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)

We can derive the formal solution of the equation as

(2)

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

(3)

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)

and is the Planck Function. Because we do not consider the emission behind the medium which is not from the dust, we simply let . Finally we can reform the formula as below,

(5)

Then we can define the opacity

(6)

and we will get

(7)

In Rayleigh-Jeans Regime, , so

(8)

In practically observational view, the relation between monochromatic specific intensity and a Plank Function at is defined using emissivity or , namely . In the same way, we consider the Rayleigh-Jeans Regime, the opacity will be . Then we often use an exponential function to fit the data as

(9)

And , is the frequency where . 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 is a constant along the path, then we can reform Eq.(8) as

(10)

where is the total volume of the cloud, is the total dust mass, is the total number of dust grains, and is the cross section of one dust grain.
Because , , and so

(11)

and

(12)

When the redshift is non-negligible, we need to use luminosity distance to replace the distance 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)

and

(14)

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)

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)

Then, we can define , and then

(17)

This equation (actually this equation contains the K-correction as the 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)

We can also derive the relation between opacity and mass-absorption coefficient, which in fact can be rewritten from Eq.(8) as: , as below

(19)

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

(20)

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.

1 comment / Add your comment below

1. Louis says:

Thanks for using the time and effort to write something so interesting.

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