2215:
1084:
2202:
828:
782:
20:
2221:
2220:
2217:
2216:
2222:
1459:
2219:
1685:
2238:
measurements of our Galaxy, researchers were able to simulate the dark matter halo distribution using Jeans equations. By comparing measured values with Jeans equation simulation results, they confirmed the need for extra dark matter and placed limits on its ellipsoid size. They estimated the ratio
1889:
2267:
The limiting factor of these simulations however, has been the data required to approximate stress tensor parameter values that dictate the Jeans equations behavior. Additionally, some constraints can be placed on Jeans equation simulations in order to produce reliable results Some of these
327:
1079:{\displaystyle n{\frac {\partial \langle {v_{j}}\rangle }{\partial t}}+\sum _{i}n\langle {v_{i}}\rangle {\frac {\partial {\langle {v_{j}}\rangle }}{\partial x_{i}}}=-n{\frac {\partial \Phi }{\partial x_{j}}}-\sum _{i}{\frac {\partial (n\sigma _{ij}^{2})}{\partial x_{i}}}\qquad (j=1,2,3.)}
201:
2192:
to the entire universe. Using measurements of stellar number density and various kinematic values, parameters within the Jeans equations can be estimated. This allows for various analyses to be made through the lens of Jeans equations. This is particularly useful when simulating
585:
2090:
578:
1244:
1196:
47:. The Jeans equations can come in a variety of different forms, depending on the structure of what is being modelled. Most utilization of these equations has been found in simulations with large number of gravitationally bound objects.
2197:
halo distributions, due to its isothermal, non-interactive behavior. Searches for structure in galaxy formation, dark matter formation, and universe formation can have observations supplemented with simulations using Jeans equations.
1465:
1691:
63:
in 1915 while working on stellar hydrodynamics. Since then, multiple solutions to the equations have been calculated analytically and numerically. Some notable solutions include a spherically symmetric solution, derived by
2218:
206:
94:
777:{\displaystyle {\frac {\partial (n\langle {v_{j}}\rangle )}{\partial t}}+n{\frac {\partial \Phi }{\partial x_{j}}}+\sum _{i}{\frac {\partial (n\langle {v_{i}v_{j}}\rangle )}{\partial x_{i}}}=0\qquad (j=1,2,3.)}
1941:
1948:
473:
381:
1454:{\displaystyle {\partial \rho \langle v_{r}\rangle \over \partial t}+{\partial \rho \langle v_{r}^{2}\rangle \over \partial r}+{\rho \over r}+\rho {\partial \Phi \over \partial r}=0}
2174:
820:
1096:
2133:
2643:
Adams, Joshua J.; Gebhardt, Karl; Blanc, Guillermo A.; Fabricius, Maximilian H.; Hill, Gary J.; Murphy, Jeremy D.; van den Bosch, Remco C. E.; van de Ven, Glenn (2012-01-03).
2302:
39:. The Jeans equations relate the second-order velocity moments to the density and potential of a stellar system for systems without collision. They are analogous to the
2257:
1223:
Notice that the Jeans equations contain 9 unknowns (3 average velocities and 6 stress tensor terms), but only 3 equations. This means that Jeans equations are not
1680:{\displaystyle {\partial \rho \langle v_{\theta }\rangle \over \partial t}+{\partial \rho \langle v_{r}v_{\theta }\rangle \over \partial r}+{\rho \over r}=0}
1884:{\displaystyle {\partial \rho \langle v_{\phi }\rangle \over \partial t}+{\partial \rho \langle v_{r}v_{\phi }\rangle \over \partial r}+{\rho \over r}=0}
322:{\displaystyle {\partial f \over \partial t}+{\vec {v}}\cdot {\vec {\nabla }}f-{\vec {\nabla }}\Phi \cdot {\partial f \over \partial {\vec {v}}}=0}
2849:
2268:
limitations include a wavelength resolution requirement, variable gravitational softening, and a minimum vertical structure particle resolution.
2578:
Loebman, Sarah R.; Ivezić, Željko; Quinn, Thomas R.; Governato, Fabio; Brooks, Alyson M.; Christensen, Charlotte R.; Jurić, Mario (2012-09-26).
822:
is the average of component 1 of the velocity of the stars at a given point and time. The second set of equations may alternately be written as
2781:
2408:
2345:
196:{\displaystyle {\partial f \over \partial t}+v{\partial f \over \partial r}-{\partial \Phi \over \partial r}{\partial f \over \partial v}=0}
2259:
0.14. This method has been applied to many other galactic halos and have produced similar results regarding dark matter halo topology.
