99:"One would make a grave mistake if one supposed that the spheroids of revolution are the only admissible figures of equilibrium even under the restrictive assumption of second-degree surfaces" (...) "In fact a simple consideration shows that ellipsoids with three unequal axes can very well be figures of equilibrium; and that one can assume an ellipse of arbitrary shape for the equatorial section and determine the third axis (which is also the least of the three axes) and the angular velocity of rotation such that the ellipsoid is a figure of equilibrium."
109:
20:
455:
1645:
1702:
The Jacobi and
Dedekind ellipsoids are both equilibrium figures for a body of rotating homogeneous self-gravitating fluid. However, while the Jacobi ellipsoid spins bodily, with no internal flow of the fluid in the rotating frame, the Dedekind ellipsoid maintains a fixed orientation, with the
688:
2347:
1871:
234:
1491:
1460:
1128:
831:
134:
The broken lines are for the
Maclaurin spheroid in the range where it has dynamic but not secular stability – it will relax into the Jacobi ellipsoid provided it can dissipate energy by virtue of a viscous constituent
2170:
2085:
510:
1977:
2246:
1758:
2235:
2015:
1912:
1746:
1169:
2105:
206:
2035:
2208:, the Jacobi and Dedekind ellipsoids (and the Maclaurin spheroid) become one and the same; bodily rotation and circular flow amount to the same thing. In this case
869:
478:
450:{\displaystyle {\frac {\Omega ^{2}}{\pi G\rho }}=2abc\int _{0}^{\infty }{\frac {u\,du}{(a^{2}+u)(b^{2}+u)\Delta }}\ ,\quad \Delta ^{2}=(a^{2}+u)(b^{2}+u)(c^{2}+u),}
166:
2206:
1640:{\displaystyle {\frac {L}{\sqrt {GM^{3}r}}}={\frac {\sqrt {3}}{10}}{\frac {a^{2}+b^{2}}{r^{2}}}{\sqrt {\frac {\Omega ^{2}}{\pi G\rho }}}\ ,\quad r={\sqrt{abc}},}
1688:
1668:
1483:
751:
731:
711:
498:
226:
186:
2240:
In the general case, the Jacobi and
Dedekind ellipsoids have the same energy, but the angular momentum of the Jacobi spheroid is the greater by a factor of
1177:
2363:
877:
2368:
759:
2113:
1704:
683:{\displaystyle a^{2}b^{2}\int _{0}^{\infty }{\frac {du}{(a^{2}+u)(b^{2}+u)\Delta }}=c^{2}\int _{0}^{\infty }{\frac {du}{(c^{2}+u)\Delta }}.}
2044:
83:
in 1811 considered the possibility of a tri-axial ellipsoid being in equilibrium, but concluded that the two equatorial axes of the
124:) semi-principal axes of a Jacobi ellipsoid and Maclaurin spheroid, as a function of normalized angular momentum, subject to
2342:{\displaystyle {\frac {L_{\mathrm {Jac} }}{L_{\mathrm {Ded} }}}={\frac {1}{2}}\left({\frac {a}{b}}+{\frac {b}{a}}\right).}
1917:
2373:
2631:
2668:
1866:{\displaystyle \mathbf {u} =\zeta {\frac {-a^{2}y\mathbf {\hat {x}} +b^{2}x\mathbf {\hat {y}} }{a^{2}+b^{2}}},}
2566:
2211:
1710:
For any given Jacobi ellipsoid, there exists a
Dedekind ellipsoid with the same semi-principal axes
88:
60:
2663:
40:
1982:
1879:
1713:
1136:
841:
501:
2619:
2658:
2575:
2176:
2107:
of the Jacobi ellipsoid and vorticity of the corresponding
Dedekind ellipsoid are related by
2090:
191:
92:
80:
2020:
2584:
2529:
2418:
847:
463:
142:
2179:
elliptical circuit in the same period in which the Jacobi spheroid performs one rotation.
8:
2501:
2185:
2588:
2533:
2520:
Darwin, G. H. (1886). "On Jacobi's figure of equilibrium for a rotating mass of fluid".
