Figure 1 illustrates a Keplerian orbit, with Cartesian coordinates (x,y) and pol

ID: 2123666 • Letter: F

Question

Figure 1 illustrates a Keplerian orbit, with Cartesian coordinates (x,y) and polar coordinates (r,%u03C6).

Parametric equations for a general Keplerian orbit are r(%u03C8) = a (1 %u2212 e cos%u03C8)

t(%u03C8) = *(T)/2%u03C0+ ( %u03C8%u2212 e sin%u03C8)

tan(%u03C6/2) = *(1+e)/(1%u2212e)+1/2 tan(%u03C8/2)

Here %u03C8 is an independent variable. The solution is periodic in %u03C8 with period %u2206%u03C8 = 2%u03C0. Also, a, e, T are constants that determine the parameters of the orbit.

Figure 2 shows the relation between %u03C8 and the spatial coordinates. In the figure, the coordinates (%u03BE,%u03B7) are defined by %u03BE = x + ae and %u03B7 = y a /b,

withb=a*1%u2212e2 ]1/2.

(A) What kind of curve is the orbit?

(D) Determine %u03BE(%u03C8) and %u03B7(%u03C8).

(E) Express r as a function %u03C6 .

(F) The angular momentum is L = m r2d%u03C6/dt. Determine L in terms of {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(d%u03C6/dt)2%u2212 GMm /r.Determine E in terms of {a,e,T}.

(H) E must be a constant of the motion. Hence determine T and E from the equation that you obtained in (F).

(I) Write T in terms of the spatial orbit parameters and the force 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.

Figure 1 illustrates a Keplerian orbit, with Cartesian coordinates (x,y) and polar coordinates (r,%u03C6). Parametric equations for a general Keplerian orbit are r(%u03C8) = a (1 %u2212 e cos%u03C8) t(%u03C8) = *(T)/2%u03C0+ ( %u03C8%u2212 e sin%u03C8) tan(%u03C6/2) = *(1+e)/(1%u2212e)+1/2 tan(%u03C8/2) Here %u03C8 is an independent variable. The solution is periodic in %u03C8 with period %u2206%u03C8 = 2%u03C0. Also, a, e, T are constants that determine the parameters of the orbit. Figure 2 shows the relation between %u03C8 and the spatial coordinates. In the figure, the coordinates (%u03BE,%u03B7) are defined by %u03BE = x + ae and %u03B7 = y a /b, with b=a*1%u2212e2 ]1/2. What kind of curve is the orbit? Determine x(%u03C8) and y(%u03C8). Determine %u03BE(%u03C8) and %u03B7(%u03C8). Express r as a function %u03C6 . The angular momentum is L = m r2d%u03C6/dt. Determine L in terms of {a,e,T} Hence verify that L is a constant of the motion. The energy isE = [1/2] m ( dr/dt )2+ [1/2] m r2(d%u03C6/dt)2%u2212 GMm /r. Determine E in terms of {a,e,T}. E must be a constant of the motion. Hence determine T and E from the equation that you obtained in (F). Write T in terms of the spatial orbit parameters and the force constant GM. Now express L and E in terms of a, e, GM. In Figure 1, determine the coordinates of the points P, A, and R In Figure 1, determine the time t at the point R.

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Figure 1 illustrates a Keplerian orbit, with Cartesian coordinates (x,y) and pol

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