4024:
gas, the adiabatic index is 5/3. However, in astrophysical objects this value is usually close to 1 (for example, in partially ionized gas at temperatures low compared to the ionization energy). More generally, the process is not really adiabatic but involves cooling by radiation that is much faster than the contraction, so that the process can be modeled by an adiabatic index as low as 1 (which corresponds to the polytropic index of an isothermal gas). So the second case is the rule rather than an exception in stars. This is the reason why stars usually form in clusters.
187:
1640:
4724:
4736:
1393:
3244:; i.e., the cloud's radius is the Jeans' length when thermal energy per particle equals gravitational work per particle. At this critical length the cloud neither expands nor contracts. It is only when thermal energy is not equal to gravitational work that the cloud either expands and cools or contracts and warms, a process that continues until equilibrium is reached.
1282:
4748:
4675:
797:
42:
1635:{\displaystyle M_{\text{J}}={\frac {4\pi }{3}}\rho R_{\text{J}}^{3}={\frac {\pi }{6}}\cdot {\frac {c_{\text{s}}^{3}}{G^{\frac {3}{2}}\rho ^{\frac {1}{2}}}}\approx 2{\text{ M}}_{\odot }\cdot \left({\frac {c_{\text{s}}}{0.2{\text{ km/s}}}}\right)^{3}\left({\frac {n}{10^{3}{\text{ cm}}^{-3}}}\right)^{-{\frac {1}{2}}}.}
1114:
940:
4023:
the Jeans mass decreases with increasing density. During gravitational collapse density always increases, thus in the second case the Jeans mass will decrease during collapse, allowing smaller overdense regions to collapse, leading to fragmentation of the giant molecular cloud. For an ideal monatomic
3676:
1662:
An alternative, arguably even simpler, derivation can be found using energy considerations. In the interstellar cloud, two opposing forces are at work. The gas pressure, caused by the thermal movement of the atoms or molecules comprising the cloud, tries to make the cloud expand, whereas gravitation
1034:
is appropriate for molecular hydrogen with 20% helium by number). When the sound-crossing time is less than the free-fall time, pressure forces temporarily overcome gravity, and the system returns to a stable equilibrium. However, when the free-fall time is less than the sound-crossing time, gravity
556:
two possible scenarios. An exactly homogeneous distribution has no clear center of mass and no clear way to define a gravitational acceleration direction. For the other case, Jeans extends what Newton wrote about: Jeans demonstrated that small deviations from exact homogeneity lead to instabilities.
543:
considered a universe without edges and pondered what would happen if the "system of the world", which exists within the universe, were finite or infinite. In the finite case, stars would gravitate towards the center, and if infinite, all the stars would be nearly in equilibrium and the stars would
1649:
It was later pointed out by other astrophysicists including Binney and
Tremaine that the original analysis used by Jeans was flawed: in his formal analysis, although Jeans assumed that the collapsing region of the cloud was surrounded by an infinite, static medium, the surrounding medium should in
555:
James Jeans extended the issue of gravitational stability to include pressure. In 1902, Jeans wrote, similarly to Halley, that a finite distribution of matter, assuming pressure does not prevent it, will collapse gravitationally towards its center. For an infinite distribution of matter, there are
519:
is the radius. The equilibrium is stable if small perturbations are damped and unstable if they are amplified. In general, the cloud is unstable if it is either very massive at a given temperature or very cool at a given mass; under these circumstances, the gas pressure gradient cannot overcome
2526:
3873:
3464:
Jeans instability can also give rise to fragmentation in certain conditions. To derive the condition for fragmentation an adiabatic process is assumed in an ideal gas and also a polytropic equation of state is taken. The derivation is shown below through a dimensional analysis:
591:. The cloud is stable for sufficiently small mass (at a given temperature and radius), but once this critical mass is exceeded, it will begin a process of runaway contraction until some other force can impede the collapse. He derived a formula for calculating this critical
3388:
679:
1653:
Remarkably, when using a more careful analysis taking into account other factors such as the expansion of the
Universe fortuitously cancel out the apparent error in Jeans' analysis, and Jeans' equation is correct, even if its derivation might have been dubious.
2729:
is the critical radius of a cloud (typically a cloud of interstellar molecular gas and dust) where thermal energy, which causes the cloud to expand, is counteracted by gravity, which causes the cloud to collapse. It is named after the
British astronomer
3136:
2827:
806:
3950:
3540:
2389:
1650:
reality also be collapsing, since all larger scales are also gravitationally unstable by the same analysis. The influence of this medium was completely ignored in Jeans' analysis. This flaw has come to be known as the "Jeans swindle".
2717:
3550:
1277:{\displaystyle \lambda _{\text{J}}={\frac {c_{\text{s}}}{(G\rho )^{\frac {1}{2}}}}\approx 0.4{\text{ pc}}\cdot {\frac {c_{\text{s}}}{0.2{\text{ km/s}}}}\cdot \left({\frac {n}{10^{3}{\text{ cm}}^{-3}}}\right)^{-{\frac {1}{2}}}.}
368:
2382:
3748:
3290:
3755:
799:
for sound waves to cross the region and attempt to push back and re-establish the system in pressure balance. At the same time, gravity will attempt to contract the system even further, and will do so on a
1388:
3048:
2740:
1080:
2603:
the mass of a particle comprising the gas. Assuming the cloud to consist of atomic hydrogen, the prefactor can be calculated. If we take the solar mass as the unit of mass, and use units of
2209:
2144:
583:
within a gaseous cloud. He was able to show that, under appropriate conditions, a cloud, or part of one, would become unstable and begin to collapse when it lacked sufficient gaseous
4021:
3988:
1663:
tries to make the cloud collapse. The Jeans mass is the critical mass where both forces are in equilibrium with each other. In the following derivation numerical constants (such as
792:{\displaystyle t_{\text{sound}}={\frac {R}{c_{\text{s}}}}\approx 0.5{\text{ Myr}}\cdot {\frac {R}{0.1{\text{ pc}}}}\cdot \left({\frac {c_{\text{s}}}{0.2{\text{ km/s}}}}\right)^{-1}}
2013:
3242:
1286:
This length scale is known as the Jeans length. All scales larger than the Jeans length are unstable to gravitational collapse, whereas smaller scales are stable. The Jeans mass
3192:
2636:
1109:
1928:
404:
3423:
3285:
2854:
2553:
2310:
1338:
1311:
1014:
674:
4596:
3879:
453:
277:
3475:
3043:
2631:
2317:
935:{\displaystyle t_{\text{ff}}={\frac {1}{(G\rho )^{\frac {1}{2}}}}\approx 2{\text{ Myr}}\cdot \left({\frac {n}{10^{3}{\text{ cm}}^{-3}}}\right)^{-{\frac {1}{2}}},}
3447:
3008:
2988:
2942:
980:
2898:
2064:
1876:
1737:
1681:
1847:
3671:{\displaystyle PV=nRT\Rightarrow P\cdot P^{-{\frac {1}{\gamma }}}=P^{\frac {\gamma -1}{\gamma }}\propto T\Rightarrow P\propto T^{\frac {\gamma }{\gamma -1}}.}
2965:
3156:
2918:
2878:
2601:
2577:
2271:
2251:
2231:
2084:
2035:
1970:
1950:
1821:
1801:
1781:
1761:
1708:
960:
644:
624:
517:
493:
473:
424:
552:, wrote that it's hard to imagine that particles in an infinite space should be able to stand in such a configuration to result in a perfect equilibrium.
