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 bound state. A gravitationally bound system has a lower (''i.e.'', more negative)
gravitational potential energy
Gravitational energy or gravitational potential energy is the potential energy a massive object has in relation to another massive object due to gravity. It is the potential energy associated with the gravitational field, which is released (conv ...
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]
where ''G'' is the
gravitational constant, ''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 surf ...
is a sphere of uniform density (which it is not, but is close enough to get an
order-of-magnitude
An order of magnitude is an approximation of the logarithm of a value relative to some contextually understood reference value, usually 10, interpreted as the base of the logarithm and the representative of values of magnitude one. Logarithmic dis ...
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), is given in the
Preliminary Reference Earth Model (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 th ...
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 held together by its gravity. The nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night, but their immense distances from Earth make ...
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, ...
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 planetar ...
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 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 defo ...
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
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
, 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 yields:
Since
is simply equal to the mass of the whole divided by its volume for objects with uniform density, therefore
And finally, plugging this into our result leads to
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 component of the system, equal, for uniformly spherical solutions, to:
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 applie ...
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:
which is smaller than
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 characteris ...
:
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'',
Given current values
*
*
*
and the star mass ''M'' expressed relative to the solar mass,
then the relativistic fractional binding energy of a neutron star is
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 stress t ...
* Stress–energy–momentum pseudotensor
* 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 us ...
Binding energy