Separate line position from line width

Specify the light before measuring the line

Wavelength $\lambda$ is the spacing between neighboring wave crests. We express it in nanometres. Frequency counts crests passing per second, and their product is light speed. Atomic energy differences select which photon energies can be absorbed.

Specific intensity $I_\lambda$ is energy per projected area, time, wavelength and solid angle along a chosen direction. It describes one beam, which is what the slab calculation follows. Divide it by the adjacent continuum $I_{\lambda,c}$ to make a dimensionless spectrum whose continuum lies at one.

Reading: Condon & Ransom, Essential Radio Astronomy, Appendix E

Turn common motion into a wavelength shift

Let $\lambda_0$ be a transition’s wavelength in the resting gas. Recession at radial velocity $v_r$ increases the arrival interval between crests. At speeds small compared with light speed, retaining only the first power of the speed ratio gives the expression below.

For $\lambda_0=500\,\mathrm{nm}$ and $v_r=12\,\mathrm{km\,s^{-1}}$, divide the speed by $299792.458\,\mathrm{km\,s^{-1}}$ and multiply by the rest wavelength. The shift is about $0.0200\,\mathrm{nm}$, placing the center near $500.0200\,\mathrm{nm}$. A longer-wavelength line moves farther in nanometres for the same velocity.

Add many atomic speeds to broaden the line

Temperature does not give every atom the same speed. The line-of-sight thermal velocity is approximately Gaussian, with standard deviation $\sigma_v$. It grows as the square root of temperature and decreases as the square root of atomic mass. Slightly different Doppler shifts then combine into a broadened profile.

The conventional Doppler parameter $b$ is square-root two times that standard deviation. A prescribed small-scale random speed $\xi$ adds in quadrature if it uses the same Gaussian convention. With mass 56 atomic mass units, temperature $4500\,\mathrm K$ and $\xi=2\,\mathrm{km\,s^{-1}}$, the model has $b\simeq2.31\,\mathrm{km\,s^{-1}}$. Its transitions are artificial so that individual effects can be separated.

Include the instrument before interpreting width

Resolving power $R$ divides wavelength by the instrument’s full width at half maximum. At $500\,\mathrm{nm}$ and $R=60000$, that width is $0.00833\,\mathrm{nm}$. It is already broader than the atomic-motion scale in this model.

A prediction of the measured profile must therefore include the calibrated instrument response. Otherwise an apparent width can be incorrectly assigned to the gas. The next chapter calculates the intensity profile before that final spreading step.