Some notes on dust emission, dust mass, etc

We start from the basic formula from the theory of radiative transfer, namely the formal solution of the radiative transfer equation,

(1)   \begin{equation*} {dI_\nu \over d\tau_\nu} = -I_\nu+S_\nu . \end{equation*}

We can derive the formal solution of the equation as

(2)   \begin{equation*}  I_\nu (\tau_\nu)=I_\nu(0)e^{-\tau_\nu}+\int^{\tau_\nu}_0 e^{-(\tau_\nu-\tau_\nu ')}S_\nu(\tau_\nu ')d\tau_\nu '. \end{equation*}

If we now consider a CONSTANT SOURCE FUNCTION S_\nu, then Eq.(2) gives the solution

(3)   \begin{equation*} I_\nu(\tau_\nu)=I_\nu(0)e^{-\tau_\nu}+S_\nu(1-e^{-\tau_\nu}). \end{equation*}

Here we use Kirchhoff’s Law in thermal radiation for further discussions, regarding the dust as a blackbody. So we can rewrite the Eq.(3) as

(4)   \begin{equation*} I_\nu(\tau_\nu)=I_\nu(0)e^{-\tau_\nu}+B_\nu(1-e^{-\tau_\nu}), \end{equation*}

and B_\nu is the Planck Function. Because we do not consider the emission behind the medium which is not from the dust, we simply let I_\nu(0)=0. Finally we can reform the formula as below,

(5)   \begin{equation*} I_\nu(\tau_\nu)=B_\nu(1-e^{-\tau_\nu}), \end{equation*}

Then we can define the opacity

(6)   \begin{equation*} \tau_\nu=\int^s_0 \kappa_\nu \rho ds, \end{equation*}

and we will get

(7)   \begin{equation*} I_\nu(\tau_\nu)=B_\nu(1-e^{\int^0_s \kappa_\nu \rho ds}). \end{equation*}

In Rayleigh-Jeans Regime, \tau_\nu \ll 1, so

(8)   \begin{equation*} I_\nu(\tau_\nu) \approx B_\nu \tau_\nu = B_\nu \int^s_0 \kappa_\nu \rho ds. \end{equation*}

In practically observational view, the relation between monochromatic specific intensity I_\nu and a Plank Function at T_d is defined using emissivity \epsilon_\nu or Q_\nu, namely I_\nu=Q_\nu B_\nu (T_d). In the same way, we consider the Rayleigh-Jeans Regime, the opacity will be \tau_\nu \approx Q_\nu. Then we often use an exponential function to fit the data as

(9)   \begin{equation*} I_\nu(\tau_\nu) = \tau_0 (\nu / \nu_0)^\beta B_\nu. \end{equation*}

And \tau_0=1, \nu_0 is the frequency where \tau_0=1. \beta is the so-called emissivity index, which is about 1.5~2, see L. Dunne et al., 2001 (1). For big dust grains, it tends to be 2 while for smaller ones, the value becomes lower.
If we assume \kappa_\nu is a constant along the path, then we can reform Eq.(8) as

(10)   \begin{equation*} I_\nu(\tau_\nu) = B_\nu \kappa_\nu \rho s = B_\nu \kappa_\nu \rho V {S \over V}={B_\nu \kappa_\nu M_d \over N \sigma}, \end{equation*}

where V is the total volume of the cloud, M_d is the total dust mass, N is the total number of dust grains, and \sigma is the cross section of one dust grain.
Because S_\nu=I_\nu \Omega_s, \Omega_s=N\sigma/D^2, and L_\nu=4 \pi D^2 S_\nu so

(11)   \begin{equation*} S_\nu=B_\nu \kappa_\nu M_d {\Omega_s \over N \sigma}=B_\nu \kappa_\nu M_d {N \sigma \over D^2 N \sigma} = {B_\nu \kappa_\nu M_d \over D^2} \end{equation*}

and

(12)   \begin{equation*} L_\nu=4 \pi D^2 S_\nu = 4 \pi B_\nu \kappa_\nu M_d. \end{equation*}

When the redshift is non-negligible, we need to use luminosity distance D_L to replace the distance D used above. Thus the flux and luminosity should be corrected (called K-correction). Because the energy of the source is constant. So we use

(13)   \begin{equation*} L_{\nu_e} {\nu_e} =L_{\nu_o} {\nu_o},\quad 1+z={\nu_e} / {\nu_o} \Rightarrow L_{\nu_o}=(1+z)L_{\nu_e}, \end{equation*}

and

(14)   \begin{equation*} \quad D_L=\sqrt{L_{\nu_o}/4\pi}, \end{equation*}

to correct the formula above, where e stands for the rest frame quantity and o stands for the observed quantity. Then we reform the formula Eq. (11) and Eq. (12) as below,

(15)   \begin{equation*} S_\nu={(1+z) \over {D_L}^2} B_\nu \kappa_\nu M_d, \quad L_\nu=4 \pi B_\nu \kappa_\nu M_d. \end{equation*}

The left one above is the formula (1) in Beelen et al., 2006 (3). IF we want to get the far-infrared (FIR hereafter) luminosity, we should simply integrate over the FIR (cover the whole dust continuum) as,

(16)   \begin{equation*} L_{FIR}=4 \pi M_{dust} \int{\kappa_\nu B_\nu(T_d) d\nu} . \end{equation*}

Then, we can define L_\nu= (L f_\nu) / ( \int f_\nu' d\nu '), and then

(17)   \begin{equation*} {L f_\nu \over \int f_\nu' d\nu '}= B_\nu \kappa_\nu M_d. \end{equation*}

This equation (actually this equation contains the K-correction as the L_\nu is defined in the way above) is the Eq. (2) in Andrew B. Blain et al., 2002 (2).
Additionally, we can use Eq. (9) to reform the monochromatic flux:

(18)   \begin{equation*} S_\nu = I_\nu \Omega_s=B_\nu {({\nu \over \nu_0})^\beta} {N \sigma \over D^2} = N {({\nu \over \nu_0})^\beta} {\sigma \over D^2} B_\nu . \end{equation*}

We can also derive the relation between opacity and mass-absorption coefficient, which in fact can be rewritten from Eq.(8) as: \kappa_\nu=\kappa_0{(\nu / \nu_0)}^\beta, as below

(19)   \begin{equation*} \tau_\nu=\kappa \rho s = Q_\nu \Rightarrow \kappa_\nu=\tau_\nu / (\rho s)= {Q_\nu \sigma \over \rho V} = {3 \over 4} {Q_\nu \over a \rho} . \end{equation*}

This Eq.(19) is often used in observation. Using Eq.(11) we can express dust mass with the quantities above as

(20)   \begin{equation*} M_d={S_\nu D^2 \over B_\nu \kappa_\nu}={4 a \rho D^2 S_\nu \over 3 Q_\nu B_\nu} \end{equation*}

This formula is often used to calculate the dust mass from FIR/submm radiation.

References

[1] L. Dunne and S. Eales, 2001, MNRAS, 327, 697,
[2] Blain, A. W., Smail, I., Ivison, R. J., Kneib, J.P., Frayer, D. T. 2002, Phys. Rep., 369, 111.
[3] Beelen, A., Cox, P., Benford, D. J., et al. 2006, ApJ, 642, 694.

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