Redshift Calculator
Compute redshift using validated scientific equations. See step-by-step derivations, unit analysis, and reference values.
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Redshift z measures how much a wavelength has been stretched by cosmic expansion. The recession velocity uses the special relativistic Doppler formula. Distances are calculated by numerically integrating the Friedmann equation for a flat Lambda-CDM cosmology with H0 = 67.4 km/s/Mpc, Omega_M = 0.315, Omega_Lambda = 0.685.
Last reviewed: December 2025
Worked Examples
Example 1: Distant Galaxy Observation
Example 2: Hydrogen-Alpha Line Shift
Background & Theory
The Redshift Calculator applies the following established principles and formulas. Astronomy and space science rely on a set of precisely defined physical relationships that allow distances, sizes, motions, and energies of celestial objects to be calculated from observational data. Kepler's three laws of planetary motion, derived empirically in the early seventeenth century, describe elliptical orbits, equal areas swept in equal times, and the harmonic law Tยฒ = aยณ, where T is the orbital period in Earth years and a is the semi-major axis in astronomical units (AU). This relationship holds for any object orbiting the Sun and can be generalized using Newton's law of gravitation. Distances in astronomy are expressed in multiple units: one light-year equals approximately 9.461 ร 10ยนโต meters, one parsec equals 3.086 ร 10ยนโถ meters or about 3.26 light-years, defined as the distance at which one AU subtends one arcsecond of parallax. Angular size is calculated as ฮธ = 206,265 ร (d / D) arcseconds, where d is the physical diameter and D is the distance. The stellar magnitude system uses Pogson's formula: m1 โ m2 = โ2.5 ร log10(F1 / F2), where F represents flux. Each magnitude step corresponds to a flux ratio of approximately 2.512, meaning a first-magnitude star is 100 times brighter than a sixth-magnitude star. Hubble's Law relates recessional velocity to distance: v = Hโd, where the Hubble constant Hโ is approximately 70 km/s/Mpc. Escape velocity from any body is given by v = โ(2GM/r), yielding 11.2 km/s for Earth. Orbital period for a circular orbit follows T = 2ฯโ(rยณ/GM). Luminosity and distance are linked by the inverse square law: F = L / (4ฯdยฒ). Stars are classified by spectral type using the mnemonic OBAFGKM, corresponding to surface temperatures from approximately 30,000 K (O-type) to under 3,500 K (M-type). Each type reflects characteristic absorption spectra tied to ionization states of elements in the stellar photosphere.
History
The history behind the Redshift Calculator traces back through the following developments. The history of astronomy is one of progressive scale โ each era expanding humanity's conception of the universe's size and structure. The Copernican revolution of 1543, when Nicolaus Copernicus published De revolutionibus orbium coelestium, displaced Earth from the center of the cosmos and placed the Sun at the center of the planetary system. Decades later, Galileo Galilei turned a Dutch-invented telescope toward the sky in 1609, discovering the moons of Jupiter, the phases of Venus, and the cratered surface of the Moon โ observations that provided compelling evidence for the heliocentric model and led to his conflict with the Catholic Church. Johannes Kepler, working from Tycho Brahe's meticulous naked-eye observations, derived his three laws of planetary motion between 1609 and 1619. Isaac Newton unified celestial and terrestrial mechanics with his law of universal gravitation in 1687, explaining the cause behind Kepler's empirical laws and enabling precise prediction of planetary positions. The eighteenth and nineteenth centuries brought systematic sky surveys, stellar parallax measurements, and the discovery that the Milky Way is itself a galaxy among many. Edwin Hubble's 1929 observations using the 100-inch Hooker Telescope at Mount Wilson demonstrated that galaxies are receding from us at velocities proportional to their distance โ the first direct evidence for an expanding universe and the empirical basis for Big Bang cosmology. NASA was founded in 1958 following the Sputnik shock, and the Apollo 11 mission landed humans on the Moon on July 20, 1969. The Hubble Space Telescope, launched in 1990, revolutionized observational astronomy by operating above Earth's atmosphere and producing imagery from ultraviolet to near-infrared wavelengths. The first confirmed exoplanet around a Sun-like star was detected in 1995 by Michel Mayor and Didier Queloz using the radial velocity method. The James Webb Space Telescope, launched in December 2021 and fully operational by 2022, extended infrared observations to probe the earliest galaxies formed after the Big Bang.
Frequently Asked Questions
Formula
z = (lambda_obs - lambda_emit) / lambda_emit | v = c ร ((1+z)ยฒ - 1) / ((1+z)ยฒ + 1)
Redshift z measures how much a wavelength has been stretched by cosmic expansion. The recession velocity uses the special relativistic Doppler formula. Distances are calculated by numerically integrating the Friedmann equation for a flat Lambda-CDM cosmology with H0 = 67.4 km/s/Mpc, Omega_M = 0.315, Omega_Lambda = 0.685.
