Vol. Science · Astronomy · Spectra

Redshift
lines on the move

A spectrum carries a ruler. The same line, measured at a longer wavelength, has redshift z = (λ_obs − λ_rest) / λ_rest. Slide z and the cited lines walk across a visible strip. Two speeds are printed beside that number: the non-relativistic product v ≈ c z, and the special-relativistic Doppler speed from 1 + z = √((1+β)/(1−β)). At z = 1, that second formula gives β = 0.6, while c z is the speed of light and is not a speed. Cosmological redshift is a third statement: 1 + z = a_now / a_then, the scale factor of the expansion. A large-z “velocity” is not a Doppler speed through space. The temperature ladder that sets the continuum under these lines is the blackbody desk. The wavelength ladder is the electromagnetic spectrum.

Classroom schematic. Lines are drawn at the cited rest wavelength times (1+z). They are not a measured spectrum of the preset object, and they have no flux. Optical rest wavelengths are NIST air values; Lyman-alpha is vacuum. A catalog redshift is reduced to vacuum. The shaded band is the NASA Imagine classroom optical window, about 400–700 nm. Hubble’s law on this page is v = H₀ d for a made-up nearby distance, not a distance to Andromeda, NGC 4258, 3C 273, or GN-z11. Not a cosmological calculator. Not an ephemeris.

Interactive · Shift the lines

Observed wavelength, log scale · dashed marks are the rest wavelengths · 400–700 nm shaded

Shifted
Rest
400–700 nm

Cited presets · each note says where the number comes from

Redshift z

Observed wavelength of each cited line at the current redshift
Line	Rest	Observed	Band
Lyα	121.567 nm vacuum	121.567 nm	Ultraviolet
Ca II K	393.366 nm air	393.366 nm	Ultraviolet
Ca II H	396.847 nm air	396.847 nm	Ultraviolet
Hβ	486.132 nm air	486.132 nm	Visible
Hα	656.280 nm air	656.280 nm	Visible

The slider needs JavaScript. The anchors are still below. z = (λ_obs − λ_rest) / λ_rest. At z = 1, β = 0.6 and c z = c. NIST rest wavelengths used here: Lyα 121.567 nm (vacuum), Ca II K 393.36614 nm and H 396.84673 nm (air), Hβ 486.132 nm and Hα 656.280 nm (air). The visible window is about 400–700 nm. Planck 2018 base-ΛCDM gives H0 = 67.4 ± 0.5 km/s/Mpc. Riess et al. 2022 give H0 = 73.04 ± 1.04 km/s/Mpc. At 10 Mpc those central values are 674 km/s and 730.4 km/s.

Nearby toy · Hubble’s law

For small redshift, z ≈ v / c = H₀ d / c and v = H₀ d. Hogg’s notes say this is only true for small redshifts. The slider stops at 100 Mpc. With the Planck central value, z at that far end is still under 0.03. The two lines are the two cited central values. The gap between them is the tension: same distance, two published H₀ figures. This is not a distance to any preset above, and it is the wrong tool for Andromeda, which is blueshifted.

Velocity versus distance · both cited H0 values · marker on the selected one

Planck 67.4
SH0ES 73.04

Select which H0 drives the marker

Distance Mpc

The number is not the speed

Three different uses of one z.

The measured redshift is the wavelength ratio. Hogg writes z = λ_obs / λ_rest − 1. That definition does not say why the wavelength changed. A laboratory source moving along the line of sight follows the special-relativistic Doppler formula, 1 + z = √((1+β)/(1−β)), with β = v/c positive when the source recedes. Solving it gives β = ((1+z)² − 1) / ((1+z)² + 1). At z = 1 that is 3/5, so β = 0.6. The familiar v ≈ c z is the small-speed limit of that formula. At z = 0.1 it is already about 5% high. At z = 1, c z equals c, which is not a speed the formula will return.

Cosmological redshift is the expansion. Hogg’s equation is 1 + z = a(t_now) / a(t_then). He also writes that the special-relativistic velocity formula is not the one used in the usual cosmological coordinates, and that the linear Hubble law z ≈ v/c = H0 d / c holds only at small redshift. Galaxy surveys still print c z as a “velocity” because it is a convenient label. Past the nearby universe that label is not a Doppler speed, and it is allowed to exceed c without anything traveling faster than light.

