1151:
20:
851:
72:. 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
605:
1488:
Neutron star relativistic equations of state include a graph of radius vs. mass for various models. 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
1074:
1342:
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:
1484:
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.
1699:
846:{\displaystyle 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}}
598:
1775:
937:
498:
1295:
1552:
1166:, as 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..
1831:
1960:
930:
419:
1470:
1888:
1604:
1394:
1135:
161:
80:
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
284:, the gravitational binding energy required for hydrostatic equilibrium approaches zero and the star becomes unstable (highly sensitive to perturbations), which may lead to a
1421:
328:
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.
873:
349:
505:
326:
424:
1614:
1207:
1500:
1897:
880:
2182:
1705:
354:
2065:
1430:
1975:
1781:
1843:
61:
54:
1556:
1346:
110:
1069:{\displaystyle U=-G{\frac {16}{15}}\pi ^{2}R^{5}\left({\frac {M}{{\frac {4}{3}}\pi R^{3}}}\right)^{2}=-{\frac {3GM^{2}}{5R}}}
1172:
242:
2157:
2100:
Katz, Joseph; Lynden-Bell, Donald; Bičák, Jiřà (27 October 2006). "Gravitational energy in stationary spacetimes".
84:
of the system, as it is manifest in its gravitational interaction with other distant systems, and the sum of the
1196:
from each other and reciprocally not moving, exert a gravitational force on a third body slightly smaller when
1084:
2102:
1328:), and if its atoms were sparse over an arbitrarily large volume the Earth would weigh its current mass plus
238:
49:
than the sum of the energies of its parts when these are completely separated—this is what keeps the system
1472:
and therefore never visible to an external observer. However this is only a
Newtonian approximation and in
875:
is simply equal to the mass of the whole divided by its volume for objects with uniform density, therefore
1321:
34:
of a system is the minimum energy which must be added to it in order for the system to cease being in a
2026:
1168:
1163:
2033:(Chicago: U. of Chicago; reprinted in New York: Dover), section 9, eqs. 90–92, p. 51 (Dover edition)
1970:
2213:
1399:
277:
81:
168:
246:
2121:
2074:
1424:
858:
334:
281:
46:
8:
93:
2125:
2078:
1339:
kilograms (and its gravitational pull over a third body would be accordingly stronger).
50:
2137:
2111:
1473:
1299:
For example, the fact that Earth is a gravitationally-bound sphere of its current size
502:
The required energy for a shell is the negative of the gravitational potential energy:
311:
289:
187:
69:
2133:
60:
The gravitational binding energy can be conceptually different within the theories of
2086:
245:(PREM). Using this, the real gravitational binding energy of Earth can be calculated
2208:
2141:
2129:
2082:
1980:
73:
1694:{\displaystyle G=6.6743\times 10^{-11}\,\mathrm {m^{3}\cdot kg^{-1}\cdot s^{-2}} }
2161:
2060:
2056:
234:, 60% of the absolute value of the potential energy per kilogram at the surface.
2174:
237:
The actual depth-dependence of density, inferred from seismic travel times (see
1999:
265:
186:
is a sphere of uniform density (which it is not, but is close enough to get an
85:
23:
2202:
1317:
1201:
593:{\displaystyle dU=-G{\frac {m_{\mathrm {shell} }m_{\mathrm {interior} }}{r}}}
303:
297:
1150:
2154:
1770:{\displaystyle c^{2}=8.98755\times 10^{16}\,\mathrm {m^{2}\cdot s^{-2}} }
38:
1204:
component of the system, equal, for uniformly spherical solutions, to:
293:
26:
are the largest known gravitationally bound structures in the universe.
2116:
1892:
then the relativistic fractional binding energy of a neutron star is
285:
493:{\displaystyle m_{\mathrm {interior} }={\frac {4}{3}}\pi r^{3}\rho }
16:
Minimum energy to remove a system from a gravitationally bound state
1312:
65:
19:
1290:{\displaystyle M_{\mathrm {binding} }=-{\frac {3GM^{2}}{5Rc^{2}}}}
100:
35:
1547:{\displaystyle BE={\frac {0.60\,\beta }{1-{\frac {\beta }{2}}}}}
1826:{\displaystyle M_{\odot }=1.98844\times 10^{30}\,\mathrm {kg} }
1955:{\displaystyle BE={\frac {885.975\,M_{x}}{R-738.313\,M_{x}}}}
1476:
conditions other factors must be taken into account as well.
