Redshift Calculator
The formula
Three velocities, one redshift
At small z everything agrees: the Doppler formula, the relativistic formula and Hubble's v = H₀d all give nearly the same answer — at z = 0.01 they differ by less than 1%. By z = 1 they disagree wildly, and that is not a flaw but a statement that "recession velocity" becomes ambiguous in general relativity. The relativistic Doppler velocity the calculator reports is the speed a special-relativity measurement would infer; the Hubble flow velocity at the object's comoving distance can exceed the speed of light for z above ~1.5 without violating anything, because it is space expanding between us, not motion through space. Galaxies beyond the Hubble sphere (~14.4 billion ly) recede faster than light, and we still see them — their ancient light was emitted inside the sphere and surfed the expansion outward.
The distance you get depends on the model
Redshift is measured directly and exactly; distance is inferred through a cosmological model. This calculator assumes flat ΛCDM with Ωm = 0.3, ΩΛ = 0.7 and your chosen H₀, integrating numerically rather than using the cz/H₀ shortcut that only works below z ≈ 0.1. Different definitions give different numbers at z = 1: comoving distance ~3.4 Gpc, luminosity distance ~6.8 Gpc (used for supernova brightnesses), angular-diameter distance ~1.7 Gpc (used for galaxy sizes — distant objects can appear larger again beyond z ≈ 1.6). Changing H₀ from 67.4 to 73 shifts all distances by about 8% — the practical face of the Hubble tension.
Blueshifts and peculiar motion
Negative z means the observed wavelength is shorter — the object approaches. Andromeda shows z ≈ −0.001 (301 km/s toward us) because its local peculiar velocity beats the Hubble flow at a mere 0.78 Mpc. Nearby galaxies always mix the two: the Virgo cluster recedes at ~1,200 km/s cosmologically while individual members scatter ±700 km/s from mutual gravity. That is why redshift surveys inside ~100 Mpc show "Fingers of God" — clusters smeared along the line of sight — and why the cz distance shortcut fails locally: the Great Attractor pulls everything within tens of Mpc at several hundred km/s regardless of expansion.
Frequently asked questions
What does a redshift of 1 mean?
Can recession velocity exceed the speed of light?
How is redshift measured in practice?
What is the highest redshift ever measured?
Source & methodology
Hubble (1929), PNAS; Planck Collaboration 2018 cosmological parameters (Ωm = 0.315, H₀ = 67.4); Riess et al. SH0ES (H₀ = 73.0); standard ΛCDM distance integrals per Hogg (1999), "Distance measures in cosmology".
Redshift: what this tool answers
You are here because you need a specific number, and you want to see how it was produced before you rely on it. Redshift is the kind of number people argue about from memory. This page ends that argument with arithmetic you can watch happen: enter your figures, get Redshift z in real time, and see every substitution printed below.
The useful way to hold Redshift is as a ratio with a story attached: z = (λ_obs − λ_emit)/λ_emit; relativistic recession v = c·((1+z)² − 1)/((1+z)² + 1); comoving distance D_C = (c/H₀)∫₀^z dz′/E(z′), E(z) = √(Ωm(1+z)³ + ΩΛ) says how the parts combine, and the 0.100 figure above is simply that sentence written in numbers. Everything that follows unpacks one clause of the sentence at a time.
A calculation like this one is worth roughly as much as a careful sentence: it forces several vague considerations into one comparable quantity. Once Redshift is on the table, the argument can stop being about impressions.
What makes a calculation page trustworthy is that its copy is derived from its own arithmetic: here redshift z = 0.100, recomputed every time a field changes, and the sensitivity section ranks the inputs that move the result — Rest (emitted) wavelength first, at ≈122.2 % of the headline for a ten-percent error either way in that input. Read the sections in order and you will be able to reproduce every figure by hand.
