1. 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

2. 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!

3. channel maps integral under line Star Forming Region

4. rotating disk line of sight to the observer

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

6. infalling envelope line of sight to the observer

7. red-shifted absorption bulk emission blue-shifted emission VLA channel maps 100-m spectra Hofner et al. (1999)

8. 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

9. 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 )

11. 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 )

12. 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

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

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

15. 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)

16. 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

17. 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!

18. 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 10-100 –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

19. (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

20. observed T B observed T B ratio T K = 20-60 K n H 2 ≈ 3 10 6 cm -3 satisfy observed values τ > 1  thermalization

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

22. H 2 densities from best fits

23. 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