Stars are formed in dense molecular cloud cores, whereas planets are formed, contemporaneously, in young circumstellar disks.
Compression of gas from a cloud size ~1018 cm down to a stellar size ~1011 cm, i.e., density increases by a factor of ~1021.
Chap 13 StarForm Draine Chap 41, 42 1 A gas-to-dust ratio ~100 (by mass) seems universal.
Predehl & Schmitt (1995)
2 Chap 13 StarForm Filamentary Molecular Clouds
Molecular clumps/ Giant Molecular Clouds clouds/condensations D=20~100 pc 3 −3 5 6 n ~ 10 cm , D ~ 5 pc, ℳ = 10 ~10 ℳ⨀ 3 −3 M ~ 10 ℳ⨀ 휌 ≈ 10~300 cm Dense molecular cores 푇 ≈ 10~30 K n 104 cm−3 , D ~ 0.1 pc, ∆퓋 ≈ 5~15 km−1 M ~ 1-2 ℳ⨀ http://www.bu.edu/galacticring/outgoing/PressRelease/ 3 Stars are formed in groups seen as star clusters if gravitationally bound. Groups of young stars are found at the densest parts of the molecular clouds.
Molecular clouds observed by different tracers … Taurus molecular cloud 4 Nearby Examples Massive Star-Forming Regions - Per OB2 (350 pc) - Orion OB Association (350--400 pc) … rich Low-Mass Star-Forming Regions - Taurus Molecular Cloud (TMC-1) (140 pc) - Rho Ophiuchi cloud (130 pc) - Lupus (140 pc) 4/5 in the southern sky … why? - Chamaeleon (160 pc) - Corona Australis (130 pc)
Chap 13 StarForm 5 Stability: The Virial Theorem In a spherically symmetric cloud of temperature 푇, for each particle, the equation of motion is 푭푖 = 푚푖 풓ሷ 푖 = 풑ሶ 푖, the momentum change with time. Sum up all particles and take time derivative 푑 풑 ∙ 풓 = 풑ሶ ∙ 풓 + 풑 ∙ 풓ሶ 푑푡 푖 푖 푖 푖 푖 i 푖 푖 푖
= 푭푖 ∙ 풓푖 + 푚푖풓ሶ 푖 ∙ 풓ሶ i 푑 푖 푖 푚푖푟푖ሶ ∙ 푟푖 푑푡 = 퐸푝 + 2퐸푘 푖 σi 푭푖 ∙ 풓푖 = virial of Clausius Chap 13 StarForm 6 2 For moment of inertia, 퐼 = σ푖 푚푖 푟푖 , 푑2퐼 푑 푚 = 푚 2 푟 푟ሶ 푣 푑푡2 푑푡 푖 푖 푖 푖 Hence 푀 1 푑2퐼 = 2퐸 + 퐸 2 푑푡2 푘 푝
To be stable, LHS = 0
2 1ൗ 푚푣2 = 퐺푚푀/푟 2퐸푘 + 퐸푝 = 0 2
Chap 13 StarForm 7 LHS = 0 stable LHS < 0 collapsing Note: LHS > 0 expanding Virial theorem governs the motion status, 푬푲 a variety of kinetic energies Kinetic energy of molecules whereas the total energy Bulk motion of clouds 푬퐭퐨퐭퐚퐥 = 푬푲 + 푬푷 Rotation = 푬푲 + 훀 퐦퐨퐬퐭퐥퐲 … governs whether the system is dynamically bound. 푬푷 a variety of potential energies Gravitation A coins flying either upward Magnetic field or downward is bound. Electrical field
… 8 휔 Cloud of mass 푀, radius 푅, rotating at 휔
푀, 푅, 휌 푃푒푥푡
Generalized virial theorem
If 휔 = 0, and 푃푒푥푡 = 0
μ 2.37 for solar This is the Jeans length. abundance with H9 2 Jeans length = critical spatial wavelength (length scale) If the perturbation length scale is longer Medium is decoupled from self-gravity stable
This is the Jeans mass … the critical mass for onset of gravitational collapse
If cloud mass 푀 > 푀Jeans cloud collapse Note the above does not consider external pressure, or other internal supporting mechanisms. 10 A non-magnetic, isothermal cloud in equilibrium with external pressure a Bonnor-Ebert sphere (Bonnor 1956, Ebert 1955)
2퐸퐾 + 퐸푃 − 3푃ext푉 = 0 The potential term may include, other than the gravity, also rotation, magnetic field, etc. At first, the cloud is optically thin. Contraction density ↑ collisions more frequent molecules excited and radiated radiation escapes cooling less resistance to the contraction cloud collapse (free fall)
Chap 13 StarForm RJ cs τff = [isothermal sound speed] * [free fall time] 11 To maintain 2EK + EP = 0, the total energy Et = EK + EP must change. The gravitational energy 퐺푀2 푑푟 Ω~ − ⟶ 푑Ω~ 푟 푟2 For contraction, 푑푟 < 0, so 푑훺 < 0, then 1 푑퐸푡 = 푑퐸푘 + 푑훺 = Τ2 푑Ω = 퐿푑푡 This means to maintain quasistatic contraction, half of the gravitation energy from the contraction is radiated away. Eventually the cloud becomes dense enough (i.e., optically thick) and contraction leads to temperature increase. The cloud’s temperature increases while energy is taken away
negative heat capacity 12 Numerically, 3 ൗ2 −1ൗ 푇 푛H 2 푀 = 1.0 2 [ℳ ] 퐽 10 K 104 cm−3 ⊙ • H I clouds
푇 ≈ 100 K, 푛퐻 ≈ 100, RJ 25 pc; MJ 300 ℳ☉ > Mobs So H I clouds are not collapsing. • Dark molecular clouds 5 푇 ≈ 15 K, 푛퐻 ≈ 10 , MJ 20 ℳ ☉ < Mobs 100−1000 ℳ☉
So H2 clouds (dense cores and Bok globules) should be collapsing. But observations show that most are not. There is additional support other than the thermal
pressure, e.g., rotation, magnetic field, turbulence, etc. 13 Chap 13 StarForm If 퓜cloud > 퓜crit supercritical Cloud collapses dynamically Massive star formation
If 퓜cloud < 퓜crit subcritical Cloud collapses quasistatically Low-mass star formation
4 Clouds tend to condense with 퓜~10 푀⊙, but the observed stellar mass ranges 0.05 ≤ 퓜/푀⊙ ≤ 100 Why is there a lower mass limit and an upper mass limit for stars?