Every input in Redshift, and what it does to the answer
One formula, published in full: z = (λ_obs − λ_emit)/λ_emit; relativistic recession v = c·((1+z)² − 1)/((1+z)² + 1); comoving distance D_C = (c/H₀)∫₀^z dz′/E(z′), E(z) = √(Ωm(1+z)³ + ΩΛ). Anything you can read, you can reproduce in a spreadsheet, and that reproducibility is the whole point of printing it rather than hiding it behind a button.
Rest (emitted) wavelength (nm) (example value 656) is a quantity rather than a category — precision here is cheap and worth having. The example value is there to demonstrate the mechanics, not to describe your situation — overwrite it.
Observed wavelength (nm) (example value 722) is a measured number in the unit named by the label; the formula does not convert for you. If you are unsure what belongs in this field, look at the printed steps: the substitution shows exactly how the engine uses it.
Hubble constant H₀ (km/s/Mpc) (example value 70) is a quantity rather than a category — precision here is cheap and worth having. The example value is there to demonstrate the mechanics, not to describe your situation — overwrite it.
Redshift on the example inputs: the complete substitution
Rather than describe the calculation, here it is end to end — every substitution in order, starting from the inputs above. Starting figures: Rest (emitted) wavelength (nm), Observed wavelength (nm), Hubble constant H₀ (km/s/Mpc). The computed redshift z comes out at 0.100, which the interpretation band labels “Receding at 28,488 km/s · 1.1000× the rest wavelength”.
- z = (λ_obs − λ_rest) ÷ λ_rest = (721.910 − 656.280) ÷ 656.280 = 0.100003.
- Relativistic Doppler: v = c·((1+z)² − 1)/((1+z)² + 1) = 299,792.458 × 0.095025 = 28,487.9 km/s (9.50% of c).
- c/H₀ = 299,792.458 ÷ 70.0 = 4,283 Mpc; integrating 1/E(z) with Ωm = 0.3, ΩΛ = 0.7 gives comoving distance 418.5 Mpc (1,364.9 Mly).
- Lookback time = (1/H₀)∫ z/((1+z)E(z)) dz with 1/H₀ = 13.97 Gyr → 64 Myr. At low z, v ≈ cz = 29,980 km/s agrees closely.
Recession velocity (relativistic Doppler) — 28,487.9 km/s. This is the number experienced users quote when the headline alone would mislead.
Non-relativistic cz — 29,980.2 km/s. Compare it against the main result of 0.100 before drawing conclusions.
Comoving distance (ΛCDM) — 418.5 Mpc. Compare it against the main result of 0.100 before drawing conclusions.
Distance in light years — 1,364,855,019 ly. Read next to the headline, this is the figure that shows the shape of the trade-off.
Lookback time — 63.5 Myr. Compare it against the main result of 0.100 before drawing conclusions.
Keep this worked example open while you enter your own figures: matching the shape of the substitution is the fastest way to spot a unit or convention mistake.
Scenario table: Redshift at ±10 % and ±20 % of the key input
The honest version of any estimate is a corridor, and this one is cheap to produce: Redshift was recomputed at 5 points along the dominant input, with every other field left untouched.
| Rest (emitted) wavelength setting | Redshift z |
|---|---|
| 80 % of the example value | 0.375 |
| 90 % of the example value | 0.222 |
| example inputs | 0.100 |
| 110 % of the example value | 0.000 |
| 120 % of the example value | -0.083 |
That spread — -0.083 at the low end, 0.375 at the high — is what a careful read of Redshift looks like: a band you can state out loud, with the reason for each edge. Reproduce it in the form above by editing Rest (emitted) wavelength alone; the substitution steps will follow your figures exactly.
What actually moves your Redshift
This list was computed for this specific engine just now: each numeric input was nudged ±10 % while every other field held its example value, and the redshift z was recalculated from scratch. It is the fastest honest answer to “what matters most in Redshift?” — and it is specific to these defaults, because sensitivity moves with the starting point.
