Atomic line formation and the curve of growth

Let N l be the column density in the absorbing lower level, with units per square metre. Define an integrated effective line cross section $\sigma_{\mathrm{int}}=\int\sigma_\nu d\nu$ in square-metre hertz units and a profile whose frequency integral is one. Then $\tau_\nu=N_l\sigma_{\mathrm{int}}\phi_\nu$. The effective cross section here assumes the net stimulated-emission correction is already specified or negligible; a lower-level column alone is not enough when the upper population changes that correction. The normalized profile has inverse-hertz units.

For the pure absorbing screen, the residual continuum-normalized intensity is the exponential of minus line optical depth. Its equivalent width is the frequency integral of the missing fraction and has hertz units. In a sufficiently narrow line, wavelength equivalent width equals frequency equivalent width times the squared central wavelength divided by c. The absolute Jacobian is required because frequency decreases when wavelength increases. Neither equivalent width is a line-centre depth, and an observed unresolved width also involves the instrumental response.

Thermal velocities produce a Gaussian core. Write its Doppler width as $\Delta\nu_D=\nu_0b/c$, where b is the Gaussian Doppler velocity parameter rather than its one-dimensional rms value. For independent thermal and Gaussian microscopic turbulent velocities, $b^2=2k_BT/m+b_{\mathrm{turb}}^2$. The turbulent parameter uses the same Doppler convention. Different ion masses help separate thermal from shared velocity broadening, but unresolved bulk components can complicate that interpretation. A single wide line does not uniquely determine its gas temperature.

Finite upper-state lifetimes produce Lorentzian natural wings; collisions can add pressure broadening under an appropriate impact approximation. Convolving Gaussian velocities with a Lorentz profile gives a Voigt profile, while macroscopic velocity structure can instead produce displaced components. These are line-shape mechanisms around a transition frequency, not the broad continuum spectral slopes produced by free-free or synchrotron radiation. Natural and collisional widths also need not be negligible merely because an observed core appears approximately Gaussian at the available spectral resolution.

In the weak regime, expanding the attenuation exponential gives equivalent width proportional to lower-level column, independent of how a normalized profile distributes its small opacity. Once the core is black, adding column mostly widens the opaque frequency interval, producing slow square-root-of-logarithm growth for a Gaussian. At still larger columns, Lorentz wings can dominate and give equivalent width proportional to square root of column at fixed atomic and broadening parameters. These regimes explain why a deep line can be much less informative about column than a shallow unsaturated transition.

Several transitions with different strengths can reveal hidden saturation because their equivalent-width ratios cease to follow their weak-line strengths. Partial covering, line emission filling the trough, uncertain continua, velocity blends, and instrumental convolution must still be modeled. Integrated equivalent width is conserved by a normalized instrumental convolution of a well-defined isolated profile, but a finite integration window or blends can spoil a practical measurement. The curve of growth is therefore a conditional inference tool: state the background geometry, atomic population assumptions, and profile family before extracting an abundance.