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Gravitational binding energy

From Academic Kids

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 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.

It is equal to minus the gravitational potential energy. For a system consisting of a celestial body and a satellite, the gravitational binding energy is more in 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 is

<math>U = \frac{(3/5)GM^2}{r}<math>

Where G is the gravitational constant, M is the mass of the sphere, and r is the radius of the sphere.

Thus it is 20 % more than the energy to separate to infinity two such spheres touching each other.

Assuming that Earth is uniform (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 kg at the surface.

According to the virial theorem, the gravitational binding energy of a star is -2 times its internal thermal energy.

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