Spectra & Composition

Section 5 of 8

Line Shifts

Part 4: The Doppler Shift — Reading Stellar Motion from Light

The Doppler Effect for Light

You’ve experienced the Doppler effect with sound: an ambulance siren rises in pitch as it approaches and drops as it recedes. Light does the same. When a source moves toward or away from an observer, the observed wavelength shifts.

Diagram of a star with an orbiting exoplanet and three corresponding spectra labeled blueshift, neutral, and redshift, showing the same absorption lines moving to shorter or longer wavelengths as the star moves toward or away from the viewer. — Figure 11The same spectral lines slide left for blueshift and right for redshift. Even tiny wavelength shifts let us infer line-of-sight motion and detect an orbiting companion.JWST/STScI

If a source has radial velocity $v_r$ (positive = receding, negative = approaching), the observed wavelength $\lambda_{\text{obs}}$ relates to the laboratory rest wavelength $\lambda_0$ by the Doppler shift formula:

Here $\Delta \lambda = \lambda_{\text{obs}} - \lambda_0$ , valid for $v_r \ll c$ (the non-relativistic limit), which covers virtually all stellar velocities in our galaxy.

Doppler shift

The change in observed wavelength caused by a source’s motion along the line of sight: $\Delta\lambda/\lambda_0 = v_r/c$ . Receding sources shift to longer wavelengths (redshift); approaching sources shift to shorter wavelengths (blueshift).

Interpretation — shifts tell you direction:

Redshift ( $\Delta \lambda \gt 0$ , $\lambda_{\text{obs}} \gt \lambda_0$ ): source receding, $v_r \gt 0$ .
Blueshift ( $\Delta \lambda \lt 0$ , $\lambda_{\text{obs}} \lt \lambda_0$ ): source approaching, $v_r \lt 0$ .

Redshift

A shift of spectral lines to longer wavelengths ( $\Delta\lambda \gt 0$ ), indicating the source is receding from the observer.

Blueshift

A shift of spectral lines to shorter wavelengths ( $\Delta\lambda \lt 0$ ), indicating the source is approaching the observer.

Typical stellar radial velocities are 10–100 km/s; the speed of light is 300,000 km/s. The ratio $v_r/c \sim 10^{-4}$ to $10^{-3}$ — tiny shifts, but measurable with precision spectrographs. For relativistic speeds (galaxies, jets, cosmology), a full relativistic formula is needed, and for the expanding universe the interpretation becomes cosmological redshift — a stretch of space itself, not simple motion. One crucial diagnostic: a real Doppler shift moves all lines by the same fractional amount $\Delta\lambda/\lambda_0$ . If only one line is shifted, it’s not motion — it’s a misidentification.

Worked Example 2A Star's Radial Velocity from Hα

Problem

A nearby star’s Hα absorption line is observed at $\lambda_{\text{obs}} = 656.5\ \text{nm}$ . The rest wavelength is $\lambda_0 = 656.3\ \text{nm}$ . Is the star approaching or receding, and at what speed?

StepDetermine direction

Since $\lambda_{\text{obs}} = 656.5\ \text{nm}$ exceeds $\lambda_0 = 656.3\ \text{nm}$ , the line is redshifted — the star is receding.

StepWavelength shift, then solve for $v_r$

$\Delta\lambda = 656.5 - 656.3 = 0.2\ \text{nm} \qquad v_r = c\,\frac{\Delta\lambda}{\lambda_0}$

StepPlug in numbers

$v_r = (3.0 \times 10^5\ \text{km/s}) \times \frac{0.2\ \text{nm}}{656.3\ \text{nm}} = (3.0 \times 10^5\ \text{km/s})(3.05 \times 10^{-4}) = 91\ \text{km/s}$

Dimensional check

$(\text{km/s}) \times (\text{nm}/\text{nm}) = \text{km/s}$ ✓.

