17:
989:
983:
1125:
1721:
are constants which will be obtained with numerical solution. This behavior of density gives rise to increase in mass with increase in radius. Thus, the model is usually valid to describe the core of the star, where the temperature is approximately constant.
1469:
Assuming isothermal sphere has some disadvantages. Though the density obtained as solution of this isothermal gas sphere decreases from the centre, it decreases too slowly to give a well-defined surface and finite mass for the sphere. It can be shown that, as
793:
1263:
162:
2149:
1459:
2419:
361:
1955:
1341:
1676:
498:
2510:
2230:
1842:
848:
1002:
2307:
251:
684:
2762:
Chandrasekhar, S. (1972). A limiting case of relativistic equilibrium. In
General Relativity (in honor of J. L. Synge), ed. L. O'Raifeartaigh. Oxford. Clarendon Press (pp. 185-199).
1163:
62:
2595:
2002:
2658:
2030:
2548:
1378:
1494:
1370:
1155:
278:
1762:
1719:
647:
2684:
553:
524:
205:
185:
2327:
2615:
2022:
1699:
836:
816:
672:
625:
603:
578:
404:
2733:
294:
1857:
1274:
1502:
2801:
370:
for a spherical vessel. The relativistic version of this spherically symmetric isothermal model was studied by
Subrahmanyan Chandrasekhar in 1972.
2723:
Chandrasekhar, Subrahmanyan, and
Subrahmanyan Chandrasekhar. An introduction to the study of stellar structure. Vol. 2. Courier Corporation, 1958.
2705:
419:
2430:
2160:
288:. The isothermal assumption is usually modeled to describe the core of a star. The equation is solved with the initial conditions,
2744:
Emden, R. (1907). Gaskugeln: Anwendungen der mechanischen Wärmetheorie auf kosmologische und meteorologische
Probleme. B. Teubner.
1770:
2753:
Kippenhahn, Rudolf, Alfred
Weigert, and Achim Weiss. Stellar structure and evolution. Vol. 282. Berlin: Springer-Verlag, 1990.
978:{\displaystyle {\frac {k_{B}T}{WH}}{\frac {1}{r^{2}}}{\frac {d}{dr}}\left(r^{2}{\frac {d\ln \rho }{dr}}\right)=-4\pi G\rho }
650:
2786:
2771:
Henrich, L. R., & Chandrasekhar, S. (1941). Stellar Models with
Isothermal Cores. The Astrophysical Journal, 94, 525.
2734:
http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1949ApJ...109..551C&defaultprint=YES&filetype=.pdf
1120:{\displaystyle \psi =\ln {\frac {\rho _{c}}{\rho }},\quad \xi =r\left({\frac {4\pi G\rho _{c}WH}{k_{B}T}}\right)^{1/2}}
2241:
49:
2796:
2732:
Chandrasekhar, S., and Gordon W. Wares. "The
Isothermal Function." The Astrophysical Journal 109 (1949): 551-554.
788:{\displaystyle {\frac {1}{r^{2}}}{\frac {d}{dr}}\left({\frac {r^{2}}{\rho }}{\frac {dp}{dr}}\right)=-4\pi G\rho }
2700:
678:
The equation for equilibrium of the star requires a balance between the pressure force and gravitational force
367:
210:
2695:
1258:{\displaystyle {\frac {1}{\xi ^{2}}}{\frac {d}{d\xi }}\left(\xi ^{2}{\frac {d\psi }{d\xi }}\right)=e^{-\psi }}
285:
157:{\displaystyle {\frac {1}{\xi ^{2}}}{\frac {d}{d\xi }}\left(\xi ^{2}{\frac {d\psi }{d\xi }}\right)=e^{-\psi }}
2791:
839:
2553:
366:
The equation appears in other branches of physics as well, for example the same equation appears in the
407:
2144:{\displaystyle {\frac {d^{2}z}{dt^{2}}}-{\frac {dz}{dt}}+e^{z}-2=0,\quad {\text{where}}\quad t=\ln x}
1963:
2623:
1454:{\displaystyle \psi ={\frac {\xi ^{2}}{6}}-{\frac {\xi ^{4}}{120}}+{\frac {\xi ^{6}}{1890}}+\cdots }
2318:
2620:
The solutions of the Emden–Chandrasekhar equation which are finite at the origin have necessarily
2524:
556:
411:
1473:
1349:
1133:
256:
1848:
1733:
1704:
632:
2663:
2414:{\displaystyle u={\frac {\xi e^{-\psi }}{d\psi /d\xi }},\quad v=\xi {\frac {d\psi }{d\xi }}}
581:
37:
531:
509:
280:
is the density of the gas at the centre. The equation has no known explicit solution. If a