2361:"Efficient solution of the anisotropic spherically aligned axisymmetric Jeans equations of stellar hydrodynamics for galactic dynamics"
2297:
2795:
Truelove, J. Kelly; Klein, Richard I.; McKee, Christopher F.; Holliman Ii, John H.; Howell, Louis H.; Greenough, Jeffrey A. (1997).
2474:
338:
44:
40:
2854:
1896:
2226:
Artistic rendition of dark matter halo around the Milky Way. Jeans
Equation simulations place limits on the size of this halo
2085:{\displaystyle {\partial (\rho \sigma _{r}^{2}) \over \partial r}+{2\rho \over r}+\rho {\partial \Phi \over \partial r}=0}
2797:"The Jeans Condition: A New Constraint on Spatial Resolution in Simulations of Isothermal Self-gravitational Hydrodynamics"
2580:"Constraints on the Shape of the Milky Way Dark Matter Halo from Jeans Equations Applied to Sloan Digital Sky Survey Data"
573:{\displaystyle {\frac {\partial n}{\partial t}}+\sum _{i}{\frac {\partial (n\langle {v_{i}}\rangle )}{\partial x_{i}}}=0,}
32:
19:
2494:
1236:
2717:
2458:
2230:
An example of such an analysis is given by the constraints that can be placed on the dark matter halo within the
1090:
2235:
2138:
790:
1191:{\displaystyle \sigma _{ij}^{2}=\langle {v_{i}v_{j}}\rangle -\langle {v_{i}}\rangle \langle {v_{j}}\rangle }
465:
86:
2422:
1224:
2775:
2402:
2339:
2097:
332:
2808:
2739:
2666:
2601:
2544:
2462:
2311:
1217:
56:
36:
2826:
2763:
2729:
2698:
2656:
2625:
2591:
2579:
2372:
787:
Here, the ⟨...⟩ notation means an average at a given point and time (x,t), so that, for example,
2277:
23:
University of Texas N-body
Simulation. Governed by Jeans equations in a gravitational potential
2755:
2690:
2617:
2560:
2470:
2390:
2327:
2185:
2816:
2747:
2682:
2674:
2609:
2552:
2531:
Haas, Brian L.; Hash, David B.; Bird, Graeme A.; Lumpkin, Forrest E.; Hassan, H. A. (1994).
2382:
2360:
2319:
2718:"Numerical requirements for simulations of self-gravitating and non-self-gravitating discs"
2242:
82:
2812:
2743:
2670:
2605:
2548:
2466:
2315:
2450:
2188:
gravitational research. The scale of these simulations can range in size from just our
2613:
2843:
2830:
2751:
2702:
2678:
2490:
2767:
2629:
2446:
2189:
65:
2201:
1227:
To solve different systems, various assumptions are made about the stress tensor.
2514:
2644:
2532:
2194:
60:
1235:
One fundamental usage of Jean's equation is in spherical gravitational bodies.
2323:
2759:
2694:
2621:
2564:
2394:
2331:
2386:
2231:
2205:
SDSS telescope that provides data for Jeans equation parameter estimation.
2734:
2686:
2533:"Rates of thermal relaxation in direct simulation Monte Carlo methods"
2556:
1893:
Using the stress tensor with the assumption that it is diagonal and
68:
in 1983 and axisymmetric solutions found in 1995 by
Richard Arnold.
2821:
2796:
2377:
2661:
2596:
2213:
1943:, can reduce these equations to a single simplified equation:
402:) is the density of stars in space, as a function of position
2176:) that require assumptions for the equation to be solved.
1209:
Some given assumptions regarding these equations include:
1936:{\displaystyle \sigma _{\theta }^{2}=\sigma _{\phi }^{2}}
1216:
The density around a given star remains the same, or is
35:
that describe the motion of a collection of stars in a
81:
2245:
2141:
2100:
1951:
1899:
1694:
1468:
1247:
1099:
831:
793:
588:
476:
341:
209:
97:
1198:and measures the velocity dispersion in components
59:. However, they were first applied to astronomy by
2645:"The Central Dark Matter Distribution of NGC 2976"
2251:
2168:
2127:
2084:
1935:
1883:
1679:
1453:
1190:
1078:
814:
776:
572:
375:
321:
195:
2722:Monthly Notices of the Royal Astronomical Society
2365:Monthly Notices of the Royal Astronomical Society
2303:Monthly Notices of the Royal Astronomical Society
2239:of minor to major axis of this halo to be 0.47
55:The Jeans equations were originally derived by
2298:"Axisymmetric solutions of the Jeans equation"
376:{\displaystyle \nabla ^{2}\phi =4\pi G\rho .}
8:
2499:. Princeton, NJ: Princeton University Press.
