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

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summary
SUMMARY
  • Statistical equilibrium and radiative transfer in molecular (H2) cloud – Derivation of physical parameters of molecular clouds
  • High-mass star formation: theoretical problems and observational results
statistical equilibrium and radiative transfer
Statistical equilibriumandradiative 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
slide3
Problem:

Calculate molecular line brightness Iνas a function ofcloud 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 << nH2always; e.g. CO most abundant species but nCO/ nH2 = 10-4 !!!

slide4

i

Aij

Bji

Cji

Bij

Cij

j

slide7
2

A21

B12

C12

B21

C21

1

3 level system
3-level system

3

A32

B23

C23

B32

C32

A31

B13

C13

B31

C31

2

A21

B12

C12

B21

C21

1

slide14
J=2

A21≈ 10 A10

A31 = 0

A21

J=1

A10

J=0

slide15
nH2~ ncr

Tex(1-0) >TK

slide16
nH2~ ncr

Tex(1-0) < 0

i.e. pop. invers.

MASER!!!

radio observations
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:
slide18
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 = TBΩS/(ΩB+ ΩS)
  • Sν = (2k/λ2) TBΩS = (2k/λ2) TMB (ΩB+ ΩS)
  • ΘS’ = (ΘS2 + ΘB2)1/2
slide19
‘‘extended’’ source:

ΩS>> ΩB TMB≈ TB

‘‘pointlike’’ source:

ΩS<< ΩB TMB ≈ TB ΩS/ΩB << TB

estimate of physical parameters of molecular clouds
Estimate of physical parametersof molecular clouds
  • Observables: TMB(orFν), ν,ΩS
  • Unknowns:V, TK, NX, MH2, nH2
    • V velocity field
    • TK kinetic temperature
    • NX column density of molecule X
    • MH2 gas mass
    • nH2gas volume density
velocity field
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!

slide22
Star Forming Region

channel maps

integral

under line

slide23
rotating disk

line of sight to the observer

slide24
GG Tau disk

13CO(2-1) channel maps

1.4 mm continuum

Guilloteau et al. (1999)

slide25
GG Tau disk

13CO(2-1) & 1.3mm cont.

near IR cont.

slide26
infalling

envelope

line of sight to the observer

slide27
VLA channel maps

100-m spectra

red-shifted

absorption

bulk emission

blue-shifted

emission

Hofner et al. (1999)

slide28
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
Kinetic temperature TKand column density NX

LTEnH2>> 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)

slide31
τ ≈ 1:τ = -ln[1-TMB(sat)/TMB(main)] e.g. NH3

TK= (hν/k)/ln(g2τ1/g1τ2)  Nu τTK 

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

slide32
If Ni is known for >2 lines TK and NX from rotation diagrams (Boltzmann plots): e.g. CH3C2H

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

slide33
CH3C2H

Fontani et al. (2002)

slide34
CH3C2H

Fontani et al. (2002)

slide36
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
slide37
Possible solutions:
  • high angular resolution  small ΩB
  • high spectral resolution  parameters of gas moving at different V’salong line profile

 line interferometry needed!

mass m h 2 and density n h 2
Mass MH2and 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
slide39
(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
slide40
τ> 1  thermalization

observed TB

observed TB ratio

TK = 20-60 K

nH2≈ 3 106 cm-3

satisfy observed

values

bibliography
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 10 3 l b3 o
The formation of high-mass stars: observations and problems (high-mass star  M*>8M⊙ L*>103L⊙ B3-O)
  • Importance of high-mass stars: their impact
  • High- and low-mass stars: differences
  • High-mass stars: observational problems
  • The formation of high-massstars: where
  • The formation of high-massstars: how
importance of high mass stars
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
slide46
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)
  • 1051 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!

slide48
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

slide49
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

?

low mass vs high mass
nR-2

nR-3/2

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)

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

slide52
nR-2

nR-3/2

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)
  • 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

nR-2

nR-3/2

slide53
Palla & Stahler (1990)

tKH=tacc

dM/dt=10-5 MO/yr

Main Sequence

Sun

slide54
Problem:

Stellar radiation pressure (+ wind + ionizing flux) halt accretion above M*=8 Msun

 how to form M*>8 M⊙ ?

slide55
Solutions:
  • Competitive accretion: boosts dM/dt by deepening potential well through cluster: dM/dt(M*>8M⊙) >> dM/dt(M* <8M⊙)
  • Monolithic collapse: accretionthroughdisk+jet; focuses dM/dt enhancing ram pressure (disk) and allows photons to escape lowering radiation pressure (jet)
  • “Merging’’ of many stars with M*< 8 M⊙: insensitive to radiation pressure … but needs >106 stars/pc3>> observed 104 stars/pc3 !!!
slide56
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.

clusternecessary to focus accretion onto OB star

high mass star forming regions observational problems
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
    • rapidevolution: tacc = 20 MO/10-3 MOyr-1 = 2104 yr
    • parental environment profoundly altered
  • Advantage:
    • very luminous (cont. & line) and rich (molecules)!
slide59
Visible:

extinction AV>100!

slide60
NIR-MIR:

mostly stars…

slide61
NIR-MIR:

… and hot dust

slide62
MIR-FIR:

poor resolution…

slide63
FIR:

…but more sensitive

to embedded stars!

 luminosity estimate

slide64
Radio (sub)mm:

dusty clumps

slide65
Radio (sub)mm:

molecular lines

slide66
Radio < 2cm:

thin free-free 

 young HII regions

slide67
Radio > 6cm:

free-free 

old HII regions

slide68
“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
possible evolutionary sequence for high mass stars
monolithic collapse

(disk accretion)?

or

competitive accretion

(with merging)?

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

slide71
IR-dark clouds (>1pc): pre-stellar phase

MSX 8 m

MSX 8 m

SCUBA 850 m

MSX 8 m

SCUBA 850 m

SCUBA 850 m

slide72
Clump

UCHII

HMC

Core

slide73
Clump

UCHII

HMC

slide74
Hot molecular core: site of high-mass star formation

rotation!

HCHII or wind

HMC

embedded

massive stars

CH3CN(12-11)

slide75
Formation of inverse P-Cyg profile

Observed inverse P Cyg profiles

(Girart et al. 2009)  infall!

H2CO(312-211)

CN(2-1)

slide76
Expanding

hypercompact HII region

Moscadelli et al. (2007)

Beltran et al. (2007)

7mm free-free & H2O masers

500 AU

slide77
Expanding

hypercompact HII region

Moscadelli et al. (2007)

Beltran et al. (2007)

7mm free-free & H2O masers

30 km/s

slide78
IRAS 20126+4104

Cesaroni et al.

Hofner et al.

Moscadelli et al.

Keplerian rotation:

M*=7 MO

Moscadelli et al. (2005)

conclusions
Conclusions
  • More or less accepted:
    • IR-dark cloudsprecursors of high-mass stars
    • Hot molecular corescradle 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
bibliography1
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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