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 From line profile: Doppler effect: V = c(ν 0 - ν)/ν 0 along line of sight in most cases line FWHM thermal < FWHM observed  thermal broadening often negligible  line profile due to turbulence & velocity field Any molecule can be used!

channel maps integral under line Star Forming Region

rotating disk line of sight to the observer

GG Tau disk 13 CO(2-1) channel maps 1.4 mm continuum Guilloteau et al. (1999)

infalling envelope line of sight to the observer

red-shifted absorption bulk emission blue-shifted emission VLA channel maps 100-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 T K and column density N X LTE n H 2 >> n cr  T K = T ex τ >> 1: T K ≈ (Ω B /Ω S ) T MB but no N X ! e.g. 12 CO τ << 1: N u  (Ω B /Ω S ) T MB e.g. 13 CO, C 18 O, C 17 O T K = (hν/k)/ln(N l g u /N u g l ) N X = (N u /g u ) P.F.(T K ) exp(E u /kT K )

τ ≈ 1: τ = -ln[1-T MB (sat) /T MB (main) ] e.g. NH 3 T K = (hν/k)/ln(g 2 τ 1 /g 1 τ 2 )  N u  τT K  N X = (N u /g u ) P.F.(T K ) exp(E u /kT K )

If N i is known for >2 lines  T K and N X from rotation diagrams (Boltzmann plots): e.g. CH 3 C 2 H P.F.= Σ g i exp(-E i /kT K ) partition function

CH 3 C 2 H Fontani et al. (2002)

CH 3 C 2 H Fontani et al. (2002)

Non-LTE numerical codes (LVG) to model T MB by varying T K, N X, n H 2 e.g. CH 3 CN Olmi et al. (1993)

Problems: calibration error at least 10-20% on T MB T MB is mean value over Ω B 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 H 2 needed

Possible solutions: high angular resolution  small Ω B high spectral resolution  parameters of gas moving at different V’s along line profile  line interferometry needed!

Mass M H 2 and density n H 2 Column density: M H 2  (d 2 /X) ∫ N X dΩ –uncertainty on X by factor –error scales like distance 2 Virial theorem: M H 2  d Θ S (ΔV) 2 –cloud equilibrium doubtful –cloud geometry unknown –error scales like distance

(Sub)mm continuum: M H 2  d 2 F ν /T K –T K changes across cloud –error scales like distance 2 –dust emissivity uncertain depending on environment Non-LTE: n H 2 from numerical (LVG) fit to T MB of lines of molecule far from LTE, e.g. C 34 S –results model dependent –dependent on other parameters (T K, X, IR field, etc.) –calibration uncertainty > 10-20% on T MB –works only for n H 2 ≈ n cr

observed T B observed T B ratio T K = K n H 2 ≈ cm -3 satisfy observed values τ > 1  thermalization

best fits to T B of four C 34 S lines (Olmi & Cesaroni 1999)

H 2 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