The gravitational force is conservative. We can see that it only depends
on the end points by considering the work it takes to go between
two points

(1.16) |

points in the radial direction, that is in the negative
direction. The dot product
.
But
is the component of in
the radial direction.
Call it . So

(1.17) |

From the definition of potential energy, we know that the potential
energy is defined in relation to some reference point, say at radius .
Let's set the potential at radius equal to zero
so

(1.18) |

But point towards the center so it is

(1.19) |

(1.20) |

(1.21) |

So after all this, the final form for the potential energy is pretty simple. The potential energy decreases as the two objects get closer together. It is inversely proportional to , unlike the force which is inversely propotional to . It is shown in figure 1.5.