2422:
2545:
2409:
2358:
1673:
1653:
1468:
736:
716:
696:
483:
211:
171:
72:
36:
2459:
Chandrasekhar, S. (1967). "Ellipsoidal figures of equilibrium—an historical account".
1455:{\displaystyle {\frac {a^{2}b^{2}}{b^{2}-a^{2}}}=c^{2}R_{J}(a^{2},b^{2},c^{2},c^{2}).}
2653:
2627:
2549:
837:
2615:
2592:
2537:
2468:
2426:
56:
44:
1123:{\displaystyle {\frac {\Omega ^{2}}{\pi G\rho }}={\frac {4abc}{3(a^{2}-b^{2})}}}
108:
2647:
2430:
1749:
95:'s demonstration is a sufficiency condition, but not necessary. He remarked:
826:{\displaystyle {\frac {1}{c^{2}}}>{\frac {1}{a^{2}}}+{\frac {1}{b^{2}}}.}
2541:
2472:
2175:
That is, each particle of the fluid of the
Dedekind ellipsoid describes a
16:
Shape taken by a self-gravitating fluid body rotating at constant velocity
2165:{\displaystyle \zeta =\left({\frac {a}{b}}+{\frac {b}{a}}\right)\Omega .}
1703:
constituent fluid circulating within it. This is a direct consequence of
2499:
Dirichlet, G. L. (1856). "Gedächtnisrede auf Carl Gustav Jacob Jacobi".
87:
must be equal, leading back to the solution of
Maclaurin spheroid. But
75:, which was formulated in 1742, was considered to be the only type of
2383:
2038:
84:
76:
2404:
2080:{\displaystyle \nabla \times \mathbf {u} =\zeta \mathbf {\hat {z}} }
2597:
2570:
2378:
1133:
and the condition on the relative size of the semi-principal axes
19:
52:
24:
2571:"The Equilibrium and the Stability of the Dedekind Ellipsoids"
2448:. Vol. 10. New Haven: Yale University Press. p. 253.
1694:, the radius of a sphere of the same volume as the ellipsoid.
48:
2626:. Houches Lecture Series. CRC Press. pp. 267–268.
1697:
2249:
2214:
2188:
2116:
2093:
2047:
2023:
1985:
1920:
1882:
1761:
1716:
1676:
1656:
1494:
1471:
1180:
1139:
880:
850:
762:
739:
719:
699:
513:
486:
466:
237:
214:
194:
174:
145:
139:
For an ellipsoid with equatorial semi-principal axes
1972:{\displaystyle {\hat {x}},\ {\hat {y}},\ {\hat {z}}}
2561:
2559:
2341:
2229:
2200:
2164:
2099:
2079:
2029:
2009:
1971:
1906:
1865:
1740:
1682:
1662:
1639:
1477:
1454:
1163:
1122:
863:
825:
745:
725:
705:
682:
492:
472:
449:
220:
200:
180:
160:
2237:, as is always true for a rigidly rotating body.
2071:
1826:
1798:
2645:
2622:. In DeWitt, C.; DeWitt, Bryce Seligman (eds.).
2620:"Rapidly Rotating Stars, Disks, and Black Holes"
2556:
2502:Journal für die reine und angewandte Mathematik
871:, the formula for the angular velocity becomes
27:, a dwarf planet with triaxial ellipsoid shape.
2461:Communications on Pure and Applied Mathematics
2565:
2458:
2443:
128: = 1 (i.e. for constant volume of 4
59:. It is named after the German mathematician
2041:, which is uniform throughout the spheroid (
836:The integrals can be expressed in terms of
2522:Proceedings of the Royal Society of London
2610:
2608:
2596:
2498:
299:
107:
18:
2614:
733:, the above condition has solution for
2646:
2605:
2519:
2402:
2405:"Ueber die Figur des Gleichgewichts"
1698:Relationship with Dedekind ellipsoid
1485:of the Jacobi ellipsoid is given by
13:
2446:Ellipsoidal figures of equilibrium
2282:
2279:
2276:
2264:
2261:
2258:
2224:
2156:
2094:
2048:
1914:are Cartesian coordinates on axes
1578:
884:
671:
633:
604:
544:
366:
352:
288:
241:
195:
14:
2680:
1670:is the mass of the ellipsoid and
103:
37:triaxial (i.e. scalene) ellipsoid
2068:
2055:
1823:
1795:
1763:
2374:Dirichlet's ellipsoidal problem
2230:{\displaystyle \zeta =2\Omega }
1607:
364:
168:and polar semi-principal axis
2513:
2492:
2479:
2452:
2437:
2396:
1979:aligned respectively with the
1963:
1945:
1927:
1446:
1394:
1368:
1365:
1313:
1297:
1245:
1232:
1117:
1114:
1062:
1036:
984:
961:
955:
929:
668:
649:
601:
582:
579:
560:
441:
422:
419:
400:
397:
378:
349:
330:
327:
308:
1:
2389:
838:incomplete elliptic integrals
79:which can be in equilibrium.
