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Satellites

Mars and eart...

Question

Mars and earth have masses in the ratio $1 : 11$ and radii in the ratio $42 : 79$ . Compare
a. their densities, assuming them to be spheres of uniform density;
b. gravitational field strengths at their surfaces;
c. escape velocities from their surfaces;
d. periods of their satellites near their surfaces.

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Solution

a) $\frac{ρ_{1}}{ρ_{2}} = \frac{\frac{M_{1}}{v_{1}}}{\frac{M_{2}}{v_{2}}} = \frac{M_{1}}{M_{2}} . \frac{v_{2}}{v_{1}} = \frac{M_{1}}{M_{2}} \times {(\frac{R_{2}}{R_{1}})}^{2} = \frac{1}{1} \times {(\frac{42}{79})}^{3} = \frac{814968}{493039}$
b) $\frac{g_{1}}{g_{2}} = \sqrt{\frac{M_{1}}{R_{1}} \times \frac{R_{2}}{M_{2}}} = \sqrt{\frac{1}{11} \times \frac{79}{42}} = 0.41$
c) $\frac{v_{1}}{v_{2}} = \frac{g_{1}}{g_{2}} = 0.4$
d) $\frac{T_{1}}{T_{2}} = \sqrt{\frac{M_{2}}{M_{1}}} \frac{R_{2}}{R_{1}} = 2.5$

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