Determination of physical properties from molecular lines

Kate Brooks

Australia Telescope National Facility

Mopra Induction Weekend May 2005

Ehrenfreund & Charnley 2000, ARA&A, 38, 427

Interstellar Molecules

137 molecules have been detected in space (205 including isotopomers, 50 in comets)

Talk Outline

Radiative Transfer

12CO(1-0): Workhorse of mm-line studies

Optically thin density tracers (LTE Mass)

Temperature tracers

Non-LTE models

Signatures for infalling gas

Bipolar outflows

Radiative Transfer

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dIv

dz= −κIv + jv

Fundamental equation of radiative transfer

Absorption emission coefficients

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τν = κν∫ (s) ds

Kirchhoff’s law valid in TE and LTE

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jv

κ v

= Bv (T)

Optical depth

Planck law

Rayleigh-Jeans approximation to Planck law

Brightness Temperature

For isothermal medium

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BRJ ν ,T( ) =2kν 2

c 2kT

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T Bν( ) =c 2

2kν 2Bν

Temperature that would result in brightness if source were a black-body in the Rayleigh-Jeans limit

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TB Iν ,s( ) = TB Bν( ) 1− e−τν( ) + TB Iν ,0( )e−τν

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Iv = Sv 1− e−τ( ) + Iv 0( )e−τ

Optically thin€

TB Iν ,s( ) = TB Bν( ) 1− e−τν( ) + TB Iν ,0( )e−τν

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τ <<1 TB = τ ν TB Bν( )

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τ >>1 TB = TB Bν( ) Optically thick

Integration along the line of sight:

Absorption coefficient -> Optical depth τ

Level population -> Level column density N

Total column density : Sum over all levels

Column Density

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dIv

dz= −κIv + jv

κ is related to the level population

In LTE there is one excitation temperature Tex that describes the level population according to the Boltzmann distribution

When collisions dominate: Level population can be described as Boltzmann distribution at kinetic gas temperature Tkin

One observed transition and adopting a value for Tkin gives all level populations

-> Total column density N

Excitation Temperature Tex

Measuring Kinetic Temperature Tkin

1. Optically thick transitions:

2. Line ratios e.g. 13CO(2-1) / 13CO(1-0)

3. Rotation Diagrams e.g. NH3, CH3CCH, CH3CN

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τ >>1 TB = TB Bν( )

Critical Density

Any spectral-line transition is only excited above a certain critical density

Critical density is the density at which:Collisional deexcitation ~ spontaneous radiative decay

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ncrit,21 =A21

< σ 21ν >

12CO(1-0) 115.27 GHZ 4 x 102 cm-3Lowest critical densityCS(2-1) 97.98 GHz 1 x 105 cm-3

HCN(1-0 88.63 GHz 1 x 105 cm-3

NH3(1,1) 23.694 GHZ 1 x 103 cm-3

12CO(1-0): Workhorse of mm-line studiesUbiquitous gas tracer - High abundance - Lowest critical density

Excellent for global cloud parameters - Temperature - Mass - Structure

Limitations - Optically thick - Complex velocity profiles - Confused towards Galactic plane - Depletion at high densities and low temperature

Example: The Carina Nebula

“ It would be manifestly impossible by verbal description to give any just idea of the capricious forms and irregular gradations of light affected by the different branches and appendages of this nebula. In this respect the figures must speak for themselves.”Sir J. F. W. Herschel 1847

Mopra observations of the Carina nebula

12CO(1-0) 115 GHz19962500 pointings0.1 K rms per channel

Brooks et al. 1998, PASA, 15, 202Example Grid

Excitation Temperature

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Tex = 5.5322 ln 1+5.532

P 12CO( ) + 0.837

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

−1

K

12CO1-0 is optically thickTB = Tex = Tkin

Use ‘xpeak’ in miriad to find P(12CO)

Excitation Temperature Map

“Treasure Cluster”

Mass estimates from CO observationsVirial Mass

Relies on the assumption that the cloud’s kinetic energy stabilizes it against gravitational collapse (Virialised)

The overall velocity width of the CO emission line reflects the motion of the gas and ultimately the underlying mass (Virial mass)

But …

Are molecular clouds virialised? What about external pressure?

