How Much Energy Is Required to Change an Orbiting Body's Radius?

[assaftolko]

A body with a mass of m is orbiting the Earth with orbiting radius of R1. We want to move the body so it will orbit the Earth with orbiting radius of R2. To achieve this we need to do 2 steps:

1. We need to give the body energy so it will reach the maximal distance from Earth which is exactly equal to the orbit radius we want to achieve.
2. We need to give the body energy at this maximal distance so it could achieve the appropriate speed that will alow it to orbit the Earth at such radius.

How much energy do we need to give the body in each one of the steps?

The orbits are circular.

I have no idea how to do this... I thought maybe to calculate the energy of m when it orbits the Earth with radius R1, and then calculate m's energy when it's at R2 for step 1 - and the difference between them is the answer. But what's the speed of m at the end of the first step? it's of course not the orbiting speed of circular orbit with radius R2...

 

[CWatters]

Part one is about changing PE, part two about changing KE.

At the beginning it has PE and KE. After step one PE is changed but it still has same KE. Do you actually need to know how the speed has changed?

 

[Filip Larsen]

You may want to check your references for the "Vis-viva equation" and "Hohmann transfer orbit".

 

[assaftolko]

Part one is about changing PE, part two about changing KE.

At the beginning it has PE and KE. After step one PE is changed but it still has same KE. Do you actually need to know how the speed has changed?

As you can see the answers are at the bottom of the picture I uploaded and they do not work out with your way:

Step 1 = $\frac{1}{2}$Gm_Em[$\frac{R2-R1}{R1(R2+R1)}$]

I don't think you can say the KE at the end of step 1 is the same KE as when m orbits the Earth in R1 radius. This KE is the result of a specific orbit velocity that corresponds to the specific circular orbit radius of R1.