Galaxies in Motion: Measuring Redshift and Estimating Recession Speed

What Redshift Measures in Galaxies

When you observe a galaxy spectrum, many recognizable spectral lines appear at wavelengths that are not exactly where they would be in a laboratory. For galaxies beyond the Local Group, the dominant effect is usually a systematic shift of the entire spectrum toward longer wavelengths. This shift is called redshift and is quantified by a dimensionless number z.

Redshift is defined by comparing an observed wavelength to a reference (rest) wavelength for the same transition:

z = (λ_obs − λ_rest) / λ_rest

Because z is dimensionless, it is convenient for comparing measurements across instruments and wavelength ranges. In practice, you rarely rely on a single line; you measure z from multiple features and combine them to reduce random error and to detect problems like misidentification or line blending.

Interpreting z: shift, not brightness

Redshift describes a stretching of wavelength (and corresponding reduction in photon energy), not a dimming. A galaxy can be highly redshifted and still appear bright if it is intrinsically luminous or gravitationally lensed. Conversely, a nearby galaxy can be faint if it is small or dust-obscured. Keep the measurement categories separate: redshift is about wavelength displacement; brightness-based methods are about flux.

From Redshift to Recession Speed: What You Can and Cannot Do

For small redshifts, it is common to convert z into an approximate recession speed along the line of sight. The simplest approximation is:

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v ≈ c z   (valid when z ≪ 1)

Here c is the speed of light. This approximation is widely used for nearby galaxies where z is a few thousandths to a few hundredths.

For larger redshifts, the relationship between z and speed is not linear. If you want a special-relativistic conversion between redshift and a Doppler speed (useful for some contexts, but not identical to cosmological expansion), you can use:

β = v/c = ((1+z)^2 − 1) / ((1+z)^2 + 1)

This formula maps z to a speed that never exceeds c. It is mathematically consistent for a pure Doppler shift in flat spacetime. However, galaxy redshifts at cosmological distances are primarily interpreted as the result of the expansion of space, not motion through space in the everyday sense. That means “recession speed” becomes model-dependent at high z. In this chapter, the practical goal is to measure z accurately and then compute a sensible recession-speed estimate appropriate to the regime you are working in.

Rule of thumb for choosing a conversion

• If z < 0.01, v ≈ cz is typically adequate for many applications.
• If 0.01 ≲ z ≲ 0.1, v ≈ cz is still often used, but you should be aware of the growing difference from the relativistic mapping; consider reporting both.
• If z ≳ 0.1, do not treat v = cz as a physical speed without stating the approximation; use a cosmology-aware interpretation if you need distances or expansion rates.

Choosing Spectral Features for Galaxy Redshift

Galaxies contain many stars and often gas. Their spectra can show both absorption lines (from stellar atmospheres) and emission lines (from ionized gas in star-forming regions or active galactic nuclei). The best features for redshift measurement are those that are strong, isolated, and easy to identify.

Common optical emission lines

• Hα (rest 656.28 nm): strong in star-forming galaxies; shifts into the near-infrared for moderate z.
• Hβ (486.13 nm): often paired with [O III].
• [O III] (495.9 nm and 500.7 nm): bright doublet; the known separation helps confirm identification.
• [O II] (372.7 nm doublet): useful at higher z when Hα is out of the optical band.
• [N II] (654.8 nm and 658.3 nm): near Hα; can blend depending on resolution.

Common absorption features

• Ca II H & K (396.85 nm, 393.37 nm): prominent in older stellar populations; useful for early-type galaxies.
• G-band (~430.4 nm): stellar absorption feature.
• Mg b triplet (~517 nm): strong in many galaxy spectra.

In real data, line visibility depends on spectral resolution, signal-to-noise ratio (SNR), and the galaxy’s type. A quiescent elliptical may show strong absorption and weak emission; a starburst galaxy may show strong emission lines and weaker absorption.

Step-by-Step: Measuring Redshift from Individual Lines

This workflow assumes you have a calibrated spectrum (wavelength axis in nm or Å) and a list of candidate rest wavelengths for lines you expect to see. The goal is to measure z from each line and then combine the results.

Step 1: Identify candidate lines

Scan the spectrum for prominent peaks (emission) or dips (absorption). Make a short list of observed wavelengths λ_obs for the most obvious features. If you suspect a line is [O III], check whether the companion line appears at the correct relative position (the 495.9/500.7 nm pair). If you suspect Hα, check for nearby [N II] lines. Using line patterns reduces the risk of misidentification.

Step 2: Measure the observed wavelength precisely

Do not rely on “by eye” estimates from a plot. Instead, fit a simple model to the line profile in a small wavelength window around the feature. Common choices include a Gaussian for emission lines or an inverted Gaussian for absorption lines. The fitted center gives λ_obs.

