Two masses/Movement/Difference between the potential energy/Example

Suppose that the distance between two masses (thought of as mass points) ${}M$ and ${}m$ is ${}R_{0}$. Because of gravitation, this system contains a certain potential energy. How is this potential energy changing, when we move these masses to a distance ${}R_{1}\geq R_{0}$?

The needed energy is force times path, where the force itself depends on the distance between the masses. Due to the gravitation law, the force, given the distance ${}r$ between the masses, equals

${}F(r)=\gamma {\frac {Mm}{r^{2}}}\,,$

where ${}\gamma$ denotes the constant of gravitation. Therefore, the energy needed to increase the distance from ${}R_{0}$ to ${}R_{1}$, equals

${}E=\int _{R_{0}}^{R_{1}}\gamma {\frac {Mm}{r^{2}}}\,dr=\gamma Mm\int _{R_{0}}^{R_{1}}{\frac {1}{r^{2}}}\,dr=\gamma Mm{\left(-{\frac {1}{r}}|_{R_{0}}^{R_{1}}\right)}=\gamma Mm{\left({\frac {1}{R_{0}}}-{\frac {1}{R_{1}}}\right)}\,.$

Hence it is possible to assign a value to the difference between the potential energies for the two distances ${}R_{0}$ and ${}R_{1}$, though it is not possible to assign an absolute value to the potential energy for a given distance.