- Rest (emitted) wavelength +10 % → Redshift = 0.000 · −10 % → 0.222 (the redshift z shifts ≈122.2 %)
- Nudge Observed wavelength by ±10 % and Redshift lands between -0.010 and 0.210 (sensitivity ≈110.0 %)
Rest (emitted) wavelength is the dominant lever here: ten percent either way moves the result by about 122.2 %, from 0.000 to 0.222. Measure that input from a document rather than from memory, and treat Observed wavelength as the second priority (≈110.0 % per ten percent). The rest of the inputs earn only a rounding-error of attention, and that asymmetry is the practical payoff of measuring rather than asserting.
Is your Redshift good — reading the result bands
The value above only becomes useful the moment you know what it is being measured against. The verdict attached to the default calculation — “Receding at 28,488 km/s · 1.1000× the rest wavelength” — is one band on a ladder your own figure may sit somewhere else on. How much of that band is luck? On Rest (emitted) wavelength alone ten percent either way covers 0.000 to 0.222. The practical reading rules: compare against your own history before comparing against other people; treat band boundaries as zones, not walls — one point on the wrong side of a threshold rarely changes a real outcome; and when a companion output contradicts the headline, believe the pair.
Set Redshift beside its neighbours and it usually plays a specific role: this figure establishes the size of something, and the neighbouring tools establish whether that size is acceptable. Skipping the first step makes the second one arbitrary.
How people get Redshift wrong
Negative z means the observed wavelength is shorter — the object approaches. Andromeda shows z ≈ −0.001 (301 km/s toward us) because its local peculiar velocity beats the Hubble flow at a mere 0.78 Mpc. Nearby galaxies always mix the two: the Virgo cluster recedes at ~1,200 km/s cosmologically while individual members scatter ±700 km/s from mutual gravity. That is why redshift surveys inside ~100 Mpc show "Fingers of God" — clusters smeared along the line of sight — and why the cz distance shortcut fails locally: the Great Attractor pulls everything within tens of Mpc at several hundred km/s regardless of expansion.
Reusing a Redshift from last season. In Astronomy & Space Science the inputs are the thing that drifts; the formula does not.
This model has its own particular trap: Rest (emitted) wavelength carries ≈122.2 % of the response per ten percent of error, so effort spent elsewhere is largely decorative.
The subtlest error is answering the wrong question precisely. Confirm that redshift z is actually the quantity your decision needs — the arithmetic is flawless either way, which is what makes the mistake invisible.
Redshift verification checklist
- Re-run the calculation with a ±10 % stress on the two most sensitive inputs.
- Confirm the example values are plausible for your own situation before replacing them.
- Put Rest (emitted) wavelength under a magnifier; the sensitivity table above is the reason why.
- Compare the result with one independent method or published benchmark.
- Record the date and the inputs alongside the number so future-you knows what was assumed.
- Run the ±10 % stress on the top two levers and check that your conclusion survives both.
Step by step: getting a trustworthy Redshift out of this tool
Using a calculator well sounds trivial, but the difference between a decision-grade figure and a throwaway one lives entirely in the routine. The routine for Redshift takes a minute to learn and applies to every input above.
Step 1 — collect, then type. Everything this tool asks for is listed above (rest (emitted) wavelength (nm), observed wavelength (nm), hubble constant h₀ (km/s/mpc)). Pull each figure from something you can point at — paperwork, a dashboard, a label — before touching a field. Guesswork at the input stage cannot be repaired at the interpretation stage. Repeating a calculation with the same numbers proves only that the arithmetic is reliable — which is worth knowing.
Step 2 — baseline before exploration. Get one defensible Redshift z on screen before you start nudging numbers. Without a baseline, every variation looks interesting and none of them means anything. Write the assumptions in the same file as the result, and future arguments take minutes instead of meetings.
