Gravitational Energy
Gravitational energy is energy contained in the gravitational field. The relationship governing gravitational energy can be simply derived by noting that gravity is a conservative force, and the work done by a body in a gravitational field is equal to the vector product of the force and the displacement. Newton's law of universal gravitation states that the gravitational force is proportional to the product of the masses of any two bodies, and inversely proportional to the square of their separation, F = G mm/ r2 , where m and m are the masses of the first and second bodies, respectively, and r is the magnitude of the vector displacement between the two bodies. It is straightforward to show, by integrating this force law against an infinitesimal outward radial vector displacement, and assuming that the total system energy is zero at infinite displacement, that the gravitational potential energy of the two bodies is U = -Gmm/r. Note that the minus sign arises because the vector gravitational force is attractive, and points oppositely to the outward radial vector displacement.
Physically, this means that gravity acts to bind matter together, and hence makes a negative contribution to the total energy of the system.
It is similarly possible to compute the total energy necessary to break up a body and remove all of its pieces, layer-by-layer, to infinite distance. This quantity is known as the gravitational binding energy of the body; its size determines how strongly the body is bound together. Although one can calculate the binding energy of any body knowing its mass distribution, for the purposes of this discussion, we only need to note that for a body of mass M, the binding energy will always be of order GM2 /R, where M and R are the mass and radius of the body, respectively. Another way of putting this is that the gravitational binding energy of the body, per unit mass is of order GM/R. Note that the gravitational binding energy of the body per unit mass is roughly the square of the escape speed of the body, as one would expect from dimensionless grounds.
Gravitational energy is the ultimate source of energy for some of the most fascinating phenomena in the universe; gravity is extremely effective in accelerating matter onto more massive bodies. For instance, mass can flow onto white dwarfs, neutron stars, or black holes; when it reaches the surface of those bodies (or the surface of surrounding material, in the case of a black hole), the gas will be rapidly brought to a halt through a shock front. The shock will then convert the bulk kinetic energy into heat, and the accreted matter will radiate away virtually all of that energy.
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