Astro Doppler Shift Calculator

Enter rest and observed spectral values to estimate z and compare classical vs relativistic velocity.

Worked example — Astro Doppler Shift Calculator

651.66 nm	664.62 nm
Relativistic velocity	1694.501 km/s
Model difference	0.2842%
Redshift z	0.00566831

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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At high redshift the shortcut v = zc returns three times the speed of light 🖖

The tool prints the relativistic velocity and the classical v = zc side by side, and the gap between them is the whole point. For the two hydrogen presets z is about 0.006 and the two agree to the digit, so the shortcut is harmless. Then load the quasar preset: Lyman-α has moved from 121.567 nm to 486.268 nm, exactly four times its rest wavelength, so z = 3. Now v = zc comes out at 3c — three times light speed — while the relativistic β = ((1+z)² − 1)/((1+z)² + 1) gives 0.882c. The shortcut overstates the speed by 240% and breaks physics doing it.

What a redshift actually measures 🖖

A spectral line is a fixed fingerprint — hydrogen always emits at 656.28 nm when at rest. Measure where you actually see that line: if it drifts to a longer (redder) wavelength the source is receding; a shorter (bluer) one means it approaches. The redshift z is simply the fractional stretch. Seeing H-alpha at 660 nm instead of 656.28 nm gives z ≈ 0.0057, a recession of roughly 1700 km/s.

Redshift can exceed 100% without breaking c 🖖

Load the quasar preset: Lyman-alpha shifts from 121.567 nm to 486.268 nm, so z = 3 and the line quadruples (1 + z = 4). The naive v = zc would claim three times light speed — impossible. The relativistic formula β = ((1+z)²−1)/((1+z)²+1) instead gives 15/17 ≈ 0.88, safely below c. No matter how large z grows, β never reaches 1.

Problem solved in full

Recession speed when a galaxy's hydrogen-alpha line sits at 660.00 nm 6 steps
A galaxy's hydrogen-alpha line sits at 660.00 nm instead of 656.28 nm. Find its recession speed properly, and settle whether the schoolbook v = cz is good enough here — and where it stops being good enough.
1. Redshift is defined as the fractional change in wavelength, and it is the only measured quantity on this page. Everything below is arithmetic on it.
  
  $z = \frac{λ _{obs} - λ _{0}}{λ _{0}} = \frac{660.00 - 656.28}{656.28} = 0.00566831$
2. The relativistic Doppler formula for a source moving directly away combines the classical shift with time dilation. Invert it for β — squaring both sides makes the rearrangement a single line.
  
  $1 + z = \frac{1 + β}{1 - β} ⟹ β = \frac{( 1 + z ) ^{2} - 1}{( 1 + z ) ^{2} + 1}$
3. Substitute. The numerator is essentially 2z and the denominator essentially 2, which is why the answer is so close to z at small redshift, and why the agreement has to break down once z² stops being negligible.
  
  $(1 + z)^{2} = 1.01136877, β = \frac{0.01136877}{2.01136877} = 0.0056522$
4. Multiply by c both ways and compare. The 4.8 km/s gap is far smaller than the peculiar velocity of a galaxy in a cluster, so at this redshift the choice of formula is not the limiting error.
  
  $v_{rel} = β c = 1694.501 km/s; v_{class} = z c = 1699.317 km/s; Δ = 0.2842%$
5. Push z upward and the two diverge. The classical expression has no ceiling; the relativistic one has β → 1 built into its algebra.
  
  $z = 0.1 \Rightarrow β = 0.09502 (classical is 5.24% high); z = 1 \Rightarrow β = 0.600 (classical gives c)$
6. One more step turns the speed into a distance. Hubble's law with H₀ = 70 km/s/Mpc puts this galaxy about 24 Mpc away — a long way outside the Local Group, and a distance nothing else in the measurement could have supplied.
  
  $d = \frac{v}{H _{0}} = \frac{1694.5}{70} = 24.2 Mpc \approx 79 million light years$
Answer

1694.501 km/s, against 1699.317 km/s from v = cz — a 0.28% overstatement. At this redshift the shortcut is harmless. It stays harmless for a while and then fails abruptly: at z = 0.1 it is 5.2% high, and at z = 1 it returns exactly c, which is not a large error but an impossible answer. The relativistic form cannot do that, because β = ((1+z)²−1)/((1+z)²+1) approaches 1 from below no matter how large z becomes. Feed it z = 5 and it returns 0.946c; feed the shortcut z = 5 and it returns five times the speed of light.

$β = \frac{( 1 + z ) ^{2} - 1}{( 1 + z ) ^{2} + 1} \Rightarrow v = 1694.501 km/s$

References (1)

The relativistic treatment this tool uses rather than the classical shortcut — the Doppler section is §7: A. Einstein, "Zur Elektrodynamik bewegter Körper." Annalen der Physik 322, 891–921, 1905.

By the 42Math team · Updated 2026-07-06

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

All Astronomy & Space lessons →

Example problems

H-alpha receding - H-alpha shifted to longer wavelength implies recession (redshift).
H-alpha approaching - Shorter observed wavelength implies approach (blueshift).
21-cm radio source - Lower observed 21-cm frequency corresponds to positive redshift.
High-z quasar - Large z case where relativistic velocity correction becomes essential.