7.3.4 Energy of Satellites

Consider a satellite of mass m in a circular orbit of radius r around the Earth. Since the orbital speed depends on the orbital radius, the KE of the satellite is also a function of r.

$\displaystyle \begin{aligned}({{F}_{{net}}}&=ma)\\\frac{{GMm}}{{{{r}^{2}}}}&=\frac{{m{{v}^{2}}}}{r}\\\frac{1}{2}\frac{{GMm}}{r}&=\frac{1}{2}m{{v}^{2}}\\KE&=\frac{1}{2}\frac{{GMm}}{r}\end{aligned}$

Needless to say, the GPE of the satellite is also a function of r.

$\displaystyle GPE=-\frac{{GMm}}{r}$

Finally the total energy of the satellite is also a function of r.

$\displaystyle \begin{aligned}TE&=KE+GPE\\&=(\frac{1}{2}\frac{{GMm}}{r})+(-\frac{{GMm}}{r})\\&=-\frac{1}{2}\frac{{GMm}}{r}\end{aligned}$

We can now sketch the graphs of the variation of a satellite’s KE, GPE and TE with orbital radius r.

Some important points to note:

At any r, $GPE=-2KE=2TE$
TE and GPE increase with r, whereas KE decreases with r.
To move a satellite from a lower orbit to a higher orbit, we must increase the TE of the satellite. This requires the burning of rocket fuel so that the propulsion force does (positive) work on the satellite.