Step 3 — isolate your variables. Change one field, observe, revert. The live sensitivity list does the ranking for the default inputs; verify it against your own, because dominance shifts when the starting point shifts. A result worth acting on survives having its two shakiest inputs halved.
Step 4 — ask what would have to be true. Take the result you want and work backwards through the inputs until you find the assumption that produces it; then decide honestly whether that assumption is defensible. Suspiciously tidy answers deserve the same scrutiny as untidy ones.
Step 5 — record the context, not just the number. Note today’s date and the inputs alongside the 0.100 output. Six months from now the context is what turns this figure into a trend. Suspiciously tidy answers deserve the same scrutiny as untidy ones.
Where Redshift matters in practice
In Stellar Physics the figure computed here is usually the first one somebody asks to see. Having it — with the formula behind it — changes your position in the conversation from participant to reference. A figure with a date beside it becomes evidence; without one it stays an anecdote.
Professionals in Stellar Physics use this style of calculation as a screening filter before spending real money or effort: a quick Redshift check eliminates weak options in minutes, and the survivors earn the deeper, slower analysis. The fastest sanity test is to guess the answer first and see how far the calculation disagrees.
There is a personal-administration use people overlook: periodic self-checking. Redshift quietly drifts as life changes — rates move, bodies change, costs compound — and re-running it a few times a year catches drift while correction is still cheap. The input you reach for first is usually the one you measured longest ago.
The questions that come up most about Redshift
What affects Redshift the most?
Measured on this engine, Rest (emitted) wavelength is the dominant lever: moving it ±10 % shifts the redshift z across 0.000 to 0.222 — about 122.2 % of the headline, with Observed wavelength next at ≈110.0 %. The ranked list in the sensitivity section is computed live, not asserted from a template.
Which matters more: observed wavelength or rest (emitted) wavelength?
Rest (emitted) wavelength, by a factor of about 1.1. Concretely: ten percent on Rest (emitted) wavelength is worth ≈122.2 % of Redshift, the same nudge on Observed wavelength only ≈110.0 %. That ranking is re-computed for each page from that page’s own defaults.
What is a typical Redshift?
With the example inputs on this page the result is 0.100, labelled “Receding at 28,488 km/s · 1.1000× the rest wavelength”. “Typical” varies with situation, which is why every result here ships with its verdict band and its sensitivity list: they turn one figure into a range you can locate yourself inside.
How do I calculate Redshift by hand?
Use the stated model — z = (λ_obs − λ_emit)/λ_emit; relativistic recession v = c·((1+z)² − 1)/((1+z)² + 1); comoving distance D_C = (c/H₀)∫₀^z dz′/E(z′), E(z) = √(Ωm(1+z)³ + ΩΛ) — and substitute in the order the tool prints (it starts from z = (λ_obs − λ_rest) ÷ λ_rest = (721.910 − 656.280) ÷ 656.280 = 0.100003.). On the example inputs the hand route lands on 0.100, exactly what the calculator shows. A spreadsheet reproduces it in two or three cells; doing it once by hand is what makes the automated answer trustworthy.
What happens to Redshift if rest (emitted) wavelength changes by 10 %?
Try it in the form above and the printed substitution will follow you; the pre-computed answer for the defaults is 0.000 to 0.222 across the ten-percent probe, i.e. ≈122.2 % sensitivity on the redshift z.
Is the Redshift result of 0.100 good or bad?
The tool answers “what”, not “whether”: 0.100 carries the verdict “Receding at 28,488 km/s · 1.1000× the rest wavelength” for the default case, which is a starting interpretation rather than a verdict on your situation. Re-run with your own figures and compare the bands.
What affects Redshift the most?
Sensitivity depends on where you start, but the inputs Rest (emitted) wavelength (nm) and Observed wavelength (nm) typically move Redshift further than the rest because they enter the formula multiplicatively. If you can only measure two things well, measure those and hold the others at conservative defaults.
How do I calculate Redshift manually?