2521:{\displaystyle M_{\text{J}}=\left({\frac {375k_{\text{B}}^{3}}{4\pi m^{4}G^{3}}}\right)^{\frac {1}{2}}\left({\frac {T^{3}}{n}}\right)^{\frac {1}{2}},}
603:. The greater the mass of the cloud, the bigger its size, and the colder its temperature, the less stable it will be against gravitational collapse.
168:
606:
The approximate value of the Jeans mass may be derived through a simple physical argument. One begins with a spherical gaseous region of radius
3689:
4564:
4402:
4316:
3868:{\displaystyle M_{\text{J}}\propto T^{\frac {3}{2}}\rho ^{-{\frac {1}{2}}}\propto \rho ^{{\frac {3}{2}}(\gamma -1)}\rho ^{-{\frac {1}{2}}}.}
4784:
4589:
1042:
2151:
4774:
4541:
4237:
2089:
230:
208:
3383:{\displaystyle \lambda _{\text{J}}={\frac {2\pi }{k_{\text{J}}}}=c_{\text{s}}\left({\frac {\pi }{G\rho }}\right)^{\frac {1}{2}},}
1343:
4699:
132:
4582:
2947:
Perhaps the easiest way to conceptualize Jeans' length is in terms of a close approximation, in which we discard the factors
1975:
161:
4789:
4779:
1881:
4556:
4124:
1743:
is released. When this energy equals the amount of work to be done on the gas, the critical mass is attained. Let
4751:
4059:
2148:
The critical mass is attained as soon as the released gravitational energy is equal to the work done on the gas:
2276:
201:
195:
117:
263:
is not strong enough to prevent the gravitational collapse of a region filled with matter. It is named after
4794:
4727:
3131:{\displaystyle \lambda _{\text{J}}\approx \left({\frac {k_{\text{B}}Tr^{3}}{GM\mu }}\right)^{\frac {1}{2}}.}
271:
154:
4664:
4440:
Abbasi, Amir (2018). "Effect of polarization force on the Jeans instability in collisional dusty plasmas".
3993:
3960:
2822:{\displaystyle \lambda _{\text{J}}=\left({\frac {15k_{\text{B}}T}{4\pi G\mu \rho }}\right)^{\frac {1}{2}},}
101:
4654:
3395:
2921:
2580:
1684:
1036:
580:
496:
252:
212:
86:
4033:
4689:
3197:
122:
3164:
4639:
4502:
4449:
4269:
4203:
4160:
4096:
1740:
1087:
76:
4709:
4619:
373:
142:
56:
4735:
4704:
4520:
4465:
4353:
4287:
4184:
2857:
2556:
137:
4659:
4560:
4537:
4408:
4398:
4373:
4322:
4312:
4233:
4176:
3945:{\displaystyle M_{\text{J}}\propto \rho ^{{\frac {3}{2}}\left(\gamma -{\frac {4}{3}}\right)}.}
3683:
3469:
3401:
3263:
2832:
2531:
1316:
1289:
985:
652:
96:
3535:{\displaystyle PV^{\gamma }={\text{constant}}\rightarrow V\propto P^{-{\frac {1}{\gamma }}}.}
4510:
4457:
4363:
4277:
4204:"Original letter from Isaac Newton to Richard Bentley, dated 17 January 1692/3 (Diplomatic)"
4168:
4104:
255:
of a cloud of gas or dust. It causes the collapse of interstellar gas clouds and subsequent
4426:
4624:
3955:
2731:
2712:{\displaystyle M_{\text{J}}=3\times 10^{4}\left({\frac {T^{3}}{n}}\right)^{\frac {1}{2}}.}
549:
528:
429:
61:
3013:
4506:
4453:
4273:
4164:
4100:
2606:
4605:
4038:
3432:
3426:
2993:
2970:
2927:
1017:
965:
801:
647:
569:
524:
33:
2883:
2040:
1852:
1713:
1666:
520:
gravitational force, and the cloud will collapse, is called Jeans collapse criterion.
4768:
4469:
1826:
540:
2950:
2734:, who concerned himself with the stability of spherical nebulae in the early 1900s.
2386:
If during the derivation all constants are taken along, the resulting expression is
4739:
4649:
4150:"Of the Infinity of the Sphere of Fix'd Stars. By Edmund Halley, L. L. D. R. S. S."
3141:
2944:
is the cloud's mass density (i.e. the cloud's mass divided by the cloud's volume).
2903:
2863:
2586:
2562:
2256:
2236:
2216:
2069:
2020:
1955:
1935:
1806:
1786:
1766:
1746:
1693:
945:
629:
609:
545:
502:
478:
458:
409:
248:
91:
4230:
Cosmology's century: an inside history of our modern understanding of the universe
3287:) below which stable oscillations rather than gravitational collapse will occur.
4634:
4629:
600:
576:
264:
71:
66:
27:
Mechanism by which the collapse of interstellar gas clouds causes star formation
4461:
4109:
4084:
3453:
363:{\displaystyle {\frac {dp}{dr}}=-{\frac {G\rho (r)M_{\text{enc}}(r)}{r^{2}}},}
4412:
4377:
4326:
4180:
4644:
4368:
4341:
3680:
3544:
2377:{\displaystyle M_{\text{J}}=\left({\frac {T^{3}}{n}}\right)^{\frac {1}{2}}.}
572:
81:
4515:
4490:
4282:
4257:
4172:
4392:
4306:
2314:
A little algebra leads to the following expression for the critical mass:
17:
3743:{\displaystyle P=K\rho ^{\gamma }\rightarrow T\propto \rho ^{\gamma -1}.}
584:
260:
4311:. Scott Tremaine (2nd ed.). Princeton: Princeton University Press.