2017:axes of the ellipsoid. Here
7:
2567:Chandrasekhar, Subrahmanyan
2352:
504:, subject to the condition
10:
2685:
2444:Chandrasekhar, S. (1969).
66:
2010:{\displaystyle a,\ b,\ c}
1907:{\displaystyle x,\ y,\ z}
1748:and same mass and with a
1741:{\displaystyle a,\ b,\ c}
1164:{\displaystyle a,\ b,\ c}
2485:Lagrange, J. L. (1811).
2431:10.1002/andp.18341090808
2087:). The angular velocity
61:Carl Gustav Jacob Jacobi
55:rotates with a constant
2182:In the special case of
2100:{\displaystyle \Omega }
201:{\displaystyle \Omega }
188:, the angular velocity
41:hydrostatic equilibrium
2669:Equations of astronomy
2542:10.1098/rspl.1886.0099
2473:10.1002/cpa.3160200203
2403:Jacobi, C. G. (1834).
2343:
2231:
2202:
2166:
2101:
2081:
2031:
2030:{\displaystyle \zeta }
2011:
1973:
1908:
1867:
1742:
1684:
1664:
1641:
1479:
1456:
1165:
1124:
865:
842:Carlson symmetric form
827:
747:
727:
707:
684:
502:gravitational constant
494:
474:
451:
222:
202:
182:
162:
136:
101:
28:
23:Artistic rendering of
2576:Astrophysical Journal
2344:
2232:
2203:
2167:
2102:
2082:
2032:
2012:
1974:
1909:
1868:
1743:
1685:
1665:
1642:
1480:
1465:The angular momentum
1457:
1166:
1125:
866:
864:{\displaystyle R_{J}}
828:
748:
728:
708:
685:
495:
475:
473:{\displaystyle \rho }
452:
223:
203:
183:
163:
161:{\displaystyle a,\ b}
111:
97:
22:
2528:(246–250): 319–336.
2487:Mécanique Analytique
2247:
2212:
2186:
2114:
2091:
2045:
2021:
1983:
1918:
1880:
1759:
1714:
1674:
1654:
1492:
1469:
1178:
1137:
878:
848:
760:
737:
717:
697:
693:For fixed values of
511:
484:
464:
235:
212:
192:
172:
143:
43:which arises when a
2589:1965ApJ...141.1043C
2534:1886RSPS...41..319D
2423:1834AnP...109..229J
2201:{\displaystyle a=b}
1750:flow velocity field
637:
548:
480:is the density and
292:
71:Before Jacobi, the
2410:Annalen der Physik
2359:Maclaurin spheroid
2339:
2227:
2198:
2162:
2097:
2077:
2027:
2007:
1969:
1904:
1863:
1738:
1705:Dedekind's theorem
1680:
1660:
1637:
1475:
1452:
1161:
1120:
861:
844:elliptic integral
840:. In terms of the
823:
743:
723:
703:
680:
623:
534:
490:
470:
447:
278:
218:
198:
178:
158:
137:
73:Maclaurin spheroid
29:
2616:Bardeen, James M.
2417:(8–16): 229–233.