Write down the inputs, then follow the printed steps in order — z = (λ_obs − λ_emit)/λ_emit; relativistic recession v = c·((1+z)² − 1)/((1+z)² + 1); comoving distance D_C = (c/H₀)∫₀^z dz′/E(z′), E(z) = √(Ωm(1+z)³ + ΩΛ). A spreadsheet replicates it with two or three cells. The manual route is worth doing once to build intuition; after that the calculator saves time and eliminates transcription slips.
What does a redshift of 1 mean?
Wavelengths are stretched to twice their emitted length (λ_obs = 2λ_rest). In ΛCDM that corresponds to ~3,400 Mpc comoving distance, a lookback time of ~7.8 Gyr, and light emitted when the universe was under 6 Gyr old — less than half its current age.
Can recession velocity exceed the speed of light?
Yes, and routinely does: the Hubble flow carries galaxies beyond ~4,400 Mpc (z ≈ 1.5) away faster than c. This violates nothing — no object moves through space faster than light, and no signal is transmitted superluminally. The relativistic Doppler formula reported here never exceeds c for any finite z.
Does this Redshift tool store my numbers?
No. The entire calculation runs in your browser session; nothing entered above is transmitted or saved, which is the correct behaviour for sensitive Astronomy & Space Science figures and the reason you can re-run this page as often as you like.
What the labels on this page mean
- Recession velocity (relativistic Doppler)
- 28,487.9 km/s in the default Redshift calculation — one of the secondary figures an experienced reader checks before trusting the primary number.
- Non-relativistic cz
- 29,980.2 km/s for the Redshift worked case; it sits next to the main result because it carries context the headline leaves out.
- Comoving distance (ΛCDM)
- 418.5 Mpc for the Redshift worked case; it sits next to the main result because it carries context the headline leaves out.
- Redshift z
- The primary output of this tool: 0.100 for the default inputs, computed exactly from z = (λ_obs − λ_emit)/λ_emit; relativistic recession v = c·((1+z)² − 1)/((1+z)² + 1); comoving distance D_C = (c/H₀)∫₀^z dz′/E(z′), E(z) = √(Ωm(1+z)³ + ΩΛ).
- Worked example
- The calculation written out in full so that nothing on this page has to be taken on trust.
- Verdict band
- The plain-language label — here “Receding at 28,488 km/s · 1.1000× the rest wavelength” — mapping the computed figure onto experience-based ranges for Astronomy & Space Science.
- Sensitivity (±10 %)
- How far the Redshift moves when a single input is nudged ten percent either way; the ranked list on this page is computed live from this engine.
Methodology, sources and limits
Everything above runs on one stated formula, applied without hidden coefficients: z = (λ_obs − λ_emit)/λ_emit; relativistic recession v = c·((1+z)² − 1)/((1+z)² + 1); comoving distance D_C = (c/H₀)∫₀^z dz′/E(z′), E(z) = √(Ωm(1+z)³ + ΩΛ). Source of record: Hubble (1929), PNAS; Planck Collaboration 2018 cosmological parameters (Ωm = 0.315, H₀ = 67.4); Riess et al. SH0ES (H₀ = 73.0); standard ΛCDM distance integrals per Hogg (1999), "Distance measures in cosmology". Specialist software layers extra coefficients on top of the same algebra; none are hidden here, and that transparency is what makes a disagreement traceable to a specific input instead of a black box.
Treat the output as a precise answer to the question you typed: exact arithmetic, transparent formula, published source, no hidden rounding. What that buys you is reproducibility — two people, two browsers, the same arithmetic, and a printed trail from inputs to result that either of them can audit.
Next steps after the calculation
Save the number together with its inputs and today’s date; a Redshift figure without context is unusable the moment an input changes.
This page was written for people who need one trustworthy number before their next decision, and the promise is that nothing above required trust. Bookmark the tool, re-run it whenever an input moves by more than a few percent, and keep the date with the number.