1325:
183:
273:
269:
89:
925:{\displaystyle \rho ={\frac {M}{{\frac {4}{3}}\pi R^{3}}}}
308:
The gravitational binding energy of a sphere with radius
227:
414:{\displaystyle m_{\mathrm {shell} }=4\pi r^{2}\rho \,dr}
304:
Derivation within newtonian gravity for a uniform sphere
2179:
The NIST Reference on
Constants, Units, and Uncertainty
2175:"2022 CODATA Value: Newtonian constant of gravitation"
1402:
674:
351:, the masses of a shell and the sphere inside it are:
1900:
1846:
1784:
1708:
1617:
1559:
1503:
1465:{\displaystyle R\leq {\frac {3}{10}}r_{\mathrm {s} }}
1433:
1349:
1210:
1087:
940:
883:
861:
608:
508:
427:
357:
337:
314:
280:
to be maintained. As the gas in a star becomes more
113:
2099:
2022:
2020:
934:
And finally, plugging this into our result leads to
2055:
1954:
1882:
1825:
1769:
1693:
1598:
1546:
1464:
1415:
1388:
1289:
1129:
1068:
924:
867:
845:
592:
492:
413:
343:
320:
155:
2031:An Introduction to the Study of Stellar Structure
2017:
2200:
1883:{\displaystyle M_{x}={\frac {M}{M_{\odot }}},}
288:in the case of a high-mass star due to strong
41:. A gravitationally bound system has a lower (
2063:(1981). "Preliminary Reference Earth Model".
107:is given in newtonian gravity by the formula
2066:Physics of the Earth and Planetary Interiors
1599:{\displaystyle \beta ={\frac {GM}{Rc^{2}}}.}
1389:{\displaystyle R\leq {\frac {3GM}{5c^{2}}}}
226:. This is roughly equal to one week of the
2167:
1140:
2115:
1938:
1916:
1814:
1738:
1643:
1519:
404:
2036:
1316:of mass (roughly one fourth the mass of
1130:{\displaystyle U=-{\frac {3GM^{2}}{5R}}}
268:, the gravitational binding energy of a
156:{\displaystyle U=-{\frac {3GM^{2}}{5R}}}
55:minimum total potential energy principle
18:
2201:
1840:expressed relative to the solar mass,
1479:
1144:
602:Integrating over all shells yields:
2049:
1976:Stress–energy–momentum pseudotensor
1489:neutron star gravitational mass of
1192:Two bodies, placed at the distance
103:, the gravitational binding energy
13:
1819:
1816:
1754:
1741:
1678:
1662:
1658:
1646:
1456:
1235:
1232:
1229:
1226:
1223:
1220:
1217:
578:
575:
572:
569:
566:
563:
560:
557:
545:
542:
539:
536:
533:
455:
452:
449:
446:
443:
440:
437:
434:
376:
373:
370:
367:
364:
14:
2225:
2046:(Berlin: Springer Verlag), p. 272
243:Preliminary Reference Earth Model
1200:is small. This can be seen as a
1149:
272:is about two times its internal
99:For a spherical body of uniform
96:of the system if disassembled.
175:is the mass of the sphere, and
2148:
2093:
1992:
230:'s total energy output. It is
47:gravitational potential energy
1:
2155:Neutron Star Masses and Radii
2103:Classical and Quantum Gravity
1986:
2087:10.1016/0031-9201(81)90046-7
1416:{\textstyle {\frac {3}{10}}}
1079:Gravitational binding energy
331:Assuming a constant density
68:'s theory of gravity called
32:gravitational binding energy
7:
2134:10.1088/0264-9381/23/23/030
1964:
1162:to comply with Knowledge's
10:
2230:
239:Adams–Williamson equation
1175:may contain suggestions.
