Gravitational binding energy

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Gravitational binding energy

The gravitational binding energy
Binding energy

Binding energy is the mechanical energy required to disassemble a whole into separate parts. A bound system has a lower potential energy than its constituent parts; this is what keeps the system together....
of an object consisting of loose material, held together by gravity alone, is the amount of energy

Energy

In physics, energy is a scalar physical quantity that describes the amount of Work_ that can be performed by a force. Energy is an attribute of objects and systems that is subject to a conservation law....
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

Heat

In physics and thermodynamics, heat is any transfer of energy from one body or thermodynamic system to another due to a difference in temperature....
) during the accretion

Accretion disc

An accretion disc is a structure formed by diffuse material in orbital motion around a central body. The central body is typically a young star, a protostar, a white dwarf, a neutron star, or a black hole....
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

Potential energy

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Encyclopedia

Energy

Heat

In physics and thermodynamics, heat is any transfer of energy from one body or thermodynamic system to another due to a difference in temperature....
) during the accretion

Accretion disc

Potential energy

Potential energy can be thought of as energy stored within a physical system. It is called potential energy because it has the potential to be converted into other forms of energy, such as kinetic energy, and to do Mechanical work in the process....
, 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

Mass

In physical science, mass refers to the degree of acceleration a body acquires when subject to a force: bodies with greater mass are accelerated less by the same force....
of uniform density

Density

The density of a material is defined as its mass per unit volume. The symbol of density is ....
, the gravitational binding energy U is given by the formula

where G is the gravitational constant

Gravitational constant

The gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitation between objects with mass....
, 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

Earth

Earth is the third planet from the Sun. Earth is the largest of the terrestrial planets in the Solar System in diameter, mass and density. It is also referred to as the World and Wiktionary:Terra.Note that by International Astronomical Union convention, the term "Terra" is used for naming extensive land masses, rather...
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

Sun

The Sun , a G V star, is the star at the center of the Solar System. The Earth and other matter orbit the Sun, which by itself accounts for about 98.6% of the Solar System's mass....
'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

Virial theorem

In mechanics, the virial theorem provides a general equation relating the average over time of the total kinetic energy, , of a stable system, bound by potential forces, with that of the total potential energy, , where angle brackets represent the average over time of the enclosed quantity....
, the gravitational binding energy of a star

Star

A star is a massive, luminous ball of Plasma that is held together by its own gravity. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth....
is about two times its internal thermal energy

Heat

In physics and thermodynamics, heat is any transfer of energy from one body or thermodynamic system to another due to a difference in temperature....
.

Derivation for a uniform sphere

The gravitational binding energy of a sphere is found by imagining that it is pulled apart by successively moving spherical shells to infinity, the outermost first, and finding the total energy needed for that.

If we assume a constant density then the masses of a shell and the sphere inside it are: and

The required energy for a shell is the negative of the gravitational potential energy:

Integrating over all shells we get:

Remembering that is simply equal to the mass of the whole divided by its volume for objects with uniform density we get:

And finally, plugging this in to our result we get: