Put absorption and emission in one equation

Choose the ray direction first

Let distance $s$ increase from the background toward the observer, from zero to $L$. The absorption coefficient $\alpha_\lambda$ specifies attenuation per length. The emission coefficient $j_\lambda$ specifies intensity added per length. The net change is emission minus absorption.

Define optical depth by $d\tau_\lambda=\alpha_\lambda ds$. It is dimensionless and increases toward the exit in our convention. The source function $S_\lambda=j_\lambda/\alpha_\lambda$ has intensity units. For a uniform LTE layer it equals $B_\lambda(T)$. LTE ties local atomic populations to local temperature; it does not require an entire star to be isothermal.

Reading: Krumholz, ASTR 4003/8003 (2019), Class 1, Radiative transfer I · Condon & Ransom, Essential Radio Astronomy, Appendix E

Integrate a uniform source

Multiply by $e^\tau$. The left side becomes the derivative of $I_\lambda e^\tau$. Integrate from entrance to exit, pull the constant source function outside the integral, then divide by the integrating factor.

The first resulting term is the attenuated background. The second is radiation supplied by the layer. The outgoing intensity lies between the incident intensity and the source function, approaching the latter as the layer becomes opaque.

Reading: Krumholz, ASTR 4003/8003 (2019), Class 1, Radiative transfer I

Check limits before calculating a spectrum

As $\tau\to0$, the incident background passes through. As $\tau\to\infty$, the output approaches the source function. In the isothermal case $S_\lambda=I_{\lambda,c}$, every optical depth gives the unchanged continuum. This is a particularly useful independent check.

For source-to-background ratio $r=0.2$, optical depths $0.1$, $1$ and $10$ give normalized intensities $0.924$, $0.494$ and $0.200$. Initial attenuation is nearly linear. The opaque line approaches the layer’s twenty-percent brightness.

Specify a consistent artificial atom

Define four levels before calculating the two line opacities. The ground level has zero energy. A common absorbing lower level lies $2\,\mathrm{eV}$ above it. Each line reaches a different upper level, separated from that lower level by its photon energy $hc/\lambda_{0,j}$, about $2.48\,\mathrm{eV}$. All statistical weights are one. The lower level’s excitation energy and the absorbed photon energy are distinct.

Add the four Boltzmann weights to form the partition function, the total weight used to normalize populations. The lower fraction is $p_l(T)$. The ion stage and total ion column remain fixed, while temperature redistributes the four populations. A real atom needs its actual level structure and transition data.

An upper-state atom can also undergo stimulated emission, adding light in the incoming direction. In LTE the upper-to-lower population ratio supplies an exponential, so net absorption includes the factor $f_j(T)$. Including this factor with the lower population makes the opacity consistent with the adopted Planck source function.

Use $T_0=4500\,\mathrm K$ as reference, $N/N_0$ for the total-ion column ratio and strengths $s_1=1$ and $s_2=0.42$. The Doppler parameter is $b(T)$. More net absorbers increase integrated opacity; spreading it over a wider velocity distribution lowers the center. These explicit ratios reproduce both reference and heated spectra.

Reading: Condon & Ransom, Essential Radio Astronomy, chapter 7.3 · Condon & Ransom, Essential Radio Astronomy, Appendix E

Integrate the missing light

For small optical depth, expand the exponential to first order and integrate the Gaussian. Equivalent width is proportional to $\tau_0\Delta\lambda_D$. At fixed temperature and velocity distribution, doubling column approximately doubles that area.

Once the center saturates, its deficit approaches $1-r$. Extra opacity mainly widens the opaque part, so equivalent width grows more slowly. This relation is the curve of growth. The model omits pressure-damped wings and therefore does not predict the damping branch of very strong real lines.