summary 1.statistical equilibrium and radiative transfer in molecular (h 2 ) cloud – derivation of...
TRANSCRIPT
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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
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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
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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 !!!
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ij
Aij Bij Bji Cij Cji
……
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Radiative transfer equation: the line case
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A21 B21 B12 C21 C12
2
1
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3-level system
A21 B21 B12 C21C12
3
1
2
A32 B32 B23 C32C23
A31 B31 B13 C31C13
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J=0
J=1
J=2
A21
A10
A21 ≈ 10 A10
A31 = 0
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nH2 ~ ncr
Tex(1-0) > TK
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nH2 ~ ncr
Tex(1-0) < 0
i.e. pop. invers.
MASER!!!
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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)(
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• 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
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‘‘extended’’ source:ΩS>> ΩB TMB ≈ TB
‘‘pointlike’’ source:ΩS<< ΩB TMB ≈ TB ΩS/ΩB << TB
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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
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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!
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channel maps
integralunder line
Star Forming Region
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rotating disk
line
of
sigh
t to
the
obse
rver
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GG Tau disk13CO(2-1) channel maps
1.4 mm continuum
Guilloteau et al. (1999)
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GG Tau disk13CO(2-1) & 1.3mm cont. near IR cont.
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infalling
envelopeli
ne o
f si
ght t
o th
e ob
serv
er
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red-shiftedabsorption
bulk emission
blue-shiftedemission
VLA channel maps100-m spectra
Hofner et al. (1999)
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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
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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
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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)
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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
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CH3C2HFontani et al. (2002)
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CH3C2H Fontani et al. (2002)
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Non-LTE numerical codes (LVG) to model TMB by varying TK, NX, nH2
e.g. CH3CN
Olmi et al. (1993)
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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
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Possible solutions:
• high angular resolution small ΩB
• high spectral resolution parameters of gas moving at different V’s along line profile
line interferometry needed!
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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
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• (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
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observed TB
observed TB ratio
TK = 20-60 K
nH2 ≈ 3 106 cm-3
satisfy observed
values
τ > 1 thermalization
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best fits to TB of four C34S lines (Olmi & Cesaroni 1999)
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H2 densities from best fits
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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
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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)
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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
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• 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 )(
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The formation of high-mass and low-mass stars: differences and
theoretical problems
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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
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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
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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
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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
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Palla & Stahler (1990)
dM/dt=10-5 MO/yr
tKH=tacc
Main Sequence
Sun
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Problem:Stellar radiation pressure (+ wind + ionizing flux) halt accretion above
M*=8 Msun
how to form M*>8 M⊙ ?
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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 !!!
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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
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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)!
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The formation of high-mass stars: where they form
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Visible:
extinction AV>100!
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NIR-MIR:
mostly stars…
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NIR-MIR:
… and hot dust
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MIR-FIR:
poor resolution…
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FIR:…but more sensitiveto embedded stars!
luminosity estimate
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Radio (sub)mm:
dusty clumps
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Radio (sub)mm:
molecular lines
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Radio < 2cm:
thin free-free
young HII regions
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Radio > 6cm:
free-free
old HII regions
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• (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
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The formation of high-mass stars: how they form
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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)?
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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
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ClumpUC HII
CoreHMC
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Clump
UC HII
HMC
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Hot molecular core: site of high-mass star formation
HC HII or wind
HMC
CH3CN(12-11)
rotation!
embedded massive stars
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Observed inverse P Cyg profiles
(Girart et al. 2009) infall!H2CO(312-211)
CN(2-1)
Formation of inverse P-Cyg
profile
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Expandinghypercompact HII regionMoscadelli et al. (2007)
Beltran et al. (2007)
7mm free-free & H2O masers
500 AU
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Expandinghypercompact HII regionMoscadelli et al. (2007)
Beltran et al. (2007)
7mm free-free & H2O masers
30 km/s
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IRAS 20126+4104Cesaroni et al.Hofner et al.
Moscadelli et al.Keplerian rotation:M*=7 MO
Moscadelli et al. (2005)
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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
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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
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