AST-23

Doppler redshift

z = Δλ/λ, v ≈ c z at z ≪ 1. Recession of a nearby galaxy.

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RadiationRedshift

Governing equation

z=\Delta\lambda/\lambda,\quad v\approx cz

$\Delta\lambda \in \left[0, 200\right]$

where

\Delta\lambda: Shift (nm)
\lambda: Rest wavelength (nm)
z: Redshift (—)
v: Recession (km/s)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

z=\Delta\lambda/\lambda,\quad v\approx cz

$\Delta\lambda \in \left[0, 200\right]$

$\Delta\lambda = 0$ $z = 0$

Along the curve

②

Domain of the variable — blue

\Delta\lambda_{\min} = 0,\quad \Delta\lambda_{\max} = 200

$\Delta\lambda \in \left[0, 200\right]$

$\Delta\lambda = 0$ $\Delta\lambda = 0$

Along the curve

③

Response curve — green

z=\Delta\lambda/\lambda,\quad v\approx cz

$\Delta\lambda \in \left[0, 200\right]$

$\Delta\lambda = 0$ $z = 0$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}z}{\mathrm{d}\Delta\lambda}

$\Delta\lambda \in \left[0, 200\right]$

$\Delta\lambda = 1.258$ $z^{\prime} = 0.002$

Along the curve

Lecture brief

Historical brief

Hubble expansion, Jeans collapse, Eddington luminosity, Bondi accretion and Stefan–Boltzmann stars are the first astrophysical budgets. The sheets scale a star, a cloud and a horizon. This sheet (AST-23 — Doppler redshift) is the form associated with Redshift. Working symbols: $\Delta\lambda$ , $\lambda$ $\rightarrow$ $z$ , $v$ . Relativistic: 1+z = √((1+β)/(1−β)). Cosmological z is expansion, not a local Doppler.

Purpose

Purpose: compute

z

v

from

\Delta\lambda

\lambda

in Astrophysics via

z=\Delta\lambda/\lambda,\quad v\approx cz

z = Δλ/λ, v ≈ c z at z ≪ 1. Recession of a nearby galaxy. Use it when a real astrophysics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

\Delta\lambda = 12.000\,\mathrm{nm}

\lambda = 656.300\,\mathrm{nm}

, the governing relation

z=\Delta\lambda/\lambda,\quad v\approx cz

yields

z = 0.0183\,\mathrm{—}

v = 5481.5\,\mathrm{km/s}

. A spectral line sliding redward. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Shift \Delta\lambda12 nmRest wavelength \lambda656.3 nm

Outputs

Redshift z0.0183 —
Recession v5481.5 km/s

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Narration of this film

A spectral line sliding redward.

Relativistic: 1+z = √((1+β)/(1−β)). Cosmological z is expansion, not a local Doppler.

Reading speed

Watch on YouTube