SUMMARY 1.Statistical equilibrium and radiative transfer in molecular (H 2 ) cloud – Derivation of physical parameters of molecular clouds 2.High-mass star formation: theoretical problems and observational results

Statistical equilibrium and radiative transfer Statistical equilibrium equations: coupling with radiation field The excitation temperature: emission, absorption, and masers The 2-level system: thermalization The 3-level system: population inversion maser

Problem: Calculate molecular line brightness I ν as a function of cloud physical parameters calculate populations n i of energy levels of given molecule X inside cloud of H 2 with kinetic temperature T K and density n H 2 plus external radiation field. Note: n X << n H 2 always; e.g. CO most abundant species but n CO / n H 2 = !!!

i j A ij B ij B ji C ij C ji … …

Radiative transfer equation: the line case

A 21 B 21 B 12 C 21 C

3-level system A 21 B 21 B 12 C 21 C A 32 B 32 B 23 C 32 C 23 A 31 B 31 B 13 C 31 C 13

J=0 J=1 J=2 A 21 A 10 A A 10 A 31 = 0

n H 2 ~ n cr T ex (1-0) > T K

n H 2 ~ n cr T ex (1-0) < 0 i.e. pop. invers. MASER!!!

Radio observations Useful definition: brightness temperature, T B In the radio regime Rayleigh- Jeans (hν << kT) holds: In practice one measures mean T B over antenna beam pattern, T MB : Flux measured inside solid angle :

Angular resolution: HPBW = 1.2 λ/D Beam almost gaussian: B = π/(4ln2) HPBW 2 One measures convolution of source with beam Example gaussian source gaussian image with: T MB = T B S /( B + S ) S ν = (2k/λ 2 ) T B S = (2k/λ 2 ) T MB ( B + S ) Θ S = (Θ S 2 + Θ B 2 ) 1/2

extended source: S >> B T MB T B pointlike source: S << B T MB T B S / B << T B

Estimate of physical parameters of molecular clouds Observables: T MB (or F ν ), ν, S Unknowns: V, T K, N X, M H 2, n H 2 –V velocity field –T K kinetic temperature –N X column density of molecule X –M H 2 gas mass –n H 2 gas volume density

Velocity field From line profile: Doppler effect: V = c(ν 0 - ν)/ν 0 along line of sight in most cases line FWHM thermal < FWHM observed thermal broadening often negligible line profile due to turbulence & velocity field Any molecule can be used!

channel maps integral under line Star Forming Region

rotating disk line of sight to the observer

GG Tau disk 13 CO(2-1) channel maps 1.4 mm continuum Guilloteau et al. (1999)

GG Tau disk 13 CO(2-1) & 1.3mm cont.near IR cont.

infalling envelope line of sight to the observer

red-shifted absorption bulk emission blue-shifted emission VLA channel maps 100-m spectra Hofner et al. (1999)

Problems: only V along line of sight position of molecule with V is unknown along line of sight line broadening also due to micro-turbulence numerical modelling needed for interpretation

Kinetic temperature T K and column density N X LTE n H 2 >> n cr T K = T ex τ >> 1: T K ( B / S ) T MB but no N X ! e.g. 12 CO τ << 1: N u ( B / S ) T MB e.g. 13 CO, C 18 O, C 17 O T K = (hν/k)/ln(N l g u /N u g l ) N X = (N u /g u ) P.F.(T K ) exp(E u /kT K )

τ 1: τ = -ln[1-T MB (sat) /T MB (main) ] e.g. NH 3 T K = (hν/k)/ln(g 2 τ 1 /g 1 τ 2 ) N u τT K N X = (N u /g u ) P.F.(T K ) exp(E u /kT K )

If N i is known for >2 lines T K and N X from rotation diagrams (Boltzmann plots): e.g. CH 3 C 2 H P.F.= Σ g i exp(-E i /kT K ) partition function

CH 3 C 2 H Fontani et al. (2002)

CH 3 C 2 H Fontani et al. (2002)

Non-LTE numerical codes (LVG) to model T MB by varying T K, N X, n H 2 e.g. CH 3 CN Olmi et al. (1993)

Problems: calibration error at least 10-20% on T MB T MB is mean value over B and line of sight τ >> 1 only outer regions seen different τ different parts of cloud seen chemical inhomogeneities different molecules from different regions for LVG collisional rates with H 2 needed

Possible solutions: high angular resolution small B high spectral resolution parameters of gas moving at different Vs along line profile line interferometry needed!

