3.4 Doppler effect and radial velocity measurements

Doppler Effect and Radial Velocity in Astrophysics

The Doppler effect describes how the observed frequency (or wavelength) of a wave changes when the source and observer are in relative motion. In astrophysics, this single phenomenon unlocks an enormous range of measurements: stellar velocities, binary orbits, galaxy rotation, exoplanet detection, and the expansion of the universe itself.

Radial velocity is the component of an object's velocity directed along your line of sight. You can't measure it with an image, but you can measure it from the wavelength shifts it imprints on spectral lines. That's what makes the Doppler effect so central to observational astrophysics.

Doppler Effect Basics

When a light source moves toward you, each successive wavefront is emitted a little closer, compressing the wavelength. When it moves away, the wavefronts stretch out. The result:

• Blueshift: wavelength decreases, frequency increases (source approaching)
• Redshift: wavelength increases, frequency decreases (source receding)

For speeds much less than $c$, the non-relativistic Doppler formula is all you need:

$\frac{\Delta\lambda}{\lambda_{rest}} = \frac{v_r}{c}$

where $\Delta\lambda = \lambda_{observed} - \lambda_{rest}$ and $v_r$ is the radial velocity. Rearranging:

$v_r = c \left(\frac{\lambda_{observed} - \lambda_{rest}}{\lambda_{rest}}\right)$

A positive $v_r$ means the source is receding (redshift); negative means it's approaching (blueshift).

For sources moving at a significant fraction of $c$, you need the relativistic Doppler formula:

$\frac{\lambda_{observed}}{\lambda_{rest}} = \sqrt{\frac{1 + \beta}{1 - \beta}}, \quad \beta = \frac{v}{c}$

This matters for distant galaxies and quasars, but for most stellar work the non-relativistic version is sufficient.

Radial Velocity and Spectral Shifts

You measure radial velocity by comparing the observed wavelengths of known spectral lines against their laboratory (rest) wavelengths. Here's the process:

1. Record a high-resolution spectrum of the target star or galaxy.
2. Identify spectral lines with well-known rest wavelengths (e.g., hydrogen Balmer lines, the sodium D doublet, or the calcium H and K lines).
3. Measure the observed wavelength of each line.
4. Compute $\Delta\lambda / \lambda_{rest}$ for each line and convert to $v_r$ using the Doppler formula.
5. Average over many lines to improve precision and reduce the effect of line blending or noise.

A few concrete examples of spectral shifts in action:

• The Andromeda galaxy (M31) shows a blueshift corresponding to $v_r \approx -300 \text{ km/s}$, meaning it's heading toward the Milky Way.
• Most distant galaxies show redshifts, consistent with the expansion of the universe. A galaxy with $z = \Delta\lambda / \lambda_{rest} = 0.01$ is receding at roughly 3,000 km/s.

Measurement Techniques and Applications

Methods for Radial Velocity Measurement

High-resolution spectroscopy is the workhorse technique. Modern échelle spectrographs can resolve spectral lines finely enough to detect velocity shifts of order 1 m/s. Two key approaches:

• Cross-correlation: the observed spectrum is compared against a template (either a synthetic spectrum or a well-characterized reference star). The shift of the cross-correlation peak gives the radial velocity. This is more robust than measuring individual lines because it uses information from thousands of lines simultaneously.
• Simultaneous calibration: instruments like HARPS feed a thorium-argon lamp or a laser frequency comb through the spectrograph alongside the starlight. This tracks instrumental drift in real time, which is critical when you're chasing signals at the m/s level.

Fabry-Pérot interferometry uses the sharp transmission peaks of an optical cavity to measure tiny wavelength shifts. It's less common for stellar radial velocities but useful in specialized applications like mapping velocity fields across extended objects (nebulae, galaxies).

Several factors limit radial velocity precision:

• Instrumental stability: temperature changes and mechanical flexure shift the spectrum on the detector, mimicking a Doppler shift.
• Stellar activity: starspots, convective motions, and pulsations produce line-profile variations that can masquerade as radial velocity changes. This is a major challenge for exoplanet surveys.
• Accurate rest wavelengths: your measurement is only as good as your reference. Laser frequency combs now provide calibration accurate to better than 0.01 m/s.

Applications of the Doppler Effect

Stellar physics

• Measuring projected rotation rates ($v \sin i$) from the broadening of spectral lines. A rapidly rotating star produces broader, shallower lines.
• Detecting radial pulsations in variable stars. Cepheid variables, for example, show periodic radial velocity curves that correlate with their brightness variations.
• Studying stellar winds: P Cygni line profiles show blueshifted absorption from outflowing material alongside emission from the surrounding envelope.

Binary star systems

Spectroscopic binaries are identified by periodic Doppler shifts in their spectral lines. If both stars' lines are visible (double-lined spectroscopic binary, or SB2), you can determine the mass ratio directly. The system of Sirius A and its white dwarf companion Sirius B was characterized partly through radial velocity measurements.

Galactic dynamics

Rotation curves of spiral galaxies are built by measuring the radial velocity of gas and stars at different distances from the galactic center. The classic result: rotation curves stay flat at large radii instead of declining as Keplerian orbits would predict. This is one of the strongest pieces of evidence for dark matter.

Exoplanet detection (the radial velocity method)

A planet doesn't just orbit a star; the star also orbits the common center of mass. This produces a small, periodic radial velocity signal in the star's spectrum. The steps:

1. Monitor the star's radial velocity over many nights or months.
2. Look for a periodic signal. The period gives you the orbital period of the planet.
3. The amplitude of the signal, combined with the star's mass, gives the planet's minimum mass ($M_p \sin i$, since you can't determine the orbital inclination from radial velocity alone).
4. The shape of the velocity curve reveals the orbital eccentricity.

The first exoplanet found around a Sun-like star, 51 Pegasi b, was discovered this way in 1995. It produced a radial velocity signal of about 59 m/s with a period of 4.23 days, revealing a Jupiter-mass planet in a surprisingly tight orbit.

Cosmology

The Hubble-Lemaître law ($v = H_0 d$) relates a galaxy's recession velocity to its distance. Recession velocities come from redshift measurements, making the Doppler effect foundational to our understanding of cosmic expansion. At cosmological distances the redshift is better understood as the stretching of space itself rather than a simple kinematic Doppler shift, but the observational technique is the same: measure $\Delta\lambda / \lambda_{rest}$.