SUMMARY

1. Statistical equilibrium and radiative transfer in molecular (H2) cloud – Derivation of physical parameters of molecular clouds

2. High-mass star formation: theoretical problems and observational results

Statistical equilibriumand

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 Iν as a function of cloud physical parameters

calculate populations ni of energy levels of given molecule X inside cloud of H2 with kinetic temperature TK and density nH2

plus

external radiation field.

Note: nX << nH2 always; e.g. CO most

abundant species but nCO/ nH2 = 10-4 !!!

ij

Aij Bij Bji Cij Cji

……

Radiative transfer equation: the line case

A21 B21 B12 C21 C12

2

1

3-level system

A21 B21 B12 C21C12

3

1

2

A32 B32 B23 C32C23

A31 B31 B13 C31C13

J=0

J=1

J=2

A21

A10

A21 ≈ 10 A10

A31 = 0

nH2 ~ ncr

Tex(1-0) > TK

nH2 ~ ncr

Tex(1-0) < 0

i.e. pop. invers.

MASER!!!

Radio observations• Useful definition: brightness

temperature, TB

• In the radio regime Rayleigh-Jeans (hν << kT) holds:

• In practice one measures mean TB over antenna beam pattern, TMB:

• Flux measured inside solid angle Ω:

)dΩ,(P

)dΩ,()P,(T),(T

n

nB

MB

00

00

ΩΩΩ

ΩTk

ΩTk

ΩIS d2

d2

d MB2B2

B2

2T

kI

TTTk

TB B 2

2)(

• Angular resolution: HPBW = 1.2 λ/D

• Beam almost gaussian: ΩB = π/(4ln2) HPBW2

One measures convolution of source with beam

Example

gaussian source gaussian image with:• TMB = TB ΩS/(ΩB+ ΩS)

• Sν = (2k/λ2) TB ΩS = (2k/λ2) TMB (ΩB+ ΩS)

• ΘS’ = (ΘS2 + ΘB

2)1/2

‘‘extended’’ source:ΩS>> ΩB TMB ≈ TB

‘‘pointlike’’ source:ΩS<< ΩB TMB ≈ TB ΩS/ΩB << TB

Estimate of physical parametersof molecular clouds

• Observables: TMB (or Fν), ν, ΩS

• Unknowns: V, TK, NX, MH2, nH2

– V velocity field

– TK kinetic temperature

– NX column density of molecule X

– MH2 gas mass

– nH2 gas volume density

)(1

eTT ex

SB

SMB ))((1

4)( 0 VeNB

hexkT

h

uul

Velocity field

From line profile:

• Doppler effect: V = c(ν0- ν)/ν0 along line of sight

• in most cases line FWHMthermal < FWHMobserved

thermal broadening often negligible line profile due to turbulence & velocity field

Any molecule can be used!

channel maps

integralunder line

Star Forming Region

rotating disk

line

of

sigh

t to

the

obse

rver

GG Tau disk13CO(2-1) channel maps

1.4 mm continuum

Guilloteau et al. (1999)

GG Tau disk13CO(2-1) & 1.3mm cont. near IR cont.

infalling

envelopeli

ne o

f si

ght t

o th

e ob

serv

er

red-shiftedabsorption

bulk emission

blue-shiftedemission

VLA channel maps100-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 TK

and column density NX

LTE nH2 >> ncr TK = Tex

τ >> 1: TK ≈ (ΩB/ΩS) TMB but no NX! e.g. 12CO

τ << 1: Nu (ΩB/ΩS) TMB e.g. 13CO, C18O, C17O

TK = (hν/k)/ln(Nlgu/Nugl)

NX = (Nu/gu) P.F.(TK) exp(Eu/kTK)

1

exSexB

SMB TTT 1

uSexB

SMB NTT

τ ≈ 1: τ = -ln[1-TMB(sat)/TMB

(main)] e.g. NH3

TK = (hν/k)/ln(g2 τ1/g1 τ2) Nu τTK

NX = (Nu/gu) P.F.(TK) exp(Eu/kTK)

If Ni is known for >2 lines TK and NX from rotation diagrams (Boltzmann plots): e.g. CH3C2H

P.F.=Σ gi exp(-Ei/kTK) partition function

K

i

K

X

i

i

kT

E

TFP

N

g

N

).(.

lnln

CH3C2HFontani et al. (2002)

CH3C2H Fontani et al. (2002)

Non-LTE numerical codes (LVG) to model TMB by varying TK, NX, nH2

e.g. CH3CN

Olmi et al. (1993)

Problems:

• calibration error at least 10-20% on TMB

• TMB is mean value over ΩB 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 H2 needed

Possible solutions:

• high angular resolution small ΩB

• high spectral resolution parameters of gas moving at different V’s along line profile

line interferometry needed!

