorbit:

tan(φ/2) = √{(1+e)/(1−e)} tan(ψ/2)

r = a ( 1 − e cosψ) and t = (T/ 2π) (ψ− e sinψ)

Here (r,φ) are plane polar coordinates, t istime, and ψ is a independent variable parameter. Also, T =period of revolution, a = semimajor axis, and e =eccentricity.

(A) From the parametric equations, express r as a functionof φ. That is, eliminate the variable ψ, and hence obtain anequation for the orbit in space. (Your answer cannot depend onψ.)

(B) Determine the Cartesian coordinates (x,y) andtime t for the point P shown on thefigure.

(C) Determine the Cartesian coordinates (x,y) andtime t for the point R shown on thefigure.