Lecture 24 Stability of Molecular Clouds

Transcription

1 Lectue 4 Stability of Molecula Clouds 1. Stability of Cloud Coes. Collapse and Fagmentation of Clouds 3. Applying the iial Theoem Refeences Oigins of Stas & Planetay Systems eds. Lada & Kylafis Myes, Physical Conditions in Molecula Clouds Lada, The Fomation of Low-Mass Stas Shu, Ch. 1 & Stahle, Appendix D fo iial Theoem

2 1. Stability of Molecula Cloud Coes Stat with the molecula cloud coe popeties fom Lec. 3 and focus on the items elating to viial equilibium. 1. associated with sta fomation. elongated (aspect atio ~ :1) 3. dynamics may be dominated by themal o tubulent motion 4. tempeatue: T ~ 10 0 K 5. size: R ~ 0.1 pc 6. ionization: x e ~ size-line width elation R ~ σ p, p = 0.5 +/ appoximate viial equilibium 9. mass spectum: simila to GMCs.

3 Coe Coelations (Myes 1999) v /R non-themal vs. themal line widths consistency with viial equilibium Fo pue themal suppot, GM/R ~ kt/m, m =.3 m H ), and R 0.1 (M / M ) (10 K / T) pc. N

4 Coe Stability iial equilibium seems an appopiate state fom which coes poceed to make stas. Howeve, as a significant faction of coes ae obseved to have embedded potostas (obseved by IRAS), they can t be completely quiescent. We ae theefoe concened with the issue of coe stability, e.g., why they ae stable (if they ae) and how they become de-stabilized and collapse unde gavity to fom stas. Although Mye s data pesentation suggests that andom motion (themal and/o non-themal) may stabilize coes against gavitational collapse, we should fist conside othe possibilities, in paticula otation and magnetic fields.

5 Effects of Rotation Molecula cloud coes have modest velocity gadients, km s -1 pc -1 and angula speeds Ω ~ ad s -1. β = E ot / E gav ~ 0.0 gad v β Coes do not appea to be suppoted by otation. J/M NB Not all of the obseved gadients coespond to the R oveall otation of the coe Goodman et al. ApJ

6 Magnetic Pessue Magnetic fields ae difficult to measue in cloud coes OH absoption measuements ae had because of the small chance of a backgound adio souce Example of dak cloud B1 B cos θ -19 ± 4 µg OH pobes n H 10 3 cm -3 B /8π ~ 3 x 10 5 K cm -3 Compaable to the themal pessue of a 10 K coe with n ~ 10 4 cm MHz 1665 MHz Stokes Stokes I Cutche 1993 ApJ Recall Lectue 13 and adio measuements of the Zeeman effect.

7 . Collapse and Fagmentation How do coes condense out of the lowe density egions of GMCs? The conventional wisdom is local gavitational instabilities in a globally stable GMC via the classical Jeans instability. Conside a unifom isothemal gas in hydostatic equilibium with gavity balanced by the pessue gadient. Now suppose a small spheical egion of size is petubed ρ Xρ whee X > 1 The ove-density Xρ geneates an outwad pessue foce pe unit mass:

8 foce pe unit mass Gavity wins out if Simple Stability Citeion The inceased density also leads > c Gρ 0 o F ~ p ~ GM c Gρ 0 / ρ ~ Xc / The ight hand side is essentially the Jeans length: λ = c J F G p > π Gρ 0 / to an inwad gavtitational ~ GXρ. 0 ρ 0 Xρ 0

9 we get a The Jeans Length & Mass Fom the Jeans length whose numeical value is J coesponding Jeans mass M J M λ = J = λ = 7.5M 3 J sun πc Gρ 0 πc ρ0 = G 3/ ρ 1/ 0 ( ) ( 4-3 T /10K 3/ n(h ) /10 cm ) c 3 ρ 0 1/ Length scales > λ J o masses > M J ae unstable against collapse.

10 Relevance of the Jeans Analysis Many assumptions have been made implicitly in the above analysis that violate the basic fact that molecula clouds ae not static but ae tubulent on lage scales. Tubulence povides an effective pessue suppot, and its supesonic motions must be dissipated by shocks befoe the collapse can poceed: Is it enough to eplace the sound velocity c in the Jeans fomulae by c = m kt +σ tub As the gas condenses the Jeans mass deceases, which suggests fagmentation (Hoyle 1953 ApJ ), a collapse cascade into smalle and smalle masses.

11 Does Fagmentation Occu The dispesion elation fo a petubation δρ ~ ei(ωt - kx) in the Jeans poblem is ω = c k -4πGρ 0 o ω = c (k - k J ), k J = 4π Gρ 0 / c k - k J < 0 makes ω imaginay: Exponential gowth occus fo k < k J Gowth ate -iω inceases monotonically with deceasing k, which implies that Longest wavelength petubations (lagest mass) gow the fastest The fast collapse of the lagest scales suggests that fagmentation is unlikely (Lason 1985 MNRAS )

12 Swindled by Jeans? Conside an altenate model, a thin sheet of suface density. The dispesion elation is ω = c k -π G k o ω = c ( k - k c k ) Exponential gowth occus fo k < k c = π G / c with gowth ate -i ω =( π G k - c k ) 1/ which is maximum at k f = k c /= π G / c and gives a pefeed mass, M f ~ (π / k f ) = 4 c 4 / G This is a diffeent esult than the pevious Jeans analysis fo an isotopic 3-d medium. The maximum gowth ate now occus on an intemediate scale that is smalle than pedicted by Jeans, possibly favoing fagmentation.