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Mvir

Msun

⎡

⎣ ⎢

⎤

⎦ ⎥=1145

ΔV

kms−1

⎡

⎣ ⎢

⎤

⎦ ⎥D

pc

⎡

⎣ ⎢

⎤

⎦ ⎥

A

deg2

⎡

⎣ ⎢

⎤

⎦ ⎥

0.5

Mass estimates from CO observations

X - Factor

CO-to-H2 Conversion factor

Galactic Value: XCO ≈ 2.8 x 1020 cm-2 K (km s-1)-1

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XCO =I CO( )N H2( )

H2 Column Density to Mass

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M

Msun

⎡

⎣ ⎢

⎤

⎦ ⎥= 6.6 ×10

−24 N(H2)

cm−2

⎡ ⎣ ⎢

⎤ ⎦ ⎥D

pc

⎡

⎣ ⎢

⎤

⎦ ⎥

2A

deg2

⎡

⎣ ⎢

⎤

⎦ ⎥

Mass = column density x spatial extent

Average H2 density€

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n(H2)

cm−3

⎡

⎣ ⎢

⎤

⎦ ⎥=1.5

M

Msun

⎡

⎣ ⎢

⎤

⎦ ⎥R

pc

⎡

⎣ ⎢

⎤

⎦ ⎥

−3

Spherical with effective radius R2R = min + maj

Mass determined this way is often called the ‘CO mass’

But …

To determine Xco we need an independent measure of the mass of the cloud and the distance D in order to work out N(H2)

Independent Mass estimate for Xco

Virial Mass

Not all clouds are virialised

Radiative Transfer method

Very difficult to do in for other galaxies (minimum 3 lines)

Extinction

Assumes standard reddening law and dust-to-gas ratio

Dust Emission

Assumes dust absorption coefficient and dust-to-gas ratio

Use Xco with cautionProblem for all determinations of the conversion factor.

All of them have factors between 2-5 in uncertainty.

Galactic:

Constant for specific regions only

Extra Galactic:

Very difficult to measure Xco

Localised values that depend on metallicity and galaxy type

Sometimes you have little choice e.g. z 6

Pre-stellar core Ions, Long Chains HC5N, DCO+

Cold envelope Simple species, Heavy depletionsCS, N2H+

Warm inner envelope Evaporated speciesCH3OH, HCN

Hot core Complex organicsCH3OCH3, CH3CN

Outflow: direct impact Si- and S-speciesSiO, SO2

Outflow: walls, entrainment Evaporated icesCH3OH

PDR, compact HII regions Ions, RadicalsCN/HCN, CO+

Massive Disk Ions, D-rich species, photoproductions

HCO+, DCN, CNDebris Disk Dust, CO

Chemical Characteristics of star-forming regions

(E. F. van Dishoeck)

Example: 12CO, 13CO and CS intensities in

the Carina nebula

Utilising other molecular-line transitions

More than 40 emission lines in the Mopra 3-mm band

Optically thin density tracers (LTE Mass)

Temperature tracers

Non-LTE models

Signatures for infalling gas

Bipolar outflows

Optically thin density tracers: Testing 13CO, C18O and CS

e.g. Alves et al., 1998 Lada et al., 1994

In the study by Lada et al. 1994“Dust extinction and molecular gas in the dark cloud IC 5146”

Direct comparison of 13CO, C18O and CS integrated intensities and column densities with Av to a range in Av between 0-32 mag of extinction.

Integrated intensities

I(13C0) = 1.88 + 0.72Av K km s-1 (Av ≤ 5 mag)

I(C180) = 0.07 + 0.10Av K km s-1 (Av ≤ 15 mag)

I(CS) = 0.10 + 0.06Av K km s-1 (Av ≤ 15 mag)Between 8 and 10 mag the 13CO emission appears saturatedUncomfortable prediction of molecular emission and 0 mag

Integrated Intensity to Column Density

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τ ≈P 13CO( )

10.58 e10.58 Tex −1( )

−1− 0.223

τ <1

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N 13CO( )T

= 4.227 ×1012e5.289 TexτΔV

1− e−10.58 Texcm−2

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Idν∫ =1.064PΔV

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N 13CO( )T

≈ F Tex( ) Idν∫ Integrated intensity W13CO

Case Study 13CO(2-1)

Only one transition is measured and an extrapolation to total column density is done by assuming a LTE population