Practical fitting approach:

• Choose a window that includes the full line and some continuum on both sides.
• Model the local continuum as a straight line (or constant) plus a Gaussian line.
• Fit parameters: continuum level/slope, line amplitude, line width, and line center.

If you cannot fit, you can compute a centroid using a flux-weighted mean after subtracting a local continuum, but fitting is usually more robust when noise is present.

Step 3: Compute z for each line

z_i = (λ_obs,i − λ_rest,i) / λ_rest,i

Compute z for each identified line. If the lines are correctly identified, the z values should agree within uncertainties. If one line gives a very different z, treat it as a warning sign: the line may be blended, affected by sky residuals, or misidentified.

Step 4: Estimate uncertainty per line

A practical uncertainty estimate comes from the fit uncertainty on the line center, σ(λ_obs). Propagate it to σ(z):

σ(z) ≈ σ(λ_obs) / λ_rest

This captures measurement noise and resolution effects. It does not include systematic calibration errors; if your wavelength calibration has an uncertainty σ_cal, combine it in quadrature with the fit uncertainty before propagating.

Step 5: Combine multiple lines into a best z

Use a weighted mean when you have per-line uncertainties:

z_best = (Σ (z_i / σ_i^2)) / (Σ (1 / σ_i^2))

and the uncertainty of the weighted mean:

σ(z_best) = 1 / sqrt(Σ (1 / σ_i^2))

As a sanity check, compute the scatter of z_i around z_best. If the scatter is much larger than expected from σ_i, you may have underestimated uncertainties or included problematic lines.

Step-by-Step: Measuring Redshift by Template Cross-Correlation

When individual lines are weak, blended, or numerous (especially in absorption-line spectra), cross-correlation with a template can be more reliable. The idea is to shift a reference spectrum (template) across a range of z values and find the shift that best aligns the overall pattern of features.

Step 1: Choose a template

Select a template spectrum representative of the galaxy type: an early-type (absorption-dominated) template for ellipticals, or an emission-line template for star-forming galaxies. The closer the template is to the target’s spectral character, the sharper the correlation peak tends to be.

Step 2: Put both spectra on a comparable wavelength grid

Cross-correlation works best when both spectra are sampled consistently. A common trick is to resample onto a grid uniform in log wavelength, because redshift corresponds to an additive shift in log(λ):

λ_obs = (1+z) λ_rest  →  log(λ_obs) = log(λ_rest) + log(1+z)

Uniform log sampling turns redshift into a simple shift, which simplifies correlation.

Step 3: Normalize and mask

• Remove or divide out the continuum shape (e.g., by fitting a low-order polynomial) so the correlation is driven by lines, not overall slope.
• Mask regions contaminated by strong sky emission residuals or telluric absorption if present.
• Optionally down-weight noisy regions using an inverse-variance array.

Step 4: Compute correlation over a z range

Evaluate a correlation score for z values spanning your plausible range. The z that maximizes the score is your best estimate. The width of the correlation peak provides an uncertainty estimate; a narrow, high peak indicates a confident match, while a broad or multi-peaked correlation suggests ambiguity.

Step 5: Validate with line checks

Even with a strong correlation peak, validate by overplotting the shifted template and confirming that major features align. This is especially important when only a few features drive the correlation, because different line combinations can sometimes mimic each other at different z values.

Worked Numerical Examples

Example 1: Single-line redshift from Hα (low z)

Suppose you observe a strong emission line at λ_obs = 669.4 nm and you identify it as Hα with λ_rest = 656.28 nm.

z = (669.4 − 656.28) / 656.28 ≈ 0.0200

Using the low-z approximation v ≈ cz:

v ≈ (3.00×10^5 km/s) × 0.0200 ≈ 6000 km/s

If your line-center fit uncertainty is σ(λ_obs)=0.05 nm:

σ(z) ≈ 0.05 / 656.28 ≈ 7.6×10^-5

and σ(v) ≈ c σ(z) ≈ 23 km/s (not including any systematic calibration terms).

Example 2: Multi-line consistency check with [O III] doublet

You detect two emission peaks at 510.7 nm and 515.6 nm. You suspect they are the [O III] lines at 495.9 nm and 500.7 nm.

Compute z from each:

z_4959 = (510.7 − 495.9) / 495.9 ≈ 0.0299

z_5007 = (515.6 − 500.7) / 500.7 ≈ 0.0298

The agreement supports the identification. If instead one of these gave z ≈ 0.12, you would suspect a misidentified line or a blend.

Example 3: Relativistic mapping vs v = cz at moderate z

Take z = 0.10.

Linear approximation:

v ≈ cz ≈ 0.10 c ≈ 30,000 km/s

Relativistic Doppler mapping:

β = ((1.1)^2 − 1) / ((1.1)^2 + 1) = (1.21 − 1) / (1.21 + 1) = 0.21 / 2.21 ≈ 0.0950

So v ≈ 0.0950 c ≈ 28,500 km/s. The difference is already about 1,500 km/s, which is large compared to typical measurement uncertainties. This is why it is important to state which conversion you used when reporting “recession speed.”