596:
588:
4188:
4149:
1739:, work must be done against the gas pressure. During the compression,
4524:
4291:
544:
eventually reach a resting place. Contrary to the writing of Halley,
2017:
The amount of energy released when the sphere contracts from radius
4358:
4574:
4340:
Falco, M.; Hansen, S. H.; Wojtak, R.; Mamon, G. A. (2013-05-01).
592:
256:
41:
4578:
180:
3990:, the Jeans mass increases with increasing density, while if
1972:
is, apart from constants, given by the following expression:
1687:) will be ignored. They will be reintroduced in the result.
1383:{\textstyle R_{\text{J}}={\frac {1}{2}}\lambda _{\text{J}}}
4346:
Monthly
Notices of the Royal Astronomical Society: Letters
4153:
1932:
The gravitational potential energy of a sphere with mass
4427:"Jeans Length -- from Eric Weisstein's World of Physics"
4397:. Chichester, West Sussex, U.K.: Wiley. pp. 46–47.
1878:, one arrives at the following expression for the work:
1039:. The condition for gravitational collapse is therefore
1690:
Consider a homogeneous spherical gas cloud with radius
3996:
3963:
3435:
3404:
3266:
3200:
3167:
3144:
3016:
2996:
2973:
2953:
2930:
2906:
2886:
2866:
2835:
2609:
2589:
2565:
2534:
2259:
2239:
2219:
2072:
2043:
2023:
1958:
1938:
1855:
1829:
1809:
1789:
1769:
1749:
1716:
1696:
1669:
1346:
1319:
1292:
1090:
988:
968:
948:
655:
632:
612:
505:
481:
461:
432:
412:
376:
3882:
3758:
3692:
3553:
3478:
3293:
3051:
2743:
2639:
2392:
2320:
2279:
2154:
2092:
1978:
1884:
1396:
1117:
1045:
809:
682:
676:. The gas is compressed slightly and it takes a time
280:
1035:
overcomes pressure forces, and the region undergoes
4682:
4612:
2233:must be expressed in terms of the particle density
274:, which in case of a spherical cloud translates to
4015:
3982:
3944:
3867:
3742:
3670:
3534:
3441:
3417:
3382:
3279:
3236:
3186:
3150:
3130:
3037:
3002:
2982:
2959:
2936:
2912:
2892:
2872:
2848:
2821:
2711:
2625:
2595:
2571:
2547:
2520:
2376:
2304:
2265:
2245:
2225:
2203:
2138:
2078:
2066:is obtained by differentiation this expression to
2058:
2029:
2007:
1964:
1944:
1922:
1870:
1841:
1815:
1795:
1775:
1755:
1731:
1702:
1675:
1634:
1382:
1332:
1305:
1276:
1103:
1075:{\displaystyle t_{\text{ff}}<t_{\text{sound}}.}
1074:
1008:
974:
954:
934:
791:
668:
638:
618:
511:
487:
467:
447:
418:
398:
362:
4495:Philosophical Transactions of the Royal Society A
4262:Philosophical Transactions of the Royal Society A
4089:Monthly Notices of the Royal Astronomical Society
1313:is just the mass contained in a sphere of radius
4232:. Princeton Oxford: Princeton University Press.
251:that describes an instability that leads to the
3247:
2900:is the mean molecular weight of the particles,
2204:{\displaystyle {\frac {M^{2}}{R^{2}}}=nTR^{2}.}
1710:. In order to compress this sphere to a radius
3045:. The formula for Jeans' length then becomes:
2139:{\displaystyle dU={\frac {M^{2}}{R^{2}}}\,dR.}
1849:. Using the ideal gas law, according to which
4590:
3256:is the oscillation wavelength (respectively,
1823:the gas pressure. The work to be done equals
523:The Jeans instability likely determines when
162:
8:
4251:
4249:
259:formation. It occurs when the internal gas
4597:
4583:
4575:
982:is the gas density within the region, and
169:
155:
29:
4514:
4367:
4357:
4281:
4108:
4003:
3995:
3970:
3962:
3922:
3901:
3900:
3887:
3881:
3850:
3846:
3815:
3814:
3795:
3791:
3776:
3763:
3757:
3725:
3706:
3691:
3646:
3608:
3589:
3585:
3552:
3517:
3513:
3495:
3486:
3477:
3434:
3409:
3403:
3366:
3347:
3336:
3321:
3307:
3298:
3292:
3271:
3265:
3226:
3205:
3199:
3172:
3166:
3143:
3114:
3090:
3077:
3070:
3056:
3050:
3029:
3020:
3015:
2995:
2972:
2952:
2929:
2905:
2885:
2865:
2840:
2834:
2805:
2772:
2762:
2748:
2742:
2695:
2680:
2674:
2663:
2644:
2638:
2614:
2608:
2588:
2564:
2539:
2533:
2504:
2489:
2483:
2467:
2454:
2444:
2426:
2421:
2411:
2397:
2391:
2360:
2345:
2339:
2325:
2319:
2293:
2278:
2258:
2238:
2218:
2192:
2171:
2161:
2155:
2153:
2126:
2118:
2108:
2102:
2091:
2071:
2042:
2022:
1991:
1985:
1977:
1957:
1937:
1910:
1904:
1883:
1854:
1828:
1808:
1788:
1768:
1748:
1715:
1695:
1668:
1617:
1613:
1597:
1592:
1585:
1575:
1564:
1552:
1542:
1536:
1522:
1517:
1496:
1481:
1470:
1465:
1459:
1446:
1437:
1432:
1410:
1401:
1395:
1374:
1360:
1351:
1345:
1324:
1318:
1297:
1291:
1259:
1255:
1239:
1234:
1227:
1217:
1201:
1191:
1185:
1177:
1157:
1137:
1131:
1122:
1116:
1095:
1089:
1063:
1050:
1044:
998:
987:
967:
962:is the universal gravitational constant,
947:
917:
913:
897:
892:
885:
875:
862:
842:
823:
814:
808:
780:
768:
758:
752:
736:
727:
719:
705:
696:
687:
681:
660:
654:
631:
611:
504:
480:
460:
431:
411:
381:
375:
349:
329:
307:
281:
279:
231:Learn how and when to remove this message
2633:for the particle density, the result is
194:This article includes a list of general
4394:An Introduction to Stellar Astrophysics
4051:
3248:Jeans' length as oscillation wavelength
1683:) and constants of nature (such as the
109:
48:
32:
4551:Clarke, Cathie; Carswell, Bob (2007).
