Total energy in orbit

In the example of the previous section, we found how much work it took to raise an object to a certain altitude. To put a satellite in orbit, we can’t follow that same method because the satellite would just fall down. For a satellite to remain in (circular) orbit, we need to have $F_{\text{g}}=F_{\text{c}}\text{.}$ The satellite will have a certain amount of total energy at all times:

$E_{\text{tot}}=E_{\text{g}}+E_{\text{k}}\text{.}$

For a stable orbit, we must have

$E_{\text{tot}}=-\frac{1}{2}G\frac{m_1{m}_2}{r}\text{.}$

Example

What amount of work does it take to put a 745 kg satellite into orbit 1108 km above the surface of the Earth?

The satellite begins on Earth, so $r_1=r_{\text{E}}\text{.}$ When it reaches the altitude of 1108 km, it will be at a radius of

$r_2=r_{\text{E}}+1108\text{km}=7.488\times {10}^6\text{m}\text{.}$

Now since $W=\mathrm{\Delta }{E}_{\text{tot}}\text{,}$ we should find the change in energy. For the initial value we simply use $E_{\text{g}}$ because the satellite begins at rest, but for the final value we use the special formula for stable orbit energy:

$\mathrm{\Delta }{E}_{\text{tot}}=\mathrm{\Delta }{E}_{\text{tot,2}}-\mathrm{\Delta }{E}_{\text{tot,1}}=(-\frac{1}{2}G\frac{m_1{m}_2}{r_2})-(-G\frac{m_1{m}_2}{r_1})\text{.}$

Substituting all the known quantities and evaluating gives us the answer: it would take 2.68 × 10¹⁰ J of work to put the satellite in orbit.