1160:may need to be rewritten
1141:Negative mass component
278:hydrostatic equilibrium
53:in accordance with the
2044:Astrophysical Formulae
1956:
1884:
1827:
1771:
1695:
1600:
1548:
1466:
1417:
1396:which is smaller than
1390:
1291:
1131:
1070:
926:
869:
847:
594:
494:
415:
345:
322:
169:gravitational constant
157:
27:
1957:
1885:
1828:
1772:
1696:
1608:Given current values
1601:
1549:
1467:
1418:
1391:
1292:
1132:
1071:
927:
870:
868:{\displaystyle \rho }
848:
595:
495:
416:
346:
344:{\displaystyle \rho }
323:
158:
22:
1971:Stress–energy tensor
1898:
1844:
1782:
1706:
1615:
1557:
1501:
1431:
1425:Schwarzschild radius
1400:
1347:
1208:
1085:
938:
881:
859:
606:
506:
425:
355:
335:
312:
111:
94:elementary particles
2126:2006CQGra..23.7111K
2079:1981PEPI...25..297D
1480:Non-uniform spheres
771:
635:
241:), is given in the
2160:2011-12-17 at the
2042:Lang, K. R. 1980,
2000:"Spot the cluster"
1952:
1880:
1836:and the star mass
1823:
1767:
1691:
1596:
1544:
1462:
1413:
1386:
1287:
1127:
1066:
922:
865:
843:
757:
683:
621:
590:
490:
411:
341:
318:
290:radiation pressure
188:order-of-magnitude
182:Assuming that the
153:
70:General Relativity
28:
2164:, p. 9/20, bottom
2110:(23): 7111–7128.
2057:Dziewonski, A. M.
2027:Chandrasekhar, S.
1950:
1875:
1591:
1542:
1539:
1448:
1411:
1384:
1285:
1190:
1189:
1164:quality standards
1125:
1064:
1018:
1002:
961:
920:
904:
807:
735:
710:
682:
588:
472:
321:{\displaystyle R}
296:in the case of a
264:According to the
151:
62:newtonian gravity
45:, more negative)
2221:
2194:
2193:
2191:
2190:
2171:
2165:
2152:
2146:
2145:
2119:
2097:
2091:
2090:
2053:
2047:
2040:
2034:
2024:
2015:
2014:
2012:
2010:
1996:
1981:Nordtvedt effect
1961:
1959:
1958:
1953:
1951:
1949:
1948:
1947:
1927:
1926:
1925:
1911:
1889:
1887:
1886:
1881:
1876:
1874:
1873:
1861:
1856:
1855:
1832:
1830:
1829:
1824:
1822:
1813:
1812:
1794:
1793:
1776:
1774:
1773:
1768:
1766:
1765:
1764:
1749:
1748:
1737:
1736:
1718:
1717:
1700:
1698:
1697:
1692:
1690:
1689:
1688:
1673:
1672:
1654:
1653:
1642:
1641:
1605:
1603:
1602:
1597:
1592:
1590:
1589:
1588:
1575:
1567:
1553:
1551:
1550:
1545:
1543:
1541:
1540:
1532:
1523:
1514:
1471:
1469:
1468:
1463:
1461:
1460:
1459:
1449:
1441:
1422:
1420:
1419:
1414:
1412:
1404:
1395:
1393:
1392:
1387:
1385:
1383:
1382:
1381:
1368:
1357:
1338:
1336:
1333:
1320:– see above for
1315:
1310:
1307:
1296:
1294:
1293:
1288:
1286:
1284:
1283:
1282:
1266:
1265:
1264:
1248:
1240:
1239:
1238:
1185:
1182:
1176:
1153:
1145:
1136:
1134:
1133:
1128:
1126:
1124:
1116:
1115:
1114:
1098:
1075:
1073:
1072:
1067:
1065:
1063:
1055:
1054:
1053:
1037:
1029:
1028:
1023:
1019:
1017:
1016:
1015:
1003:
995:
989:
982:
981:
972:
971:
962:
954:
931:
929:
928:
923:
921:
919:
918:
917:
905:
897:
891:
874:
872:
871:
866:
852:
850:
849:
844:
842:
841:
832:
831:
826:
820:
819:
814:
808:
800:
783:
782:
781:
770:
765:
756:
755:
746:
745:
736:
728:
711:
706:
705:
701:
697:
696:
684:
675:
667:
663:
659:
658:
637:
634:
629:
599:
597:
596:
591:
589:
584:
583:
582:
581:
550:
549:
548:
525:
499:
497:
496:
491:
486:
485:
473:
465:
460:
459:
458:
420:
418:
417:
412:
400:
399:
381:
380:
379:
350:
348:
347:
342:
327:
325:
324:
319:
260:
258:
233:
225:
223:
213:
211:
201:
199:
162:
160:
159:
154:
152:
150:
142:
141:
140:
124:
76:property of the
2229:
2228:
2224:
2223:
2222:
2220:
2219:
2218:
2199:
2198:
2197:
2188:
2186:
2173:
2172:
2168:
2162:Wayback Machine
2153:
2149:
2098:
2094:
2061:Anderson, D. L.