Worked Examples
Example 1: Distant Galaxy Observation
Problem: A galaxy has a measured redshift of z = 2.0. Calculate its recession velocity, comoving distance, and lookback time.
Solution: z = 2.0\nRelativistic velocity: v = c ร ((1+z)ยฒ - 1)/((1+z)ยฒ + 1)\nv = 299792 ร (9-1)/(9+1) = 299792 ร 0.8 = 239,834 km/s\nv/c = 80%\nComoving distance โ 5,200 Mpc (numerical integration)\nLookback time โ 10.3 Gyr (universe was 3.5 Gyr old)
Result: v = 239,834 km/s (80% c) | Distance: ~5.2 Gpc | Lookback: ~10.3 Gyr
Example 2: Hydrogen-Alpha Line Shift
Problem: The hydrogen-alpha emission line (656.3 nm) is observed at 722 nm from a galaxy. Calculate the redshift and distance.
Solution: z = (722 - 656.3) / 656.3 = 65.7 / 656.3 = 0.1001\nVelocity โ 29,553 km/s\nComoving distance โ 420 Mpc โ 1.37 billion light-years\nLookback time โ 1.29 Gyr\nScale factor at emission: a = 1/1.1 = 0.909
Result: z = 0.1001 | Distance: ~420 Mpc | Lookback: ~1.29 Gyr
Frequently Asked Questions
What is cosmological redshift?
Cosmological redshift is the stretching of light wavelengths caused by the expansion of the universe. As light travels from a distant galaxy toward us, the space through which it travels expands, stretching the wavelength of the light. This makes the light appear redder (shifted toward longer wavelengths) compared to when it was emitted. The redshift parameter z is defined as z = (lambda_observed - lambda_emitted) / lambda_emitted. A redshift of z = 1 means the wavelength has doubled during its journey, indicating the universe has expanded by a factor of 2 since that light was emitted. Cosmological redshift is distinct from Doppler redshift (caused by relative motion) and gravitational redshift (caused by strong gravitational fields), though all three produce similar observational effects.
What is the Hubble constant and why does it matter for redshift?
The Hubble constant (H0) describes the current rate of expansion of the universe, typically expressed in kilometers per second per megaparsec (km/s/Mpc). The current best estimate from the Planck satellite is approximately 67.4 km/s/Mpc, meaning a galaxy 1 Mpc (3.26 million light-years) away is receding from us at about 67.4 km/s. For nearby objects (z < 0.1), Hubble's Law provides a simple linear relationship: velocity = H0 ร distance. For higher redshifts, the relationship becomes non-linear and requires integrating over the expansion history of the universe. The precise value of H0 is actively debated in modern cosmology, with local measurements yielding about 73 km/s/Mpc, creating the so-called Hubble tension.
What is the highest redshift ever observed?
The highest spectroscopically confirmed galaxy redshifts have been pushed beyond z = 13 by the James Webb Space Telescope (JWST), with JADES-GS-z14-0 confirmed at z = 14.32 in 2024, meaning its light was emitted when the universe was only about 290 million years old. The cosmic microwave background (CMB) has a redshift of z = 1089, representing the earliest observable light from when the universe became transparent about 380,000 years after the Big Bang. At z = 1089, the CMB photons have been stretched by a factor of 1,090, shifting from visible/infrared light at ~3,000 K to the microwave radiation at 2.725 K we observe today. Gravitational waves could potentially detect signals from even earlier epochs.
Can redshift exceed z = 1, and does that mean faster than light?
Yes, redshift values routinely exceed z = 1 for distant galaxies. Thousands of galaxies have been observed with z > 1, and the most distant known objects have z > 10. While z > 1 corresponds to recession velocities exceeding the speed of light when calculated using the relativistic Doppler formula naively, this does not violate special relativity. The recession is due to the expansion of space itself, not motion through space. General relativity allows space to expand at any rate; it is only local motion through space that cannot exceed c. The recession velocity at z = 1.5 is approximately 0.724c using the special relativistic formula, but the cosmological recession velocity (Hubble flow) actually exceeds c for objects beyond the Hubble sphere, approximately 4,400 Mpc from us.
What is redshift and how does it indicate motion?
Redshift occurs when light from an object moving away from us is stretched to longer (redder) wavelengths. It is measured as z = (observed wavelength - emitted wavelength) / emitted wavelength. Cosmological redshift indicates the expansion of the universe. A galaxy at z=1 is seen as it was when the universe was half its current size.
Is my data stored or sent to a server?
No. All calculations run entirely in your browser using JavaScript. No data you enter is ever transmitted to any server or stored anywhere. Your inputs remain completely private.
References
Reviewed by Daniel Agrici, Founder & Lead Developer ยท Editorial policy