Practical Complications and How to Handle Them

Peculiar velocities and local motions

Galaxies are not only carried by cosmic expansion; they also move within groups and clusters. These additional motions (often called peculiar velocities) can be hundreds to over a thousand km/s in rich clusters. For nearby galaxies, peculiar velocities can be a significant fraction of cz, so redshift is not a clean proxy for distance or expansion speed in that regime. Practically, this means:

• Do not over-interpret small differences in z among nearby galaxies as purely “expansion.”
• When studying clusters, expect a spread in measured redshifts around the cluster mean.

Line broadening and blended features

Galaxy lines can be broadened by internal velocity dispersion (random stellar motions) or by gas kinematics. Broad lines make the line center harder to measure, increasing σ(λ_obs). Blends (e.g., Hα with [N II]) can bias the fitted center if you fit a single Gaussian. A practical fix is to fit multiple Gaussians with fixed or constrained separations based on known rest wavelengths, allowing you to separate components and recover a less biased redshift.

Instrumental resolution and sampling

If your spectral resolution is low, the line profile is dominated by the instrument. The smallest achievable σ(λ_obs) is limited by resolution and SNR. A useful operational mindset is: if the line spans only a few pixels, you can still measure the center, but you must expect larger uncertainty and be cautious about systematic shifts from imperfect calibration.

Wavelength calibration and reference frames

Even if you measure λ_obs precisely, the absolute accuracy depends on calibration. Calibration errors can introduce a systematic offset in z. Additionally, the observed wavelength is affected by the motion of the observer (Earth’s rotation and orbit). Many data products provide wavelengths already corrected to a standard frame (often heliocentric or barycentric). If not, you must apply a correction before interpreting the galaxy’s redshift. Practically:

• Check metadata for whether wavelengths are in the observed frame, heliocentric, or barycentric.
• If combining data from different nights or instruments, ensure consistent frame corrections.

Quality control checklist

• Do multiple lines give consistent z within uncertainties?
• Do line ratios and separations match the proposed identification (e.g., [O III] doublet)?
• Is the spectrum affected by sky residuals near the lines used?
• Is the continuum removal or normalization influencing your cross-correlation peak?
• Are you reporting the conversion used for v (linear vs relativistic vs cosmology-based interpretation)?

Turning Redshift into a Recession-Speed Estimate with Uncertainty

Once you have z_best and σ(z_best), you can compute a recession-speed estimate appropriate to your regime.

Low-z speed estimate

v = c z_best

σ(v) = c σ(z_best)

This is straightforward and often sufficient for nearby-galaxy kinematics, cluster membership screening, and many survey summaries where z is small.

Relativistic Doppler-mapped speed estimate

If you choose to report the special-relativistic mapping, compute v(z) using β(z) and propagate uncertainty numerically by evaluating v at z±σ(z):

v_best = c * ((1+z_best)^2 − 1) / ((1+z_best)^2 + 1)

σ(v) ≈ (v(z_best + σz) − v(z_best − σz)) / 2

This avoids algebraic mistakes and works even when the relationship is nonlinear.

Mini-Lab: A Complete Redshift Measurement Workflow

This mini-lab outlines a full, practical procedure you can apply to a galaxy spectrum from an archive or your own observations.

Inputs

• Wavelength array λ
• Flux array F(λ)
• Optional uncertainty array σ_F(λ) or inverse variance
• List of rest wavelengths for candidate lines

Procedure

• 1) Preprocess: mask obvious bad pixels; optionally smooth lightly for visualization (do not smooth for fitting unless necessary).
• 2) Find features: locate prominent peaks/dips; record approximate λ positions.
• 3) Propose identifications: test whether multiple features match a consistent z using known line patterns (doublets, triplets, neighborhoods like Hα+[N II]).
• 4) Fit lines: for each selected feature, fit a local continuum + Gaussian (or multi-Gaussian for blends) to obtain λ_obs and σ(λ_obs).
• 5) Compute z per line: calculate z_i and σ(z_i) for each line.
• 6) Combine: compute weighted mean z_best and σ(z_best); check for outliers and refit or exclude problematic lines with justification.
• 7) Convert to speed: compute v and σ(v) using your chosen mapping; record which mapping you used.
• 8) Report: provide z_best ± σ(z_best), the list of lines used, and a brief quality note (e.g., “based on [O III] doublet and Hβ; consistent within uncertainties”).

Suggested reporting format

Galaxy redshift: z = 0.02985 ± 0.00006 (weighted mean of [O III] 4959, 5007 and Hβ 4861)  Recession-speed estimate: v = cz = 8950 ± 18 km/s (linear approximation)