4016:{\textstyle \gamma <{\frac {4}{3}}}
3983:{\textstyle \gamma >{\frac {4}{3}}}
2273:. This can be done using the relation
4491:"The Stability of a Spherical Nebula"
4258:"The Stability of a Spherical Nebula"
2008:{\displaystyle U={\frac {M^{2}}{R}}.}
455:is the density of the gas (at radius
7:
4747:
270:For stability, the cloud must be in
3456:would travel in the collapse time.
3237:{\textstyle k_{\text{B}}T=GM\mu /r}
4342:"Why does the Jeans Swindle work?"
3187:{\textstyle \lambda _{\text{J}}=r}
200:it lacks sufficient corresponding
25:
2880:is the temperature of the cloud,
2737:The formula for Jeans length is:
4746:
4734:
4723:
4722:
4673:
1104:{\textstyle \lambda _{\text{J}}}
587:support to balance the force of
579:, who considered the process of
185:
40:
1923:{\displaystyle dW=nTR^{2}\,dR.}
4125:"The Jeans Collapse Criterion"
4085:"1957MNRAS.117..104B Page 104"
3837:
3825:
3712:
3633:
3572:
3500:
3449:is the enclosed mass density.
1154:
1144:
839:
829:
442:
436:
399:{\textstyle M_{\text{enc}}(r)}
393:
387:
341:
335:
322:
316:
1:
4442:Plasma Science and Technology
4553:Astrophysical Fluid Dynamics
4532:Longair, Malcolm S. (1998).
4148:Halley, Edmund (1720–1721).
3161:It follows immediately that
3158:is the radius of the cloud.
1783:the (absolute) temperature,
1020:for mean mass per particle (
4785:Fluid dynamic instabilities
4041:(similar waves in a plasma)
1390:is half the Jeans length):
1084:The resultant Jeans length
4811:
4700:Kelvin–Helmholtz mechanism
4557:Cambridge University Press
4228:Peebles, P. J. E. (2022).
4208:www.newtonproject.ox.ac.uk
3452:It is also the distance a
1803:the particle density, and
1763:be the mass of the cloud,
133:Kelvin–Helmholtz mechanism
4718:
4671:
4391:LeBlanc, Francis (2010).
3418:{\textstyle c_{\text{s}}}
3280:{\textstyle k_{\text{J}}}
2990:and in which we rephrase
2849:{\textstyle k_{\text{B}}}
2548:{\textstyle k_{\text{B}}}
2305:{\displaystyle M=nR^{3}.}
1333:{\textstyle R_{\text{J}}}
1306:{\textstyle M_{\text{J}}}
1009:{\textstyle n=\rho /\mu }
669:{\textstyle c_{\text{s}}}
4775:Concepts in astrophysics
4462:10.1088/2058-6272/aa96fa
548:, in a 1692/3 letter to
4110:10.1093/mnras/117.1.104
1658:Energy-based derivation
272:hydrostatic equilibrium
215:more precise citations.
4655:Pre-main-sequence star
4516:10.1098/rsta.1902.0012
4305:Binney, James (2008).
4283:10.1098/rsta.1902.0012
4173:10.1098/rstl.1720.0006
4083:Bonnor, W. B. (1957).
4017:
3984:
3946:
3869:
3744:
3672:
3536:
3443:
3419:
3396:gravitational constant
3384:
3281:
3238:
3188:
3152:
3132:
3039:
3004:
2984:
2961:
2938:
2922:gravitational constant
2914:
2894:
2874:
2850:
2823:
2713:
2627:
2597:
2581:gravitational constant
2573:
2549:
2522:
2378:
2306:
2267:
2247:
2227:
2205:
2140:
2080:
2060:
2031:
2009:
1966:
1946:
1924:
1872:
1843:
1817:
1797:
1777:
1757:
1733:
1704:
1685:gravitational constant
1677:
1636:
1384:
1334:
1307:
1278:
1105:
1076:
1037:gravitational collapse
1010:
976:
956:
936:
793:
670:
640:
620:
581:gravitational collapse
513:
497:gravitational constant
489:
469:
449:
420:
406:is the enclosed mass,
400:
364:
253:gravitational collapse
87:Pre-main-sequence star
4690:Initial mass function
4489:Jeans, J. H. (1902).
4369:10.1093/mnrasl/sls051
4256:Jeans, J. H. (1902).
4018:
3985:
3947:
3870:
3745:
3673:
3537:
3444:
3420:
3385:
3282:
3239:
3189:
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3133:
3040:
3005:
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2939:
2915:
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2875:
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2721:
2714:
2628:
2598:
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2550:
2523:
2379:
2307:
2268:
2248:
2228:
2206:
2141:
2081:
2061:
2032:
2010:
1967:
1947:
1925:
1873:
1844:
1818:
1798:
1778:
1758:
1734:
1705:
1678:
1637:
1385:
1335:
1308:
1279:
1106:
1077:
1011:
977:
957:
937:
794:
671:
646:, and with a gaseous
641:
621:
595:as a function of its
514:
490:
470:
450:
448:{\textstyle \rho (r)}
421:
401:
365:
123:Initial mass function
4683:Theoretical concepts
4640:Young stellar object
4536:. Berlin: Springer.
