Motion Along the Line of Sight: Doppler Shifts and Radial Velocities

Why line-of-sight motion matters

Many astronomical motions are not directly visible as objects sliding across the sky. What we often measure most precisely is motion toward or away from us along the line of sight. This component is called the radial velocity. Radial velocity is central to problems like: measuring binary star orbits, detecting exoplanets via stellar “wobble,” mapping the rotation of galaxies, and determining how gas moves in stellar winds or accretion disks.

The key tool is the Doppler effect: motion along the line of sight shifts the wavelengths (or frequencies) of spectral features. If we can measure how far a known spectral line is shifted relative to its rest wavelength, we can infer the radial velocity that produced that shift. This chapter focuses on how Doppler shifts translate into radial velocities, what assumptions are involved, and how to compute and interpret results in practical data workflows.

Doppler shifts: wavelength, frequency, and sign conventions

Light can be described by wavelength (λ) or frequency (ν). They are related by c = λν, where c is the speed of light. A Doppler shift changes the observed wavelength and frequency when the source has a velocity component along the observer’s line of sight.

In astronomy, the most common convention is:

• Redshift: lines move to longer wavelengths (Δλ > 0). Typically indicates the source is receding (positive radial velocity).
• Blueshift: lines move to shorter wavelengths (Δλ < 0). Typically indicates the source is approaching (negative radial velocity).

Be careful: some subfields define positive velocity differently, and some software packages may use opposite signs. Always check the convention in your dataset or tool.

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Non-relativistic approximation (small velocities)

For speeds much smaller than c (a good approximation for most stars in the Milky Way and many exoplanet applications), the Doppler shift can be written as:

v_r ≈ c * (Δλ / λ0) = c * ((λ_obs - λ0) / λ0)

Here λ0 is the rest wavelength of the spectral line (measured in the laboratory), and λ_obs is the observed wavelength. This approximation is accurate when |v_r| ≪ c (for example, tens of km/s or even a few hundred km/s).

Relativistic Doppler formula (higher velocities)

When velocities become a non-negligible fraction of c (e.g., some galaxies, jets, or compact-object systems), use the relativistic relation. For purely radial motion:

λ_obs / λ0 = sqrt((1 + β) / (1 - β))   where β = v_r / c

Solving for v_r gives:

β = ( (λ_obs/λ0)^2 - 1 ) / ( (λ_obs/λ0)^2 + 1 )

In many practical stellar radial-velocity contexts, the non-relativistic formula is sufficient, but it is good practice to know when it might fail. A quick check: if |v_r| > 0.01c (~3000 km/s), relativistic corrections are no longer tiny.

From a shifted line to a velocity: a concrete example

Suppose you observe the Hα line (rest wavelength λ0 = 656.28 nm) in a star’s spectrum. You measure the line center at λ_obs = 656.58 nm.

Compute the shift:

Δλ = 656.58 - 656.28 = 0.30 nm

Compute radial velocity (non-relativistic):

v_r ≈ c * (Δλ/λ0) ≈ 299792 km/s * (0.30/656.28)

Numerically:

0.30/656.28 ≈ 4.57e-4  →  v_r ≈ 137 km/s

The line is redshifted, so the star is receding at about +137 km/s (using the common sign convention).

What you actually measure: line centers, profiles, and templates

In real data, spectral lines are not infinitely thin. They have shapes (profiles) broadened by temperature, pressure, rotation, turbulence, and instrumental resolution. Measuring “the wavelength of a line” therefore means choosing a method to estimate the line’s central wavelength or overall shift.

Method 1: centroid or minimum/maximum location

For an absorption line, you might take the wavelength of minimum flux as the line center. For an emission line, you might take the wavelength of maximum flux. This is simple but can be sensitive to noise, blending with nearby lines, and asymmetries in the profile.

Method 2: fit a model profile

A common approach is to fit a Gaussian (or Voigt) profile to the line and take the fitted center parameter as λ_obs. This is more robust than picking the minimum/maximum pixel, and it naturally provides an uncertainty estimate from the fit covariance.

However, a Gaussian is not always physically correct. Rotational broadening, pressure broadening, and instrumental line spread functions can produce non-Gaussian shapes. The goal is not perfect physics; it is a stable estimate of the shift.

Method 3: cross-correlation with a template (multi-line RV)

High-precision radial velocities rarely rely on a single line. Instead, you compare the observed spectrum to a template spectrum (either a synthetic model or a high signal-to-noise observation of a similar star). You shift the template in velocity space and compute a cross-correlation; the shift that maximizes the correlation gives the radial velocity.

This method uses information from many lines at once, improving precision and reducing sensitivity to any single problematic line. It also naturally handles crowded spectra where individual lines are blended.