2364:Riemann ellipsoid
2329:
2316:
2301:
2288:
2149:
2136:
2074:
2003:
1994:
1966:
1956:
1948:
1938:
1930:
1900:
1891:
1858:
1829:
1801:
1734:
1725:
1683:{\displaystyle r}
1663:{\displaystyle M}
1632:
1603:
1599:
1598:
1572:
1533:
1529:
1518:
1517:
1478:{\displaystyle L}
1230:
1157:
1148:
959:
904:
818:
798:
778:
746:{\displaystyle c}
726:{\displaystyle b}
706:{\displaystyle a}
675:
608:
493:{\displaystyle G}
360:
356:
261:
221:{\displaystyle c}
181:{\displaystyle c}
154:
2676:
2638:
2637:
2612:
2603:
2602:
2600:
2563:
2554:
2553:
2517:
2511:
2510:
2496:
2490:
2483:
2477:
2476:
2456:
2450:
2449:
2441:
2435:
2434:
2400:
2348:
2346:
2345:
2340:
2335:
2331:
2330:
2322:
2317:
2309:
2302:
2294:
2289:
2287:
2286:
2285:
2269:
2268:
2267:
2251:
2236:
2234:
2233:
2228:
2207:
2205:
2204:
2199:
2171:
2169:
2168:
2163:
2155:
2151:
2150:
2142:
2137:
2129:
2106:
2104:
2103:
2098:
2086:
2084:
2083:
2078:
2076:
2075:
2067:
2058:
2036:
2034:
2033:
2028:
2016:
2014:
2013:
2008:
2001:
1992:
1978:
1976:
1975:
1970:
1968:
1967:
1959:
1954:
1950:
1949:
1941:
1936:
1932:
1931:
1923:
1913:
1911:
1910:
1905:
1898:
1889:
1872:
1870:
1869:
1864:
1859:
1857:
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1855:
1843:
1842:
1832:
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1822:
1816:
1815:
1803:
1802:
1794:
1788:
1787:
1774:
1766:
1747:
1745:
1744:
1739:
1732:
1723:
1689:
1687:
1686:
1681:
1669:
1667:
1666:
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1646:
1644:
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1633:
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1626:
1615:
1601:
1600:
1597:
1586:
1585:
1576:
1575:
1573:
1571:
1570:
1561:
1560:
1559:
1547:
1546:
1536:
1534:
1525:
1524:
1519:
1513:
1512:
1500:
1496:
1484:
1482:
1481:
1476:
1461:
1459:
1458:
1453:
1445:
1444:
1432:
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1419:
1418:
1406:
1405:
1393:
1392:
1383:
1382:
1364:
1363:
1351:
1350:
1338:
1337:
1325:
1324:
1312:
1311:
1296:
1295:
1283:
1282:
1270:
1269:
1257:
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1244:
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1231:
1229:
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1215:
1214:
1204:
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1202:
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1192:
1182:
1170:
1168:
1167:
1162:
1155:
1146:
1129:
1127:
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1121:
1113:
1112:
1100:
1099:
1087:
1086:
1074:
1073:
1061:
1060:
1051:
1050:
1035:
1034:
1022:
1021:
1009:
1008:
996:
995:
983:
982:
973:
972:
960:
958:
954:
953:
941:
940:
924:
910:
905:
903:
892:
891:
882:
870:
868:
867:
862:
860:
859:
832:
830:
829:
824:
819:
817:
816:
804:
799:
797:
796:
784:
779:
777:
776:
764:
752:
750:
749:
744:
732:
730:
729:
724:
712:
710:
709:
704:
689:
687:
686:
681:
676:
674:
661:
660:
647:
639:
636:
631:
622:
621:
609:
607:
594:
593:
572:
571:
558:
550:
547:
542:
533:
532:
523:
522:
499:
497:
496:
491:
479:
477:
476:
471:
456:
454:
453:
448:
434:
433:
412:
411:
390:
389:
374:
373:
358:
357:
355:
342:
341:
320:
319:
306:
294:
291:
286:
262:
260:
249:
248:
239:
227:
225:
224:
219:
207:
205:
204:
199:
187:
185:
184:
179:
167:
165:
164:
159:
152:
131:
112:The equatorial (
57:angular velocity
51:body of uniform
45:self-gravitating
33:Jacobi ellipsoid
2684:
2683:
2679:
2678:
2677:
2675:
2674:
2673:
2644:
2643:
2642:
2641:
2634:
2613:
2606:
2564:
2557:
2518:
2514:
2497:
2493:
2489:sect. IV 2 vol.