3994:
3961:
3880:
3756:
3690:
3551:
3476:
3433:
3402:
3291:
3264:
3198:
3165:
3142:
3049:
3038:{\textstyle M/r^{3}}
3014:
2994:
2971:
2951:
2928:
2904:
2884:
2864:
2833:
2741:
2637:
2607:
2587:
2563:
2532:
2390:
2318:
2277:
2257:
2237:
2217:
2152:
2090:
2070:
2041:
2021:
1976:
1956:
1936:
1882:
1853:
1827:
1807:
1787:
1767:
1747:
1741:gravitational energy
1714:
1694:
1667:
1394:
1344:
1317:
1290:
1115:
1088:
1043:
986:
966:
946:
807:
680:
653:
630:
610:
503:
479:
459:
430:
410:
374:
278:
110:Theoretical concepts
77:Young stellar object
4710:Planetary migration
4620:Interstellar medium
4507:1902RSPTA.199....1J
4454:2018PlST...20c5301A
4274:1902RSPTA.199....1J
4165:1720RSPT...31...22H
4129:csep10.phys.utk.edu
4101:1957MNRAS.117..104B
4060:"Jeans instability"
3470:adiabatic processes
2626:{\textstyle m^{-3}}
2431:
1475:
1442:
568:is named after the
143:Planetary migration
57:Interstellar medium
4790:Interstellar media
4780:Effects of gravity
4705:Nebular hypothesis
4665:Herbig–Haro object
4013:
3980:
3942:
3865:
3740:
3668:
3532:
3442:{\textstyle \rho }
3439:
3415:
3380:
3277:
3234:
3184:
3148:
3128:
3035:
3003:{\textstyle \rho }
3000:
2983:{\textstyle 4\pi }
2980:
2957:
2937:{\textstyle \rho }
2934:
2910:
2890:
2870:
2858:Boltzmann constant
2846:
2819:
2709:
2623:
2593:
2569:
2557:Boltzmann constant
2545:
2518:
2417:
2374:
2302:
2263:
2243:
2223:
2201:
2136:
2076:
2056:
2027:
2005:
1962:
1942:
1920:
1868:
1839:
1813:
1793:
1773:
1753:
1729:
1700:
1673:
1632:
1461:
1428:
1380:
1330:
1303:
1274:
1101:
1072:
1006:
975:{\textstyle \rho }
972:
952:
932:
789:
666:
636:
616:
509:
485:
465:
445:
416:
396:
360:
138:Nebular hypothesis
102:Herbig–Haro object
4762:
4761:
4695:Jeans instability
4660:Herbig Ae/Be star
4566:978-0-521-85331-6
4501:(312–320): 1–53.
4404:978-0-470-69957-7
4318:978-0-691-13026-2
4308:Galactic dynamics
4268:(312–320): 1–53.
4034:Bonnor–Ebert mass
4011:
3978:
3930:
3909:
3890:
3858:
3823:
3803:
3784:
3766:
3684:equation of state
3662:
3624:
3597:
3525:
3498:
3412:
3374:
3360:
3339:
3327:
3324:
3301:
3274:
3258:Jeans' wavenumber
3208:
3175:
3122:
3108:
3080:
3059:
2893:{\textstyle \mu }
2843:
2813:
2799:
2775:
2751:
2703:
2689:
2647:
2542:
2512:
2498:
2475:
2461:
2424:
2400:
2368:
2354:
2328:
2213:Next, the radius
2177:
2124:
2059:{\textstyle R-dR}
2000:
1871:{\textstyle p=nT}
1732:{\textstyle R-dR}
1676:{\textstyle \pi }
1625:
1607:
1595:
1558:
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1545:
1520:
1508:
1504:
1489:
1468:
1454:
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1377:
1368:
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1327:
1300:
1267:
1249:
1237:
1207:
1204:
1194:
1180:
1169:
1165:
1140:
1125:
1111:is approximately
1098:
1066:
1053:
925:
907:
895:
865:
854:
850:
817:
774:
771:
761:
742:
739:
722:
711:
708:
690:
663:
426:is the pressure,
384:
355:
332:
299:
245:Jeans instability
241:
240:
233:
179:
178:
128:Jeans instability
97:Herbig Ae/Be star
16:(Redirected from
4802:
4750:
4749:
4738:
4726:
4725:
4677:
4676:
4599:
4592:
4585:
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4570:
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4534:Galaxy Formation
4528:
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4070:
4064:Oxford Reference
4056:
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2014:
2012:
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1909:
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1877:
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1848:
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1845:
1842:{\textstyle pdV}
1840:
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645:
643:
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637:
625:
623:
622:
617:
529:molecular clouds
518:
516:
515:
510:
494:
492:
491:
486:
474:
472:
471:
466:
454:
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451:
446:
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397:
386:
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369:
367:
366:
361:
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354:
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344:
334:
333:
330:
308:
300:
298:
290:
282:
247:is a concept in
236:
229:
225:
222:
216:
211:this article by
202:inline citations
189:
188:
181:
171:
164:
157:
44:
30:
21:
4810:
4809:
4805:
4804:
4803:
4801:
4800:
4799:
4765:
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4763:
4758:
4714:
4678:
4674:
4669:
4625:Molecular cloud
4608:
4603:
4573:
4567:
4550:
4544:
4531:
4488:
4484:
4483:
4477:
4439:
4438:
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4405:
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4385:
4339:
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4304:
4303:
4299:
4255:
4254:
4247:
4240:
4227:
4226:
4222:
4212:
4210:
4202:Newton, Isaac.