Andromeda is the local counterexample. SIMBAD’s adopted heliocentric redshift is negative: the Local Group companion is approaching, and a Hubble-flow distance is the wrong reading. NGC 4258 is redshifted and nearby enough that peculiar motion still matters; its place in the SH0ES ladder is a geometric maser anchor, not this slider. 3C 273 and GN-z11 are expansion redshifts. GN-z11’s published Lyman-alpha line is offset from the systemic redshift; this page shifts every line by the systemic z only.

Literacy · the words on the slider

Spectroscopic z: z = (λ_obs − λ_rest) / λ_rest, so λ_obs = λ_rest (1+z). A blueshift is a negative z. This desk multiplies the cited NIST rest wavelength by (1+z). It does not convert air wavelengths to vacuum.
c · z: The non-relativistic product, with c = 299,792.458 km/s exactly. Useful while |z| is tiny. It is the number many catalogs print as a velocity. When |z| reaches 1 it equals c, and beyond that it is larger than c. That does not mean a faster-than-light object.
Relativistic Doppler: 1 + z = √((1+β)/(1−β)). β = ((1+z)² − 1) / ((1+z)² + 1). Positive β recedes. This is a special-relativistic line-of-sight speed. Applying it to a cosmological redshift is a math exercise, not the expansion.
Scale factor: 1 + z = a_now / a_then. The readout is a_then / a_now = 1/(1+z). At z = 1 the universe was half its present scale factor. At GN-z11’s z = 10.6034 the ratio is 1/11.6034.
Hubble’s law: v = H0 d, and z ≈ v/c, nearby only. This toy offers two cited central values: Planck 2018 base-ΛCDM, 67.4 ± 0.5 km/s/Mpc, and the SH0ES 2022 baseline, 73.04 ± 1.04 km/s/Mpc. The uncertainties are printed. The marker uses the central value.
Visible window: 400–700 nm, the NASA Imagine classroom optical band, the same cut as the blackbody desk. Ca II H and K sit just short of 400 nm at rest, so a small redshift can carry them in. Hα leaves the window once (1+z) × 656.280 nm passes 700 nm. Lyman-alpha, at 121.567 nm, needs z around 2.3 before it reaches 400 nm, and it has left again before z = 5.

Sources · NIST, Hogg, Planck, SH0ES, SIMBAD

Cited, not invented

NIST — Atomic spectroscopy compendium, hydrogenic series — customary unresolved wavelengths: Lyman-alpha 1215.67 Å, Hβ 4861.32 Å, Hα 6562.80 Å. Wavelengths below 2000 Å are vacuum; values above 2000 Å are air. This desk uses 121.567 nm, 486.132 nm, and 656.280 nm.
NIST — Persistent lines of Ca II — 3933.6614 Å (4s ²S_1/2 – 4p ²P°_3/2) and 3968.4673 Å (4s ²S_1/2 – 4p ²P°_1/2), air. This desk uses 393.36614 nm and 396.84673 nm.
Linsky and Avrett 1970, PASP 82, 485 — the solar Ca II resonance lines near 3933 Å and 3968 Å were designated K and H by Fraunhofer. The wavelengths on the chart are the NIST figures, not the 1970 rounding.
NIST CODATA — speed of light in vacuum — c = 299,792,458 m/s exactly, so 299,792.458 km/s. Every c z and every β c on this page uses that value.
Hogg 1999, “Distance measures in cosmology,” arXiv:astro-ph/9905116 — eq. 8, z = λ_obs/λ_rest − 1; eq. 9, the special-relativistic 1+z = √((1+β)/(1−β)); eq. 11, z ≈ v/c = d/D_H only for small redshift, and the note that surveys still print v = c z; eq. 12, 1+z = a(t_now)/a(t_then). He writes that eq. 9 is not the relation in the usual cosmological coordinates.
Planck Collaboration 2020, A&A 641, A6 (arXiv:1807.06209) — abstract, base-ΛCDM: H0 = (67.4 ± 0.5) km s⁻¹ Mpc⁻¹. The toy uses the central value 67.4.
Riess et al. 2022, ApJL 934, L7 (arXiv:2112.04510) — baseline SH0ES result H0 = 73.04 ± 1.04 km s⁻¹ Mpc⁻¹, including systematics. The toy uses the central value 73.04. The same paper names masers in NGC 4258 as one of three geometric anchors. This desk does not copy a distance from that anchor.
NASA Imagine the Universe — Wavelength, frequency, and energy — classroom optical band about 400–700 nm. Boundaries are conventions. The same chart is the visible slice on the EM spectrum and blackbody desks.
SIMBAD — M31 — adopted heliocentric redshift −0.001000 ± 0.000013, cz = −299.85 ± 4.00 km/s, radial velocity −300.0 ± 4.0 km/s, referenced to McConnachie 2012, AJ 144, 4. This desk sets z = −0.001. Its own c z is −299.79 km/s, inside the ±4 km/s. A Hubble-flow distance does not apply.
BASS DR2 data table — NGC 4258 listed at redshift 0.00158. The catalog paper is Koss et al. 2022, ApJS 261, 2. This desk sets z = 0.00158 and does not invent a distance. c z at that redshift is 473.67 km/s.
SIMBAD — 3C 273 — adopted redshift 0.15756751 ± 0.0005, radial velocity 43,555 ± 150 km/s, cz = 47,237.5511 ± 150 km/s, referenced to Koss et al. 2022, ApJS 261, 2. This desk prints a relativistic speed of 43,555.49 km/s and c z of 47,237.55 km/s, both inside that ±150 km/s.
Bunker et al. 2023, arXiv:2302.07256 — GN-z11 systemic redshift z = 10.6034 ± 0.0013 from emission lines other than Lyman-alpha. The paper’s Lyman-alpha centroid is offset by 555 ± 32 km/s from that systemic redshift. This slider uses 10.6034 for every line and does not add the offset.