13 Pefeed Length & Mass Apply the thin sheet model to the Tauus dak cloud. T = 10 K, c = 0.19 km/s λ c ~ 0.05 pc A() 5 mag. o 0.03 g cm - λ f ~ 0.1 pc, M f ~ M t f = λ f /(π c) ~ x 10 5 y Moe ealistic analyses tend to show that, when themal pessue povides the dominant suppot against gavity, thee is a minimum length & mass scale which can gow If the cloud is non-unifom thee is a pefeed scale Typically ~ few times the chaacteistic length of the backgound, e.g., the scale height, H =c /πg, fo a gaseous equilibium sheet (Lason 1985 MNRAS )

14 3.The iial Theoem and Stability A. Geneal Consideations The deivation of viial theoem stats with the equation of motion fo the macoscopic velocity of the fluid afte aveaging ove the andom themal velocities. The mathematics is given by Shu Ch. 4 and Stahle Appendix D. Dv t ρ = P ρ φ + T Dt Tij = Bi B j / 4π B δij / 8π t 1 T B ( 1 = + B )B 8π 4π The pespective of the iial Theoem is that these equations contain too much infomation. Hence, take moments, specifically the fist moment:

15 iial Theoem: LHS Analysis Take the dot poduct with and integate the equation of motion ove a finite volume, and analyze the ight and left hand sides sepaately. The LHS side yields two tems since: ( ) + = = Dt D d v v d Dt D d and ) ( ) ( 1 ρ ρ ρ K E Dt I D d v d Dt D 1 1 and finally = ρ ρ The 1 st tem vanishes fo a static cloud.

16 iial Theoem: RHS Analysis The RHS yields volume & suface tems 3P d + S P + B 8π B d 8π ds + 1 4π φ ( ) S B dm B d ( )( S ) The 3 d tem becomes the gavitational enegy. The 1 st and 4 th tems measue the diffeence between the extenal and mean intenal pessues. The othe tems ae tansfomed into magnetic pessue and tension.

17 The iial Theoem ( ) and then inset appoximate expessions in the RHS pessue we can solve fo the extenal Fo a static cloud, 8 1 ) ( whee 3 K ext ext K M W E P P B ds B B ds B d M M W P P E Dt I D + + = + = = π π π P ex t becomes a function of the global vaiables of the volume and the gas inside. If thee is no otation, E K = 0.

18 Simplest Application of the iial Theoem Conside a spheical, isothemal, un-magnetized & nontubulent cloud in equilibium of mass M and adius R 3 3 4πR Pext = 3Mc 5 GM R whee we have used P = ρc. If we now also ignoe gavity, we check that the extenal and intenal pessues ae the same. In the pesence of gavity, the second tem seves to lowe the extenal pessue needed to confine the gas Inside volume. Next, divide by 4πR 3 and plot the RHS vs. R

19 Simplest Application of the iial Theoem P ext P = 3c M 4π R c 1 R 3 3GM 0π Thee is a minimum adius below which this model cloud cannot suppot itself against gavity. States along the left segment, whee P inceases with R, ae unstable. 1 R 4 P ~ 1/R 3 R

20 Stable and Unstable iial Equlibia Fo a given extenal pessue, thee ae two equilibia, but only one is stable. P A B P ex t R max R Squeeze the cloud at B (decease R) Requies moe pessue to confine it: e-expands (stable) Squeeze the cloud at A (decease R) Requies less pessue to confine it: contacts (unstable)

21 The Citical Pessue Diffeentiating the extenal pessue fo fixed M and c P ext = 3c M 4π gives the citical adius, R c 1 R 3 GM c 3GM 0π 1 R The coesponding density and mass ae 6 3 c c ρc M 3 c M This last esult is essentially the same as gotten fom the Jeans analysis. ρ c 4

22 Fagmentation vs. Stability (o Jeans vs. iial) Both methods yield a simila chaacteistic Jeans mass The diffeence is in the initial conditions Cloud scenaio assumes a self-gavitating cloud coe in hydostatic equilibium The cloud may o may not be close to the citical condition fo collapse In the gavitational fagmentation pictue thee is no cloud It cannot be distinguished fom the backgound until it has stated to collapse Both pespectives may be applicable in diffeent pats of GMCs Thee ae pessue confined clumps in GMCs Not stongly self-gavitating Coes make stas Must be self-gavitating If fagmentation poduces coes these must be collapsing and it is too late to apply viial equilibium

23 Fagmentation vs. Stability A vey elevant question is how quickly coes fom (elative to the dynamical time scale) NH 3 coes do seem to be close to viial equilibium but they could be contacting (o expanding) slowly Statistics of coes with and without stas should eveal thee ages If coes ae stable, thee should be many moe coes without stas than with young stas Compaison of NH 3 coes and IRAS souces suggests ~ 1/4 of coes have young stella objects (Wood et al. ApJS ), but Jijina et al. give a lage faction The coesponding life time is < 1 My (Onishi et al ApJ )