4201:
4200:
4196:
4147:
4146:
4142:
4133:
4131:
4123:
4122:
4118:
4082:
4081:
4077:
4068:
4066:
4058:
4057:
4053:
4048:
4030:
3992:
3991:
3959:
3958:
3956:adiabatic index
3952:
3915:
3911:
3896:
3883:
3878:
3877:
3842:
3810:
3787:
3772:
3759:
3754:
3753:
3721:
3702:
3688:
3687:
3651:
3642:
3610:
3604:
3581:
3549:
3548:
3509:
3482:
3474:
3473:
3462:
3431:
3430:
3405:
3400:
3399:
3352:
3343:
3342:
3332:
3317:
3309:
3294:
3289:
3288:
3267:
3262:
3261:
3250:
3201:
3196:
3195:
3168:
3163:
3162:
3140:
3139:
3097:
3086:
3073:
3072:
3066:
3065:
3052:
3047:
3046:
3025:
3012:
3011:
2992:
2991:
2969:
2968:
2960:{\textstyle 15}
2949:
2948:
2926:
2925:
2902:
2901:
2882:
2881:
2862:
2861:
2836:
2831:
2830:
2782:
2768:
2764:
2758:
2757:
2744:
2739:
2738:
2732:Sir James Jeans
2724:
2676:
2670:
2669:
2659:
2640:
2635:
2634:
2610:
2605:
2604:
2585:
2584:
2561:
2560:
2535:
2530:
2529:
2485:
2479:
2478:
2450:
2440:
2433:
2413:
2407:
2406:
2393:
2388:
2387:
2341:
2335:
2334:
2321:
2316:
2315:
2289:
2275:
2274:
2255:
2254:
2235:
2234:
2215:
2214:
2188:
2167:
2157:
2150:
2149:
2114:
2104:
2088:
2087:
2068:
2067:
2039:
2038:
2019:
2018:
1987:
1974:
1973:
1954:
1953:
1934:
1933:
1900:
1880:
1879:
1851:
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1805:
1804:
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1765:
1764:
1745:
1744:
1712:
1711:
1692:
1691:
1665:
1664:
1660:
1647:
1645:"Jeans swindle"
1591:
1581:
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1571:
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1548:
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748:
747:
732:
701:
683:
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651:
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628:
627:
608:
607:
562:
550:Richard Bentley
537:
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372:
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345:
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237:
226:
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217:
207:Please help to
206:
190:
186:
175:
62:Molecular cloud
28:
23:
22:
15:
12:
11:
5:
4808:
4806:
4798:
4797:
4795:Star formation
4792:
4787:
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4777:
4767:
4766:
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4757:
4756:
4744:
4732:
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4707:
4702:
4697:
4692:
4686:
4684:
4680:
4679:
4672:
4670:
4668:
4667:
4662:
4657:
4652:
4647:
4642:
4637:
4632:
4627:
4622:
4616:
4614:
4613:Object classes
4610:
4609:
4606:Star formation
4604:
4602:
4601:
4594:
4587:
4579:
4572:
4571:
4565:
4548:
4542:
4529:
4485:
4482:
4481:
4475:
4432:
4418:
4403:
4383:
4332:
4317:
4297:
4245:
4238:
4220:
4194:
4159:(364): 22–24.
4140:
4116:
4075:
4050:
4049:
4047:
4044:
4043:
4042:
4039:Langmuir waves
4036:
4029:
4026:
4010:
4007:
4002:
3999:
3977:
3974:
3969:
3966:
3941:
3935:
3929:
3926:
3921:
3918:
3914:
3908:
3905:
3899:
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3886:
3864:
3857:
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3845:
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3833:
3830:
3827:
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3813:
3809:
3802:
3799:
3794:
3790:
3783:
3780:
3775:
3771:
3762:
3739:
3734:
3731:
3728:
3724:
3720:
3717:
3714:
3709:
3705:
3701:
3698:
3695:
3667:
3660:
3657:
3654:
3650:
3645:
3641:
3638:
3635:
3632:
3629:
3623:
3619:
3616:
3613:
3607:
3603:
3596:
3593:
3588:
3584:
3580:
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3565:
3562:
3559:
3556:
3531:
3524:
3521:
3516:
3512:
3508:
3505:
3502:
3494:
3489:
3485:
3481:
3467:
3461:
3458:
3438:
3408:
3379:
3373:
3370:
3364:
3358:
3355:
3351:
3346:
3335:
3331:
3320:
3315:
3312:
3306:
3297:
3270:
3249:
3246:
3233:
3229:
3225:
3222:
3219:
3216:
3213:
3204:
3183:
3180:
3171:
3151:{\textstyle r}
3147:
3127:
3121:
3118:
3112:
3106:
3103:
3100:
3093:
3089:
3085:
3076:
3069:
3064:
3055:
3032:
3028:
3023:
3019:
2999:
2979:
2976:
2956:
2933:
2913:{\textstyle G}
2909:
2889:
2873:{\textstyle T}
2869:
2839:
2818:
2812:
2809:
2803:
2797:
2794:
2791:
2788:
2785:
2780:
2771:
2767:
2761:
2756:
2747:
2723:
2720:
2708:
2702:
2699:
2693:
2688:
2683:
2679:
2673:
2666:
2662:
2658:
2655:
2652:
2643:
2620:
2617:
2613:
2596:{\textstyle m}
2592:
2572:{\textstyle G}
2568:
2538:
2517:
2511:
2508:
2502:
2497:
2492:
2488:
2482:
2474:
2471:
2465:
2457:
2453:
2447:
2443:
2439:
2436:
2429:
2420:
2416:
2410:
2405:
2396:
2373:
2367:
2364:
2358:
2353:
2348:
2344:
2338:
2333:
2324:
2301:
2296:
2292:
2288:
2285:
2282:
2266:{\textstyle M}
2262:
2246:{\textstyle n}
2242:
2226:{\textstyle R}
2222:
2200:
2195:
2191:
2187:
2184:
2181:
2174:
2170:
2164:
2160:
2135:
2132:
2129:
2121:
2117:
2111:
2107:
2101:
2098:
2095:
2079:{\textstyle R}
2075:
2055:
2052:
2049:
2046:
2030:{\textstyle R}
2026:
2004:
1999:
1994:
1990:
1984:
1981:
1965:{\textstyle R}
1961:
1945:{\textstyle M}
1941:
1919:
1916:
1913:
1907:
1903:
1899:
1896:
1893:
1890:
1887:
1867:
1864:
1861:
1858:
1838:
1835:
1832:
1816:{\textstyle p}
1812:
1796:{\textstyle n}
1792:
1776:{\textstyle T}
1772:
1756:{\textstyle M}
1752:
1728:
1725:
1722:
1719:
1703:{\textstyle R}
1699:
1672:
1659:
1656:
1646:
1643:
1631:
1624:
1621:
1616:
1611:
1603:
1600:
1588:
1584:
1579:
1574:
1567:
1562:
1551:
1541:
1535:
1530:
1525:
1515:
1512:
1503:
1500:
1495:
1488:
1485:
1480:
1473:
1464:
1458:
1453:
1450:
1445:
1440:
1431:
1427:
1422:
1418:
1415:
1409:
1400:
1373:
1367:
1364:
1359:
1350:
1323:
1296:
1273:
1266:
1263:
1258:
1253:
1245:
1242:
1230:
1226:
1221:
1216:
1211:
1200:
1190:
1184:
1176:
1173:
1164:
1161:
1156:
1152:
1149:
1146:
1136:
1130:
1121:
1094:
1071:
1062:
1058:
1049:
1018:number density
1005:
1001:
997:
994:
991:
971:
955:{\textstyle G}
951:
931:
924:
921:
916:
911:
903:
900:
888:
884:
879:
874:
869:
861:
858:
849:
846:
841:
837:
834:
831:
827:
822:
813:
802:free-fall time
786:
783:
778:
767:
757:
751:
746:
735:
731:
726:
718:
715:
704:
700:
695:
686:
659:
639:{\textstyle M}
635:
619:{\textstyle R}
615:
561:
558:
536:
533:
525:star formation
512:{\textstyle r}
508:
488:{\textstyle G}
484:
468:{\textstyle r}
464:
444:
441:
438:
435:
419:{\textstyle p}
415:
395:
392:
389:
380:
359:
352:
348:
343:
340:
337:
328:
324:
321:
318:
315:
312:
306:
303:
297:
294:
289:
286:
239:
238:
193:
191:
184:
177:
176:
174:
173:
166:
159:
151:
148:
147:
146:
145:
140:
135:
130:
125:
120:
112:
111:
107:
106:
105:
104:
99:
94:
89:
84:
79:
74:
69:
64:
59:
51:
50:
49:Object classes
46:
45:
37:
36:
34:Star formation
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4807:
4796:
4793:
4791:
4788:
4786:
4783:
4781:
4778:
4776:
4773:
4772:
4770:
4755:
4754:
4745:
4743:
4742:
4737:
4733:
4731:
4730:
4721:
4720:
4717:
4711:
4708:
4706:
4703:
4701:
4698:
4696:
4693:
4691:
4688:
4687:
4685:
4681:
4666:
4663:
4661:
4658:
4656:
4653:
4651:
4648:
4646:
4643:
4641:
4638:
4636:
4633:
4631:
4628:
4626:
4623:
4621:
4618:
4617:
4615:
4611:
4607:
4600:
4595:
4593:
4588:
4586:
4581:
4580:
4577:
4568:
4562:
4558:
4555:. Cambridge:
4554:
4549:
4545:
4543:3-540-63785-0
4539:
4535:
4530:
4526:
4522:
4517:
4512:
4508:
4504:
4500:
4496:
4492:
4487:
4486:
4479:
4476:
4471:
4467:
4463:
4459:
4455:
4451:
4448:(3): 035301.
4447:
4443:
4436:
4433:
4428:
4422:
4419:
4414:
4410:
4406:
4400:
4396:
4395:
4387:
4384:
4379:
4375:
4370:
4365:
4360:
4355:
4351:
4347:
4343:
4336:
4333:
4328:
4324:
4320:
4314:
4310:
4309:
4301:
4298:
4293:
4289:
4284:
4279:
4275:
4271:
4267:
4263:
4259:
4252:
4250:
4246:
4241:
4239:9780691196022
4235:
4231:
4224:
4221:
4209:
4205:
4198:
4195:
4190:
4186:
4182:
4178:
4174:
4170:
4166:
4162:
4158:
4154:
4151:
4144:
4141:
4130:
4126:
4120:
4117:
4111:
4106:
4102:
4098:
4094:
4090:
4086:
4079:
4076:
4065:
4061:
4055:
4052:
4045:
4040:
4037:
4035:
4032:
4031:
4027:
4025:
4008:
4005:
4000:
3997:
3975:
3972:
3967:
3964:
3957:
3939:
3933:
3927:
3924:
3919:
3916:
3912:
3906:
3903:
3897:
3893:
3884:
3875:
3862:
3855:
3852:
3847:
3843:
3834:
3831:
3828:
3820:
3817:
3811:
3807:
3800:
3797:
3792:
3788:
3781:
3778:
3773:
3769:
3760:
3750:
3737:
3732:
3729:
3726:
3722:
3718:
3715:
3707:
3703:
3699:
3696:
3693:
3685:
3682:
3678:
3665:
3658:
3655:
3652:
3648:
3643:
3639:
3636:
3630:
3627:
3621:
3617:
3614:
3611:
3605:
3601:
3594:
3591:
3586:
3582:
3578:
3575:
3569:
3566:
3563:
3560:
3557:
3554:
3546:
3529:
3522:
3519:
3514:
3510:
3506:
3503:
3492:
3487:
3483:
3479:
3471:
3466:
3460:Fragmentation
3459:
3457:
3455:
3450:
3436:
3428:
3406:
3397:
3393:
3377:
3371:
3368:
3362:
3356:
3353:
3349:
3344:
3333:
3329:
3318:
3313:
3310:
3304:
3295:
3268:
3259:
3255:
3254:Jeans' length
3245:
3231:
3227:
3223:
3220:
3217:
3214:
3211:
3202:
3181:
3178:
3169:
3159:
3145:
3125:
3119:
3116:
3110:
3104:
3101:
3098:
3091:
3087:
3083:
3074:
3067:
3062:
3053:
3030:
3026:
3021:
3017:
2997:
2977:
2974:
2954:
2945:
2931:
2923:
2907:
2887:
2867:
2859:
2837:
2816:
2810:
2807:
2801:
2795:
2792:
2789:
2786:
2783:
2778:
2769:
2765:
2759:
2754:
2745:
2735:
2733:
2728:
2727:Jeans' length
2722:Jeans' length
2719:
2706:
2700:
2697:
2691:
2686:
2681:
2677:
2671:
2664:
2660:
2656:
2653:
2650:
2641:
2618:
2615:
2611:
2590:
2582:
2566:
2558:
2536:
2515:
2509:
2506:
2500:
2495:
2490:
2486:
2480:
2472:
2469:
2463:
2455:
2451:
2445:
2441:
2437:
2434:
2427:
2418:
2414:
2408:
2403:
2394:
2384:
2371:
2365:
2362:
2356:
2351:
2346:
2342:
2336:
2331:
2322:
2312:
2299:
2294:
2290:
