Figure 1 illustrates a Keplerian orbit, with Cartesiancoordinates (x,y) and polar coordinates (r,Ï).
Parametric equations for a general Keplerian orbit are r(Ï) = a(1 â e cosÏ)
t(Ï) = *(T)/2Ï+ ( Ïâ e sinÏ)
tan(Ï/2) = *(1+e)/(1âe)+1/2 tan(Ï/2)
Here Ï is an independent variable. The solution is periodic in Ïwith period âÏ = 2Ï. Also, a, e, T are constants that determine theparameters of the orbit.
Figure 2 shows the relation between Ï and the spatialcoordinates. In the figure, the coordinates (ξ,η) are defined by ξ= x + ae and η = y a /b,
withb=a*1âe2 ]1/2.
(A) What kind of curve is the orbit?
(C) Determine x(Ï) and y(Ï).
(D) Determine ξ(Ï) and η(Ï).
(E) Express r as a function Ï .
(F) The angular momentum is L = m r2dÏ/dt. Determine L in termsof {a,e,T} Hence verify that L is a constant of the motion.
(G) The energy isE = [1/2] m ( dr/dt )2+ [1/2] m r2(dÏ/dt)2â GMm/r.Determine E in terms of {a,e,T}.
(H) E must be a constant of the motion. Hence determine T and Efrom the equation that you obtained in (F).
(I) Write T in terms of the spatial orbit parameters and theforce constant GM.
(J) Now express L and E in terms of a, e, GM.
(K) In Figure 1, determine the coordinates of the points P, A,and R
(L) In Figure 1, determine the time t at the point R.
