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The gravitational binding energy of a system is the minimum energy which must be added to it in order for the system to cease being in a
gravitationally In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the strong ...
bound state. A gravitationally bound system has a lower (''i.e.'', more negative) gravitational potential energy than the sum of the energies of its parts when these are completely separated—this is what keeps the system aggregated in accordance with the minimum total potential energy principle. For a spherical body of uniform
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
, the gravitational binding energy ''U'' is given by the formula 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)Lang, K. R. 1980, ''Astrophysical Formulae'' (Berlin: Springer Verlag), p. 272 U = -\frac where ''G'' is the
gravitational constant The gravitational constant (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant), denoted by the capital letter , is an empirical physical constant involved in ...
, ''M'' is the mass of the sphere, and ''R'' is its radius. Assuming that the
Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's sur ...
is a sphere of uniform density (which it is not, but is close enough to get an order-of-magnitude estimate) with ''M'' = and ''r'' = , then ''U'' = . This is roughly equal to one week of the Sun's total energy output. It is , 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 The Adams–Williamson equation, named after Leason H. Adams and E. D. Williamson, is an equation used to determine density as a function of radius, more commonly used to determine the relation between the velocities of seismic waves and the den ...
), is given in the
Preliminary Reference Earth Model The preliminary reference Earth model (PREM) plots the average of Earth's properties by depth. It includes a table of Earth properties, including elastic properties, attenuation, density, pressure, and gravity. PREM has been widely used as ...
(PREM). Using this, the real gravitational binding energy of Earth can be calculated
numerically Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods t ...
as ''U'' = . According to the
virial theorem In mechanics, the virial theorem provides a general equation that relates the average over time of the total kinetic energy of a stable system of discrete particles, bound by potential forces, with that of the total potential energy of the system. ...
, the gravitational binding energy of a
star A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...
is about two times its internal
thermal energy The term "thermal energy" is used loosely in various contexts in physics and engineering. It can refer to several different well-defined physical concepts. These include the internal energy or enthalpy of a body of matter and radiation; heat, de ...
in order for
hydrostatic equilibrium In fluid mechanics, hydrostatic equilibrium (hydrostatic balance, hydrostasy) is the condition of a fluid or plastic solid at rest, which occurs when external forces, such as gravity, are balanced by a pressure-gradient force. In the planeta ...
to be maintained. As the gas in a star becomes more relativistic, the gravitational binding energy required for hydrostatic equilibrium approaches zero and the star becomes unstable (highly sensitive to perturbations), which may lead to a
supernova A supernova is a powerful and luminous explosion of a star. It has the plural form supernovae or supernovas, and is abbreviated SN or SNe. This transient astronomical event occurs during the last evolutionary stages of a massive star or whe ...
in the case of a high-mass star due to strong
radiation pressure Radiation pressure is the mechanical pressure exerted upon any surface due to the exchange of momentum between the object and the electromagnetic field. This includes the momentum of light or electromagnetic radiation of any wavelength that is a ...
or to a
black hole A black hole is a region of spacetime where gravity is so strong that nothing, including light or other electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts that a sufficiently compact mass can def ...
in the case of a
neutron star A neutron star is the collapsed core of a massive supergiant star, which had a total mass of between 10 and 25 solar masses, possibly more if the star was especially metal-rich. Except for black holes and some hypothetical objects (e.g. w ...
.


Derivation for a uniform sphere

The gravitational binding energy of a sphere with radius R 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. Assuming a constant density \rho, the masses of a shell and the sphere inside it are: m_\mathrm = 4\pi r^\rho\,dr and m_\mathrm = \frac\pi r^3 \rho The required energy for a shell is the negative of the gravitational potential energy: dU = -G\frac Integrating over all shells yields: U = -G\int_0^R dr = -G\pi^2 \rho^2 \int_0^R dr = -G^2^2 R^5 Since \rho is simply equal to the mass of the whole divided by its volume for objects with uniform density, therefore \rho=\frac And finally, plugging this into our result leads to U = -G\frac \pi^2 R^5 \left(\frac\right)^2 = -\frac


Negative mass component

Two bodies, placed at the distance ''R'' from each other and reciprocally not moving, exert a gravitational force on a third body slightly smaller when ''R'' is small. This can be seen as a
negative mass In theoretical physics, negative mass is a type of exotic matter whose mass is of opposite sign to the mass of normal matter, e.g. −1 kg. Such matter would violate one or more energy conditions and show some strange properties such as t ...
component of the system, equal, for uniformly spherical solutions, to: M_\mathrm=-\frac For example, the fact that Earth is a gravitationally-bound sphere of its current size ''costs'' of mass (roughly one fourth the mass of Phobos – see above for the same value in
Joule The joule ( , ; symbol: J) is the unit of energy in the International System of Units (SI). It is equal to the amount of work done when a force of 1 newton displaces a mass through a distance of 1 metre in the direction of the force appli ...
s), and if its atoms were sparse over an arbitrarily large volume the Earth would weigh its current mass plus kilograms (and its gravitational pull over a third body would be accordingly stronger). It can be easily demonstrated that this negative component can never exceed the positive component of a system. A negative binding energy greater than the mass of the system itself would indeed require that the radius of the system be smaller than: R\leq\frac which is smaller than \frac its
Schwarzschild radius The Schwarzschild radius or the gravitational radius is a physical parameter in the Schwarzschild solution to Einstein's field equations that corresponds to the radius defining the event horizon of a Schwarzschild black hole. It is a characteri ...
: R\leq\frac r_\mathrm and therefore never visible to an external observer. However this is only a Newtonian approximation and in relativistic conditions other factors must be taken into account as well.


Non-uniform spheres

Planets and stars have radial density gradients from their lower density surfaces to their much denser compressed cores. Degenerate matter objects (white dwarfs; neutron star pulsars) have radial density gradients plus relativistic corrections. Neutron star relativistic equations of state include a graph of radius vs. mass for various models.Neutron Star Masses and Radii
, p. 9/20, bottom
The most likely radii for a given neutron star mass are bracketed by models AP4 (smallest radius) and MS2 (largest radius). BE is the ratio of gravitational binding energy mass equivalent to observed neutron star gravitational mass of ''M'' with radius ''R'', BE = \frac \beta = \frac . Given current values *G = 6.6743\times10^\, \mathrm *c^2 = 8.98755\times10^\, \mathrm *M_\odot = 1.98844\times10^\, \mathrm and the star mass ''M'' expressed relative to the solar mass, M_x = \frac , then the relativistic fractional binding energy of a neutron star is BE = \frac{R - 738.313\,M_x}


See also

*
Stress–energy tensor The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the str ...
*
Stress–energy–momentum pseudotensor In the theory of general relativity, a stress–energy–momentum pseudotensor, such as the Landau–Lifshitz pseudotensor, is an extension of the non-gravitational stress–energy tensor that incorporates the energy–momentum of gravity. It allo ...
* Nordtvedt effect


References

Binding energy In physics and chemistry, binding energy is the smallest amount of energy required to remove a particle from a system of particles or to disassemble a system of particles into individual parts. In the former meaning the term is predominantly use ...
Binding energy