1854:
1831:
1822:
1799:
1766:
1743:
1717:
1704:
1647:
1629:
1623:
1605:
1596:
1573:
1540:
1517:
1491:
1478:
1413:
1395:
1389:
1371:
1365:
1347:
1314:
1296:
1270:
1257:
1185:
1170:
1167:
1152:
1146:
1121:
926:
911:
901:
886:
856:
841:
809:
794:
716:
691:
616:
601:
537:
522:
2184:Jeans equation have found great utility in
43:for fluid flow and may be derived from the
1213:The flow in phase space must conserve mass
2820:
2733:
2660:
2595:
2496:Dynamics and Evolution of Galactic Nuclei
2376:
2244:
2151:
2146:
2140:
2110:
2105:
2099:
2056:
2041:
2036:
2023:
2018:
1996:
1973:
1968:
1952:
1950:
1927:
1922:
1909:
1904:
1898:
1848:
1838:
1816:
1806:
1783:
1760:
1750:
1734:
1711:
1695:
1693:
1641:
1636:
1617:
1612:
1590:
1580:
1557:
1534:
1524:
1508:
1485:
1469:
1467:
1425:
1407:
1402:
1383:
1378:
1359:
1354:
1331:
1308:
1303:
1287:
1264:
1248:
1246:
1178:
1173:
1160:
1155:
1139:
1129:
1124:
1112:
1104:
1098:
1039:
1021:
1013:
997:
991:
975:
957:
939:
919:
914:
910:
904:
894:
889:
877:
849:
844:
835:
830:
802:
797:
792:
731:
709:
699:
694:
679:
673:
657:
639:
609:
604:
589:
587:
552:
530:
525:
510:
504:
477:
475:
468:, the Jeans equations may be written as:
346:
340:
299:
298:
284:
267:
266:
249:
248:
234:
233:
210:
208:
167:
147:
124:
98:
96:
2200:
2169:{\displaystyle \sigma _{\theta }^{2}(r)}
2094:Again, there are two unknown functions (
18:
2288:
815:{\displaystyle \langle {v_{1}}\rangle }
331:Combining the Vlasov equation with the
2773:
2423:"The Collisionless Boltzmann Equation"
2400:
2337:
7:
2508:
2506:
2780:: CS1 maint: unflagged free DOI (
2407:: CS1 maint: unflagged free DOI (
2359:Cappellari, Michele (2020-06-01).
2344:: CS1 maint: unflagged free DOI (
2128:{\displaystyle \sigma _{r}^{2}(r)}
2067:
2062:
2059:
1984:
1955:
1771:
1737:
1722:
1698:
1545:
1511:
1496:
1472:
1436:
1431:
1428:
1319:
1290:
1275:
1251:
1032:
1000:
968:
963:
960:
932:
907:
861:
838:
724:
682:
650:
645:
642:
624:
592:
545:
513:
488:
480:
343:
295:
287:
278:
269:
251:
221:
213:
178:
170:
158:
153:
150:
135:
127:
109:
101:
77:Derivation from Boltzmann equation
14:
2752:10.1111/j.1365-2966.2006.11119.x
45:collisionless Boltzmann equation
16:System of differential equations
1048:
746:
2850:Partial differential equations
2515:"Yale Astronomy Lecture Notes"
2163:
2157:
2122:
2116:
2047:
2011:
1979:
1958:
1872:
1869:
1863:
1793:
1668:
1665:
1659:
1650:
1602:
1567:
1416:
1341:
1089:where the spatial part of the
1073:
1049:
1027:
1003:
771:
747:
719:
685:
619:
595:
540:
516:
304:
272:
254:
239:
33:partial differential equations
1:
456:) is the velocity, and Φ = Φ(
2614:10.1088/2041-8205/758/1/l23
383:gives the Jeans equations.
2871:
2716:Nelson, Andrew F. (2006).
2679:10.1088/0004-637x/745/1/92
2459:Princeton University Press
2210:Milky Way dark matter halo
2801:The Astrophysical Journal
2649:The Astrophysical Journal
2584:The Astrophysical Journal
1231:Spherical Jeans equations
2296:Arnold, Richard (1995).