Step-by-step: measuring radial velocity from one absorption line

This workflow is appropriate for moderate-resolution spectra and educational datasets where you can clearly identify a line.

Step 1: choose a line with a known rest wavelength

Select a line that is strong, isolated, and well within the wavelength range of your spectrum. Avoid regions with heavy telluric absorption (Earth’s atmosphere) if possible. Record λ0 from a trusted line list.

Step 2: define a local wavelength window

Extract a small region around the expected line position. The window should include continuum on both sides of the line. If the window is too wide, unrelated features can bias the fit; too narrow, and you lose continuum reference.

Step 3: normalize the local continuum

Fit a simple continuum model (often a straight line) to the edges of the window and divide the flux by that model. This makes the line depth and shape easier to fit and reduces sensitivity to broad spectral slopes.

Step 4: fit a line profile

For an absorption line, a simple Gaussian model in normalized flux might look like:

F(λ) = 1 - A * exp( - (λ - μ)^2 / (2σ^2) )

Here μ is the fitted line center (your λ_obs estimate). Fit parameters (A, μ, σ) by least squares or another optimizer.

Step 5: compute v_r from the fitted center

Use:

v_r ≈ c * (μ - λ0) / λ0

Keep track of units. If λ is in nm or Å, the ratio is unitless, so v_r comes out in the same units as c (commonly km/s).

Step 6: estimate uncertainty

If your fit provides an uncertainty σ_μ on μ, propagate it to velocity:

σ_v ≈ c * (σ_μ / λ0)

This captures statistical uncertainty from the fit. It does not include systematic effects like wavelength calibration errors or template mismatch.

Step-by-step: measuring radial velocity by cross-correlation

This workflow is typical when you want a single RV from many lines.

Step 1: prepare the observed spectrum

• Mask bad pixels and strong telluric regions if known.
• Continuum-normalize (often with a low-order polynomial or spline).
• Optionally resample to a uniform grid in ln(λ), which makes Doppler shifts correspond to constant pixel shifts in velocity space.

Step 2: choose a template

The template should match the star’s spectral type and resolution as closely as possible. If the template has much higher resolution, convolve it to match the instrument’s line spread function; if it has different sampling, resample it to the same grid.

Step 3: compute the cross-correlation function (CCF)

Shift the template by trial velocities v and compute a similarity metric with the observed spectrum. Conceptually:

CCF(v) = Σ [ F_obs(λ_i) * F_temp(λ_i shifted by v) ]

The velocity that maximizes CCF(v) is the best-fit radial velocity.

Step 4: fit the CCF peak

The CCF is sampled at discrete trial velocities. Fit a parabola or Gaussian around the peak to estimate the maximum more precisely than the grid spacing.

Step 5: uncertainty and quality checks

CCF peak width and height provide information about precision and reliability. A broad, shallow peak can indicate low signal-to-noise, fast rotation (broadened lines), or template mismatch. Inspect residuals and consider repeating with a different template or masking problematic regions.

Wavelength calibration and reference frames: what “radial velocity” is relative to

A measured Doppler shift is only meaningful if the wavelength scale is accurate. In practice, the wavelength solution is established using calibration sources (e.g., arc lamps) or stable reference techniques. Even with a good calibration, the velocity you compute is initially relative to the observatory and the time of observation.

Topocentric, heliocentric, and barycentric velocities

The Earth rotates and orbits the Sun, so the observer has a time-dependent velocity relative to the Solar System barycenter. If you do not correct for this, you can introduce apparent velocity shifts of up to about ±30 km/s from Earth’s orbital motion, plus up to ~0.5 km/s from Earth’s rotation (depending on latitude and target position).

Common frames:

• Topocentric: as observed at the telescope, no correction for Earth’s motion.
• Heliocentric: corrected to the Sun’s center (older but still used).
• Barycentric: corrected to the Solar System barycenter (standard for precision work).

In many datasets, the barycentric correction is provided or can be computed from observation time, observatory location, and target coordinates. When comparing radial velocities from different nights or instruments, ensure they are in the same reference frame.

Systemic velocity vs. relative velocity

In binary stars or exoplanet RV searches, you often care about changes in velocity over time more than the absolute value. The measured RV time series can be modeled as:

v(t) = γ + Δv_orbit(t)

Here γ is the systemic velocity (the center-of-mass velocity relative to the chosen reference frame), and Δv_orbit(t) is the time-varying orbital component. Relative RV precision can be much better than absolute accuracy if the instrument is stable.

Interpreting radial velocities in common scenarios

Single star kinematics

A single RV measurement contributes one component of the star’s 3D space motion. By itself, it does not tell you the full velocity vector, but it is essential for mapping motions in the Milky Way and identifying moving groups. When combined with proper motion and distance, you can reconstruct the full space velocity; the radial component is often the limiting factor in accuracy if the RV is poorly measured.

