ASTR 420 · Lecture 02

Lecture 02: Relativistic Redshift and Distance

Cosmology

At low redshift several distance definitions nearly agree; at cosmological redshift they separate because the scale factor changes during photon travel.

Objectives and route

Distinguish velocity, redshift, luminosity distance, and angular-diameter distance.
Define and distinguish special-relativistic redshift and cosmological distance measures.
Derive or audit one quantitative relation with units and assumptions.
Connect the calculation to an observational test and a limitation.

Terms for This Lecture

Doppler redshift

Definition: A wavelength shift produced by relative motion through spacetime between emitter and observer.

Why it matters here: In Relativistic Redshift and Distance, Doppler redshift is needed to reason about special-relativistic redshift and cosmological distance measures rather than merely label the phenomenon.

Relationship: It is interpreted alongside Cosmological redshift and the lecture relation below; identify which quantities are observed, calibrated, or model-derived.

Limitation or common confusion: The difference is about 1.5%; cosmological redshift is not generally a simple Doppler velocity.

\(1+z=\\sqrt{(1+\\beta)/(1-\\beta)}\)

Motivating evidence

At low redshift several distance definitions nearly agree; at cosmological redshift they separate because the scale factor changes during photon travel.

Cosmological claims are indirect: an observable is calibrated, placed in a model, and compared against alternatives. Keep those layers separate throughout the lecture.

special-relativistic redshift and cosmological distance measures. The axes, labels, and geometry carry the physical inference rather than serving as decoration.

Physical framework

Start from the governing relation and define its domain:

\(1+z=\\sqrt{(1+\\beta)/(1-\\beta)}\)

State the coordinate choice, units, boundary conditions, and approximation before substituting numbers. The same symbol can mean a different quantity in Newtonian, special-relativistic, and expanding-space contexts.

Read the visual

Use the figure introduced on the evidence slide as an argument: identify the measured axis, the inferred axis, the comparison model, and the parameter that controls the conclusion.

The paired lab asks you to reproduce the visual from a supplied data extract and test one modeling choice.

Worked calculation

For z=0.10, the special-relativistic formula gives v = 28487.1 km s^-1, while cz = 29979.2 km s^-1.

The difference is about 1.5%; cosmological redshift is not generally a simple Doppler velocity.

Estimate the scale first, then check dimensions and comparative direction. The displayed value is generated from shared ASTR420 constants and data.

Evidence versus inference

Raw data are not the same as calibrated observables.
A fitted parameter is conditional on the likelihood, prior, and selection function.
Independent probes can agree because the model is useful, or disagree because an assumption is hidden.

Boundary condition

Limit of the model: The difference is about 1.5%; cosmological redshift is not generally a simple Doppler velocity.

Ask what happens if the dominant assumption fails. In cosmology, approximation honesty matters because distances, times, and abundances accumulate model dependence.

Active reasoning

Prompt: Find the special-relativistic recession speed associated with z=0.10 and compare it with cz.

Predict the direction before calculating. Then name one observation that could discriminate between two explanations.

Connection to the seven-unit arc

Earlier tools: ASTR310 radiation, dynamics, and statistical mechanics.
Paired data activity: use the corresponding CSV and reproduce the calculation with a perturbation test.
Next step: combine this probe with an independent epoch or distance measure.

Takeaways

Distinguish velocity, redshift, luminosity distance, and angular-diameter distance.
A cosmological parameter is meaningful only with its model and uncertainty context.
State whether a number is live-verified, a published value awaiting spot-check, or an instructor-provided model value.
Carry the limitation forward into the lab and problem set.

Reading and provenance

OpenStax Astronomy 2e, Chapter 29, section 29.2; Chapter 26, section 26.5; embedded Figures 26.16 and 29.5.

OpenStax Astronomy 2e Chapter 29 embedded figure numbers were checked against the extracted text; the source index has chapter-label offsets near the Chapter 29 boundary, so the adopted PDF remains the final exercise-number authority.

Additional sources and dataset status are recorded in reference-log.md. Standard published values not re-fetched this session are level 2 and require human spot-checking.