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Astro Doppler Shift Calculator

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

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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

  1. 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.

    0.1 1 z β = 1 1.000 0.600 β = z

    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=λobs−λ0λ0=660.00−656.28656.28=0.00566831z=\frac{\lambda_{\text{obs}}-\lambda_0}{\lambda_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=1+β1−β⟹β=(1+z)2−1(1+z)2+11+z=\sqrt{\frac{1+\beta}{1-\beta}}\quad\Longrightarrow\quad \beta=\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=0.011368772.01136877=0.0056522(1+z)^{2}=1.01136877,\qquad \beta=\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.

      vrel=βc=1694.501 km/s;vclass=zc=1699.317 km/s;Δ=0.2842%v_{\text{rel}}=\beta c=1694.501\ \mathrm{km/s};\quad v_{\text{class}}=zc=1699.317\ \mathrm{km/s};\quad \Delta=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⇒β=0.09502 (classical is 5.24% high);z=1⇒β=0.600 (classical gives c)z=0.1\Rightarrow\beta=0.09502\ (\text{classical is }5.24\%\ \text{high});\qquad z=1\Rightarrow\beta=0.600\ (\text{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=vH0=1694.570=24.2 Mpc≈79 million light yearsd=\frac{v}{H_0}=\frac{1694.5}{70}=24.2\ \mathrm{Mpc}\approx79\ \text{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=(1+z)2−1(1+z)2+1⇒v=1694.501 km/s\beta=\frac{(1+z)^{2}-1}{(1+z)^{2}+1}\Rightarrow v=1694.501\ \mathrm{km/s}
References (1)

· Reviewed against published sources. Corrections welcome.