## 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.