---
title: "Separate line position from line width"
description: "Define vacuum wavelength, radial velocity and intensity, then derive a first Doppler estimate and calculate thermal and instrumental widths."
canonical_url: "https://sandboxphysics.com/stories/stellar-spectroscopy/wavelength-motion-and-width/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/stellar-spectroscopy/wavelength-motion-and-width/"
last_modified: "2026-10-28"
---

[Stellar spectra from dark lines to abundance models](https://sandboxphysics.com/stories/stellar-spectroscopy/)

AC03 · L1

# Separate line position from line width

Sandbox Physics · 2026-10-28

A layer moving together shifts its whole line pattern. Atoms with different random speeds spread that pattern out. Separating these motions gives us two distinct questions: where the line sits and how wide it is.

## Before you begin

Use wavelength, frequency, ratios and square roots. The companion concept article introduces absorption. All wavelengths here are in vacuum; positive radial velocity means motion away from the observer.

## 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.

$$
c=\lambda\nu,\qquad E=h\nu=hc/\lambda
$$

Planck’s constant converts frequency into photon energy. A fixed atomic energy difference selects a fixed rest wavelength.

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.

$$
\frac{\lambda_c-\lambda_0}{\lambda_0}\simeq\frac{v_r}{c},\qquad |v_r|\ll c
$$

This first-order form is adequate for the teaching model’s tens of kilometres per second. High speeds need the relativistic relation. Observations also require a stated reference frame and observer-motion correction.

## 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.

$$
\sigma_v^2=\frac{kT}{m_a},\quad b^2=\frac{2kT}{m_a}+\xi^2,\quad \Delta\lambda_D=\frac{\lambda_0 b}{c}
$$

Use kilograms and kelvin to obtain speed in SI units before converting. Doppler width is the scale in the Gaussian exponent; full width at half maximum uses a different numerical convention.

## 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.

$$
\Delta\lambda_{\rm FWHM}=\frac{\lambda}{R},\quad \sigma_{\rm inst}=\frac{\Delta\lambda_{\rm FWHM}}{\sqrt{8\ln2}}
$$

The second equality assumes a Gaussian instrument profile. Real wings require an appropriate calibration.

Saved default prediction for the teaching slab

Uniform layer, fixed ion stage and artificial transitions; no random noise. The saved calculation uses 721 wavelengths and a normalized Gaussian instrument response.

| Layer temperature | $4500\,\mathrm K$ |
| --- | --- |
| Column / reference | $N/N_0=1$ |
| Receding velocity | $12\,\mathrm{km\,s^{-1}}$ |
| Resolving power | $R=60000$ |
| Strong-line center | $500.0200\,\mathrm{nm}$ |
| Total equivalent width | $11.952\,\mathrm{pm}$ |
| Layer / background source ratio | $0.201$ |

## [Separate common motion from spreading](https://sandboxphysics.com/labs/stellar-absorption-lines/)

1. Hold temperature and column fixed. Drag the motion handle and follow both line centers.
2. Switch to the low-resolution preset and compare the center with the full shape.
3. Predict the effect of changing temperature on the center and width, then try it.

## Exercises with explanations

1. Two lines at five hundred and six hundred nanometres recede together. Which moves farther in nanometres?

   Read the explanation

   The six-hundred-nanometre line moves twenty percent farther because the fractional wavelength shift is shared.

2. If thermal motion dominates, how does quadrupling temperature change Doppler width?

   Read the explanation

   It doubles the width. A fixed microturbulent component or dominant instrumental response reduces the relative change in the observed width.

[Previous: What dark lines in starlight tell us](https://sandboxphysics.com/stories/stellar-spectroscopy/why-starlight-has-dark-lines/) [Next: Put absorption and emission in one equation](https://sandboxphysics.com/stories/stellar-spectroscopy/absorption-emission-and-growth/)

## Sources and calculation scope

1. [Krumholz, ASTR 4003/8003 (2019), Class 1, Radiative transfer I](https://www.mso.anu.edu.au/~krumholz/teaching/astr8003_s1_2019/notes1.pdf)

   Equations 24–28: integrating factor, formal solution and constant-source slab. Independently derived and checked by numerical transport here.

2. [Condon & Ransom, Essential Radio Astronomy, Appendix E](https://www.cv.nrao.edu/~sransom/web/A5.html)

   Transfer equation 2.27, LTE Kirchhoff relation 2.30 and Planck law 2.86; used at optical wavelengths with the full Planck function.