A minimum-energy transfer orbit to an outer planet consists of putting a spacecr

Question

A minimum-energy transfer orbit to an outer planet consists of putting a spacecraft on
an elliptical trajectory with the departure planet corresponding to the perihelion of the ellipse, or the closest point to the Sun, and the arrival planet at the aphelion, or the farthest point from the Sun. (Assume the orbital radius of the Earth is 1.50 1011 m, and the orbital radius of Mars is 2.28 1011 m.)

a) Use Kepler's third law to calculate how long it would take to go from Earth to Mars on such an orbit as shown in the figure above.
1 yr

(b) Can such an orbit be undertaken at any time?

Explain.

3

Explanation / Answer

a) a=(1.5e11+2.28e11)/2=1.89e11(m). kepler's 3rd law. a^3/T^2=const. use this with earth. 1.5e11^3/365^2=1.89e11^3/T^2 so T=516(days). b) no, because we only can land on mar when its at a appropriate position so that imediately after 516 days, the mar is at the opposite position with the initial position of the earth.

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