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.

Tagged: Astronomy, dust, ISM, Notes, radiation, Radiative transfer, Submm

Rhapsody of the Universe

Some notes on dust emission, dust mass, etc

References

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