We need a value for Tex

-use value determined from 12CO-assume a value (e.g. 35 K)

The value of Tex has a large impact on optical depthbut not on column density

f(35 K) = 0.64

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N 13CO( )T

≈ F Tex( ) ×1015W13CO cm−2

Back to the study by Lada et al. 1994

Assuming LTEFor 13CO and C18O: Based on 12CO data: Tex = 10 K

For CS:

Subthermal excitation: Tex = 5 K

Column DensitiesN(13C0)LTE/Av = 2.18 x 1015 cm-2 mag-1 (Av ≤ 5

mag)N(C180) LTE/Av = 2.29 x 1014 cm-2 mag-1 (Av ≤ 15

mag)N(CS)LTE/Av = 4.5 x 1011 cm-2 mag-1 (Av ≤ 15

mag)

Column density to H2 density

Not there yet!

Gas-to-dust ratio of Savage & Drake (1978)N(H2) = 0.94 x 1021 Av cm-2

Which leads to:N(13C0)/N(H2) = 4 x 105 (Av 5 mag)

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Mass determined this way is often called the ‘LTE mass’

Depletion

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C18O Dust Emission

Bianchi et al.

Dust Extinction

0.1

pc

Alves et al.

T < 15 K and n > 105 cm-3

CO and CS freeze out onto the dust grains

Species linked to molecular nitrogen are less affectedE.g. NH3, N2H+, N2D+

Simple Line Ratio Analysis

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Beam filling factor:

Ratio of lines with similar frequency (and hence similar ) -> cancels out

Ratio of different species -> Optical Depth(if Tex and the isotopic abundance ratio is known)e.g. 12CO(1-0) / 13CO(1-0) [12CO/13CO] ≈ 89

Ratio of different transitions (τ << 1) -> Excitation temperaturee.g. C18O(2-1) / C18O(1-0)

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TB Iν ,s( ) = φ TB Bν( ) 1− e−τν( ) + TB Iν ,0( )e−τν( )

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R21 =TB ,21

TB ,10

= 4e−E21 Tex,21

Note: Different species and different transitions of one species arising in different parts of a region with different beam filling factors

Good Thermometers: Molecules with many transitions with a large range of energy levels in a small frequency interval

Symmetric top molecules:e.g. Ammonia NH3

Methyl Acetylene CH2C2HMethyl Cyanide CH3CN

NH3(1,1): 18 hyperfine components mixed into 5 linesFitting all 18 components -> optical depth

Rotation Diagrams

Integrated line intensity versus energy above ground

If LTE plot is a straight line with slope ~ (-1/T)

Trot = Tkin

Garay, Brooks et al., 2002

Non-LTE Modelling

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Additional Considerations- Stimulated emission- Radiative (photon) trapping

Large Velocity Gradient (LVG) approximation- assume large-scale velocity gradient exists in cloud- photons are absorbed locally, then immediately

escape

Maximum Escape Probability models

Static envelope

R2 R1B2B1

Optically thin line

Infall asymmetry Optically thick line

Constant line-of-sightvelocity

Tex (R2) > Tex (R1)

Tex (B2) > Tex (B1)

Infall region

Infall Protostar SMM4 in Serpens

Narayanan et al., 2002, ApJ, 565, 319

16272-4837

evidence for infall infall velocities of 0.5 km s-

1

are obtained using model of Myers et al. (1996)- Minfall 10-3 Msun yr-1

evidence for outflow - voutflow = 15 km s-1

.

Garay, Brooks, et al. 2003

Outflows

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Bourke et al. 1997

Outflows

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Belloche et al., 2002, A&A, 393, 972

Protostar IRAM 04191 in Taurus

Integrated Intensity to Column Density

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τ ≈P 13CO( )

10.58 e10.58 Tex −1( )

−1− 0.223

τ <1

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N 13CO( )T

= 4.227 ×1012e5.289 TexτΔV

1− e−10.58 Texcm−2

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Idν∫ =1.064PΔV

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N 13CO( )T

≈ F Tex( ) Idν∫

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F Tex( ) =Q8kπν 2

1.064hc 3

Ag2eErot kTex

1− e−hν kTex

10−6

ehν kTex −1( )−1

− ehν kTB −1( )−1 cm−3 sK−1

Integrated intensity W13CO

Case Study 13CO(2-1)