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Relativistic Aberration of Light Calculator

Enter the emission angle and speed ratio β to get the observed (aberrated) angle.

How it is calculated

cos θ′ = (cos θ − β) / (1 − β·cos θ), then θ′ = arccos(cos θ′) in degrees. Here θ is the angle in degrees between the direction the light travels and the direction of the observer's velocity, as measured in the original (rest) frame, β = v/c is the observer's speed as a fraction of the speed of light, and θ′ is the same direction as measured by the moving observer.

The angle must be from 0 to 180 degrees and β from 0 up to but not including 1; other values give no result. With β = 0 nothing changes, and light travelling exactly along or against the motion (0° or 180°) keeps its angle. For all angles in between, the observed angle comes out larger than the original, and the effect grows as β approaches 1. The tool gives one angle only, not the Doppler shift or the intensity change.

Worked example

Light travels at θ = 90° to the observer's direction of motion, and the observer moves at β = 0.6. cos 90° = 0, so cos θ′ = (0 − 0.6) / (1 − 0.6 × 0) = −0.6. θ′ = arccos(−0.6) = 126.87°. The light's direction is tilted 36.87° beyond the right angle. The source's apparent position is 180° − 126.87° = 53.13° from the direction of motion, so the source looks shifted toward that direction.

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Common mistakes

  • Entering β as a speed in km/s or m/s: β is the ratio v/c, so 0.6 means 60% of the speed of light; a value of 1 or more gives no result.
  • Using radians: the angle field is in degrees, from 0 to 180.
  • Measuring the angle from the wrong axis, for example from the sideways direction: it is measured from the direction of the observer's motion.
  • Reading θ′ as the apparent position of the source: it is the travel direction of the light, so the apparent source direction is 180° − θ′ from the line of motion.

About this calculator

This calculator computes the relativistic aberration of light: how the direction in which light travels, θ, becomes θ′ for an observer moving at speed ratio β = v/c, using cos θ′ = (cos θ − β) / (1 − β·cos θ). This differs from classical (non-relativistic) aberration by the Lorentz-related denominator term, which becomes significant as β approaches 1.

Aberration explains why, at relativistic speeds, the apparent position of a light source (180° − θ′, where θ′ is the direction the light travels in the observer's frame) shifts toward the direction of motion in an observer's field of view — an effect called relativistic beaming or the "headlight effect." Astrophysicists rely on it to interpret observations of jets from active galactic nuclei and pulsars, and it's a standard topic in special relativity for connecting an object's rest-frame emission pattern to what a fast-moving observer actually sees.

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Comments (1)

  • Editorial team

    A quick sanity check: light at 90 degrees with beta 0.5 comes out at exactly 120 degrees, because the result is arccos of minus beta. Results show two decimals of a degree, so tiny speeds barely register. Beta 0.0001, roughly Earth's orbital speed, gives 90.01 degrees. Which situation are you modelling, a jet, a starship or stellar aberration?

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