1. Mellinger Lesson 7 LVG model & X CO Toshihiro Handa Dept. of Phys. & Astron., Kagoshima University Kagoshima Univ./ Ehime Univ. Galactic radio astronomy

2. Mellinger Multi-line observations(1) ▶ LTE approximation ■ T ex is constant between any two levels ■ Line intensities differ due to  T B =T ex (1-e -  ) ■ Compare lines with  ≫ 1 and  ≪ 1 T B,thick =T ex, T B,thin =T ex , ▶ Optical depth from intensity→column density ▶ Optically thick line→excitation temperature

3. Mellinger Multi-line observations(2) ▶ Multi-levels (allow  j=±1: diatomic mol.) dn j =n j+1 A j+1,j -n j B j, j+1 I j+1,j +n j+1 B j+1,j I j+1,j -n j C j,j+1 +n j+1 C j+1,j n=  n j total number is const. ■ Solve it under steady state dn j =0 ▶ Change of I j+1,j :simliar to the 2 level model   = (h  )/(4  )  ( ) n j A j+1,j   = (h  )/(4  )  ( ) (n j B j,j+1 -n j+1 B j+1,j ) ▶ Change of intensity dI =(   –   I )dx ■ Depend on the large scale structure of the cloud

4. Mellinger Multi-level model ▶ Transitions between multi-levels ■ n i  i  P i,j =  j  n j P j,i ■ P i,j =A i,j +B i,j I i,j +C i,j (i>j) ■ P i,j =B i,j I i,j +C i,j (j>i) ▶ I i,j is transfered source funct. ■ I i,j = ∫ K i,j (|r-r'|) S i,j (r') dr' ■ Boundary cond. ： cloud edge in CMB; B (T CMB )

5. Mellinger LVG model(1) ▶ Assume: large, monotonic vel. grad. ■ Radiative coupling is bounded in local. ▶ Assume: abs. and rad. are thermally coupled. ■ escape probability  ▶ I i,j = (1-  i,j ) S i,j +  i,j B (T CMB )

6. Mellinger LVG model(2) ▶ Optical depth  →photon escape prob.  controlled by geometrical structure “Abs. & rad. are bounded in a small space” →Vel. structure has large gradient. (LVG) ■  =[1-exp(-  )]/  : 1D model = slab ■  =[1-exp(-  )]/(3  ): spherical symmetric ▶ LVG model ■ Under this structure, derive the all level population ■ T ex for each trans. are fixed.→intensity of each line

7. Mellinger LVG model (3) ▶ Three input parameters ■ Kinetic temp. of H 2 T k ■ Gas density of H 2 n(H 2 ) ■ Mol. Numb per depth & velocity span n(X)/(d v /dr) ▶ Solve equations numerically ■ Goldreich & Kwan (1974) ApJ 189, 441 ■ Scoville & Solomon (1974) ApJ 187, L67 Scoville&Solomon (1974)

8. Mellinger Features of a molecular cloud ▶ Features of an actually observed emission line ▶ Gaussian like profile ▶ width: much wider than thermal motion ■ Larger scale motion than thermal ■ Turbulence? ▶ Intensity: much colder than gas temperature ■ beam filling factor

9. Mellinger Turbulent model ▶ Motion in a beam (observed pixel) ■ Gaussian like velocity field: random motion ■ wider than thermal width→supersonic turbulence Problem: rapid dissipation What supplies the turbulent motion energy? ■ Super high reso. obs: thermal width is observed!

10. Mellinger Feasibility of the LVG model ▶ The line width is finite! ■ Disconnect if velocity difference is large. ■ Only a small region is connected by radiation. ▶ Order is OK with LVG, even diff. geometry. ■ We cannot know the detail geom. structure! ▶ If you want to calculate more precisely, ■ e.g. photon tracing using Monte-Carlo simulation

11. Mellinger Beam filling factor(1) ▶ Observed resolution is poor. ▶ Inhomogeneous gas in a beam ■ First approx. : all or nothing obs. Beam size gas is located only here.

12. Mellinger Beam filling factor(2) ▶ Beam filling factor ■ The more parameters, the more freedom. ■ Filling factor may be different for diff. lines. Oh! More freedom!! We need the simplest model ■ The same factors give no effect on line ratio! The effective critical densities are close.

13. Mellinger Geometrical structure of a cloud ▶ No information  assume spherical symmetry ■ “Common sense” in astronomy, 1D approx. ▶ Actually far from a spherical geometry ■ “infinite” fine structure.  fractal structure ■ filaments

14. Mellinger Volume filling factor ▶ In the “outer boundary” of a cloud ▶ Inhomogeneous gas in a cloud ■ First approx. : all or nothing ■ clumpy model gas is located only here =clump Volume of a cloud

15. Mellinger Conversion factor X (1) ▶ CO intensity ∝ gas column density ■ Why? CO is optically thick!! Intensity ratio is far from abundance 13 CO/ 12 CO intensity ratio ～ 10 -1 13 CO/ 12 CO abundance ～ 1/89( 太陽系 ) 、 1/67(MWG) ■ Empirical relation originally Line profiles are similar in 12 CO and 13 CO. ▶ N(H 2 )=X ∫ T B (CO,J=1-0) d v ■ X=2.3×10 20 cm -2 /(K km s -1 )

16. Mellinger Conversion factor X (2) ■ Gamma ray: interaction between CR & proton ■ Correlation between gamma, HI, and CO CGRO, NASADicky&Lockman HI AMANOGAWA CO

17. Mellinger Conversion factor X (3) ▶ Why it works well? ■ Cloud property is similar over the galaxy(?) ■ T B gives beam filling factor(?) ←small beam filling factor in general ▶ “theoretical” model ■ Virial equiv.→ m ∝ R v 2, optically thick→T B ∝ T ex ■ In this case, X=N(H 2 )/(∫T B d v ) ∝ n(H 2 ) 1/2 r 3/2 /T ex ■ subthermally excited→T ex ∝ n(H 2 ) -1/2 ： LVG model ■ ∴ If clump size is const., X is const.