Is there a logical reason why we see no back and forth aberration effects when observing binary stars?

Question

I read this article, Questioning Einstein: Is Relativity Necessary? (Bethell (2010), Proc. Natural Philosophy Alliance, v.6, no. 2), regarding the 2 stars of Mizar A and wondering if it is true that we do not see back and forth aberration effects when observing binary star systems? If true, is there a known explanation?

In 1889 Edward Pickering had discovered spectroscopic binary stars, orbiting a common center of mass so closely that they appear as a single star even in the most powerful telescopes. The two stars of Mizar A, in the handle of the Big Dipper, are only 18 million miles apart. Their separate character is apparent only from the alternating Doppler shifts in their spectral lines. They orbit one another at a velocity of 50 kilometers per second, or 1.7 times the Earth’s orbital velocity.

It follows that even if we still don’t know the true velocity of the binary systems with respect to the sun, we do know that there is a difference, sometimes large, between the separate stars of the binary system relative to us. In the case of Mizar A, at a given moment in its 104-day cycle, one star is approaching us at the high velocity of 50 kilometers per second (relative to the center of mass of the system) while the other star is moving away from us at the same velocity. Irrespective of the movement of the system as a whole, then, the twin elements of the orbiting pair are moving at velocities that exceed the Earth’s orbital velocity by quite a large margin.

By Einstein’s formula, alternating back and forth aberrations should therefore be observed, corresponding to the period of their orbits. The aberration angle of each star should increase and then decrease as they circle each other. This would be easily observable by modern instruments. The differential aberration would be so large that the stars would become visibly separate in the sky before closing again. In the case of Mizar A, the angular separation of the binary components would be more than a minute of arc. But they always remain as an unresolvable point in the sky. And because binary systems are so common, we should be seeing this back and forth apparent motion of binary components all over the heavens. But we never do

Answer 1

There is no dependence of aberration on the velocity of the source. None is seen, and none is predicted by special relativity.

The error in the text that you quoted seems to be the implicit assumption that because of the principle of relativity, the amount of aberration can only depend on the relative velocity of the emitter and receiver. Actually, there is a third relevant quantity in this problem—the spacetime separation of the emitter and receiver—and the principle of relativity only tells you that the answer must be some frame-independent function of those three quantities. It turns out to depend only on the receiver's velocity and the direction of the separation vector.

Answer 2

This is easy to understand if you consider the problem in terms of classical plane wave optics:

the apparent position of the star is defined by the orientation of the wavefront we receive. Ignoring any changes of this due to physical effects like refraction etc. along its path, this can only change if the position of the source (the star) changes. If you assume the star is displaced by a distance $\Delta x$ transversely to the direction of observation, then the orientation of the wave front will change by an angle

$$\Delta \alpha = arctan(\frac{\Delta x}{R})$$

where $R$ is the distance of the star.

For Mizar, the distance R is 83 light years = $4.9 \cdot 10^{14} miles$, so if it is displaced by $\Delta x = 18 \cdot 10^6 miles$, the apparent position will change only by about

$$\Delta \alpha \approx 2 \cdot 10^{-6} deg = 7.6 \cdot 10^{-3} arcsec$$

This is completely negligible compared to the aberration due to the Earth's orbital motion of about 20 arcsec.

Answer 3

You can measure red and blue shift of binary stars, but you can't see it.

The Hydrogen spectrum peak taken on the east side of the sun (left side) is a wavelength of 434.044nm and from the west-side (right side) is a 434.050nm. For aligned binary stars the shift if far higher.

It cant be always be seen for binary because: The period of change is too slow, 5+ minutes

The binary rotation has to be aligned with the viewer.

The chances that the stars occlude one another is minimal.

http://www.alevelphysicsnotes.com/astrophysics/doppler_effect.php