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 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 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 !!!
…
Aij Bij Bji Cij Cji
i
j
…
Radiative transfer equation: the line case
2
A21 B21 B12 C21 C12
1
3-level system
3
A32
B32
B23
C32
C23
A31
A21
B21
B12
C21
B31
B13
C31
C13
2
C12
1
J=2
A21 ≈ 10 A10
A31 = 0
A21
J=1
A10
J=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 RayleighJeans (hν << kT) holds:
I 
2k
TB

2k
B (T )  2 T  TB  T

2
TB(,)Pn(  0 ,  0 )dΩ
• In practice one measures

mean TB over antenna beam TMB(0 ,0 ) 
Pn(,)dΩ

pattern, TMB:
• Flux measured inside solid
angle Ω:
S   I dΩ 
Ω
2k

2
2k
 T dΩ    T
B
Ω
MB
2
Ω
dΩ
• 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 + ΘB2)1/2
‘‘extended’’ source:
ΩS>> ΩB  TMB ≈ TB
‘‘pointlike’’ source:
ΩS<< ΩB  TMB ≈ TB ΩS/ΩB << TB
Estimate of physical parameters
of molecular clouds
TMB
S

Tex 1  e  ( )
B  S


 kTh

h
ex

 ( ) 
Bul N u e  1 (  0 (V ))


4


• 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
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!
Star Forming Region
channel maps
integral
under line
line of sight to the observer
rotating disk
GG Tau disk
13CO(2-1)
channel maps
1.4 mm continuum
Guilloteau et al. (1999)
GG Tau disk
13CO(2-1)
& 1.3mm cont.
near IR cont.
line of sight to the observer
infalling
envelope
100-m spectra
VLA channel maps
red-shifted
absorption
bulk emission
blue-shifted
emission
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
TMB
S

Tex   S Tex    1
B
TMB
S

Tex   S Nu    1
B
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)
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
 Ni 
 N X  Ei
 
ln   ln
 P.F .(TK )  kTK
 gi 
P.F.=Σ gi exp(-Ei/kTK) partition function
CH3C2H
Fontani 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
τ > 1  thermalization
observed TB
observed TB ratio
TK = 20-60 K
nH2 ≈ 3 106 cm-3
satisfy observed
values
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
The formation of high-mass stars:
observations and problems
(high-mass star  M*>8M⊙  L*>103L⊙  B3-O)
1)
2)
3)
4)
5)
Importance of high-mass stars: their impact
High- and low-mass stars: differences
High-mass stars: observational problems
The formation of high-mass stars: where
The formation of high-mass stars: how
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( M 1  M  M 2 )  
M2
M1
M2 M
M2
dN
dN
tL
dM  
L
dM   M 1.35 dM
M1 L
M1
dM
dM
 L(10-100MO) = 0.3 L(1-10MO)
 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 < 8MO
sub-mm
isothermal unstable clump
far-IR
accretion onto protostar
near-IR
disk & outflow formation
visible+NIR
disk without accretion
visible
protoplanetary disk
stars > 8MO
sub-mm
isothermal unstable clump
far-IR
accretion onto protostar
near-IR
disk & outflow formation
visible+NIR
disk without accretion
visible
protoplanetary disk
Low-mass VS High-mass
Two mechanisms at work:
Accretion onto protostar:
Static envelope: nR-2
Free-falling core: nR-3/2
tacc= M*/(dMacc/dt)
nR-2
nR-3/2
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
Two mechanisms at work:
Accretion onto protostar:
Static envelope: nR-2
Free-falling core: nR-3/2
tacc= M*/(dMacc/dt)
nR-2
nR-3/2
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
Palla & Stahler (1990)
tKH=tacc
dM/dt=10-5 MO/yr
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 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 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 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
“Typical’’ star forming region
• (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
The formation of high-mass stars:
how they form
Possible evolutionary sequence for high-mass stars
IR-dark (cold) cloud
fragmentation
(hot) molecular core
infall+rotation
(proto)star+disk+outflow
accretion
hypercompact HII region
expansion
extended HII region
monolithic collapse
(disk accretion)?
or
competitive accretion
(with merging)?
IR-dark clouds (>1pc): pre-stellar phase
MSX 8 m
MSX 8 m
MSX 8 m
SCUBA 850 m
SCUBA 850 m
SCUBA 850 m
Clump
UC HII
HMC
Core
Clump
UC HII
HMC
Hot molecular core: site of high-mass star formation
rotation!
HC HII or wind
embedded
HMC
massive stars
CH3CN(12-11)
Formation of
inverse P-Cyg
profile
Observed inverse P Cyg profiles
(Girart et al. 2009)  infall!
H2CO(312-211)
CN(2-1)
Expanding
hypercompact HII region
Moscadelli et al. (2007)
Beltran et al. (2007)
7mm free-free & H2O masers
500 AU
Expanding
hypercompact HII region
Moscadelli et al. (2007)
Beltran et al. (2007)
7mm free-free & H2O masers
30 km/s
IRAS 20126+4104
Cesaroni et al.
Hofner et al.
Keplerian
Moscadelli etrotation:
al.
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