   Chapter 7, Problem 67AP

Chapter
Section
Textbook Problem

A minimum-energy 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 closest point to the Sun, and the arrival planet corresponding to the aphelion of the ellipse, or farthest point from the Sun. (a) Use Kepler’s third law to calculate how long it would take to go from Earth to Mars on such an orbit. (Answer in years.) (b) Can such an orbit be undertaken at any time? Explain.

(a)

To determine
The time taken to go from earth to mars.

Explanation

In the diagram,

• rp is the perihelion distance.
• ra is the aphelion distance.

The semi-major axis of the ellipse is,

a=rp+ra2

From Kepler’s 3rd law, the time period is,

T=a3/2

From the above equations,

T=(rp+ra2)3/2

Substitute 1.496×1011m for rp and 2.8×1011m for ra .

a=[1.496×1011m+2

(b)

To determine
The timing of the trip.

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