2236:Sloan Digital Sky Survey
1237:In spherical coordinates
2324:10.1093/mnras/276.1.293
466:gravitational potential
203:Or in vector form:
2855:Equations of astronomy
2513:Van Den Bosch, Frank.
2263:Simulation limitations
2253:
2227:
2206:
2170:
2129:
2086:
1937:
1885:
1681:
1455:
1192:
1080:
816:
778:
574:
377:
323:
197:
24:
2387:10.1093/mnras/staa959
2254:
2225:
2204:
2171:
2130:
2087:
1938:
1886:
1682:
1456:
1239:, the equations are:
1193:
1081:
817:
779:
575:
378:
324:
198:
85:is a special form of
22:
2461:. pp. 195–197.
2252:{\displaystyle \pm }
2243:
2139:
2098:
1949:
1897:
1692:
1466:
1245:
1097:
1091:stress–energy tensor
829:
791:
586:
474:
386:More explicitly, If
339:
207:
95:
2813:1997ApJ...489L.179T
2744:2006MNRAS.373.1039N
2671:2012ApJ...745...92A
2606:2012ApJ...758L..23L
2549:1994PhFl....6.2191H
2467:1988gady.book.....B
2316:1995MNRAS.276..293A
2156:
2115:
2046:
2028:
1978:
1932:
1914:
1646:
1622:
1412:
1388:
1364:
1313:
1117:
1026:
87:Liouville' equation
57:James Clerk Maxwell
37:gravitational field
2249:
2228:
2207:
2166:
2142:
2125:
2101:
2082:
2032:
2014:
1964:
1933:
1918:
1900:
1881:
1677:
1632:
1608:
1451:
1398:
1374:
1350:
1299:
1206:at a given point.
1188:
1100:
1076:
1009:
996:
882:
812:
774:
678:
570:
509:
373:
319:
193:
25:
2537:Physics of Fluids
2455:Galactic Dynamics
2223:
2186:N-body simulation
2074:
2009:
1991:
1791:
1778:
1729:
1565:
1552:
1503:
1443:
1339:
1326:
1282:
1046:
987:
982:
946:
873:
868:
738:
669:
664:
631:
559:
500:
495:
311:
307:
275:
257:
242:
228:
185:
165:
142:
116:
89:and is given by:
2862:
2835:
2834:
2824:
2807:(2): L179–L183.
2792:
2786:
2785:
2779:
2771:
2737:
2735:astro-ph/0609493
2728:(3): 1039–1073.
2713:
2707:
2706:
2664:
2640:
2634:
2633:
2599:
2575:
2569:
2568:
2557:10.1063/1.868221
2543:(6): 2191–2201.
2528:
2522:
2521:
2519:
2510:
2501:
2500:
2487:
2481:
2480:
2443:
2437:
2436:
2434:
2433:
2419:
2413:
2412:
2406:
2398:
2380:
2371:(4): 4819–4837.
2356:
2350:
2349:
2343:
2335:
2293:
2258:
2256:
2255:
2250:
2224:
2175:
2173:
2172:
2167:
2155:
2150:
2134:
2132:
2131:
2126:
2114:
2109:
2091:
2089:
2088:
2083:
2075:
2073:
2065:
2057:
2045:
2040:
2027:
2022:
2010:
2005:
1997:
1992:
1990:
1982:
1977:
1972:
1953:
1942:
1940:
1939:
1934:
1931:
1926:
1913:
1908:
1890:
1888:
1887:
1882:
1853:
1852:
1843:
1842:
1821:
1820:
1811:
1810:
1792:
1784:
1779:
1777:
1769:
1765:
1764:
1755:
1754:
1735:
1730:
1728:
1720:
1716:
1715:
1696:
1686:
1684:
1683:
1678:
1645:
1640:
1621:
1616:
1595:
1594:
1585:
1584:
1566:
1558:
1553:
1551:
1543:
1539:
1538:
1529:
1528:
1509:
1504:
1502:
1494:
1490:
1489:
1470:
1460:
1458:
1457:
1452:
1444:
1442:
1434:
1426:
1411:
1406:
1387:
1382:
1363:
1358:
1340:
1332:
1327:
1325:
1317:
1312:
1307:
1288:
1283:
1281:
1273:
1269:
1268:
1249:
1197:
1195:
1194:
1189:
1184:
1183:
1182:
1166:
1165:
1164:
1145:
1144:
1143:
1134:
1133:
1116:
1111:
1085:
1083:
1082:
1077:
1047:
1045:
1044:
1043:
1030:
1025:
1020:
998:
995:
983:
981:
980:
979:
966:
958:
947:
945:
944:
943:
930:
929:
925:
924:
923:
905:
900:
899:
898:
881:
869:
867:
859:
855:
854:
853:
836:
821:
819:
818:
813:
808:
807:
806:
783:
781:
780:
775:
739:
737:
736:
735:
722:
715:
714:
713:
704:
703:
680:
677:
665:
663:
662:
661:
648:
640:
632:
630:
622:
615:
614:
613:
590:
579:
577:
576:
571:
560:
558:
557:
556:
543:
536:
535:
534:
511:
508:
496:
494:
486:
478:
382:
380:
379:
374:
351:
350:
333:Poisson equation
328:
326:
325:
320:
312:
310:
309:
308:
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293:
285:
277:
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268:
259:
258:
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202:
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199:
194:
186:
184:
176:
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164:
156:
148:
143:
141:
133:
125:
117:
115:
107:
99:
2870:
2869:
2865:
2864:
2863:
2861:
2860:
2859:
2840:
2839:
2838:
2794:
2793:
2789:
2772:
2715:
2714:
2710:
2642:
2641:
2637:
2577:
2576:
2572:
2530:
2529:
2525:
2517:
2512:
2511:
2504:
2489:
2488:
2484:
2477:
2453:(1988). "4.2".
2451:Tremaine, Scott
2445:
2444:
2440:
2431:
2429:
2427:www.cv.nrao.edu
2421:
2420:
2416:
2399:
2358:
2357:
2353:
2336:
2295:
2294:
2290:
2286:
2278:Jeans's theorem
2274:
2265:
2241:
2240:
2214:
2212:
2182:
2137:
2136:
2096:
2095:
2066:
2058:
1998:
1983:
1954:
1947:
1946:
1895:
1894:
1844:
1834:
1812:
1802:
1770:
1756:
1746:
1736:
1721:
1707:
1697:
1690:
1689:
1586:
1576:
1544:
1530:
1520:
1510:
1495:
1481:
1471:
1464:
1463:
1435:
1427:
1318:
1289:
1274:
1260:
1250:
1243:
1242:
1233:
1174:
1156:
1135:
1125:
1095:
1094:
1093:is defined as:
1035:
1031:
999:
971:
967:
959:
935:
931:
915:
906:
890:
860:
845:
837:
827:
826:
798:
789:
788:
727:
723:
705:
695:
681:
653:
649:
641:
623:
605:
591:
584:
583:
548:
544:
526:
512:
487:
479:
472:
471:
455:
448:
441:
426:
419:
412:
342:
337:
336:
294:
286:
220:
212:
205:
204:
177:
169:
157:
149:
134:
126:
108:
100:
93:
92:
83:Vlasov Equation
79:
74:
53:
41:Euler equations
29:Jeans equations
17:
12:
11:
5:
2868:
2866:
2858:
2857:
2852:
2842:
2841:
2837:
2836:
2822:10.1086/310975
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1218:incompressible
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2476:0-691-08445-9
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2452:
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2447:Binney, James
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1014:
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1006:
992:
988:
984:
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972:
954:
951:
948:
940:
936:
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916:
895:
891:
883:
878:
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870:
864:
850:
846:
832:
825:
824:
823:
803:
799:
768:
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762:
759:
756:
753:
750:
743:
740:
732:
728:
710:
706:
700:
696:
688:
674:
670:
666:
658:
654:
636:
633:
627:
610:
606:
598:
582:
581:
580:
567:
564:
561:
553:
549:
531:
527:
519:
505:
501:
497:
491:
483:
469:
467:
463:
459:
452:
445:
438:
434:
430:
423:
416:
409:
405:
401:
397:
393:
389:
384:
370:
367:
364:
361:
358:
355:
352:
347:
334:
329:
316:
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301:
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263:
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245:
236:
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224:
216:
190:
187:
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173:
161:
144:
138:
130:
121:
118:
112:
104:
90:
88:
84:
76:
71:
69:
67:
62:
58:
50:
48:
46:
42:
38:
34:
31:are a set of
30:
21:
2804:
2800:
2790:
2776:cite journal
2725:
2721:
2711:
2652:
2648:
2638:
2587:
2583:
2573:
2540:
2536:
2526:
2495:
2485:
2454:
2441:
2430:. Retrieved
2426:
2417:
2403:cite journal
2368:
2364:
2354:
2340:cite journal
2307:
2301:
2291:
2266:
2229:
2190:solar system
2183:
2180:Applications
2093:
1945:
1892:
1688:
1462:
1241:
1234:
1222:
1208:
1203:
1199:
1088:
786:
470:
461:
457:
450:
443:
436:
432:
428:
421:
414:
407:
403:
399:
395:
391:
387:
385:
335:for gravity:
330:
91:
80:
66:James Binney
54:
28:
26:
2310:: 293–300.
