Gravitaional potential energy

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Question

A satellite of mass m is in circular orbit of radius $3 R_{E}$ about earth (mass of earth $M_{E}$ , radius of earth $R_{E}$ ). How much additional energy is required to transfer the satellite to an orbit of radius $9 R_{E}$ ?

Moderate

Solution

Total Energy of a satellite $E = - \frac{G M_{E} m}{2 r}$
The difference in Energy of the orbits is
$\begin{array}{l} E_{2} - E_{1} = - \frac{G M_{E} m}{2 \times 9 R_{E}} - (- \frac{G M_{E} m}{2 \times 3 R_{E}}) \\ Δ E = \frac{G M_{E} m}{9 R_{E}} \end{array}$

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