Measuring the dust mass from long wavelength observations

We begin with Tielens equation 5.1 which defines the dust optical depth, $$\tau = n_{\rm d}\,C_{\rm ext}\,L,$$ where $C_{\rm ext}$ is the extinction cross section, $n_{\rm d}$ is the number of dust grains per unit volume, and $L$ the pathlength. The book then goes on to define the extinction efficiency, $Q_{\rm ext}$, as the ratio of extinction corss section to geometric cross section and derive its wavelength dependence based on the optical properties of the dust (Mie theory).

The observational notation you will see in the literature is different and instead defines $$\tau = \rho_{\rm d}\kappa\,L$$ where $\rho_{\rm d}$ is the dust mass density and $\kappa$ is the mass absorption coefficient. $\kappa$ has cgs units of ${\rm cm}^2\,{\rm g}^{-1}$ and expresses the effective surface area for extinction per unit mass$^1$.

Dust grains interact most effectively with light of about the same wavelength of their size. As most grains are sub-micron in size, the extinction is high in the UV and optical but decreases toward the infrared. Longward of several hundred microns, dust emission is generally optically thin (a notable exception being the inner regions of circumstellar disks), and long wavelength observations provide a good measure of the dust mass.

For optically thin emission, the specific intensity is $$I_\nu = B_\nu(T_{\rm d})\,(1-e^{-\tau_\nu})\approx B_\nu(T_{\rm d})\,\tau_\nu,$$ where $B_\nu(T_{\rm d})$ is the Planck function at frequency $\nu$ for dust temperature $T_{\rm d}$. The flux is therefore $$ \begin{align} F_\nu &= \int_\Omega I_\nu\,d\Omega \\ &= \int_A B_\nu\,\tau_\nu\,dA/D^2 \\ &= \int_V B_\nu\,\rho_{\rm d}\kappa_\nu dV/D^2 \\ &= \kappa_\nu\,B_\nu\,M_{\rm d}/D^2.\\ \end{align} $$ The steps follow by converting the integral over solid angle to one over area (where $D$ is the distance) and then to include the pathlength to convert from area to volume. The final step assumes that $\kappa_\nu$ and $T_{\rm d}$ are uniform over the observed region.

As the emission is optically thin, the flux basically counts the number of grains or, rather, their surface area and we then convert to mass via $\kappa$. There is the usual inverse square dependence on distance and a dependence on dust temperature. Dust mass measurements are generally made at (sub-)millimeter wavelengths (e.g. SCUBA on the JCMT) and the Planck function is near the Rayleigh-Jeans regime, $B_\nu(T_{\rm d})\approx 2kT_{\rm d}\nu^2/c^2$, so the flux is only linearly dependent on temperature. The main uncertainty in the mass determination is most commonly in the dust opacity. A classic reference that is I often use is Hildebrand 1983 which gives the handy prescription for $\lambda > 250\,\mu$m, $$\kappa_\nu = 0.1\left(\nu\over 1200\,{\rm GHz}\right)\,{\rm cm}^2\,{\rm g}^{-1}.$$ Note that this implicitly includes a gas-to-dust ratio, $\rho_{\rm gas}/\rho_{\rm d}=100$, which is the typical usage except in (gas-poor) debris disks. More modern references with detailed modeling of different grain compositions and growth of icy mantles are Pollack et al. 1984 and Ossenkopf & Henning 1994.

The Hildebrand $\kappa$ has a linear dependence on frequency but this can also be parameterized (see Tielens equation 5.32), $$\kappa_\nu = \kappa_0\left(\nu\over \nu_0\right)^\beta.$$ This can be observationally determined by fitting a power law slope to the long wavelength SED, $F_\nu\propto \nu^{2+\beta}$ (assuming $kT_{\rm d}\ll h\nu$ for all frequencies) and provides a diagnostic of grain growth. A significant population of large grains means more emission at long wavelengths and a shallower slope. The Pollack reference above provides some model values and a more recent reference is Draine 2006. The typical value in the diffuse ISM is $\beta\approx 1.7$ but significantly lower values are observed in circumstellar disks, $\beta\approx 1$, showing the first step toward planets! [This was a subject of Sean Andrew's recent PhD thesis -- see Andrews & Williams 2005.]



$^1$ The conversion between the two definitions is $\kappa = n_{\rm d}\,C_{\rm ext}/\rho_{\rm d} = Q_{\rm ext}\,\pi a^2/m_{\rm gr} = 4Q_{\rm ext}/3a\rho_{\rm gr}$ where $m_{\rm gr}=4/3\pi a^3\rho_{\rm gr}$ is the mass of a dust grain, $a$ is its radius and $\rho_{\rm gr}$ its density. Note that the density here, $\rho_{\rm gr}$ is of an individual grain not the space density of dust, $\rho_{\rm d}$.

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