Cloud collapse (local) density increase (local) 푀퐽 decrease easier to satisfy M > MJ, i.e., cloud becomes more unstable fragmentation Formation of a cluster of stars ~~
Chap 13 StarForm 14 푇 휌 1 Recall 푀 ≈ 1.2 × 105 ( )3Τ2 ( 0 )−1Τ2 [푀 ] 퐽 100 퐾 10−24 g cm−3 휇3Τ2 ⨀ 푇3Τ2 ∝ ൗ휌1Τ2
A small/decreasing 푀퐽 favors cloud collapse.
If during collapse, local 푀퐽 ↓ subregions become unstable and continue to collapse to ever smaller (fragmentation).
Since during collapse ρ always ↑, the behavior of 푀퐽 depends on T.
If gravitational energy is radiated away, i.e., 휏cooling ≪ 휏ff and −1Τ2 collapse is isothermal, T = const, so 푀퐽 ∝ 휌 collapse continues 15 Equation of motion for a spherical surface at 푟 is 푑2푟 퐺푀 = − 푑푡2 푟2 Dimensional analysis yields 푅 퐺푀 1 ~ ⟹ 푡푓푓 ~ 푡2 푅2 퐺휌
1 3휋 2 3.4 × 107 푡 = More accurately, ff = yr = 35ൗ 휌cgs [min] 32 퐺휌0 푛0
It takes the Sun ~30 minutes to collapse (the free-fall time scale). 16 Ex: How long does a typical dense molecular core take to collapse?
Chap 13 StarForm 17 Carroll & Ostlie Chap 13 StarForm 18 Carroll & Ostlie Chap 13 StarForm 19 1 Note that 푡ff ∝ has no dependence on 푟0. 퐺휌0
If 휌0 is uniform, all m collapse to the center at the same time homologous collapse
In reality, 휌0 is somewhat centrally condensed, as observed, −1 −2 e.g., 휌0 ∝ 푟 to 푟 , inner region (small 푟), 푡ff ↓↓ inside-out collapse A protostellar core is formed, followed by material “raining down” accretion Gravitational energy kinetic energy heat 퐿acc~ 퐺푀∗ 푀ሶ /푅∗ Cores form within molecular clouds.
A core collapse inside- out and form a protostar with a toroid.
A stellar wind with a A star is form with a bipolar flow forms. circumstellar disk.
21 Chap 13 StarForm Central condensed protostar, 푟 ~ a few 푅⨀
Circumstellar disk, 푟 ~ 100 au
Surrounding envelope, 푟 ~ 5000 au
Matter accretes from the envelope via the disk onto the protostar
Ward-Thomson (2002)
22 Spectral energy distribution 퐹휆 vs 휆 or log 휆퐹휆 vs log 휆
Chap 13 StarForm 23 Spectral index useful to classify a young stellar object (YSO) 푑 log (휆F휆) Where λ=wavelength, between 훼 = 푑 log (휆) 2.2 and 20 μm; Fλ=flux density Class 0 sources --- undetectable at λ < 20 μm Class I sources --- α > 0.3 Flat spectrum sources --- 0.3 > α > − 0.3 Class II sources --- 0.3 > α > − 1.6 Class III sources --- α < − 1.6
Evolutionary sequence in decreasing amounts of circumstellar material (disk clearing)
Chap 13 StarForm 24 Class 0 Submilimeter cores
Class I Protostars
Class II Classical T Tauri stars
Class III Weak-lined T Tauri stars
Chap 13 StarForm 25 Basic Questions in Star Formation • The rate and efficiency of SF as a function of time and position in the Galaxy, and in external galaxies? How are these quantities measured? • Cloud fragmentation to form clusters? • Triggered SF? • Different processes for high-mass and low-mass? • Mass spectrum? Typical 0.1 − 1 M☉, why? • Formation of multiple systems? • What is a protostar observationally? • Evolution of disks? • Origin of bipolar outflows? • Environments for planet formation? • Role of rotation and magnetic field? Bodenheimer 26 Chap 13 StarForm