Annual aberration, also known as stellar aberration, is an astronomical phenomenon caused by the motion of the Earth in its orbit around the Sun. It is a type of apparent motion observed in the positions of stars as seen from Earth.

When an observer on Earth looks at a star, they perceive its position slightly shifted from its true location due to the finite speed of light and the motion of the observer. This effect is similar to how raindrops appear slanted when viewed from a moving car.

The annual aberration occurs because the Earth is in motion relative to the fixed stars. As the Earth orbits the Sun, its velocity creates an apparent change in the direction from which starlight arrives. This change causes a slight displacement in the apparent position of the star.

The magnitude of annual aberration is dependent on the speed of the Earth in its orbit (about 30 kilometers per second) and the speed of light. It is approximately 20.5 arcseconds (0.0057 degrees) in magnitude.

To illustrate the concept, imagine standing in the middle of a rainstorm and extending an umbrella straight up. Even though the rain is falling vertically, from your perspective, the raindrops appear to be coming in at a slight angle because of your own motion. Similarly, due to the Earth’s motion, starlight appears to come from slightly different directions.

The discovery of annual aberration is credited to the English astronomer James Bradley, who first measured it in 1728. Annual aberration is an important correction factor that needs to be considered in precise astronomical observations and calculations to accurately determine the positions of celestial objects.

In this example of 2D frame moving only in the $x$-direction

$$ \tan\phi = \frac{c_y'}{c_x'} = \frac{c_y}{\gamma(c_x+v)} $$

where $\gamma=(1-v^2/c^2)^{-1/2}$ and $v$ is the velocity of the observer. The velocity components of light in the rest and moving frame are $(c_x,c_y)$ and $(c_x',c_y')$, respectively. So

$$ \tan\phi = \frac{c\sin\theta}{\gamma(c\cos\theta+v)} = \frac{\sin\theta}{\gamma(v/c+\cos\theta)}. $$

If $v \ll c$, $\gamma=1$. and if $\theta=90^\circ$ then

$$ \tan(\theta-\phi) = \cot\phi = \frac{\gamma(v/c+\cos\theta)}{\sin\theta} = \frac{v}{c} $$

and, finally, if $\theta-\phi$ is very small then aberration $\alpha = \theta-\phi \approx v/c$.