Editing Gravitational binding energy

{{Short description|Minimum energy to remove a system from a gravitationally bound state}}
[[File:Spot the cluster.jpg|thumb|300px|[[Galaxy cluster]]s are the largest known gravitationally bound structures in the universe.<ref>{{cite web|title=Spot the cluster|url=https://www.eso.org/public/images/potw1731a/|website=www.eso.org|access-date=31 July 2017}}</ref>]]

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 [[Gravity|gravitationally]] [[bound state]]. A gravitationally bound system has a lower (''i.e.'', more negative) [[Gravitational energy|gravitational potential energy]] than the sum of the energies of its parts when these are completely separated—this is what keeps the system [[Wiktionary:aggregation|aggregated]] in accordance with the [[minimum total potential energy principle]].  

The gravitational binding energy can be conceptually different within the theories of [[Newton's law of universal gravitation|Newtonian gravity]] and [[Einstein|Albert Einstein]]'s theory of gravity called [[General Relativity]].  In Newtonian gravity, the binding energy can be considered to be the linear sum of the interactions between all pairs of microscopic components of the system, while in General Relativity, this is only approximately true if the gravitational fields are all weak.  When stronger fields are present within a system, the binding energy is a [[Nonlinear system|nonlinear]] property of the {{em|entire}} system, and it cannot be conceptually attributed among the elements of the system.  In this case the binding energy can be considered to be the (negative) difference between the [[ADM formalism#ADM_energy_and_mass|ADM mass]] of the system, as it is manifest in its gravitational interaction with other distant systems, and the sum of the [[Invariant mass|energies]] of all the [[atom]]s and other [[elementary particle]]s of the system if disassembled. 

For a spherical body of uniform [[density]], the gravitational binding energy ''U'' is given in Newtonian gravity by the formula<ref name="Chandrasekhar 1939">[[Subrahmanyan Chandrasekhar|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)</ref><ref name="Lang 1980">Lang, K. R. 1980, ''Astrophysical Formulae'' (Berlin: Springer Verlag), p. 272</ref>

<math display="block">U = -\frac{3GM^2}{5R}</math>
where ''G'' is the [[gravitational constant]], ''M'' is the mass of the sphere, and ''R'' is its radius.

Assuming that the [[Earth]] is a sphere of uniform density (which it is not, but is close enough to get an [[order-of-magnitude]] estimate) with ''M'' = {{val|5.97|e=24|u=kg}} and ''r'' = {{val|6.37|e=6|u=m}}, then ''U'' = {{val|2.24|e=32|u=J}}. This is roughly equal to one week of the [[Sun]]'s total energy output. It is {{val|37.5|u=MJ/kg}}, 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).<ref name=PREM>{{cite journal | last1 = Dziewonski | first1 = A. M. | author-link = Adam Dziewonski | last2 = Anderson | first2 = D. L. | author2-link = Don L. Anderson | title = Preliminary Reference Earth Model | journal = [[Physics of the Earth and Planetary Interiors]] | year = 1981 | volume = 25 | issue = 4 | pages = 297–356 | doi=10.1016/0031-9201(81)90046-7 | bibcode = 1981PEPI...25..297D }}</ref> Using this, the real gravitational binding energy of Earth can be calculated [[Numerical integration|numerically]] as ''U'' = {{val|2.49|e=32|u=J}}.

According to the [[virial theorem]], the gravitational binding energy of a [[star]] is about two times its internal [[heat|thermal energy]] in order for [[hydrostatic equilibrium]] to be maintained.<ref name="Chandrasekhar 1939"/>  As the gas in a star becomes more [[Theory of relativity|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]] or to a [[black hole]] in the case of a [[neutron star]].

