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_"g" = F_"c"`. The satellite will have a certain amount of total energy at all times:

`E_"tot" = E_"g" + E_"k"`.

For a stable orbit, we must have

`E_"tot" = -1/2G(m_1m_2)/r`.

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_"E"`. When it reaches the altitude of 1108 km, it will be at a radius of

`r_2 = r_"E" + 1108\ "km" = 7.488xx10^6\ "m"`.

Now since `W = Delta E_"tot"`, we should find the change in energy. For the initial value we simply use `E_"g"` because the satellite begins at rest, but for the final value we use the special formula for stable orbit energy:

`Delta E_"tot" = Delta E_"tot,2" - Delta E_"tot,1" = (-1/2G(m_1m_2)/r_2) - (-G(m_1m_2)/r_1)`.

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.