2054:
2050:
2041:
2037:
2025:
2018:
2008:
2006:
1998:
1997:
1993:
1989:
1967:
1943:
1939:
1928:
1921:
1917:
1912:
1910:
1899:
1896:
1895:
1869:
1865:
1860:
1851:
1847:
1845:
1842:
1841:
1815:
1808:
1804:
1789:
1785:
1783:
1780:
1779:
1757:
1753:
1744:
1740:
1739:
1732:
1728:
1713:
1709:
1707:
1704:
1703:
1681:
1677:
1665:
1661:
1649:
1645:
1644:
1634:
1630:
1616:
1613:
1612:
1584:
1580:
1576:
1568:
1566:
1558:
1555:
1554:
1531:
1524:
1515:
1513:
1502:
1499:
1498:
1482:
1455:
1454:
1450:
1440:
1432:
1429:
1428:
1403:
1401:
1398:
1397:
1377:
1373:
1369:
1358:
1356:
1348:
1345:
1344:
1334:
1331:
1329:
1308:
1305:
1303:
1278:
1274:
1267:
1260:
1256:
1249:
1247:
1216:
1215:
1211:
1209:
1206:
1205:
1186:
1180:
1177:
1167:
1154:
1143:
1138:
1117:
1110:
1106:
1099:
1097:
1086:
1083:
1082:
1056:
1049:
1045:
1038:
1036:
1024:
1011:
1007:
994:
993:
988:
984:
983:
977:
973:
967:
963:
953:
939:
936:
935:
913:
909:
896:
895:
890:
882:
879:
878:
860:
857:
856:
837:
833:
827:
822:
821:
815:
810:
809:
799:
777:
773:
772:
766:
761:
751:
747:
741:
737:
727:
692:
688:
673:
672:
668:
654:
650:
643:
639:
638:
636:
630:
625:
607:
604:
603:
556:
555:
551:
532:
531:
527:
526:
524:
507:
504:
503:
481:
477:
464:
433:
432:
428:
426:
423:
422:
395:
391:
363:
362:
358:
356:
353:
352:
336:
333:
332:
313:
310:
309:
306:
256:
254:
232:37.5 MJ/kg
231:
221:
219:
209:
207:
197:
195:
190:estimate) with
179:is its radius.
143:
136:
132:
125:
123:
112:
109:
108:
66:Albert Einstein
36:gravitationally
24:Galaxy clusters
17:
12:
11:
5:
2227:
2217:
2216:
2214:Binding energy
2211:
2196:
2195:
2166:
2147:
2092:
2073:(4): 297–356.
