Chapter 6 Problem 34

Problem

(a) Show that if a satellite orbits very near the surface of a planet with period $T$, the density (= mass per unit volume) of the planet is $\rho =m/V=3\pi /GT^{2}$.

(b) Estimate the density of the Earth, given that a satellite near the surface orbits with a period of 85 min. Approximate the Earth as a uniform sphere.

Solution

(a)

Since the satellite is orbiting very near the surface of the planet, $r_{s}=R$, where $R$ is the radius of the planet. Then,

$G{\frac {m_{\text{sat}}M}{R^{2}}}=m_{\text{sat}}{\frac {v^{2}}{R}}$ $\rightarrow v={\sqrt {G{\frac {M}{R}}}}$

also

$v={\frac {2\pi R}{T}}$

thus,

${\sqrt {G{\frac {M}{R}}}}={\frac {2\pi R}{T}}$ $\rightarrow {\frac {M}{R^{3}}}={\frac {4\pi ^{2}}{GT^{2}}}$

Assuming the planet is a sphere $V={\frac {4}{3}}\pi R^{3}$

$\rho ={\frac {3M}{4\pi R^{3}}}={\frac {3}{4\pi }}{\frac {4\pi ^{2}}{GT^{2}}}={\frac {3\pi }{GT^{2}}}$

(b)

For Earth,

$\rho ={\frac {3\pi }{GT^{2}}}={\frac {3\pi }{\left(6.67\times 10^{-11}{\text{Nm}}^{2}/{\text{kg}}^{2}\right)\left[\left(85{\text{min}}\right)\left(60{\text{s}}/{\text{min}}\right)\right]}}=5.4\times 10^{3}{\text{kg}}/{\text{m}}^{3}$