190:
170:
2600:
2007:
1684:
821:
801:
657:
610:
588:
563:
389:
16:
2780:
356:{\displaystyle \psi =0,\quad {\frac {d\psi }{d\xi }}=0\quad {\text{at}}\quad \xi =0.}
33:
1950:{\displaystyle e^{-\psi _{s}}=2x^{2},\quad {\text{or}}\quad -\psi _{s}=2\ln x+\ln 2}
1336:{\displaystyle \psi =0,\quad {\frac {d\psi }{d\xi }}=0\quad {\text{at}}\quad \xi =0}
53:
45:
25:
1671:{\displaystyle {\frac {\rho }{\rho _{c}}}=e^{-\psi }={\frac {2}{\xi ^{2}}}\left}
379:
281:
41:
493:{\displaystyle p=\rho {\frac {k_{B}}{WH}}T+{\frac {4\sigma }{3c}}T^{4}}
2505:{\displaystyle {\frac {u}{v}}{\frac {dv}{du}}=-{\frac {u-1}{u+v-3}}}
988:
44:
gas sphere subjected to its own gravitational force, named after
2225:{\displaystyle y={\frac {dz}{dt}}=\xi {\frac {d\psi }{d\xi }}-2}
383:
284:
fluid is used instead of an isothermal fluid, one obtains the
1837:{\displaystyle x^{4}{\frac {d^{2}\psi }{dx^{2}}}=e^{-\psi }}
2154:
This equation can be reduced to first order by introducing
40:
for the density distribution of a spherically symmetric
2666:
2626:
2603:
2556:
2527:
2433:
2330:
2244:
2163:
2033:
2010:
1966:
1860:
1773:
1736:
1707:
1687:
1505:
1476:
1381:
1352:
1277:
1166:
1136:
1005:
851:
824:
804:
687:
660:
635:
613:
591:
566:
534:
512:
422:
392:
297:
259:
213:
193:
173:
65:
2550:
is a solution to Emden–Chandrasekhar equation, then
207:is the related to the density of the gas sphere as
2678:
2652:
2609:
2589:
2542:
2504:
2413:
2301:
2224:
2143:
2016:
1996:
1949:
1836:
1756:
1713:
1693:
1670:
1488:
1453:
1364:
1335:
1257:
1149:
1119:
977:
830:
810:
787:
666:
641:
619:
597:
572:
547:
518:
492:
398:
355:
272:
245:
199:
179:
156:
20:Numerical solution of Emden–Chandrasekhar equation
1960:Therefore, a new variable can be introduced as
2302:{\displaystyle y{\frac {dy}{dz}}-y+e^{z}-2=0}
1157:is the central density of the star, leads to
8:
818:is the radius measured from the center and
2597:is also a solution of the equation, where
2665:
2633:
2625:
2602:
2555:
2526:
2470:
2444:
2434:
2432:
2391:
2364:
2347:
2337:
2329:
2281:
2248:
2243:
2196:
2170:
2162:
2120:
2098:
2071:
2059:
2041:
2034:
2032:
2009:
1965:
1914:
1901:
1891:
1873:
1865:
1859:
1825:
1809:
1791:
1784:
1778:
1772:
1746:
1735:
1706:
1686:
1651:
1606:
1583:
1579:
1570:
1551:
1542:
1530:
1515:
1506:
1504:
1475:
1434:
1428:
1414:
1408:
1394:
1388:
1380:
1351:
1318:
1291:
1276:
1246:
1214:
1208:
1184:
1176:
1167:
1165:
1141:
1135:
1107:
1103:
1087:
1069:
1053:
1024:
1018:
1004:
926:
920:
896:
888:
879:
859:
852:
850:
823:
803:
742:
731:
725:
705:
697:
688:
686:
659:
634:
612:
590:
565:
539:
533:
511:
484:
460:
438:
432:
421:
391:
338:
311:
296:
264:
258:
246:{\displaystyle \rho =\rho _{c}e^{-\psi }}
234:
224:
212:
192:
172:
145:
113:
107:
83:
75:
66:
64:
987:
15:
2716:
992:Actual solution and asymptotic solution
52:. The equation was first introduced by
7:
2706:Chandrasekhar's white dwarf equation
2590:{\displaystyle \psi (A\xi )-2\ln A}
2317:There is another reduction due to
368:Frank-Kamenetskii explosion theory
14:
842:. The equation is re-written as
187:is the dimensionless radius and
2802:Ordinary differential equations
2381:
2125:
2119:
1997:{\displaystyle -\psi =2\ln x+z}
1906:
1900:
1730:Introducing the transformation
1323:
1317:
1290:
1038:
996:Introducing the transformation
343:
337:
310:
2653:{\displaystyle d\psi /d\xi =0}
2569:
2560:
2537:
2531:
1660:
1644:
627:is the temperature of the star
1:
1268:The boundary conditions are
30:Emden–Chandrasekhar equation
2543:{\displaystyle \psi (\xi )}
1764:transforms the equation to
56:in 1907. The equation reads
2818:
50:Subrahmanyan Chandrasekhar
2617:is an arbitrary constant.