2484:
2480:
2457:
2453:
2442:
2438:
2401:
2397:
2392:
2369:Roche ellipsoid
2355:
2321:
2308:
2307:
2303:
2293:
2275:
2274:
2270:
2257:
2256:
2252:
2250:
2248:
2245:
2244:
2213:
2210:
2209:
2187:
2184:
2183:
2141:
2128:
2127:
2123:
2115:
2112:
2111:
2092:
2089:
2088:
2066:
2065:
2054:
2046:
2043:
2042:
2022:
2019:
2018:
1984:
1981:
1980:
1958:
1957:
1940:
1939:
1922:
1921:
1919:
1916:
1915:
1881:
1878:
1877:
1851:
1847:
1838:
1834:
1833:
1821:
1820:
1811:
1807:
1793:
1792:
1783:
1779:
1775:
1773:
1762:
1760:
1757:
1756:
1715:
1712:
1711:
1700:
1675:
1672:
1671:
1655:
1652:
1651:
1627:
1616:
1614:
1587:
1581:
1577:
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1555:
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1542:
1538:
1537:
1535:
1523:
1508:
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1495:
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1427:
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1401:
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1388:
1384:
1378:
1374:
1359:
1355:
1346:
1342:
1333:
1329:
1320:
1316:
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1303:
1291:
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1278:
1274:
1265:
1261:
1252:
1248:
1239:
1235:
1223:
1219:
1210:
1206:
1205:
1198:
1194:
1188:
1184:
1183:
1181:
1179:
1176:
1175:
1138:
1135:
1134:
1108:
1104:
1095:
1091:
1082:
1078:
1069:
1065:
1056:
1052:
1046:
1042:
1030:
1026:
1017:
1013:
1004:
1000:
991:
987:
978:
974:
968:
964:
949:
945:
936:
932:
925:
911:
909:
893:
887:
883:
881:
879:
876:
875:
855:
851:
849:
846:
845:
812:
808:
803:
792:
788:
783:
772:
768:
763:
761:
758:
757:
738:
735:
734:
718:
715:
714:
698:
695:
694:
656:
652:
648:
640:
638:
632:
627:
617:
613:
589:
585:
567:
563:
559:
551:
549:
543:
538:
528:
524:
518:
514:
512:
509:
508:
485:
482:
481:
465:
462:
461:
429:
425:
407:
403:
385:
381:
369:
365:
337:
333:
315:
311:
307:
295:
293:
287:
282:
250:
244:
240:
238:
236:
233:
232:
213:
210:
209:
193:
190:
189:
173:
170:
169:
144:
141:
140:
133:
129:
106:
69:
17:
12:
11:
5:
2682:
2672:
2671:
2666:
2664:Fluid dynamics
2661:
2656:
2640:
2639:
2632:
2604:
2598:10.1086/148195
2555:
2512:
2491:
2478:
2467:(2): 251–265.
2451:
2436:
2394:
2393:
2391:
2388:
2387:
2386:
2381:
2376:
2371:
2366:
2361:
2354:
2351:
2350:
2349:
2338:
2334:
2328:
2325:
2320:
2315:
2312:
2306:
2300:
2297:
2292:
2284:
2281:
2278:
2273:
2266:
2263:
2260:
2255:
2226:
2223:
2220:
2217:
2197:
2194:
2191:
2173:
2172:
2161:
2158:
2154:
2148:
2145:
2140:
2135:
2132:
2126:
2122:
2119:
2096:
2073:
2070:
2064:
2061:
2057:
2053:
2050:
2026:
2006:
2000:
1997:
1991:
1988:
1965:
1962:
1953:
1947:
1944:
1935:
1929:
1926:
1903:
1897:
1894:
1888:
1885:
1874:
1873:
1862:
1854:
1850:
1846:
1841:
1837:
1828:
1825:
1819:
1814:
1810:
1806:
1800:
1797:
1791:
1786:
1782:
1778:
1772:
1769:
1765:
1737:
1731:
1728:
1722:
1719:
1699:
1696:
1679:
1659:
1648:
1647:
1636:
1630:
1625:
1622:
1619:
1613:
1610:
1606:
1596:
1593:
1590:
1584:
1580:
1569:
1565:
1558:
1554:
1550:
1545:
1541:
1532:
1528:
1522:
1516:
1511:
1507:
1503:
1499:
1474:
1463:
1462:
1451:
1448:
1443:
1439:
1435:
1430:
1426:
1422:
1417:
1413:
1409:
1404:
1400:
1396:
1391:
1387:
1381:
1377:
1373:
1370:
1367:
1362:
1358:
1354:
1349:
1345:
1341:
1336:
1332:
1328:
1323:
1319:
1315:
1310:
1306:
1302:
1299:
1294:
1290:
1286:
1281:
1277:
1273:
1268:
1264:
1260:
1255:
1251:
1247:
1242:
1238:
1234:
1226:
1222:
1218:
1213:
1209:
1201:
1197:
1191:
1187:
1160:
1154:
1151:
1145:
1142:
1131:
1130:
1119:
1116:
1111:
1107:
1103:
1098:
1094:
1090:
1085:
1081:
1077:
1072:
1068:
1064:
1059:
1055:
1049:
1045:
1041:
1038:
1033:
1029:
1025:
1020:
1016:
1012:
1007:
1003:
999:
994:
990:
986:
981:
977:
971:
967:
963:
957:
952:
948:
944:
939:
935:
931:
928:
923:
920:
917:
914:
908:
902:
899:
896:
890:
886:
858:
854:
834:
833:
822:
815:
811:
807:
802:
795:
791:
787:
782:
775:
771:
767:
742:
722:
702:
691:
690:
679:
673:
670:
667:
664:
659:
655:
651:
646:
643:
635:
630:
626:
620:
616:
612:
606:
603:
600:
597:
592:
588:
584:
581:
578:
575:
570:
566:
562:
557:
554:
546:
541:
537:
531:
527:
521:
517:
489:
469:
458:
457:
446:
443:
440:
437:
432:
428:
424:
421:
418:
415:
410:
406:
402:
399:
396:
393:
388:
384:
380:
377:
372:
368:
363:
354:
351:
348:
345:
340:
336:
332:
329:
326:
323:
318:
314:
310:
305:
302:
298:
290:
285:
281:
277:
274:
271:
268:
265:
259:
256:
253:
247:
243:
217:
197:
177:
157:
151:
148:
105:
104:Jacobi formula
102:
91:realized that
68:
65:
15:
9:
6:
4:
3:
2:
2681:
2670:
2667:
2665:
2662:
2660:
2657:
2655:
2652:
2651:
2649:
2635:
2633:9780677156101
2629:
2625:
2621:
2617:
2611:
2609:
2599:
2594:
2590:
2586:
2583:: 1043–1055.
2582:
2578:
2577:
2572:
2568:
2562:
2560:
2551:
2547:
2543:
2539:
2535:
2531:
2527:
2523:
2516:
2508:
2505:(in German).
2504:
2503:
2495:
2488:
2482:
2474:
2470:
2466:
2462:
2455:
2447:
2440:
2432:
2428:
2424:
2420:
2416:
2413:(in German).
