Determination of physical properties from molecular lines

Kate Brooks Australia Telescope National Facility Mopra Induction Weekend May 2005

Interstellar Molecules

Ehrenfreund & Charnley 2000, ARA&A, 38, 427 137 molecules have been detected in space (205 including isotopomers, 50 in comets)

Talk Outline

Radiative Transfer 12 CO(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

Fundamental equation of radiative transfer

dI v dz

  

I v



j v

Absorption emission coefficients Optical depth       (

s

)

ds

Kirchhoff’s law valid in TE and LTE

j v



v



B v

(

T

) Planck law 

  Rayleigh-Jeans approximation to Planck law

B RJ

   2

k

 2

c

2

kT

Brightness Temperature   

c

2 2

k

 2

B

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

I v



S v

 1 

e

   

I v

 

e

 

T B

  

T B

   1 

e

    

T B

 

e

    

  

T B

  

T B

   1 

e

    

T B

 

e

     1

T B

  

T B

  1

T B



T B

Optically thin Optically thick



Column Density

dI v dz

  

I v



j v

 is related to the level population Integration along the line of sight: Absorption coefficient -> Optical depth  Level population -> Level column density N Total column density : Sum over all levels

Excitation Temperature T

ex In LTE there is one excitation temperature T ex 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 T kin One observed transition and adopting a value for T kin gives all level populations -> Total column density N

Measuring Kinetic Temperature T

kin  1. Optically thick transitions:   1

T B



T B

2. Line ratios e.g. 13 CO(2-1) / 13 CO(1-0) 3. Rotation Diagrams e.g. NH 3 , CH 3 CCH, CH 3 CN

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

n crit

,21  

A

21  21   12 CO(1-0) 115.27 GHZ CS(2-1) 97.98 GHz HCN(1-0 88.63 GHz NH 3 (1,1) 23.694 GHZ 4 x 10 2 1 x 10 5 cm -3 cm -3 1 x 10 3 1 x 10 5 cm -3 Lowest critical density cm -3

12

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

Ubiquitous 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

12 CO(1-0) 115 GHz 1996 2500 pointings 0.1 K rms per channel Example Grid Brooks et al. 1998, PASA, 15, 202

Excitation Temperature

12 CO1-0 is optically thick T B = T ex = T kin

T ex

      

P

12 5.532

   0.837

      1

K

Use ‘xpeak’ in miriad to find P( 12 CO) 

Excitation Temperature Map

“Treasure Cluster”



Mass estimates from CO observations

Virial 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) 

M vir

 But … 1145   

V



kms

 1    

D pc

   

A

deg 2   0.5

Are molecular clouds virialised? What about external pressure?



Mass estimates from CO observations

X - Factor CO-to-H 2 Conversion factor

X CO

     Galactic Value: X CO ≈ 2.8 x 10 20 cm -2 K (km s -1 ) -1

 

H

2

Column Density to Mass

Mass = column density x spatial extent  

M M sun

  6.6

 10  24  

N

(

H

2 )

cm

 2    

D pc

  2  

A

deg 2   Average H 2 density  

n

(

H

2 )

cm

 3   1.5

 

M M sun

   

R pc

   3  Spherical with effective radius R 2R =  min +  maj Mass determined this way is often called the ‘CO mass’

But … To determine X co we need an independent measure of the mass of the cloud and the distance D in order to work out N(H 2 )

Independent Mass estimate for X

co 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 X

co

with caution

Problem 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 X co Localised values that depend on metallicity and galaxy type Sometimes you have little choice e.g. z  6

Chemical Characteristics of star-forming regions Pre-stellar core Cold envelope Warm inner envelope Hot core Outflow: direct impact Outflow: walls, entrainment PDR, compact HII regions Massive Disk Debris Disk Ions, Long Chains HC 5 N, DCO + Simple species, Heavy depletions CS, N 2 H + Evaporated species CH 3 OH, HCN Complex organics CH 3 OCH 3 , CH 3 CN Si- and S-species SiO, SO2 Evaporated ices CH 3 OH Ions, Radicals CN/HCN, CO + Ions, D-rich species, photoproductions HCO + , DCN, CN Dust, CO (E. F. van Dishoeck)

Example:

12

CO,

13

CO 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 13 CO, C 18 O 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 13 CO, C 18 O and CS integrated intensities and column densities with A between 0-32 mag of extinction.

