Estimate of physical parameters of molecular clouds
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Estimate of physical parameters of molecular clouds. Observables: T MB (or F ν ), ν , Ω 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.

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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!


Star Forming Region

channel maps

integral

under line


rotating disk

line of sight to the observer


GG Tau disk

13CO(2-1) channel maps

1.4 mm continuum

Guilloteau et al. (1999)


infalling

envelope

line of sight to the observer


VLA channel maps

100-m spectra

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 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)


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)


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


CH3C2H

Fontani et al. (2002)


CH3C2H

Fontani et al. (2002)


Non-LTE numerical codes (LVG) to model TMB by varying TK, NX, nH2e.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’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


  • (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
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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