2048:
2035:
2016:
1990:
1988:
1985:
1984:
1983:
1978:
1973:
1966:
1963:
1946:
1942:
1937:
1934:
1931:
1924:
1920:
1915:
1909:
1906:
1903:
1879:
1872:
1868:
1864:
1859:
1854:
1850:
1834:
1833:
1821:
1818:
1811:
1807:
1803:
1800:
1797:
1792:
1788:
1777:
1763:
1760:
1756:
1752:
1747:
1743:
1735:
1731:
1727:
1724:
1721:
1716:
1712:
1701:
1687:
1684:
1680:
1676:
1671:
1668:
1664:
1660:
1657:
1652:
1648:
1640:
1637:
1633:
1629:
1626:
1623:
1620:
1595:
1587:
1583:
1579:
1574:
1571:
1565:
1562:
1538:
1535:
1530:
1527:
1522:
1518:
1512:
1509:
1506:
1481:
1478:
1458:
1453:
1447:
1444:
1439:
1436:
1410:
1407:
1380:
1376:
1372:
1367:
1364:
1361:
1355:
1352:
1322:the same value
1281:
1277:
1273:
1270:
1263:
1259:
1255:
1252:
1246:
1243:
1237:
1234:
1231:
1228:
1225:
1222:
1219:
1214:
1188:
1187:
1157:
1155:
1148:
1142:
1139:
1123:
1120:
1113:
1109:
1105:
1102:
1096:
1093:
1090:
1077:
1062:
1059:
1052:
1048:
1044:
1041:
1035:
1032:
1027:
1022:
1014:
1010:
1006:
1001:
998:
992:
987:
980:
976:
970:
966:
960:
957:
952:
949:
946:
943:
916:
912:
908:
903:
900:
894:
889:
886:
864:
840:
836:
830:
825:
818:
813:
806:
803:
798:
795:
792:
789:
786:
780:
776:
769:
764:
760:
754:
750:
744:
740:
734:
731:
726:
723:
720:
717:
714:
709:
704:
700:
695:
691:
687:
681:
678:
671:
666:
662:
657:
653:
649:
646:
642:
633:
628:
624:
620:
617:
614:
611:
587:
580:
577:
574:
571:
568:
565:
562:
559:
554:
547:
544:
541:
538:
535:
530:
523:
520:
517:
514:
511:
489:
484:
480:
476:
471:
468:
463:
457:
454:
451:
448:
445:
442:
439:
436:
431:
410:
407:
403:
398:
394:
390:
387:
384:
378:
375:
372:
369:
366:
361:
340:
317:
305:
302:
274:thermal energy
266:virial theorem
149:
146:
139:
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131:
128:
122:
119:
116:
79:
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2117:gr-qc/0610052
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2088:
2084:
2080:
2076:
2072:
2068:
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2052:
2045:
2039:
2032:
2028:
2023:
2021:
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1995:
1991:
1982:
1979:
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1944:
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1932:
1929:
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1893:
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1877:
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291:
287:
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279:
276:in order for
275:
271:
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189:
185:
180:
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282:relativistic
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176:
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2004:www.eso.org
1181:August 2024
247:numerically
88:of all the
39:bound state
2203:Categories
2189:2024-05-18
2185:. May 2024
1987:References
1337:10 kg
294:black hole
200:10 kg
92:and other
51:aggregated
1933:−
1871:⊙
1802:×
1791:⊙
1759:−
1751:⋅
1726:×
1683:−
1675:⋅
1667:−
1656:⋅
1636:−
1628:×
1561:β
1534:β
1529:−
1521:β
1438:≤
1354:≤
1245:−
1173:talk page
1095:−
1034:−
1005:π
965:π
948:−
907:π
885:ρ
863:ρ
824:ρ
812:π
794:−
759:∫
749:ρ
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722:−
699:ρ
686:π
661:ρ
648:π
623:∫
616:−
519:−
488:ρ
475:π
402:ρ
389:π
339:ρ
286:supernova
259:10 J
224:10 J
212:10 m
121:−
74:nonlinear
2158:Archived
1965:See also
1311:10
292:or to a
86:energies
82:ADM mass
2209:Gravity
2142:1375765
2122:Bibcode
2075:Bibcode
2009:31 July
1936:738.313
1914:885.975
1799:1.98844
1723:8.98755
214:, then
167:is the
101:density
2140:
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1625:6.6743
1326:Joules
1318:Phobos
1171:. The
855:Since
163:where
78:entire
2138:S2CID
2112:arXiv
1330:2.494
1304:2.494
1301:costs
184:Earth
90:atoms
2183:NIST
2011:2017
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1423:its
421:and
270:star
255:2.49
220:2.24
208:6.37
202:and
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253:=
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2114::
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377:l
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257:Ă—
251:U
222:Ă—
216:U
210:Ă—
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165:G
148:R
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127:3
118:=
115:U
105:U
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