How did Bradley arrive at the +/- correct speed of light when he calculated it?

Question

After Bradley discovered stellar aberration and the corresponding constant of aberration he was able to calculate the speed of light, since he knew the speed of the Earth around the Sun.

As far as I think I understand the math behind his calculations. Yet, I believe that Bradley (and anyone else until Relativity) didn't consider the velocity of the Solar System itself. It seems (to me, that is) that Bradley assumed a Solar System at rest. If the Solar System is moving, this certainly affects the result of the calculation (especially so since at that time the speed of light was not considered constant).

I'd appreciate if someone pointed out to me where my thinking is wrong.

Answer 1

He didn't need to know the absolute position of the stars. The change of the appearent position has been sufficient to get a good estimate. For velocites small in comparison to the speed of light, the shift of the angle is still small.

For small angles the sine is proportional to the angle with 2nd order precision. The first Taylor summands are $$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - + ... $$

The 2nd order approximation $$\sin x\approx x$$ can be applied on stars perpendicular, or at least apearently perpendicular, to the plane of the Earth orbit, where the method is most accurate.

Hence the underlying reason, for which it works, is the smoothness of sine.

This argument can be adjusted to less optimal conditions.

Answer 2

The velocity of the Earth going around the Sun changes such that the sign changes every six months. Therefore, the part of the aberration due solely to the Earth's motion about the Sun also changes sign. In fact, the aberration from the Earth's orbit makes the positions of stars go into ellipses that take 1 year to complete. The motion of the Sun about the Galaxy also makes galaxy positions move around in an ellipse, but one that takes 250 million years to complete.