==Derivation within Newtonian gravity for a uniform sphere==
The gravitational binding energy of a sphere with radius <math>R</math> 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 <math>\rho</math>, the masses of a shell and the sphere inside it are:
<math display="block">m_\mathrm{shell} = 4\pi r^{2}\rho\,dr</math>
and
<math display="block">m_\mathrm{interior} = \frac{4}{3}\pi r^3 \rho</math>

The required energy for a shell is the negative of the gravitational potential energy:
<math display="block">dU = -G\frac{m_\mathrm{shell} m_\mathrm{interior}}{r}</math>

Integrating over all shells yields:
<math display="block">U = -G\int_0^R {\frac{\left(4\pi r^2\rho\right)\left(\tfrac{4}{3}\pi r^{3}\rho\right)}{r}} dr = -G{\frac{16}{3}}\pi^2 \rho^2 \int_0^R {r^4} dr = -G{\frac{16}{15}}{\pi}^2{\rho}^2 R^5</math>

Since <math>\rho</math> is simply equal to the mass of the whole divided by its volume for objects with uniform density, therefore

<math display="block">\rho=\frac{M}{\frac{4}{3}\pi R^3}</math>

And finally, plugging this into our result leads to
<math display="block">U = -G\frac{16}{15} \pi^2 R^5 \left(\frac{M}{\frac{4}{3}\pi R^3}\right)^2 = -\frac{3GM^2}{5R}</math>

{{Equation box 1|title='''Gravitational binding energy'''|equation=<math>U = -\frac{3GM^2}{5R}</math>}}

==Negative mass component==
{{Cleanup rewrite|there appears to be a serious conceptual inconsistency between the newtonian formula for binding energy and the relativistic concept of Schwarzschild radius.  Perhaps the section would be best deleted.|section|date=August 2024}}

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:
<math display="block">M_\mathrm{binding}=-\frac{3GM^2}{5Rc^2}</math>

For example, the fact that Earth is a gravitationally-bound sphere of its current size ''costs'' {{val|2.49421|e=15|ul=kg}} of mass (roughly one fourth the mass of [[Phobos (moon)|Phobos]] – see above for [[Mass–energy equivalence|the same value]] in [[Joule]]s), and if its atoms were sparse over an arbitrarily large volume the Earth would weigh its current mass plus {{val|2.49421|e=15|u=kg}} 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:
<math display="block">R\leq\frac{3GM}{5c^2}</math>
which is smaller than <math display="inline">\frac{3}{10}</math> its [[Schwarzschild radius]]:
<math display="block">R\leq\frac{3}{10} r_\mathrm{s}</math>
and therefore never visible to an external observer. However this is only a Newtonian approximation and in [[General Relativity|relativistic]] conditions other factors must be taken into account as well.<ref>{{cite journal | last1 = Katz | first1 = Joseph | last2 = Lynden-Bell | first2 = Donald | last3 = Bičák | first3 = Jiří | date = 27 October 2006 | title = Gravitational energy in stationary spacetimes | journal = [[Classical and Quantum Gravity]] | volume = 23 | issue = 23 | pages = 7111–7128 | doi = 10.1088/0264-9381/23/23/030 | arxiv = gr-qc/0610052 | bibcode = 2006CQGra..23.7111K | s2cid = 1375765 }}</ref>

==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.<ref>[http://www.ns-grb.com/PPT/Lattimer.pdf Neutron Star Masses and Radii] {{Webarchive|url=https://web.archive.org/web/20111217102314/http://www.ns-grb.com/PPT/Lattimer.pdf |date=2011-12-17 }}, p. 9/20, bottom</ref> 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'', 
<math display="block">BE = \frac{0.60\,\beta}{1 - \frac{\beta}{2}}</math>
<math display="block">\beta = \frac{G M}{R c^2} .</math>

Given current values
*<math>G = 6.6743\times10^{-11}\, \mathrm{m^3 \cdot kg^{-1} \cdot s^{-2}}</math>{{physconst|G|ref=only}}
*<math>c^2 = 8.98755\times10^{16}\, \mathrm{m^2 \cdot s^{-2}}</math>
*<math>M_\odot = 1.98844\times10^{30}\, \mathrm{kg}</math>
and the star mass ''M'' expressed relative to the solar mass,
<math display="block">M_x = \frac{M}{M_\odot} ,</math>

then the relativistic fractional binding energy of a neutron star is

<math display="block">BE = \frac{885.975\,M_x}{R - 738.313\,M_x}</math>

==See also==
*[[Stress–energy tensor]]
*[[Stress–energy–momentum pseudotensor]]
*[[Nordtvedt effect]]

==References==
<references/>

[[Category:Gravity|Binding energy]]
[[Category:Binding energy]]