2195:dark matter
427:) and time
72:Mathematics
61:James Jeans
2844:Categories
2687:2152/35153
2590:(1): L23.
2432:2022-04-23
2378:1907.09894
2284:References
2831:120393398
2760:0035-8711
2703:118429887
2695:0004-637X
2662:1110.5951
2655:(1): 92.
2622:2041-8205
2597:1209.2708
2565:1070-6631
2395:0035-8711
2332:1365-2966
2247:±
2232:Milky Way
2148:θ
2144:σ
2103:σ
2068:∂
2063:Φ
2060:∂
2054:ρ
2038:θ
2034:σ
2030:−
2016:σ
2003:ρ
1985:∂
1966:σ
1962:ρ
1956:∂
1924:ϕ
1920:σ
1906:θ
1902:σ
1867:θ
1861:
1855:⟩
1850:ϕ
1840:θ
1832:⟨
1823:⟩
1818:ϕ
1800:⟨
1786:ρ
1772:∂
1767:⟩
1762:ϕ
1744:⟨
1741:ρ
1738:∂
1723:∂
1718:⟩
1713:ϕ
1705:⟨
1702:ρ
1699:∂
1663:θ
1657:
1648:⟩
1638:ϕ
1630:⟨
1627:−
1624:⟩
1614:θ
1606:⟨
1597:⟩
1592:θ
1574:⟨
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1536:θ
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1515:ρ
1512:∂
1497:∂
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1437:∂
1432:Φ
1429:∂
1423:ρ
1414:⟩
1404:ϕ
1396:⟨
1393:−
1390:⟩
1380:θ
1372:⟨
1369:−
1366:⟩
1348:⟨
1334:ρ
1320:∂
1315:⟩
1297:⟨
1294:ρ
1291:∂
1276:∂
1271:⟩
1258:⟨
1255:ρ
1252:∂
1186:⟩
1171:⟨
1168:⟩
1153:⟨
1150:−
1147:⟩
1122:⟨
1102:σ
1033:∂
1011:σ
1001:∂
989:∑
985:−
969:∂
964:Φ
961:∂
952:−
933:∂
927:⟩
912:⟨
908:∂
902:⟩
887:⟨
875:∑
862:∂
857:⟩
842:⟨
839:∂
810:⟩
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725:∂
717:⟩
692:⟨
683:∂
671:∑
651:∂
646:Φ
643:∂
625:∂
617:⟩
602:⟨
593:∂
546:∂
538:⟩
523:⟨
514:∂
502:∑
489:∂
481:∂
464:) is the
368:ρ
362:π
353:ϕ
344:∇
305:→
296:∂
288:∂
282:⋅
279:Φ
273:→
270:∇
264:−
255:→
252:∇
246:⋅
240:→
222:∂
214:∂
179:∂
171:∂
159:∂
154:Φ
151:∂
145:−
136:∂
128:∂
110:∂
102:∂
2768:17175349
2630:18220516
2493:(2013).
2272:See also
2234:. Using
2809:Bibcode
2740:Bibcode
2667:Bibcode
2602:Bibcode
2545:Bibcode
2463:Bibcode
2312:Bibcode
1225:closed.
51:History
2829:
2766:
2758:
2701:
2693:
2628:
2620:
2563:
2473:
2393:
2330:
2827:S2CID
2764:S2CID
2730:arXiv
2699:S2CID
2657:arXiv
2626:S2CID
2592:arXiv
2518:(PDF)
2373:arXiv
2782:link
2756:ISSN
2691:ISSN
2618:ISSN
2561:ISSN
2471:ISBN
2409:link
2391:ISSN
2346:link
2328:ISSN
2135:and
1202:and
27:The
2817:doi
2805:489
2748:doi
2726:373
2683:hdl
2675:doi
2653:745
2610:doi
2588:758
2553:doi
2383:doi
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