Result

The star recedes at about 91 km/s — typical for the galactic disk. A shift of only 0.2 nm out of 656.3 nm (0.03%) translates to nearly 100 km/s because light is so fast. Modern spectrographs detect shifts 1,000 times smaller, reaching velocities of $\sim 1\ \text{m/s}$ — precise enough to feel the tug of an orbiting exoplanet.

Observable

Lines shifted from their laboratory wavelengths

Absorption lines in a stellar spectrum appear at slightly different wavelengths than laboratory measurements — and all lines shift by the same fractional amount.

Model

The non-relativistic Doppler effect

Motion along the line of sight shifts wavelengths by $\Delta\lambda/\lambda_0 = v_r/c$ , valid for $v_r \ll c$ .

Inference

The star's radial velocity

The measured shift gives $v_r$ — the component of motion directly toward or away from us. Blueshift means approach; redshift means recession.

What Doppler Shifts Cannot Tell Us

The Doppler effect measures only the radial velocity — motion along the line of sight. A star moving sideways across the sky (transverse velocity, or “proper motion”) doesn’t shift its spectral lines at all. To get the full 3D velocity we need both Doppler (radial) and astrometric (transverse) measurements — the latter requires tracking the star’s position over years.

Radial velocity

The component of a star’s velocity along the observer’s line of sight, measured from the Doppler shift of its spectral lines. The perpendicular (transverse) motion produces no shift and must be measured astrometrically.

Also, if a star is rotating, one limb moves toward you and the other away. This doesn’t shift the line center — it broadens the line symmetrically. Line broadening tells us about rotation speed and turbulence, but it’s a separate measurement from the Doppler shift of the line center.

Problem

A star’s Hβ line (rest 486.1 nm) is observed at 485.9 nm. Is the star approaching or receding?
Calculate the radial velocity for the star in question 1.
Can we determine a star’s rotation rate from a single Doppler shift? Why or why not?

Doppler Applied: Detecting an Unseen Companion

So far we’ve measured a single Doppler shift — one velocity at one moment. But suppose you observe the same star night after night and its radial velocity changes periodically: the Hα line oscillates back and forth around the rest wavelength on a regular cycle. What could cause that? If the star has an orbiting companion — another star or a planet — gravitational tugs pull it toward us during part of the orbit and away during the rest. The result is a periodic Doppler oscillation whose amplitude tells you how fast the star moves and whose period tells you the orbital period.

Worked Example 3A Star with a Periodic Wobble

Problem

You monitor a star over several weeks. Its Hα line ( $\lambda_0 = 656.3\ \text{nm}$ ) oscillates between 656.25 nm and 656.35 nm with a period of 4.0 days. (a) What is the velocity amplitude of the wobble? (b) What does the periodicity tell you?

Step(a) Convert the wavelength swing to a velocity

The line oscillates by $\Delta\lambda = \pm 0.05\ \text{nm}$ around rest, so $v_r = c\,\frac{\Delta\lambda}{\lambda_0} = (3.0 \times 10^5\ \text{km/s})\frac{0.05\ \text{nm}}{656.3\ \text{nm}} = \pm 23\ \text{km/s}$

Dimensional check

$(\text{km/s}) \times (\text{nm}/\text{nm}) = \text{km/s}$ ✓.

Result

(a) The radial velocity oscillates between $+23\ \text{km/s}$ (receding) and $-23\ \text{km/s}$ (approaching) — an amplitude of 23 km/s. (b) A periodic oscillation means the star orbits an unseen companion; the 4.0-day period is the orbital period. This is how the first exoplanet around a Sun-like star, 51 Pegasi b, was found in 1995 — its host wobbled with a 56 m/s amplitude over 4.23 days, 400 times smaller than this example. Repeated Doppler measurements reveal orbits, and orbits — through Kepler’s laws — give masses. This is the bridge to Lecture 4.

Part 4 takeaway: the Doppler shift formula $\Delta\lambda/\lambda_0 = v_r/c$ converts tiny wavelength shifts into stellar velocities. Blueshifts mean approach; redshifts mean recession. This tool unlocks binary star masses in Lecture 4 and, on cosmological scales, reveals the expansion of the universe.