A satellite of mass m is in a circular orbit of Radius 2R_E around the Earth. How much energy is required to transfer it to a circular orbit of radius 4R_E?

(g - acceleration due to gravity)

Concept:

• The total mechanical energy of a satellite is the sum of its kinetic energy (always positive) and potential energy (may be negative).
• At infinity, the gravitational potential energy of the satellite is zero.
• As the Earth-satellite system is a bound system, the total energy of the satellite is negative.
• Generally, the total energy of the planet and satellite system is negative. This means the satellite cannot escape from the earth’s gravity.​

The energy of the satellite is calculated as = \(\frac{GMm}{2R}\)  .............(1)

Where G = gravitation constant, M = mass of Earth

m = mass of the satellite, R = orbital of radius of the satellite

Calculation:

 Now we need to find the energy required to move the satellite from 2R_E to 4R_E.

∴ Total energy at 2R_E position = \(\frac{GMm}{2(2R_E)}\) = \(\frac{GMm}{4R_E}\) (∵ using eq.(1))

∴ Total energy at 4RE position = \(\frac{{GMm}}{{2\left( {4R_E} \right)}}\) = \(\frac{{GMm}}{{8R_E}}\)

∴ difference energy of two orbits 2R_E - 4R_E = \(\frac{GMm}{4R_E}\) - \(\frac{{GMm}}{{8R_E}}\) = \(\frac{{GMm}}{{8R_E}}\)