2004:, where the equation for
1489:{\displaystyle \xi \gg 1}
1372:, the solution goes like
1365:{\displaystyle \xi \ll 1}
1150:{\displaystyle \rho _{c}}
651:Stefan–Boltzmann constant
605:is the mass of the proton
273:{\displaystyle \rho _{c}}
2701:Frank-Kamenetskii theory
1757:{\displaystyle x=1/\xi }
1465:Limitations of the model
1714:{\displaystyle \delta }
642:{\displaystyle \sigma }
2680:
2679:{\displaystyle \xi =0}
2654:
2611:
2591:
2544:
2506:
2415:
2303:
2226:
2145:
2018:
1998:
1951:
1838:
1758:
1715:
1695:
1672:
1490:
1455:
1366:
1337:
1259:
1151:
1121:
993:
979:
840:gravitational constant
832:
812:
789:
668:
643:
621:
599:
574:
549:
520:
494:
400:
357:
274:
247:
201:
181:
158:
21:
2681:
2655:
2612:
2592:
2545:
2507:
2416:
2304:
2227:
2146:
2019:
1999:
1952:
1839:
1759:
1716:
1696:
1673:
1491:
1456:
1367:
1338:
1260:
1152:
1122:
991:
980:
833:
813:
790:
674:is the speed of light
669:
644:
622:
600:
575:
550:
548:{\displaystyle k_{B}}
521:
519:{\displaystyle \rho }
495:
401:
358:
275:
248:
202:
200:{\displaystyle \psi }
182:
159:
19:
2787:Equations of physics
2664:
2624:
2601:
2554:
2525:
2431:
2328:
2242:
2161:
2031:
2008:
1964:
1858:
1771:
1734:
1705:
1685:
1503:
1474:
1379:
1350:
1275:
1164:
1134:
1003:
849:
822:
802:
685:
658:
633:
611:
589:
564:
532:
510:
420:
390:
295:
257:
211:
191:
180:{\displaystyle \xi }
171:
63:
2696:Lane–Emden equation
2319:Edward Arthur Milne
1847:The equation has a
286:Lane–Emden equation
2676:
2650:
2607:
2587:
2540:
2502:
2411:
2299:
2222:
2141:
2014:
1994:
1947:
1834:
1754:
1711:
1691:
1668:
1486:
1451:
1362:
1333:
1255:
1147:
1117:
994:
975:
828:
808:
785:
664:
639:
617:
595:
570:
557:Boltzmann constant
545:
516:
490:
412:radiation pressure
396:
353:
270:
243:
197:
177:
154:
22:
2610:{\displaystyle A}
2500:
2462:
2442:
2409:
2376:
2266:
2214:
2188:
2123:
2089:
2066:
2017:{\displaystyle z}
1904:
1849:singular solution
1816:
1726:Singular solution
1694:{\displaystyle A}
1616:
1612:
1593:
1557:
1521:
1443:
1423:
1403:
1321:
1309:
1232:
1197:
1182:
1097:
1033:
950:
909:
894:
877:
831:{\displaystyle G}
811:{\displaystyle r}
760:
740:
718:
703:
667:{\displaystyle c}
620:{\displaystyle T}
598:{\displaystyle H}
573:{\displaystyle W}
478:
452:
399:{\displaystyle p}
341:
329:
131:
96:
81:
2809:
2797:Stellar dynamics
2772:
2769:
2763:
2760:
2754:
2751:
2745:
2742:
2736:
2730:
2724:
2721:
2685:
2683:
2682:
2677:
2659:
2657:
2656:
2651:
2637:
2616:
2614:
2613:
2608:
2596:
2594:
2593:
2588:
2549:
2547:
2546:
2541:
2511:
2509:
2508:
2503:
2501:
2499:
2482:
2471:
2463:
2461:
2453:
2445:
2443:
2435:
2420:
2418:
2417:
2412:
2410:
2408:
2400:
2392:
2377:
2375:
2368:
2356:
2355:
2354:
2338:
2321:. Let us define
2308:
2306:
2305:
2300:
2286:
2285:
2267:
2265:
2257:
2249:
2231:
2229:
2228:
2223:
2215:
2213:
2205:
2197:
2189:
2187:
2179:
2171:
2150:
2148:
2147:
2142:
2124:
2121:
2103:
2102:
2090:
2088:
2080:
2072:
2067:
2065:
2064:
2063:
2050:
2046:
2045:
2035:
2024:can be derived,
2023:
2021:
2020:
2015:
2003:
2001:
2000:
1995:
1956:
1954:
1953:
1948:
1919:
1918:
1905:
1902:
1896:
1895:
1880:
1879:
1878:
1877:
1843:
1841:
1840:
1835:
1833:
1832:
1817:
1815:
1814:
1813:
1800:
1796:
1795:
1785:
1783:
1782:
1763:
1761:
1760:
1755:
1750:
1720:
1718:
1717:
1712:
1700:
1698:
1697:
1692:
1677:
1675:
1674:
1669:
1667:
1663:
1659:
1658:
1637:
1633:
1617:
1608:
1607:
1594:
1592:
1591:
1587:
1571:
1558:
1556:
1555:
1543:
1538:
1537:
1522:
1520:
1519:
1507:
1495:
1493:
1492:
1487:
1460:
1458:
1457:
1452:
1444:
1439:
1438:
1429:
1424:
1419:
1418:
1409:
1404:
1399:
1398:
1389:
1371:
1369:
1368:
1363:
1342:
1340:
1339:
1334:
1322:
1319:
1310:
1308:
1300:
1292:
1264:
1262:
1261:
1256:
1254:
1253:
1238:
1234:
1233:
1231:
1223:
1215:
1213:
1212:
1198:
1196:
1185:
1183:
1181:
1180:
1168:
1156:
1154:
1153:
1148:
1146:
1145:
1126:
1124:
1123:
1118:
1116:
1115:
1111:
1102:
1098:
1096:
1092:
1091:
1081:
1074:
1073:
1054:
1034:
1029:
1028:
1019:
984:
982:
981:
976:
956:
952:
951:
949:
941:
927:
925:
924:
910:
908:
897:
895:
893:
892:
880:
878:
876:
868:
864:
863:
853:
837:
835:
834:
829:
817:
815:
814:
809:
794:
792:
791:
786:
766:
762:
761:
759:
751:
743:
741:
736:
735:
726:
719:
717:
706:
704:
702:
701:
689:
673:
671:
670:
665:
648:
646:
645:
640:
626:
624:
623:
618:
604:
602:
601:
596:
582:molecular weight
579:
577:
576:
571:
554:
552:
551:
546:
544:
543:
525:
523:
522:
517:
499:
497:
496:
491:
489:
488:
479:
477:
469:
461:
453:
451:
443:
442:
433:
408:kinetic pressure
405:
403:
402:
397:
362:
360:
359:
354:
342:
339:
330:
328:
320:
312:
279:
277:
276:
271:
269:
268:
252:
250:
249:
244:
242:
241:
229:
228:
206:
204:
203:
198:
186:
184:
183:
178:
163:
161:
160:
155:
153:
152:
137:
133:
132:
130:
122:
114:
112:
111:
97:
95:
84:
82:
80:
79:
67:
38:Poisson equation
2817:
2816:
2812:
2811:
2810:
2808:
2807:
2806:
2777:
2776:
2775:
2770:
2766:
2761:
2757:
2752:
2748:
2743:
2739:
2731:
2727:
2722:
2718:
2714:
2692:
2662:
2661:
2622:
2621:
2599:
2598:
2552:
2551:
2523:
2522:
2518:
2483:
2472:
2454:
2446:
2429:
2428:
2401:
2393:
2357:
2343:
2339:
2326:
2325:
2315:
2277:
2258:
2250:
2240:
2239:
2206:
2198:
2180:
2172:
2159:
2158:
2094:
2081:
2073:
2055:
2051:
2037:
2036:
2029:
2028:
2006:
2005:
1962:
1961:
1910:
1887:
1869:
1861:
1856:
1855:
1821:
1805:
1801:
1787:
1786:
1774:
1769:
1768:
1732:
1731:
1728:
1703:
1702:
1683:
1682:
1647:
1605:
1601:
1575:
1563:
1559:
1547:
1526:
1511:
1501:
1500:
1472:
1471:
1467:
1430:
1410:
1390:
1377:
1376:
1348:
1347:
1301:
1293:
1273:
1272:
1242:
1224:
1216:
1204:
1203:
1199:
1189:
1172:
1162:
1161:
1137:
1132:
1131:
1083:
1082:
1065:
1055:
1049:
1048:
1020:
1001:
1000:
942:
928:
916:
915:
911:
901:
884:
869:
855:
854:
847:
846:
820:
819:
800:
799:
752:
744:
727:
724:
720:
710:
693:
683:
682:
656:
655:
631:
630:
609:
608:
587:
586:
562:
561:
535:
530:
529:
508:
507:
480:
470:
462:
444:
434:
418:
417:
388:
387:
386:, the pressure
376:
321:
313:
293:
292:
260:
255:
254:
230:
220:
209:
208:
189:
188:
169:
168:
165:
141:
123:
115:
103:
102:
98:
88:
71:
61:
60:
12:
11:
5:
2815:
2813:
2805:
2804:
2799:
2794:
2792:Fluid dynamics
2789:
2779:
2778:
2774:
2773:
2764:
2755:
2746:
2737:
2725:
2715:
2713:
2710:
2709:
2708:
2703:
2698:
2691:
2688:
2687:
2686:
2675:
2672:
2669:
2649:
2646:
2643:
2640:
2636:
2632:
2629:
2618:
2606:
2586:
2583:
2580:
2577:
2574:
2571:
2568:
2565:
2562:
2559:
2539:
2536:
2533:
2530:
2517:
2514:
2513:
2512:
2498:
2495:
2492:
2489:
2486:
2481:
2478:
2475:
2469:
2466:
2460:
2457:
2452:
2449:
2441:
2438:
2422:
2421:
2407:
2404:
2399:
2396:
2390:
2387:
2384:
2380:
2374:
2371:
2367:
2363:
2360:
2353:
2350:
2346:
2342:
2336:
2333:
2314:
2311:
2310:
2309:
2298:
2295:
2292:
2289:
2284:
2280:
2276:
2273:
2270:
2264:
2261:
2256:
2253:
2247:
2233:
2232:
2221:
2218:
2212:
2209:
2204:
2201:
2195:
2192:
2186:
2183:
2178:
2175:
2169:
2166:
2152:
2151:
2140:
2137:
2134:
2131:
2128:
2118:
2115:
2112:
2109:
2106:
2101:
2097:
2093:
2087:
2084:
2079:
2076:
2070:
2062:
2058:
2054:
2049:
2044:
2040:
2013:
1993:
1990:
1987:
1984:
1981:
1978:
1975:
1972:
1969:
1958:
1957:
1946:
1943:
1940:
1937:
1934:
1931:
1928:
1925:
1922:
1917:
1913:
1909:
1899:
1894:
1890:
1886:
1883:
1876:
1872:
1868:
1864:
1845:
1844:
1831:
1828:
1824:
1820:
1812:
1808:
1804:
1799:
1794:
1790:
1781:
1777:
1753:
1749:
1745:
1742:
1739:
1727:
1724:
1710:
1690:
1679:
1678:
1666:
1662:
1657:
1654:
1650:
1646:
1643:
1640:
1636:
1632:
1629:
1626:
1623:
1620:
1615:
1611:
1604:
1600:
1597:
1590:
1586:
1582:
1578:
1574:
1569:
1566:
1562:
1554:
1550:
1546:
1541:
1536:
1533:
1529:
1525:
1518:
1514:
1510:
1485:
1482:
1479:
1466:
1463:
1462:
1461:
1450:
1447:
1442:
1437:
1433:
1427:
1422:
1417:
1413:
1407:
1402:
1397:
1393:
1387:
1384:
1361:
1358:
1355:
1344:
1343:
1332:
1329:
1326:
1316:
1313:
1307:
1304:
1299:
1296:
1289:
1286:
1283:
1280:
1266:
1265:
1252:
1249:
1245:
1241:
1237:
1230:
1227:
1222:
1219:
1211:
1207:
1202:
1195:
1192:
1188:
1179:
1175:
1171:
1144:
1140:
1128:
1127:
1114:
1110:
1106:
1101:
1095:
1090:
1086:
1080:
1077:
1072:
1068:
1064:
1061:
1058:
1052:
1047:
1044:
1041:
1037:
1032:
1027:
1023:
1017:
1014:
1011:
1008:
986:
985:
974:
971:
968:
965:
962:
959:
955:
948:
945:
940:
937:
934:
931:
923:
919:
914:
907:
904:
900:
891:
887:
883:
875:
872:
867:
862:
858:
827:
807:
796:
795:
784:
781:
778:
775:
772:
769:
765:
758:
755:
750:
747:
739:
734:
730:
723:
716:
713:
709:
700:
696:
692:
676:
675:
663:
653:
638:
628:
616:
606:
594:
584:
569:
559:
542:
538:
527:
526:is the density
515:
501:
500:
487:
483:
476:
473:
468:
465:
459:
456:
450:
447:
441:
437:
431:
428:
425:
406:is due to the
395:
375:
372:
364:
363:
352:
349:
346:
336:
333:
327:
324:
319:
316:
309:
306:
303:
300:
267:
263:
240:
237:
233:
227:
223:
219:
216:
196:
176:
151:
148:
144:
140:
136:
129:
126:
121:
118:
110:
106:
101:
94:
91:
87:
78:
74:
70:
58:
13:
10:
9:
6:
4:
3:
2:
2814:
2803:
2800:
2798:
2795:
2793:
2790:
2788:
2785:
2784:
2782:
2768:
2765:
2759:
2756:
2750:
2747:
2741:
2738:
2735:
2729:
2726:
2720:
2717:
2711:
2707:
2704:
2702:
2699:
2697:
2694:
2693:
2689:
2673:
2670:
2667:
2647:
2644:
2641:
2638:
2634:
2630:
2627:
2619:
2604:
2584:
2581:
2578:
2575:
2572:
2566:
2563:
2557:
2534:
2528:
2520:
2519:
2515:
2496:
2493:
2490:
2487:
2484:
2479:
2476:
2473:
2467:
2464:
2458:
2455:
2450:
2447:
2439:
2436:
2427:
2426:
2425:
2405:
2402:
2397:
2394:
2388:
2385:
2382:
2378:
2372:
2369:
2365:
2361:
2358:
2351:
2348:
2344:
2340:
2334:
2331:
2324:
2323:
2322:
2320:
2312:
2296:
2293:
2290:
2287:
2282:
2278:
2274:
2271:
2268:
2262:
2259:
2254:
2251:
2245:
2238:
2237:
2236:
2235:then we have
2219:
2216:
2210:
2207:
2202:
2199:
2193:
2190:
2184:
2181:
2176:
2173:
2167:
2164:
2157:
2156:
2155:
2138:
2135:
2132:
2129:
2126:
2116:
2113:
2110:
2107:
2104:
2099:
2095:
2091:
2085:
2082:
2077:
2074:
2068:
2060:
2056:
2052:
2047:
2042:
2038:
2027:
2026:
2025:
2011:
1991:
1988:
1985:
1982:
1979:
1976:
1973:
1970:
1967:
1944:
1941:
1938:
1935:
1932:
1929:
1926:
1923:
1920:
1915:
1911:
1907:
1897:
1892:
1888:
1884:
1881:
1874:
1870:
1866:
1862:
1854:
1853:
1852:
1850:
1829:
1826:
1822:
1818:
1810:
1806:
1802:
1797:
1792:
1788:
1779:
1775:
1767:
1766:
1765:
1751:
1747:
1743:
1740:
1737:
1725:
1723:
1708:
1688:
1664:
1655:
1652:
1648:
1641:
1638:
1634:
1630:
1627:
1624:
1621:
1618:
1613:
1609:
1602:
1598:
1595:
1588:
1584:
1580:
1576:
1572:
1567:
1564:
1560:
1552:
1548:
1544:
1539:
1534:
1531:
1527:
1523:
1516:
1512:
1508:
1499:
1498:
1497:
1483:
1480:
1477:
1464:
1448:
1445:
1440:
1435:
1431:
1425:
1420:
1415:
1411:
1405:
1400:
1395:
1391:
1385:
1382:
1375:
1374:
1373:
1359:
1356:
1353:
1330:
1327:
1324:
1314:
1311:
1305:
1302:
1297:
1294:
1287:
1284:
