Toy radiative transfer

Two-layer atmosphere

A Planckian background radiation field passes through a single isothermal gas layer. The layer has a Gaussian opacity profile, so the line depth is controlled by the line-center optical depth.

gas layer 5000 K

background field 6000 K

Parameters

Thermal structure Background temperature 6000 K Gas temperature 5000 K

Opacity profile Line depth, τ₀ 1.20 Continuum depth, τ_c 0.02 Line center 589 nm Gaussian width, σ 24 nm

Emergent spectrum

Cool gas removes light near line center.

Equivalent width: 0 nm

Optical depth

τ_λ = τ_c + τ₀ exp[-(λ - λ₀)² / 2σ²]

Model notes

The radiation intensity at the surface of our simple stellar atmosphere model is computed as I_lambda = I_bg exp(-tau_lambda) + B_lambda(T_gas) (1 - exp(-tau_lambda)). With a cooler upper layer the line appears in absorption; if the gas is hotter than the background, the same model turns the line into emission.

This is the simplest stellar model that illustrates how spectral lines can change character between absorption and emission.