When moving a probe from the radius of a planet to a new orbit of radius 2R, do we find the work done by subtracting the final potential energy from the initial potential energy? That would give GMm/2R. But then in other sources they would find the difference in total energy to find the work done on the probe, which would give GMm/4R. These are different, so which one is the correct one?
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The change in the gravitational potential energy is ${GMm \over 2R}$.
The probe is moved to the higher orbit by firing of it's engine.
The total work done $W_{total} = \Delta KE$ where $\Delta KE$ is the change in kinetic energy. Considering the work done by gravity as $-\Delta PE$, $W_{other} = \Delta KE + \Delta PE$ where $W_{other}$ is the work done by a force other than gravity. ${GMm \over 4R}$ is based on the change in energy considering the other force from firing the probe's engine to move to the newer orbit.
See Work done to change circular orbit and orbital speed on this exchange; a response by @Alfred Centauri points out "But, to change from one circular orbit to another (in the same plane) requires that the kinetic energy change by (negative) 1/2 the change in gravitational potential energy". This explains the factor of 2 difference.
John Darby
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