How do you calculate redshift z from wavelengths?

Redshift z is defined as z = (λobs − λemit) / λemit, where λemit is the rest (laboratory) wavelength of a spectral line and λobs is the observed wavelength from the astronomical object. A positive z means redshift; a negative z means blueshift.

How is redshift related to recessional velocity?

For small velocities (v ≪ c), the Doppler redshift is approximately z ≈ v / c. For higher velocities, you should use the relativistic Doppler formula: 1 + z = √((1 + v/c) / (1 − v/c)). This calculator uses the relativistic formula when converting between z and velocity.

How do you estimate distance from redshift?

For nearby galaxies with small redshift (z ≲ 0.1), you can use Hubble’s law: v ≈ H0 × d, where H0 is the Hubble constant and d is the distance. Combining this with v ≈ zc gives d ≈ (zc) / H0. This is a linear approximation and breaks down at high redshift, where full cosmological models are needed.

What is the difference between Doppler and cosmological redshift?

Doppler redshift is due to the relative motion of a source through space, like a galaxy moving away from us. Cosmological redshift is due to the expansion of space itself stretching the wavelength of light as it travels. Mathematically they can look similar at low redshift, but cosmological redshift encodes information about the history of the universe’s expansion.

Astronomical Redshift Calculator (Doppler & Cosmological)

How this astronomical redshift calculator works

This tool is designed for astronomy and cosmology students, observers, and researchers who need quick conversions between redshift, wavelength, recessional velocity, and approximate distance. It supports:

Computing redshift from rest and observed wavelength.
Converting between redshift and relativistic Doppler velocity.
Estimating distance from redshift using Hubble’s law.

1. Redshift from wavelengths

The basic definition of redshift is:

\( z = \frac{\lambda_{\text{obs}} - \lambda_{\text{emit}}}{\lambda_{\text{emit}}} \)

where:

\(\lambda_{\text{emit}}\) is the rest (laboratory) wavelength of a spectral line.
\(\lambda_{\text{obs}}\) is the observed wavelength from the astronomical source.

A positive \(z\) means redshift (lines shifted to longer wavelengths); a negative \(z\) means blueshift.

2. Relativistic Doppler redshift and velocity

For small velocities (\(v \ll c\)), the Doppler redshift is approximately:

\( z \approx \frac{v}{c} \)

However, many astronomical objects move at a significant fraction of the speed of light, so this calculator uses the relativistic Doppler formula for motion along the line of sight:

\( 1 + z = \sqrt{\frac{1 + \beta}{1 - \beta}} \quad \text{where } \beta = \frac{v}{c} \)

Solving for \(\beta\) gives:

\( \beta = \frac{(1+z)^2 - 1}{(1+z)^2 + 1} \quad\Rightarrow\quad v = \beta c \)

The calculator uses this relation in both directions:

From \(z\) to velocity \(v\).
From velocity \(v\) (km/s or fraction of \(c\)) to redshift \(z\).

3. Distance from redshift (Hubble’s law)

For nearby galaxies (small redshift, typically \(z \lesssim 0.1\)), Hubble’s law relates recessional velocity and distance:

\( v \approx H_0 \, d \)

where:

\(H_0\) is the Hubble constant (in km/s/Mpc).
\(d\) is the distance in megaparsecs (Mpc).

Combining this with the low-redshift approximation \(v \approx zc\) gives:

\( d \approx \frac{zc}{H_0} \)

This calculator:

Computes \(v\) from \(z\) using the relativistic formula.
Then estimates distance using \(d \approx v / H_0\).

This is a simple linear approximation. For high redshift (e.g. \(z \gtrsim 0.3\)), you should use a full cosmological model with \(\Omega_m\), \(\Omega_\Lambda\), etc.

4. Units and conversions

Wavelengths can be entered in nm, Å, or µm. Internally they are converted to meters.
Velocity can be entered in km/s or as a fraction of \(c\).
Distances are reported in Mpc, light-years, and million light-years (Mly).

Worked example

Suppose you observe the Hα line (rest wavelength 656.28 nm) at 721.9 nm in a galaxy spectrum.

Enter λ_emit = 656.28 nm and λ_obs = 721.9 nm.
The calculator finds \( z = (721.9 - 656.28) / 656.28 \approx 0.1 \).
Using the relativistic Doppler formula, it computes \(v \approx 28{,}400\) km/s.
With \(H_0 = 70\) km/s/Mpc, the distance is \(d \approx 405\) Mpc.

FAQ

What is astronomical redshift?

Astronomical redshift is the increase in wavelength (or decrease in frequency) of light from distant objects. It is usually interpreted as evidence that the source is moving away from us or that the universe is expanding. Redshift is a key observable in cosmology because it encodes information about distance, velocity, and the history of cosmic expansion.

What are the main types of redshift?

Doppler redshift: due to relative motion through space.
Cosmological redshift: due to expansion of space itself.
Gravitational redshift: due to light climbing out of a strong gravitational field.

When is Hubble’s law a good approximation?

Hubble’s law is accurate for relatively nearby galaxies where the universe’s expansion history has not changed dramatically over the light-travel time, and where peculiar velocities are small compared to the Hubble flow. As a rule of thumb, it is commonly used up to \(z \sim 0.1\), but precision cosmology uses full Friedmann–Lemaître–Robertson–Walker (FLRW) models at any redshift.

Why does this calculator use the relativistic Doppler formula?

At velocities above a few percent of the speed of light, the classical approximation \(z \approx v/c\) becomes inaccurate. The relativistic formula remains valid up to arbitrarily high velocities (approaching \(c\)), so it is safer and more accurate for astronomical applications.

Astronomical Redshift Calculator (Doppler & Hubble)

Redshift Calculator

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