Gravitational binding energy

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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 gravitational potential energy. 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^[1][2]

where G is the gravitational constant, M is the mass of the sphere, and r is its radius. This is 20% greater than the energy required to separate to infinity two such spheres touching each other.

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.

According to the virial theorem, the gravitational binding energy of a star is about two times its internal thermal energy.^[1]

[edit] References

1. ^ ^{a b} Chandrasekhar, S. 1939, An Introduction to the Study of Stellar Structure (Chicago: U. of Chicago; reprinted in New York: Dover), section 9, eqs. 90-92, p. 51 (Dover edition)
2. ^ Lang, K. R. 1980, Astrophysical Formulae (Berlin: Springer Verlag), p. 272