Stellar aberration is an astronomical phenomenon "which produces an apparent motion of celestial objects". It can be proven mathematically that stellar aberration is due to the change of the astronomer's inertial frame of reference. The formula is derived with the use of Lorentz transformation of the star's coordinates.
As the astronomer John Herschel has already explained in 1844, the stellar aberration does not depend on the relative velocity of the star towards Earth.[1] Otherwise eclipsing binary stars would appear to be separated, in stark contrast to observation: both stars are rotating with high speed —and ever changing and different velocity vectors— around each other, but they appear as one spot all the time.
In the year 1926 the astrophysicist Robert Emden published the article Aberration und Relativitätstheorie in the journal Naturwissenschaften.[2] In this article he states that the direction of a light ray isn't influenced by the motion of the star or by the motion of Earth.[Notes 1] At that time, the opponents of the special theory of relativity reasoned that the theory must be flawed, because it would state that the stellar aberration would depend on the relative velocity of the star — which would be in contradiction to observation — and R. Emden's article explains that the special theory of relativity does not predict this. Today, the special theory of relativity isn't contested anymore but there are still articles that suggest that the aberration would depend on the relative velocity of the star.[3]
Although a (relativistic) velocity-addition formula can be used to explain stellar aberration, (see Aberration of light), another (relativistic) explanation using only the Lorentz transformation is also possible, as will be demonstrated. This derivation only uses the star's coordinates at the time of emission, and therefore has the formal advantage there is no place for the relative velocity of the star towards the astronomer and therefore it is evident that the observed position doesn't depend on the star's velocity — provided that the resultant change of position is much smaller than the distance between star and Earth.[Notes 2] The observed position of the star wouldn't depend on Earth's motion either, if the astronomer could use the same inertial frame of reference all the time. But of course that is technically impossible,[Notes 3] the astronomer uses his current rest frame and these current rest frames are different at different times as Earth orbits around the Sun. It is mathematically convenient to declare the position of star in a rest frame of the Sun (more exactly: the center of mass of the Solar System) as the "real" position and that the difference to this "real" position derives form the "aberration".[Notes 4]
For the mathematical proofs, read more:
Stellar aberration (derivation from Lorentz transformation - Wikipedia)