Mass M H 2 and density n H 2 Column density: M H 2 (d 2 /X) N X d –uncertainty on X by factor –error scales like distance 2 Virial theorem: M H 2 d Θ S (ΔV) 2 –cloud equilibrium doubtful –cloud geometry unknown –error scales like distance

(Sub)mm continuum: M H 2 d 2 F ν /T K –T K changes across cloud –error scales like distance 2 –dust emissivity uncertain depending on environment Non-LTE: n H 2 from numerical (LVG) fit to T MB of lines of molecule far from LTE, e.g. C 34 S –results model dependent –dependent on other parameters (T K, X, IR field, etc.) –calibration uncertainty > 10-20% on T MB –works only for n H 2 n cr

observed T B observed T B ratio T K = K n H cm -3 satisfy observed values τ > 1 thermalization

best fits to T B of four C 34 S lines (Olmi & Cesaroni 1999)

H 2 densities from best fits

Bibliography Walmsley 1988, in Galactic and Extragalactic Star Formation, proc. of NATO Advanced Study Institute, Vol. 232, p.181 Wilson & Walmsley 1989, A&AR 1, 141 Genzel 1991, in The Physics of Star Formation and Early Stellar Evolution, p. 155 Churchwell et al. 1992, A&A 253, 541 Stahler & Palla 2004, The Formation of Stars

1)Importance of high-mass stars: their impact 2)High- and low-mass stars: differences 3)High-mass stars: observational problems 4)The formation of high-mass stars: where 5)The formation of high-mass stars: how The formation of high-mass stars: observations and problems (high-mass star M * >8M L * >10 3 L B3-O)

Importance of high-mass stars Bipolar outflows, stellar winds, HII regions destroy molecular clouds but may also trigger star formation Supernovae enrich ISM with metals affect star formation Sources of: energy, momentum, ionization, cosmic rays, neutron stars, black holes, GRBs OB stars luminous and short lived excellent tracers of spiral arms

Stellar initial mass function (Salpeter IMF): dN/dM M N(10M O ) = N(1M O ) Stellar lifetime: t Mc 2 /L M -3 t(10M O ) = t(1M O ) M O stars per 10 M O star! Total mass dominated by low-mass stars. However… Stellar luminosity: L M 4 L(10M O ) = 10 4 L(1M O ) Luminosity of stars with mass between M 1 and M 2 : L(10-100M O ) = 0.3 L(1-10M O ) Luminosity of OB stars is comparable to luminosity of solar-type stars!

The formation of high-mass and low-mass stars: differences and theoretical problems

stars < 8M O isothermal unstable clump accretion onto protostar disk & outflow formation disk without accretion protoplanetary disk sub-mm far-IR near-IR visible+NIR visible

stars > 8M O isothermal unstable clump accretion onto protostar disk & outflow formation disk without accretion protoplanetary disk sub-mm far-IR near-IR visible+NIR visible

Two mechanisms at work: Accretion onto protostar: Static envelope: n R -2 Free-falling core: n R -3/2 t acc = M * /(dM acc /dt) Contraction of protostar: t KH =GM 2 /R * L * –Stars t acc –Stars > 8 M sun : t KH < t acc High-mass stars form still in accretion phase Low-mass VS High-mass n R -3/2 n R -2

Two mechanisms at work: Accretion onto protostar: Static envelope: n R -2 Free-falling core: n R -3/2 t acc = M * /(dM acc /dt) Contraction of protostar: t KH =GM 2 /R * L * –Stars t acc –Stars > 8 M sun : t KH < t acc High-mass stars form still in accretion phase n R -2 n R -3/2 n R -2 Low-mass VS High-mass

Palla & Stahler (1990) dM/dt=10 -5 M O /yr t KH =t acc Main Sequence Sun

Problem : Stellar radiation pressure (+ wind + ionizing flux) halt accretion above M * =8 M sun how to form M * >8 M ?