2412:
2411:
2406:
2399:
2395:
2385:
2382:
2380:
2377:
2375:
2372:
2370:
2367:
2365:
2362:
2360:
2357:
2356:
2336:
2332:
2326:
2323:
2318:
2313:
2310:
2304:
2298:
2295:
2290:
2271:
2253:
2243:
2242:
2241:
2238:
2221:
2218:
2215:
2195:
2192:
2189:
2180:
2178:
2159:
2152:
2146:
2143:
2138:
2133:
2130:
2124:
2120:
2117:
2110:
2109:
2108:
2062:
2059:
2051:
2040:
2024:
2004:
1998:
1995:
1989:
1986:
1960:
1951:
1942:
1933:
1924:
1901:
1895:
1892:
1886:
1883:
1860:
1852:
1848:
1844:
1839:
1835:
1817:
1812:
1808:
1804:
1789:
1784:
1780:
1776:
1770:
1767:
1755:
1754:
1753:
1751:
1735:
1729:
1726:
1720:
1717:
1708:
1706:
1695:
1693:
1677:
1657:
1634:
1628:
1623:
1620:
1617:
1611:
1608:
1604:
1594:
1591:
1588:
1582:
1567:
1563:
1556:
1552:
1548:
1543:
1539:
1530:
1526:
1520:
1514:
1509:
1505:
1501:
1497:
1488:
1487:
1486:
1472:
1449:
1441:
1437:
1433:
1428:
1424:
1420:
1415:
1411:
1407:
1402:
1398:
1389:
1385:
1379:
1375:
1371:
1360:
1356:
1352:
1347:
1343:
1339:
1334:
1330:
1326:
1321:
1317:
1308:
1304:
1300:
1292:
1288:
1284:
1279:
1275:
1271:
1266:
1262:
1258:
1253:
1249:
1240:
1236:
1224:
1220:
1216:
1211:
1207:
1199:
1195:
1189:
1185:
1174:
1173:
1172:
1158:
1152:
1149:
1143:
1140:
1109:
1105:
1101:
1096:
1092:
1088:
1083:
1079:
1075:
1070:
1066:
1057:
1053:
1047:
1043:
1039:
1031:
1027:
1023:
1018:
1014:
1010:
1005:
1001:
997:
992:
988:
979:
975:
969:
965:
950:
946:
942:
937:
933:
926:
921:
918:
915:
912:
906:
900:
897:
894:
888:
874:
873:
872:
856:
852:
843:
839:
820:
813:
809:
805:
800:
793:
789:
785:
780:
773:
769:
765:
756:
755:
754:
740:
720:
700:
677:
665:
662:
657:
653:
644:
641:
628:
624:
618:
614:
610:
598:
595:
590:
586:
576:
573:
568:
564:
555:
552:
539:
535:
529:
525:
519:
515:
507:
506:
505:
503:
487:
467:
444:
438:
435:
430:
426:
416:
413:
408:
404:
394:
391:
386:
382:
375:
370:
361:
346:
343:
338:
334:
324:
321:
316:
312:
303:
300:
296:
283:
279:
275:
272:
269:
266:
263:
257:
254:
251:
245:
231:
230:
229:
215:
175:
155:
149:
146:
127:
123:
120:) and polar (
119:
115:
110:
100:
96:
94:
90:
86:
82:
78:
74:
64:
62:
58:
54:
50:
46:
42:
38:
34:
26:
21:
2659:Astrophysics
2623:
2580:
2574:
2525:
2521:
2515:
2506:
2500:
2494:
2486:
2481:
2464:
2460:
2454:
2445:
2439:
2414:
2408:
2398:
2239:
2181:
2174:
1875:
1709:
1701:
1691:
1649:
1464:
1132:
835:
692:
459:
228:is given by
138:
125:
121:
117:
113:
98:
70:
32:
30:
2624:Black Holes
1692:mean radius
2648:Categories
2509:: 193–217.
2390:References
753:such that
2550:121948418
2384:Ellipsoid
2225:Ω
2216:ζ
2157:Ω
2118:ζ
2095:Ω
2072:^
2063:ζ
2052:×
2049:∇
2039:vorticity
2025:ζ
1964:^
1946:^
1928:^
1827:^
1799:^
1777:−
1771:ζ
1595:ρ
1589:π
1579:Ω
1301:−
1217:−
1040:−
943:−
901:ρ
895:π
885:Ω
672:Δ
634:∞
625:∫
605:Δ
545:∞
536:∫
468:ρ
367:Δ
353:Δ
289:∞
280:∫
258:ρ
252:π
242:Ω
196:Ω
85:ellipsoid
77:ellipsoid
2654:Quadrics
2618:(1973).
2569:(1965).
2379:Spheroid
2353:See also
93:Lagrange
81:Lagrange
2585:Bibcode
2530:Bibcode
2419:Bibcode
2177:similar
2037:is the
1690:is the
500:is the
67:History
53:density
2630:
2548:
2002:
1993:
1955:
1937:
1899:
1890:
1876:where
1733:
1724:
1650:where
1602:
1156:
1147:
460:where
359:
208:about
153:
135:fluid.
89:Jacobi
39:under
25:Haumea
2546:S2CID
49:fluid
35:is a
2628:ISBN
781:>
713:and
132:/3).
