Radiative Transfer

Constructing synthetic observations enables direct comparison between modeled results and observational data (see review by Haworth et al., 2018). Both RT-tool and RT-synth algorithms are based on the method presented in Bisbas et al. (2017; 2021).

We solve the radiative transfer equation along the line-of-sight element \(dz\):

\[\frac{dI_{\nu}}{dz} = -\alpha_{\nu}I_{\nu} + \alpha_{\nu}S_{\nu} + \rho\kappa_{\nu,\text{dust}}(B_{\nu}(T_{\text{dust}}) - I_{\nu})\]

where \(I_{\nu}\) is the intensity, \(\alpha_{\nu}\) the absorption coefficient, \(S_{\nu}\) the source function, \(\rho\) the density, \(\kappa_{\nu,\text{dust}}\) the dust opacity, and \(B_{\nu}(T_{\text{dust}})\) the Planck function at dust temperature \(T_{\text{dust}}\).

For dust opacity at solar metallicity, we adopt:

\[\kappa_{\nu,{\rm dust}} = 0.1\times {\cal D}\left(\frac{\nu}{1000\,{\rm GHz}^2}\right)\,{\rm cm}^2\,{\rm g}^{-1}\]

per unit total gas and dust density (see Arzoumanian et al. 2011), where \(\cal D\) is the dust-to-gas mass ratio normalized to \(10^{-2}\).

The source function and absorption coefficient for transition \(i \rightarrow j\) are:

\[\begin{split}S_{\nu} &= \frac{2h\nu_{0}^{3}}{c^{2}}\frac{n_{i}g_{j}}{n_{j}g_{i} - n_{i}g_{j}} \\ \alpha_{\nu} &= \frac{c^{2}n_{i}A_{ji}}{8\pi\nu_{0}^{2}}\left\{\frac{n_{j}g_{i}}{n_{i}g_{j}} - 1\right\}\phi_{\nu}\end{split}\]

where \(\nu_{0}\) is the line center frequency, \(A_{ji}\) the Einstein A coefficient, \(n_{i}, n_{j}\) level populations, \(g_{i}, g_{j}\) statistical weights, and \(\phi_{\nu}\) the line profile.

The line profile assumes a Maxwellian velocity distribution:

\[\phi_{\nu} = \frac{1}{\sqrt{2\pi\sigma_{\nu}^{2}}}\exp\left\{-\frac{[(1+v_{\text{los}}/c)\nu - \nu_{0}]^{2}}{2\sigma_{\nu}^{2}}\right\}\]

with \(v_{\text{los}}\) the gas velocity along the line of sight and \(\sigma_{\nu}\) the dispersion:

\[\sigma_{\nu} = \frac{\nu_{0}}{c}\sqrt{\frac{k_{\text{B}}T_{\text{gas}}}{m_{\text{mol}}} + v_{\text{turb}}^{2}}\]

The formal solution is:

\[I_{\nu}(z) = I_{\nu}(0)e^{-\tau_{\nu}(z)} + \int_{0}^{\tau_{\nu}(z)}S_{\nu}(z')e^{-(\tau_{\nu}(z)-\tau_{\nu}(z'))}d\tau_{\nu}(z')\]

where \(\tau_{\nu} = \int_{0}^{z}\alpha_{\nu}(z')dz'\) is the optical depth.

The line intensity is obtained by subtracting dust continuum: \(I_{\nu,\text{line}} = I_{\nu} - I_{\nu,\text{dust}}\), as well as background emission.

The antenna temperature is:

\[T_{A} = \frac{c^{2}I_{\nu,\text{line}}}{2k_{\text{B}}\nu^{2}}\]

in units of [K], and the velocity-integrated intensity is:

\[W = \int T_{A}dv_{\text{los}}\]

in units of [K km/s].

For further reading on radiative transfer, the notes of C.P. Dullemond are recommended.