# Gravitational Binding Energy

Gravitational Binding Energy

The gravitational binding energy of an object consisting of loose material, held together by gravity alone, is the amount of energy required to pull all of the material apart, to infinity. It is also the amount of energy that is liberated (usually in the form of heat) during the accretion of such an object from material falling from infinity.

The gravitational binding energy of a system is equal to the negative of the total gravitational potential energy, considering the system as a set of small particles. For a system consisting of a celestial body and a satellite, the gravitational binding energy will have a larger absolute value than the potential energy of the satellite with respect to the celestial body, because for the latter quantity, only the separation of the two components is taken into account, keeping each intact.

For a spherical mass of uniform density, the gravitational binding energy U is given by the formula

where G is the gravitational constant, M is the mass of the sphere, and r is its radius. This is 80% greater than the energy required to separate to infinity the two hemispheres of the spherical mass.

Assuming that the Earth is a uniform sphere (which is not correct, but is close enough to get an order-of-magnitude estimate) with M = 5.97 · 1024kg and r = 6.37 · 106m, U is 2.24 · 1032J. This is roughly equal to one week of the Sun's total energy output. It is 37.5 MJ/kg, 60% of the absolute value of the potential energy per kilogram at the surface.

The actual depth-dependence of density, inferred from seismic travel times (see Adams–Williamson equation), is given in the Preliminary Reference Earth Model (PREM). Using this, the real gravitational binding energy of Earth can be calculated numerically to U = 2.487 · 1032 J

According to the virial theorem, the gravitational binding energy of a star is about two times its internal thermal energy.

Read more about Gravitational Binding Energy:  Derivation For A Uniform Sphere, Non-uniform Spheres

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