2286:
2283:
2280:
2260:
2253:and the mass
2240:
2220:
2211:
2198:
2193:
2189:
2185:
2182:
2179:
2172:
2168:
2162:
2158:
2146:
2133:
2130:
2127:
2119:
2115:
2109:
2105:
2099:
2096:
2093:
2073:
2053:
2050:
2047:
2044:
2024:
2015:
2002:
1997:
1992:
1988:
1982:
1979:
1959:
1939:
1930:
1917:
1914:
1911:
1905:
1901:
1897:
1894:
1891:
1888:
1885:
1865:
1862:
1859:
1856:
1836:
1833:
1830:
1810:
1790:
1770:
1750:
1742:
1726:
1723:
1720:
1717:
1697:
1688:
1686:
1670:
1657:
1655:
1651:
1644:
1642:
1629:
1622:
1619:
1614:
1609:
1601:
1598:
1586:
1582:
1577:
1572:
1565:
1560:
1549:
1539:
1533:
1528:
1523:
1513:
1510:
1501:
1498:
1493:
1486:
1483:
1478:
1471:
1462:
1456:
1451:
1448:
1443:
1438:
1429:
1425:
1420:
1416:
1413:
1407:
1398:
1371:
1365:
1362:
1357:
1348:
1321:
1294:
1284:
1271:
1264:
1261:
1256:
1251:
1243:
1240:
1228:
1224:
1219:
1214:
1209:
1198:
1188:
1182:
1174:
1171:
1162:
1159:
1150:
1147:
1134:
1128:
1119:
1092:
1082:
1069:
1060:
1056:
1047:
1038:
1024:
1019:
1003:
999:
995:
992:
989:
969:
949:
929:
922:
919:
914:
909:
901:
898:
886:
882:
877:
872:
867:
859:
856:
847:
844:
835:
832:
825:
820:
811:
803:
784:
781:
776:
765:
755:
749:
744:
733:
729:
724:
716:
713:
702:
698:
693:
684:
657:
649:
633:
613:
604:
602:
598:
594:
590:
586:
582:
578:
574:
571:
567:
559:
557:
553:
551:
547:
542:
541:Edmund Halley
534:
532:
530:
526:
521:
506:
498:
482:
462:
439:
433:
413:
390:
378:
357:
350:
346:
338:
326:
319:
313:
310:
304:
301:
295:
292:
287:
284:
273:
268:
266:
262:
258:
254:
250:
246:
235:
232:
224:
214:
210:
204:
203:
197:
192:
183:
182:
172:
167:
165:
160:
158:
153:
152:
150:
149:
144:
141:
139:
136:
134:
131:
129:
126:
124:
121:
119:
116:
115:
114:
113:
108:
103:
100:
98:
95:
93:
90:
88:
85:
83:
80:
78:
75:
73:
70:
68:
65:
63:
60:
58:
55:
54:
53:
52:
47:
43:
39:
38:
35:
31:
19:
4752:
4741:Stars portal
4740:
4728:
4694:
4650:T Tauri star
4552:
4533:
4498:
4494:
4478:
4445:
4441:
4435:
4421:
4393:
4386:
4352:(1): L6–L9.
4349:
4345:
4335:
4307:
4300:
4265:
4261:
4229:
4223:
4211:. Retrieved
4207:
4197:
4156:
4152:
4143:
4132:. Retrieved
4128:
4119:
4092:
4088:
4078:
4067:. Retrieved
4063:
4054:
3953:
3752:Jeans mass,
3751:
3679:
3542:
3463:
3451:
3391:
3257:
3253:
3251:
3160:
2946:
2736:
2726:
2725:
2385:
2313:
2212:
2147:
2016:
1931:
1689:
1661:
1652:
1648:
1285:
1083:
1022:
605:
565:
563:
554:
546:Isaac Newton
538:
522:
269:
249:astrophysics
244:
242:
227:
218:
199:
127:
92:T Tauri star
4635:Dark nebula
4630:Bok globule
4213:11 November
3427:sound speed
1952:and radius
1016:is the gas
648:sound speed
601:temperature
577:James Jeans
265:James Jeans
213:introducing
72:Dark nebula
67:Bok globule
4769:Categories
4134:2024-01-05
4069:2024-01-05
4046:References
3681:Polytropic
3454:sound wave
2037:to radius
1554: km/s
1203: km/s
770: km/s
566:Jeans mass
560:Jeans mass
527:occurs in
221:April 2022
196:references
18:Jeans mass
4645:Protostar
4470:103819409
4413:475440765
4378:1745-3933
4359:1210.3363
4327:195749071
4181:0260-7085
3998:γ
3965:γ
3920:−
3917:γ
3898:ρ
3894:∝
3848:−
3844:ρ
3832:−
3829:γ
3812:ρ
3808:∝
3793:−
3789:ρ
3770:∝
3730:−
3727:γ
3723:ρ
3719:∝
3713:→
3708:γ
3704:ρ
3656:−
3653:γ
3649:γ
3640:∝
3634:⇒
3628:∝
3622:γ
3615:−
3612:γ
3595:γ
3587:−
3579:⋅
3573:⇒
3545:ideal gas
3523:γ
3515:−
3507:∝
3501:→
3488:γ
3437:ρ
3357:ρ
3350:π
3314:π
3296:λ
3224:μ
3170:λ
3105:μ
3063:≈
3054:λ
2998:ρ
2978:π
2932:ρ
2888:μ
2796:ρ
2793:μ
2787:π
2746:λ
2657:×
2616:−
2438:π
2048:−
1721:−
1671:π
1615:−
1599:−
1529:⋅
1524:⊙
1511:≈
1494:ρ
1457:⋅
1449:π
1426:ρ
1417:π
1372:λ
1257:−
1241:−
1210:⋅
1183:⋅
1172:≈
1151:ρ
1120:λ
1093:λ
1031:10 g
1004:μ
996:ρ
970:ρ
915:−
899:−
868:⋅
864: Myr
857:≈
836:ρ
782:−
745:⋅
725:⋅
721: Myr
714:≈
573:physicist
539:In 1720,
434:ρ
314:ρ
305:−
118:Accretion
82:Protostar
4729:Category
4028:See also
3497:constant
1594: cm
1236: cm
1179: pc
894: cm
738: pc
585:pressure
261:pressure
4753:Commons
4503:Bibcode
4450:Bibcode
4270:Bibcode
4161:Bibcode
4097:Bibcode
4095:: 104.
3954:If the
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597:density
589:gravity
570:British
535:History
495:is the
209:improve
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3429:, and
3390:where
3138:where
2829:where
2583:, and
2528:where
942:where
499:, and
370:where
198:, but
4525:90845
4521:JSTOR
4466:S2CID
4354:arXiv
4292:90845
4288:JSTOR
4185:JSTOR
3194:when
2086:, so
1065:sound
689:sound
4561:ISBN
4538:ISBN
4409:OCLC
4399:ISBN
4374:ISSN
4323:OCLC
4313:ISBN
4234:ISBN
4215:2023
4177:ISSN
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593:mass
575:Sir
564:The
257:star
243:The
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4458:doi
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