1281:
1278:
1271:
1270:
1269:
1250:
1247:
1243:
1239:
1235:
1228:
1225:
1220:
1217:
1209:
1205:
1200:
1193:
1190:
1186:
1177:
1173:
1169:
1160:
1159:
1158:
1142:
1138:
1112:
1108:
1104:
1099:
1093:
1088:
1084:
1078:
1075:
1070:
1066:
1062:
1059:
1056:
1050:
1045:
1042:
1039:
1035:
1030:
1025:
1021:
1015:
1012:
1009:
1006:
999:
998:
997:
990:
972:
969:
966:
963:
960:
957:
953:
946:
943:
938:
935:
932:
929:
921:
917:
912:
905:
902:
898:
889:
885:
881:
873:
870:
865:
860:
856:
845:
844:
843:
841:
825:
805:
782:
779:
776:
773:
770:
767:
763:
756:
753:
748:
745:
737:
732:
728:
721:
714:
711:
707:
698:
694:
690:
681:
680:
679:
661:
654:
652:
636:
629:
614:
607:
592:
585:
583:
567:
560:
558:
540:
536:
528:
513:
506:
505:
504:
485:
481:
474:
471:
466:
463:
457:
454:
448:
445:
439:
435:
429:
426:
423:
416:
415:
414:
413:
409:
393:
385:
381:
373:
371:
369:
350:
347:
344:
334:
331:
325:
322:
317:
314:
307:
304:
301:
298:
291:
290:
289:
287:
283:
265:
261:
238:
235:
231:
225:
221:
217:
214:
194:
174:
164:
149:
146:
142:
138:
134:
127:
124:
119:
116:
108:
104:
99:
92:
89:
85:
76:
72:
68:
57:
55:
51:
47:
43:
39:
35:
34:dimensionless
31:
27:
18:
2767:
2758:
2749:
2740:
2728:
2719:
2423:
2316:
2234:
2153:
1959:
1846:
1729:
1680:
1468:
1345:
1267:
1129:
995:
797:
677:
580:is the mean
502:
377:
365:
166:
59:
54:Robert Emden
46:Robert Emden
36:form of the
29:
26:astrophysics
23:
2781:Categories
2712:References
2516:Properties
380:isothermal
374:Derivation
282:polytropic
42:isothermal
2668:ξ
2642:ξ
2631:ψ
2582:
2573:−
2567:ξ
2558:ψ
2535:ξ
2529:ψ
2494:−
2477:−
2468:−
2406:ξ
2398:ψ
2389:ξ
2373:ξ
2362:ψ
2352:ψ
2349:−
2341:ξ
2313:Reduction
2288:−
2269:−
2217:−
2211:ξ
2203:ψ
2194:ξ
2136:
2105:−
2069:−
1983:
1971:ψ
1968:−
1942:
1930:
1912:ψ
1908:−
1871:ψ
1867:−
1851:given by
1830:ψ
1827:−
1798:ψ
1752:ξ
1709:δ
1653:−
1649:ξ
1631:δ
1625:ξ
1622:
1599:
1577:ξ
1549:ξ
1535:ψ
1532:−
1513:ρ
1509:ρ
1481:≫
1478:ξ
1449:⋯
1432:ξ
1412:ξ
1406:−
1392:ξ
1383:ψ
1357:≪
1354:ξ
1325:ξ
1306:ξ
1298:ψ
1279:ψ
1251:ψ
1248:−
1229:ξ
1221:ψ
1206:ξ
1194:ξ
1174:ξ
1139:ρ
1067:ρ
1060:π
1040:ξ
1031:ρ
1022:ρ
1016:
1007:ψ
973:ρ
967:π
961:−
939:ρ
936:
783:ρ
777:π
771:−
738:ρ
637:σ
514:ρ
467:σ
430:ρ
345:ξ
326:ξ
318:ψ
299:ψ
262:ρ
239:ψ
236:−
222:ρ
215:ρ
195:ψ
175:ξ
150:ψ
147:−
128:ξ
120:ψ
105:ξ
93:ξ
73:ξ
2690:See also
382:gaseous
253:, where
838:is the
649:is the
555:is the
378:For an
1681:where
1130:where
798:where
503:where
167:where
28:, the
2424:then
2122:where
32:is a
1701:and
1441:1890
1346:For
410:and
384:star
48:and
2660:at
2521:If
1596:cos
1421:120
24:In
2783::
2579:ln
2133:ln
1980:ln
1939:ln
1927:ln
1903:or
1619:ln
1496:,
1320:at
1013:ln
933:ln
351:0.