Solutions : i.Competitive accretion: boosts dM/dt by deepening potential well through cluster: dM/dt(M * >8M ) >> dM/dt(M * <8M ) ii.Monolithic collapse: accretion through disk+jet; focuses dM/dt enhancing ram pressure (disk) and allows photons to escape lowering radiation pressure (jet) iii.Merging of many stars with M * 10 6 stars/pc 3 >> observed 10 4 stars/pc 3 !!!

Discriminate between different models requires detailed observational study of environment: structure (size, mass of cores) and kinematics (rotating disks, infall) on scales < 0.1 pc Monolithic collapse: disks (+jets) necessary for accretion onto OB star cluster natural outcome of s.f. process Competitive accretion (+merging): disks natural outcome of infall+ang.mom.cons. cluster necessary to focus accretion onto OB star

High-mass star forming regions: Observational problems Deeply embedded in dusty clumps high extinction IMF high-mass stars are rare: N(1 M O ) = 100 N(10 M O ) large distance: >400 pc, typically a few kpc formation in clusters confusion rapid evolution: t acc = 20 M O /10 -3 M O yr -1 = yr parental environment profoundly altered Advantage: very luminous (cont. & line) and rich (molecules)!

The formation of high-mass stars: where they form

Visible: extinction A V >100!

NIR-MIR: mostly stars…

NIR-MIR: … and hot dust

MIR-FIR: poor resolution…

FIR: …but more sensitive to embedded stars! luminosity estimate

Radio (sub)mm: dusty clumps

Radio (sub)mm: molecular lines

Radio < 2cm: thin free-free young HII regions

Radio > 6cm: free-free old HII regions

(IR-dark) Clouds: pc; 10 K; cm -3 ; Av=1-10; CO, 13 CO; n CO /n H 2 =10 -4 Clumps: 1 pc; 50 K; 10 5 cm -3 ; A V =100; CS, C 34 S; n CS /n H 2 =10 -8 Cores: 0.1 pc; 100 K; 10 7 cm -3 ; A V =1000; CH 3 CN, exotic molecules; n CH 3 CN /n H 2 = Outflows >1pc Disks??? (proto)stars: IR sources, maser lines, compact HII regions Typical star forming region

The formation of high-mass stars: how they form

IR-dark (cold) cloud fragmentation (hot) molecular core infall+rotation (proto)star+disk+outflow accretion hypercompact HII region expansion extended HII region Possible evolutionary sequence for high-mass stars monolithic collapse (disk accretion)? or competitive accretion (with merging)?

MSX 8 m SCUBA 850 m IR-dark clouds (>1pc): pre-stellar phase MSX 8 m SCUBA 850 m

Clump UC HII Core HMC

Clump UC HII HMC

Hot molecular core: site of high-mass star formation HC HII or wind HMC CH 3 CN(12-11) rotation! embedded massive stars

Observed inverse P Cyg profiles (Girart et al. 2009) infall! H 2 CO( ) CN(2-1) Formation of inverse P-Cyg profile

Expanding hypercompact HII region Moscadelli et al. (2007) Beltran et al. (2007) 7mm free-free & H 2 O masers 500 AU

Expanding hypercompact HII region Moscadelli et al. (2007) Beltran et al. (2007) 7mm free-free & H 2 O masers 30 km/s

IRAS Cesaroni et al. Hofner et al. Moscadelli et al. Keplerian rotation: M * =7 M O Moscadelli et al. (2005)

Conclusions More or less accepted: –IR-dark clouds precursors of high-mass stars –Hot molecular cores cradle of OB (proto)stars –Disk (+jet) natural outcome of OB S.F. process Still controversus: –Monolithic collapse (like solar-type stars) or competitive accretion (in cluster)? –Role of magnetic field and turbulence

Bibliography Beuther et al in Protostars and Planets V, p. 165 Bonnell et al in Protostars and Planets V, p. 149 Cesaroni et al in Protostars and Planets V, p. 197 Stahler & Palla 2004, The Formation of Stars