v to a range in A v Integrated intensities I( 13 C0) = 1.88 + 0.72A

v I(C 18 0) = 0.07 + 0.10A

v I(CS) = 0.10 + 0.06A

v K km s -1 K km s -1 K km s -1 (A v ≤ 5 mag) (A v ≤ 15 mag) (A v ≤ 15 mag) Between 8 and 10 mag the 13 CO emission appears saturated Uncomfortable prediction of molecular emission and 0 mag

Integrated Intensity to Column Density

Only one transition is measured and an extrapolation to total column density is done by assuming a LTE population   Case Study 13 CO(2-1) 

N



P

10.58



e

10.58

 

T ex

 1   1  0.223

 

T

  1  4.227

 10 12

e

5.289

T ex

 

V

1 

e

 10.58

T ex



Id

  1.064

P



V N

 

T

 

Id

 cm  2 Integrated intensity W 13CO  

 We need a value for T ex -use value determined from 12 CO -assume a value (e.g. 35 K)

N

 

T

  

ex

 10 15

W

13

CO

cm  2 f(35 K) = 0.64

The value of T ex has a large impact on optical depth but not on column density

Back to the study by Lada et al. 1994 Assuming LTE For 13 CO and C 18 O: Based on 12 CO data: T ex = 10 K For CS: Subthermal excitation: T ex = 5 K Column Densities N( 13 C0) LTE /A v = 2.18 x 10 15 cm -2 mag -1 (A v ≤ 5 mag) N(C 18 N(CS) 0) LTE LTE /A v /A v = 2.29 x 10 = 4.5 x 10 11 14 cm cm -2 -2 mag -1 mag -1 (A (A v v ≤ 15 mag) ≤ 15 mag)

Not there yet!

Column density to H 2 density Gas-to-dust ratio of Savage & Drake (1978) N(H 2 ) = 0.94 x 10 21 Av cm -2 Which leads to: N( 13 C0)/N(H 2 ) = 4 x 10 5 (A v  5 mag)  Mass determined this way is often called the ‘LTE mass’

Depletion

Dust Extinction C 18 O Dust Emission Bianchi et al.

 T < 15 K and

n

> 10 5 cm -3 CO and CS freeze out onto the dust grains Species linked to molecular nitrogen are less affected E.g. NH 3 , N 2 H + , N 2 D + Alves et al.



Simple Line Ratio Analysis

Beam filling factor:

T B

    

T B

 1 

e

    

T B

 

e

    Ratio of lines with similar frequency (and hence similar  ) ->  cancels out Ratio of different species -> Optical Depth (if T ex e.g. 12 CO(1-0) / 13 CO(1-0) [ 12 CO/ 13 CO] ≈ 89 Ratio of different transitions (  e.g. C 18 O(2-1) / C 18 O(1-0) << 1) -> Excitation temperature

R

21 

T B

,21

T B

,10  4

e



E

21

T ex

,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 NH 3 Methyl Acetylene CH 2 C Methyl Cyanide CH 3 CN 2 H NH 3 (1,1): 18 hyperfine components mixed into 5 lines Fitting 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) T rot = T kin Garay, Brooks et al., 2002

Non-LTE Modelling

Additional Considerations - Stimulated emission - Radiative (photon) trapping Large Velocity Gradient (LVG) approximation - assume large-scale velocity gradient exists in cloud immediately escape Maximum Escape Probability models

Infall asymmetry

T ex (R 2 ) > T ex (R 1 ) T ex (B 2 ) > T ex (B 1 ) Optically thick line Optically thin line B 1 B 2 R 2 R 1 Infall region Static envelope Constant line-of-sight velocity

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) .

- M infall 10 -3 M sun yr -1 evidence for outflow - v outflow = 15 km s -1 Garay, Brooks, et al. 2003

Outflows

Bourke et al. 1997 

Outflows



Protostar IRAM 04191 in Taurus

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

Integrated Intensity to Column Density

  Case Study 13 CO(2-1)  

P

10.58



e

10.58

 

T ex

 1   1  0.223

  1

N

 

T

 4.227

 10 12

e

5.289

T ex

 

V

1 

e

 10.58

T ex



Id

  1.064

P



V N

 

T

 

Id

 cm  2 Integrated intensity W 13CO     

Q

8

k

 2 1.064

hc

3

Ag

2

e E rot

1 

e



h



kT ex kT ex



e h



kT ex

 1 10  6   1  

e h



kT B

 1   1 cm  3 sK  1 