Context and Why This Matters
The Doppler shift is one of the most productive tools in observational astronomy: it reveals orbital motion in binary stars, radial velocities in exoplanet detection, and cosmological redshift in distant galaxies. Its reliability depends entirely on the wavelength calibration developed in the previous lecture.
Learning Goals
- Apply the non-relativistic Doppler formula to measure radial velocity from a wavelength shift.
- Cross-check a velocity measurement using multiple independent spectral lines.
- Explain why line blends or calibration drift can masquerade as a velocity signal.
Opening Phenomenon
Three lines in the same spectrum — Hα, Hβ, and [O III] — are all shifted redward of their laboratory wavelengths. If the shift is a genuine bulk radial velocity, all three lines should imply the same velocity.
Terms Developed in Context
These terms form the measurement chain for Radial Velocity and Redshift: each connects an observable to a model choice, and each has a boundary that must be respected when making a scientific claim.
- rest wavelength
The laboratory or source-frame wavelength assigned to a spectral transition before Doppler shifting. It is the reference \(\lambda_0\) against which an observed displacement is converted to velocity or redshift.
The measured shift is \(\Delta\lambda=\lambda_{obs}-\lambda_0\), entering \(v/c\approx\Delta\lambda/\lambda_0\) at low speed. Air and vacuum wavelength conventions differ slightly, so the convention must match the calibration.
Developed in this lecture: The term helps interpret this evidence: Hα: rest 656.3 nm, observed 656.9 nm. It also constrains the working model: For velocities much less than the speed of light, v/c ≈ Δλ/λ₀, with positive Δλ (redshift) meaning recession.
- observed wavelength
The calibrated wavelength at which a spectral feature appears in the recorded observer-frame spectrum. It is the direct spectral measurement paired with a rest wavelength to infer motion.
Its uncertainty combines centroiding, wavelength-solution residuals, line shape, and instrumental drift. A measured feature is not automatically the intended transition; identification and blending must be checked.
Developed in this lecture: The term helps interpret this evidence: Hα: rest 656.3 nm, observed 656.9 nm. It also constrains the working model: For velocities much less than the speed of light, v/c ≈ Δλ/λ₀, with positive Δλ (redshift) meaning recession.
- redshift
A positive fractional wavelength increase \(z=(\lambda_{obs}-\lambda_0)/\lambda_0\). It quantifies the consistent displacement of multiple lines toward longer wavelengths.
For small speeds \(v\approx cz\); at high speeds or cosmological distances the physical interpretation requires relativistic or cosmological models. Redshift is an observed ratio, not always a simple peculiar velocity.
Developed in this lecture: The term helps interpret this evidence: Hα: rest 656.3 nm, observed 656.9 nm. It also constrains the working model: For velocities much less than the speed of light, v/c ≈ Δλ/λ₀, with positive Δλ (redshift) meaning recession.
- blueshift
A negative wavelength shift in which a feature is observed at shorter wavelength than its rest value. It is the sign expected for approaching radial motion in the low-speed Doppler interpretation.
With the lecture convention, \(\Delta\lambda<0\) gives \(v<0\). Calibration offsets can shift every line blueward, so sign alone does not establish physical approach.
Developed in this lecture: The term helps interpret this evidence: Hα: rest 656.3 nm, observed 656.9 nm. It also constrains the working model: For velocities much less than the speed of light, v/c ≈ Δλ/λ₀, with positive Δλ (redshift) meaning recession.
- radial velocity
The component of an object’s velocity along the observer’s line of sight. It is inferred from calibrated line shifts and is the physical quantity tested by the three-line agreement.
At \(v\ll c\), \(v/c\approx\Delta\lambda/\lambda_0\); multiple lines should yield a common value. It excludes transverse motion and may include systemic, orbital, rotational, and observer-motion contributions.
Developed in this lecture: The term helps interpret this evidence: Hα: rest 656.3 nm, observed 656.9 nm. It also constrains the working model: For velocities much less than the speed of light, v/c ≈ Δλ/λ₀, with positive Δλ (redshift) meaning recession.
- line blend
A spectral feature formed by two or more unresolved transitions whose profiles overlap. A blend can shift the apparent centroid and imitate a radial-velocity change.
Whether components blend depends on their separation relative to the line-spread function and resolving power. Treating a blended centroid as a single isolated line yields a model-dependent, potentially biased velocity.
Developed in this lecture: The term helps interpret this evidence: Hα: rest 656.3 nm, observed 656.9 nm. It also constrains the working model: For velocities much less than the speed of light, v/c ≈ Δλ/λ₀, with positive Δλ (redshift) meaning recession.
- systemic velocity
The center-of-mass line-of-sight velocity of a source system relative to a stated reference frame. It provides the common baseline around which orbital or internal line velocities vary.
Repeated multi-line measurements can separate a stable systemic component from time-dependent motion. The value depends on the adopted reference frame and is not necessarily the velocity of every emitting component.
Developed in this lecture: The term helps interpret this evidence: Hα: rest 656.3 nm, observed 656.9 nm. It also constrains the working model: For velocities much less than the speed of light, v/c ≈ Δλ/λ₀, with positive Δλ (redshift) meaning recession.
Evidence
- Hα: rest 656.3 nm, observed 656.9 nm.
- Hβ: rest 486.1 nm, observed 486.55 nm.
- [O III]: rest 500.7 nm, observed 501.16 nm.
Model
- For velocities much less than the speed of light, v/c ≈ Δλ/λ₀, with positive Δλ (redshift) meaning recession.
- A genuine bulk-motion signal produces a consistent velocity across multiple, unrelated spectral lines.
- A velocity estimate from a single line is far less trustworthy than one confirmed by several lines of different species.
Working Equation
\[ \dfrac{v}{c} \approx \dfrac{\Delta\lambda}{\lambda_0} \]
Worked Example
- Hα: Δλ = 0.60 nm, λ₀ = 656.3 nm, v = (0.60/656.3)×299792 km/s = 274.1 km/s.
- Hβ: Δλ = 0.45 nm, λ₀ = 486.1 nm, v = (0.45/486.1)×299792 km/s = 277.5 km/s.
- [O III]: Δλ = 0.46 nm, λ₀ = 500.7 nm, v = (0.46/500.7)×299792 km/s = 275.4 km/s.
- All three independent lines agree to within about 1.9 km/s of the mean, v ≈ 275.7 km/s — strong evidence for a genuine, single bulk radial velocity rather than a line-specific artifact.
Common Misconception
Ascribing an observed wavelength shift to physical velocity without checking for calibration drift, line blending with a neighboring feature, or an error in identifying which transition produced the line.
Active-Learning Guidance
Given a fourth line with rest wavelength 434.0 nm, predict its observed wavelength if the source has the same radial velocity found above, then discuss what a large discrepancy from that prediction would suggest.
Lab / Observing Connection
Lab 06 continues into a multi-line radial-velocity determination using the wavelength solution derived earlier in the same lab.
Synthesis Questions
- What was measured directly in this lecture’s worked example, and what was inferred from the model?
- Which uncertainty or systematic would most change the interpretation?
- How does this technique connect to the technical observing report due at the end of the term?
References
- OpenStax Astronomy 2e, Chapter 5.6 (the Doppler effect).
- Local reference copy:
references/openstax-astronomy-2e.pdf.