Classroom picture of a shift. z = (λ_obs − λ_rest)/λ_rest. At z = 1, β = 0.6. c z is not a cosmological speed. 1+z = a_now/a_then. The Hubble toy is v = H0 d with two cited central values, nearby only. Not a catalog. Earthrise preview: NASA / Bill Anders (public domain).

Keep going

Continuum

Axis	Observed wavelength, logarithmic, from 80 nm to 10,000 nm. Dashed marks are the rest wavelengths. Solid marks are those wavelengths times (1+z).
Shade	400–700 nm, the NASA Imagine classroom optical window.
Lines	Lyα, Ca II K, Ca II H, Hβ, Hα. Identity colors are labels, not a flux calibration. A line past the edge of the axis is marked off-chart in the table.

Redshift	Slider from −0.01 to 11, stretched so small z and GN-z11 are both reachable. The number field accepts the same range. Arrow keys nudge the slider.
Presets	Rest, Andromeda, NGC 4258, 3C 273, GN-z11. Each button sets a cited z. The note names the source. Left and Right move through the buttons when one of them has focus.
Hubble toy	Separate from the spectrum. Distance 0 to 100 Mpc. Planck 67.4 or SH0ES 73.04. Both lines stay on the chart. The marker follows the selected central value.

c · z	299,792.458 km/s times z. Not a cosmological speed.
β and v	The special-relativistic inversion. At z = 1, β = 0.6. Still not the expansion speed.
Scale factor	a_then / a_now = 1/(1+z). The cosmological reading of the same number.
Tension	(73.04 − 67.4) times the distance. The difference of the two cited central values. Not a new measurement.

Sliders	Arrow keys nudge the focused slider. The number field is where a cited figure is typed exactly.
Buttons	Enter or Space selects. Left and Right step through a group. The pressed button is aria-pressed.
Touch	Buttons, the question mark, Close, and the fields are at least 44 pixels on a side.
This dialog	The question mark, or ?, opens it. Escape closes it. Tab stays inside until you close it. Click the dim page behind the card to close it.

Redshift
lines on the move

Three different uses of one z.

Cited, not invented

Blackbody

Galaxy

Exoplanets

Stars

EM spectrum

Waves

Solar map

Redshift desk

The chart

Controls

Readouts

Keyboard and touch

Redshiftlines on the move

Three different uses of one z.

Cited, not invented

Blackbody

Galaxy

Exoplanets

Stars

EM spectrum

Waves

Solar map

Redshift
lines on the move