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Appendix C

Radiative Transfer for a Two-Level System

This appendix details how to derive the total column density of a linear rigid-rotor molecule, such as CO, from the observed brightness of its rotational transition. Whilst all the ma jor steps in the calculation have been presented, some intermediate results are omitted and the reader is directed to Richer (1991) or Bence (1996) for fuller derivations.

C.1

Rotational Molecular Emission

Line emission from linear rotor molecules, such as CO, is simple to analyse. The energies of the rotational eigenstates are specified by a single quantum number, J , and are quantized according to: EJ = J (J + 1) L2 = 2I 2I
2

=

J (J + 1)h10 , 2

(C.1)

where L is the quantized angular momentum of the molecule (L2 = J (J + 1) 2 ), I is the moment of inertia about the rotation axis (assumed constant i.e. there is no molecular distortion) and 10 is the frequency of lowest transition (from the J = 1 to ground state). Each energy state has a degeneracy gJ = (2J + 1). Purely rotational transitions with J = 1 are forbidden.

C.2

Radiative Transfer for a Two-Level System

I now consider emission in a two-level system from an upper J + 1 (denoted `u') to lower J (denoted `l') state. The upper and lower states have degeneracies, gu and gl respectively, and likewise number densities nu and nl . Three radiative processes can occur, spontaneous emission, stimulated emission and absorption, which are governed by the Einstein A and B coefficients. Spontaneous emission is controlled by the A coefficient, with a rate Aul nu . A
ul

quantifies the

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C. RADIATIVE TRANSFER FOR A TWO-LEVEL SYSTEM

coupling of the molecule to the incident radiation field and is related to µe , the molecule's rotational electric dipole moment, Aul = 16 3 2 µ 3 0 hc3 e J +1 2J + 3
3 ul .

(C.2)

Bul controls the stimulated emission process, where a photon induces an excited molecule to emit a photon coherently with the first. Likewise Blu moderates the absorption process that removes photons from the system. The propagation of radiation in such a system is governed by the equation of radiative transfer, which describes the variation in specific intensity, I , along the line of sight (with an element of path length ds) in a medium characterized by emissivity and opacity, and respectively: dI = - I + . ds We can relate and to the Einstein coefficients through


(C.3)

= =

hul Aul nu ( ) 4 hul (nl Blu - nu Bul )( ), c

(C.4) (C.5)

where ( ) is the normalized intrinsic linewidth. If we use the relations, Blu gl = Bul gu 3 8 hul Aul = Bul , c3 (C.6) (C.7)

required for detailed balancing in equilibrium and consistency with the Planck law, whilst defining the excitation temperature, Tex , of the transition through nu gu = exp nl gl we may obtain (following Richer, 1991):


-hul k Tex

,

(C.8)

= =



hc Aul nu (v ) 4 Aul c3 2J + 3 3 8 ul 2J + 1

(C.9) 1 - exp hul k Tex nl (v ), (C.10)

where the number densities are now written per unit velocity. Collisional processes also affect the level populations, which are moderated by the collisional coefficients, ul and lu (for the upwards and downwards transitions respectively), equal to the Maxwellian-averaged value of the collisional cross section times the particle velocity. The kinetic

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C.2 Radiative Transfer for a Two-Level System

temperature, T

kin

, relates the collisional coefficients to each other: lu gu hul = exp - ul gl k Tkin . (C.11)

In equilibrium and ignoring the stimulated emission and absorption processes, the upwards transition rate (per unit volume), R = n
tot nl lu

,

(C.12)

equals the downwards transition rate (also per unit volume), R = Aul nu + n where n
tot tot nu ul

,

(C.13)

is the H2 number density. Hence, nu (gu /gl ) exp(-hul /k T = nl 1 + (Aul /ul ntot )
kin

)

=

(gu /gl ) exp(-hul /k T 1 + (nc /ntot )

kin

)

,

(C.14)

where nc = Aul /ul is the so-called critical density, where the collisional de-excitation rate equals the spontaneous emission rate (see also Section 4.1). On integrating the equation of radiative transfer (C.3), we obtain: I = I0 e-


+ (1 - e- )S ,

(C.15)

where = L is the optical depth, S = / is the source function and I0 is the background radiation (i.e. the microwave background, which I neglect hereafter). Assuming local thermodynamic equilibrium (LTE, see Section 8.1), S B (T ) (the Planck function) and T =T C.10: T
mb ex

=T

kin

, in the optically-thin limit (

1), I = S . Hence, we can form an ex-

pression for the line brightness, incorporating the form of and from equations C.9 and = hAul 2 c Nu (v ), 8 k
mb

(C.16) is the main-

where Nu (u) is the column density per unit velocity in the upper J + 1 state and T

beam brightness temperature, the radiation brightness temperature received by the antenna, when corrected for atmospheric and telescope losses and corrected for the coupling to the main beam (i.e. our best estimate of the true source brightness temperature for sources approximately beam-sized). Integrating equation C.16 over velocity and the solid angle of the source (at a distance, d), we obtain an expression for the total column density in the upper state: N
tot u

=

3 0 k d2 2 2 µ2 10 e

2J + 3 (J + 1)2

T

mb

(v , ) dv d.
tot

(C.17)

To obtain the total number of molecules in all the different levels (N

) we require the fractional

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C. RADIATIVE TRANSFER FOR A TWO-LEVEL SYSTEM

occupancy of the upper state of interest, which in LTE is given by Boltzmann statistics: Nu gu exp(-Eu /k Tex ) = , Ntot Z where Z is the partition function: Z=
all states

(C.18)

gJ exp(-EJ /k Tex ).

(C.19)

Provided T

ex

h10 /k , the sum can be approximated by an integral and Z 2k Tex /h10 .

Thus, the column density in all the molecule's states becomes: N
tot all states

=

3 0 k 2 Tex d2 2 hµ2 10 (J + 1)2 e

exp[(J + 1)(J + 2)h10 /2k Tex ]

T

mb

(v , ) dv d. (C.20)

Following this derivation and using the transition frequencies from Table 4.1, the column densities for the HARP CO J = 3 2 transitions become: N (C18 O) = 5.21 в 10 N (13 CO) = 5.17 в 10 N (12 CO) = 4.71 в 10
12

cm cm cm

-2

12

-2

12

-2

Tex exp(-31. Tex exp(-31. Tex exp(-33.

/ 6 / 8 / 7

K K/Tex ) K K/Tex ) K K/Tex )

Tmb K km Tmb K km Tmb K km

dv s-1 dv s-1 dv s-1

, , .

(C.21) (C.22) (C.23)

292