2593:doi
2581:141
2538:doi
2469:doi
2427:doi
2415:109
1752:of
1171:is
126:abc
2650::
2607:^
2591:.
2579:.
2573:.
2558:^
2544:.
2536:.
2526:41
2524:.
2507:52
2465:20
2463:.
2425:.
2407:.
1707:.
1531:10
116:,
63:.
47:,
31:A
2636:.
2601:.
2595::
2587::
2552:.
2540::
2532::
2475:.
2471::
2433:.
2429::
2421::
2337:.
2333:)
2327:a
2324:b
2319:+
2314:b
2311:a
2305:(
2299:2
2296:1
2291:=
2283:d
2280:e
2277:D
2272:L
2265:c
2262:a
2259:J
2254:L
2222:2
2219:=
2196:b
2193:=
2190:a
2160:.
2153:)
2147:a
2144:b
2139:+
2134:b
2131:a
2125:(
2121:=
2069:z
2060:=
2056:u
2005:c
1999:,
1996:b
1990:,
1987:a
1961:z
1952:,
1943:y
1934:,
1925:x
1902:z
1896:,
1893:y
1887:,
1884:x
1861:,
1853:2
1849:b
1845:+
1840:2
1836:a
1824:y
1818:x
1813:2
1809:b
1805:+
1796:x
1790:y
1785:2
1781:a
1768:=
1764:u
1736:c
1730:,
1727:b
1721:,
1718:a
1678:r
1658:M
1635:,
1629:3
1624:c
1621:b
1618:a
1612:=
1609:r
1605:,
1592:G
1583:2
1568:2
1564:r
1557:2
1553:b
1549:+
1544:2
1540:a
1527:3
1521:=
1515:r
1510:3
1506:M
1502:G
1498:L
1473:L
1450:.
1447:)
1442:2
1438:c
1434:,
1429:2
1425:c
1421:,
1416:2
1412:b
1408:,
1403:2
1399:a
1395:(
1390:J
1386:R
1380:2
1376:c
1372:=
1369:]
1366:)
1361:2
1357:b
1353:,
1348:2
1344:c
1340:,
1335:2
1331:b
1327:,
1322:2
1318:a
1314:(
1309:J
1305:R
1298:)
1293:2
1289:a
1285:,
1280:2
1276:c
1272:,
1267:2
1263:b
1259:,
1254:2
1250:a
1246:(
1241:J
1237:R
1233:[
1225:2
1221:a
1212:2
1208:b
1200:2
1196:b
1190:2
1186:a
1159:c
1153:,
1150:b
1144:,
1141:a
1118:]
1115:)
1110:2
1106:b
1102:,
1097:2
1093:c
1089:,
1084:2
1080:b
1076:,
1071:2
1067:a
1063:(
1058:J
1054:R
1048:2
1044:b
1037:)
1032:2
1028:a
1024:,
1019:2
1015:c
1011:,
1006:2
1002:b
998:,
993:2
989:a
985:(
980:J
976:R
970:2
966:a
962:[
956:)
951:2
947:b
938:2
934:a
930:(
927:3
922:c
919:b
916:a
913:4
907:=
898:G
889:2
857:J
853:R
821:.
814:2
810:b
806:1
801:+
794:2
790:a
786:1
774:2
770:c
766:1
741:c
721:b
701:a
678:.
669:)
666:u
663:+
658:2
654:c
650:(
645:u
642:d
629:0
619:2
615:c
611:=
602:)
599:u
596:+
591:2
587:b
583:(
580:)
577:u
574:+
569:2
565:a
561:(
556:u
553:d
540:0
530:2
526:b
520:2
516:a
488:G
445:,
442:)
439:u
436:+
431:2
427:c
423:(
420:)
417:u
414:+
409:2
405:b
401:(
398:)
395:u
392:+
387:2
383:a
379:(
376:=
371:2
362:,
350:)
347:u
344:+
339:2
335:b
331:(
328:)
325:u
322:+
317:2
313:a
309:(
304:u
301:d
297:u
284:0
276:c
273:b
270:a
267:2
264:=
255:G
246:2
216:c
176:c
156:b
150:,
147:a
130:π
122:c
118:b
114:a
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