340:at
2674:0
2671:=
2648:0
2645:=
2639:d
2635:/
2628:d
2605:A
2585:A
2576:2
2570:)
2564:A
2561:(
2538:)
2532:(
2497:3
2491:v
2488:+
2485:u
2480:1
2474:u
2465:=
2459:u
2456:d
2451:v
2448:d
2440:v
2437:u
2403:d
2395:d
2386:=
2383:v
2379:,
2370:d
2366:/
2359:d
2345:e
2335:=
2332:u
2297:0
2294:=
2291:2
2283:z
2279:e
2275:+
2272:y
2263:z
2260:d
2255:y
2252:d
2246:y
2220:2
2208:d
2200:d
2191:=
2185:t
2182:d
2177:z
2174:d
2168:=
2165:y
2139:x
2130:=
2127:t
2117:,
2114:0
2111:=
2108:2
2100:z
2096:e
2092:+
2086:t
2083:d
2078:z
2075:d
2061:2
2057:t
2053:d
2048:z
2043:2
2039:d
2012:z
1992:z
1989:+
1986:x
1977:2
1974:=
1945:2
1936:+
1933:x
1924:2
1921:=
1916:s
1898:,
1893:2
1889:x
1885:2
1882:=
1875:s
1863:e
1823:e
1819:=
1811:2
1807:x
1803:d
1793:2
1789:d
1780:4
1776:x
1748:/
1744:1
1741:=
1738:x
1689:A
1665:]
1661:)
1656:1
1645:(
1642:O
1639:+
1635:)
1628:+
1614:2
1610:7
1603:(
1589:2
1585:/
1581:1
1573:A
1568:+
1565:1
1561:[
1553:2
1545:2
1540:=
1528:e
1524:=
1517:c
1484:1
1446:+
1436:6
1426:+
1416:4
1401:6
1396:2
1386:=
1360:1
1331:0
1328:=
1315:0
1312:=
1303:d
1295:d
1288:,
1285:0
1282:=
1244:e
1240:=
1236:)
1226:d
1218:d
1210:2
1201:(
1191:d
1187:d
1178:2
1170:1
1143:c
1113:2
1109:/
1105:1
1100:)
1094:T
1089:B
1085:k
1079:H
1076:W
1071:c
1063:G
1057:4
1051:(
1046:r
1043:=
1036:,
1026:c
1010:=
970:G
964:4
958:=
954:)
947:r
944:d
930:d
922:2
918:r
913:(
906:r
903:d
899:d
890:2
886:r
882:1
874:H
871:W
866:T
861:B
857:k
826:G
806:r
780:G
774:4
768:=
764:)
757:r
754:d
749:p
746:d
733:2
729:r
722:(
715:r
712:d
708:d
699:2
695:r
691:1
662:c
615:T
593:H
568:W
541:B
537:k
486:4
482:T
475:c
472:3
464:4
458:+
455:T
449:H
446:W
440:B
436:k
427:=
424:p
394:p
348:=
335:0
332:=
323:d
315:d
308:,
305:0
302:=
266:c
232:e
226:c
218:=
143:e
139:=
135:)
125:d
117:d
109:2
100:(
90:d
86:d
77:2
69:1
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