Mass MH2 and density nH2

• Column density: MH2 (d2/X) ∫ NX dΩ

– uncertainty on X by factor 10-100– error scales like distance2

• Virial theorem: MH2 d ΘS (ΔV)2

– cloud equilibrium doubtful– cloud geometry unknown– error scales like distance

• (Sub)mm continuum: MH2 d2 Fν /TK

– TK changes across cloud

– error scales like distance2

– dust emissivity uncertain depending on environment

• Non-LTE: nH2 from numerical (LVG) fit to TMB

of lines of molecule far from LTE, e.g. C34S– results model dependent

– dependent on other parameters (TK, X, IR field, etc.)

– calibration uncertainty > 10-20% on TMB

– works only for nH2 ≈ ncr

observed TB

observed TB ratio

TK = 20-60 K

nH2 ≈ 3 106 cm-3

satisfy observed

values

τ > 1 thermalization

best fits to TB of four C34S lines (Olmi & Cesaroni 1999)

H2 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*>103L⊙ 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-2.35 N(10MO) = 10-2 N(1MO)

• Stellar lifetime: t Mc2/L M-3 t(10MO) = 10-3 t(1MO)

105 1 MO stars per 10 MO star! Total mass dominated by low-mass stars. However…• Stellar luminosity:

L M4 L(10MO) = 104 L(1MO) Luminosity of stars with mass between M1 and M2:

L(10-100MO) = 0.3 L(1-10MO) Luminosity of OB stars is comparable to luminosity of

solar-type stars!

dMMdMdM

dNL

L

MdM

dM

dNLtMMML

M

M

M

M

M

M 2

1

2

1

2

1

35.121 )(

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

theoretical problems

stars < 8MO

isothermal unstable clump

accretion onto protostar

disk & outflow formation

disk without accretion

protoplanetary disk

sub-mm

far-IR

near-IR

visible+NIR

visible

stars > 8MO

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: nR-2

Free-falling core: nR-3/2

tacc= M*/(dMacc/dt)

Contraction of protostar:

tKH=GM2/R*L*

– Stars < 8 Msun: tKH > tacc

– Stars > 8 Msun: tKH < tacc

High-mass stars form still in accretion phase

Low-mass VS High-mass

nR-3/2

nR-2

Two mechanisms at work:Accretion onto protostar:Static envelope: nR-2

Free-falling core: nR-3/2

tacc= M*/(dMacc/dt)

Contraction of protostar:

tKH=GM2/R*L*

– Stars < 8 Msun: tKH > tacc

– Stars > 8 Msun: tKH < tacc

High-mass stars form still in accretion phase

nR-2

nR-3/2nR-3/2

nR-2

Low-mass VS High-mass

Palla & Stahler (1990)

dM/dt=10-5 MO/yr

tKH=tacc

Main Sequence

Sun

Problem:Stellar radiation pressure (+ wind + ionizing flux) halt accretion above

M*=8 Msun

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*< 8 M⊙: insensitive to radiation pressure … but needs >106 stars/pc3 >> observed 104 stars/pc3 !!!

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 starcluster 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 MO) = 100 N(10 MO)

large distance: >400 pc, typically a few kpc formation in clusters confusion

rapid evolution: tacc = 20 MO/10-3 MOyr-1 = 2 104 yr parental environment profoundly altered

• Advantage: very luminous (cont. & line) and rich (molecules)!

The formation of high-mass stars: where they form

Visible:

extinction AV>100!

NIR-MIR:

mostly stars…

NIR-MIR:

… and hot dust

MIR-FIR:

poor resolution…

FIR:…but more sensitiveto 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: 10-100 pc; 10 K; 102-103 cm-3; Av=1-10; CO, 13CO; nCO/nH2

=10-4

• Clumps: 1 pc; 50 K; 105 cm-3; AV=100; CS, C34S; nCS/nH2

=10-8

• Cores: 0.1 pc; 100 K; 107 cm-3; AV=1000; CH3CN, exotic molecules; nCH3CN/nH2

=10-10

• 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) cloudfragmentation

(hot) molecular coreinfall+rotation

(proto)star+disk+outflowaccretion

hypercompact HII regionexpansion

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 MSX 8 m

SCUBA 850 m SCUBA 850 m

ClumpUC HII

CoreHMC

Clump

UC HII

HMC

Hot molecular core: site of high-mass star formation

HC HII or wind

HMC

CH3CN(12-11)

rotation!

embedded massive stars

Observed inverse P Cyg profiles

(Girart et al. 2009) infall!H2CO(312-211)

CN(2-1)

Formation of inverse P-Cyg

profile

Expandinghypercompact HII regionMoscadelli et al. (2007)

Beltran et al. (2007)

7mm free-free & H2O masers

500 AU

Expandinghypercompact HII regionMoscadelli et al. (2007)

Beltran et al. (2007)

7mm free-free & H2O masers

30 km/s

IRAS 20126+4104Cesaroni et al.Hofner et al.

Moscadelli et al.Keplerian rotation:M*=7 MO

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. 2007 in Protostars and Planets V, p. 165

• Bonnell et al. 2007 in Protostars and Planets V, p. 149

• Cesaroni et al. 2007 in Protostars and Planets V, p. 197

• Stahler & Palla 2004, The Formation of Stars