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Lesson 7LVG model & XCO

Toshihiro HandaDept. of Phys. & Astron., Kagoshima University

Kagoshima Univ./ Ehime Univ.Galactic radio astronomy

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Multi-line observations(1)

▶ LTE approximation■ Tex is constant between any two levels

■ Line intensities differ due to TB=Tex (1-e-)

■ Compare lines with ≫1 and≪1

TB,thick=Tex, TB,thin=Tex,

▶ Optical depth from intensity→column density

▶ Optically thick line→excitation temperature

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Multi-line observations(2)

▶ Multi-levels (allow j=±1: diatomic mol.)dnj=nj+1Aj+1,j-njBj, j+1Ij+1,j+nj+1Bj+1,jIj+1,j-njCj,j+1 +nj+1Cj+1,j

n=nj 　　 total number is const.

■ Solve it under steady state dnj=0

▶ Change of Ij+1,j:simliar to the 2 level model

= (h)/(4) () nj Aj+1,j

= (h)/(4) () (nj Bj,j+1-nj+1 Bj+1,j)

▶ Change of intensity dI=(–I)dx■ Depend on the large scale structure of the cloud

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LVG model(1)

▶ Assume: large, monotonic vel. grad.■ Radiative coupling is bounded in local.

▶ Assume: abs. and rad. are thermally coupled.■ escape probability ▶ Ii,j = (1-i,j ) Si,j+i,j B(TCMB)

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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■ Tex for each trans. are fixed.→intensity of each line

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LVG model (3)

▶ Three input parameters■ Kinetic temp. of H2 Tk

■ Gas density of H2 n(H2)

■ Mol. Numb per depth & velocity span n(X)/(dv/dr)

▶ Solve equations numerically■ Goldreich & Kwan (1974) ApJ 189, 441■ Scoville & Solomon (1974) ApJ 187, L67

Scoville&Solomon (1974)

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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

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Turbulent model

▶ Motion in a beam (observed pixel)■ Gaussian like velocity field: random motion■ wider than thermal width→supersonic turbulence

Problem: rapid dissipationWhat supplies the turbulent motion energy?

■ Super high reso. obs: thermal width is observed!

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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

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Beam filling factor(1)

▶ Observed resolution is poor.

▶ Inhomogeneous gas in a beam■ First approx. : all or nothing

obs. Beam sizegas is located only here.

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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.

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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

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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 =clumpVolume of a cloud

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Conversion factor X (1)

▶ CO intensity ∝ gas column density■ Why? CO is optically thick!!

Intensity ratio is far from abundance13CO/12CO intensity ratio ～ 10-1

13CO/12CO abundance ～ 1/89( 太陽系 ) 、 1/67(MWG)

■ Empirical relation originallyLine profiles are similar in 12CO and 13CO.

▶ N(H2)=X ∫ TB(CO,J=1-0) dv■ X=2.3×1020 cm-2/(K km s-1)

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Conversion factor X (2)

■ Gamma ray: interaction between CR & proton■ Correlation between gamma, HI, and CO

CGRO, NASADicky&Lockman HI

AMANOGAWA CO

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Conversion factor X (3)

▶ Why it works well?■ Cloud property is similar over the galaxy(?)■ TB gives beam filling factor(?)

←small beam filling factor in general

▶ “theoretical” model■ Virial equiv.→ m∝ Rv2, optically thick→TB∝ Tex

■ In this case, X=N(H2)/(∫TB dv)∝ n(H2)1/2r3/2/Tex

■ subthermally excited→Tex∝ n(H2)-1/2： LVG model■ ∴ If clump size is const., X is const.