Astronomy and astrophysics Spherical astronomy and astrometry Proposition

Aberration of light

Statement

Let $S$ be an inertial frame, and let an observer move with constant velocity $v$ relative to $S$ . Let $c$ be the speed of light, and let $\beta = \abs{v}/c$ , with $0 \le \beta < 1$ . Let light from a distant source arrive at the observer. Let $\theta \in [0, \pi]$ be the angle in $S$ between $v$ and the direction towards the source. Then the observer sees the source at the angle $\theta'$ from the direction of motion, where $\cos\theta' = \frac{\cos\theta + \beta}{1 + \beta\cos\theta}.$ In particular $\theta' \le \theta$ , with equality only when $\beta = 0$ or $\sin\theta = 0$ , and $\theta - \theta' = \beta\sin\theta + O(\beta^2)$ .

Remarks

The light travels in the direction $-n$ , where $n$ is the unit vector towards the source. Apply the Lorentz transformation to the photon's energy $E$ and momentum component $p_\parallel = -(E/c)\cos\theta$ along $v$ . This gives $E' = \gamma E (1 + \beta\cos\theta)$ and $c\,p'_\parallel = -\gamma E(\cos\theta + \beta)$ , with $\gamma = (1 - \beta^2)^{-1/2}$ , and the formula follows. So a moving observer sees all sources shifted towards the point on the celestial sphere towards which it moves. Example: the Earth's orbital speed is about $29.8$ km/s, so $\beta \approx 9.9 \times 10^{-5}$ . In radians this is about $20.5''$ , the constant of annual aberration. Over a year each star traces a small ellipse of semi-major axis about $20.5''$ .

Going deeper

For a student

The exact formula is the relativistic transformation of directions of light rays between two inertial frames. To first order in $\beta$ it agrees with the classical picture of adding the observer's velocity to the velocity of light. In astrometry one distinguishes annual aberration, from the orbital motion of the Earth, and diurnal aberration, from its rotation, which is at most about $0.32''$ . Unlike parallax, aberration does not depend on the distance of the star, only on its direction. James Bradley discovered annual aberration while he was looking for stellar parallax.

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Celestial sphere concept
Aberration of light statement

Tag	`WBCV84VHEH`
URI	`m2://WBCV84VHEH`
Kind	statement, proposition
Writing	W0 stub
MSC	85A04, 83A05
Status	This copy of the site has no build data. Run `m2 status WBCV84VHEH`.
About	Celestial sphere
Uses	Celestial sphere
Changed	3 November 2026