Spectroscopic binaries

If a star is in a binary system and the orbit is oriented such that there is line-of-sight motion, the RV varies periodically. In a single-lined spectroscopic binary (SB1), you see lines from only one star; in a double-lined system (SB2), you see two sets of lines shifting in opposite directions.

For circular orbits, the RV curve is approximately sinusoidal. For eccentric orbits, the curve becomes asymmetric, with faster changes near periastron. The amplitude of the RV variation depends on the companion mass, orbital period, and inclination (the tilt of the orbit relative to the sky plane). Because inclination is often unknown, RV alone typically yields a minimum companion mass.

Exoplanet radial velocity signals

Planets induce small stellar RV variations: from a few m/s for Earth-like planets around Sun-like stars to tens or hundreds of m/s for close-in giant planets. The same Doppler principles apply, but the practical challenge is precision and controlling systematics.

Two important interpretation points:

• Stellar activity (spots, plages, oscillations) can distort line profiles and mimic RV shifts. This is not just extra noise; it can create coherent signals near the stellar rotation period.
• Instrument stability and wavelength reference are critical. Small drifts can look like long-period planets if not tracked.

Galaxy rotation and gas kinematics

In galaxies, Doppler shifts of emission lines from gas (e.g., Hα, [O III]) can map rotation curves and internal motions. Here, you often measure velocities across many spatial positions (a long-slit spectrum or an integral field unit). The observed velocity field depends on the galaxy’s inclination: an edge-on disk shows large line-of-sight velocities; a face-on disk shows small ones even if the true rotation speed is high.

Common pitfalls and how to avoid them

Misidentifying spectral lines

If you choose the wrong line identification, the inferred velocity can be wildly incorrect. Use multiple lines when possible: a consistent velocity from several independent lines is a strong check. In crowded spectra, prefer cross-correlation or multi-line fitting.

Blended lines and asymmetries

Two nearby lines can merge into one feature at moderate resolution, shifting the apparent center. Similarly, stellar winds or disks can produce asymmetric emission profiles. In these cases, a single Gaussian center may not represent a physical bulk velocity. Consider fitting multiple components or using a template approach.

Telluric contamination

Earth’s atmosphere imprints absorption features that do not share the star’s Doppler shift. If a stellar line overlaps a telluric band, the measured center can be biased. Mask known telluric regions or use telluric correction methods when available.

Wavelength solution errors

A small wavelength calibration error translates directly into a velocity error. A useful scale: at 500 nm, a 0.001 nm error corresponds to about 0.6 km/s. For high-precision RV, calibration and instrumental drift tracking dominate the error budget.

Using air vs. vacuum wavelengths inconsistently

Some line lists quote wavelengths in air, others in vacuum. The difference is small but can matter for precision work. Ensure λ0 and λ_obs are defined in the same convention, especially in the optical where many instruments historically used air wavelengths.

Worked mini-exercise: sanity checks on an RV measurement

After computing an RV, apply quick checks before trusting it:

• Magnitude check: Is the velocity plausible for the object class? A nearby star with +20,000 km/s would be suspicious; a galaxy with +20,000 km/s might be plausible depending on context.
• Multi-line consistency: Measure RV from several lines. Do they agree within uncertainties?
• Frame check: Is the value topocentric, heliocentric, or barycentric? If comparing to catalog values, match frames.
• Residual inspection: If you fit a line or template, inspect residuals. Structured residuals can indicate blends, cosmic rays, or continuum issues.

Velocity resolution: how instrument resolution limits RV precision

Spectral resolving power is R = λ/Δλ, where Δλ is the smallest wavelength difference the instrument can separate. A rough velocity scale associated with resolution is:

Δv_res ≈ c / R

For example:

• R = 2,000 → Δv_res ~ 150 km/s
• R = 20,000 → Δv_res ~ 15 km/s
• R = 100,000 → Δv_res ~ 3 km/s

Importantly, RV precision can be better than Δv_res when many lines are used and signal-to-noise is high, because you can localize line centers to a fraction of a resolution element. But resolution sets the scale of how sharp the information is, and low resolution quickly limits what you can do.

Connecting Doppler shifts to time-domain measurements

Radial velocity becomes especially powerful when measured as a function of time. A sequence of spectra yields v(t), which can be analyzed for periodicities (planets, binaries), trends (long-period companions), or stochastic variability (pulsations, activity). The key difference from photometric time series is that the “signal” is encoded in line positions and shapes rather than brightness. In practice, you often track both: RV shifts (line position changes) and line-shape indicators (asymmetry metrics) to separate true center-of-mass motion from stellar surface effects.