These are parametric equations for a Keplerian orbit: tan(/2) = {(1+ e )/(1 e )}

Question

These are parametric equations for a Keplerian orbit:

tan(/2) = {(1+e)/(1e)} tan(/2)

r = a ( 1 e cos)    and     t = ( T/ 2) ( e sin)

Here (r,) are plane polar coordinates, t is time, and 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 function of . That is, eliminate the variable , and hence obtain an equation for the orbit in space. (Your answer cannot depend on .)

(B) Determine the Cartesian coordinates (x,y) and time t for the point P shown on the figure.

(C) Determine the Cartesian coordinates (x,y) and time t for the point R shown on the figure.

Explanation / Answer

www.math.harvard.edu/~knill/teaching/math21a/s.pdf

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