How is it possible for binary stars to not show changes in aberration angles?

[TrickyDicky]

What matters is relative velocity between Earth one date and Earth another date. The sun, or anything else are involved only for convenience. All visible aberration is derivable considering only Earth frames.

In SR, the simplest formulation is based on nothing but the transform of a null vector from one frame to another (Einstein took the more physical approach of Lorentz transform of wave fronts, but the result is the same).

In these two quotes I interpret you are stating explicitly that the formula for aberration says that the relative velocity v of Earth is referred to the rest frame of the sun and the inciding photon's null vector coming from the distant star. If not please correct the part that is not well interpreted.

Please, don't impute such ridiculous ideas to Bill_k. Bill_k is simply noting that for a binary, the angle between the momentary source frame and Earth is constantly changing.

What ridiculous ideas?

You can refer the Earth's motion to any other frame you want and get the same result. It is just needless complexity.

You mean that inserting a different v in the aberration formula would still give the same change in aberration angle for a specific star image?

 

[PAllen]

In these two quotes I interpret you are stating explicitly that the formula for aberration says that the relative velocity v of Earth is referred to the rest frame of the sun and the inciding photon's null vector coming from the distant star. If not please correct the part that is not well interpreted.

That's the opposite of what both quotes say. They state that all you really care about the motion of Earth on one time relative to Earth at another. You don't need the sun's frame to state that Earth in March is moving with speed v and some direction relative to Earth in January.

What ridiculous ideas?

The rest frame of a photon. What Bill_k wrote was fine, and I do not comprehend how you could interpret it as you did.

You mean that inserting a different v in the aberration formula would still give the same change in aberration angle for a specific star image?

I mean that any of the following would give the same result:1) Use the general angular aberration formula with v as relative velocity of Earth at t1 compared to Earth at t2.

2) Convert observed angle at t1 to a solar frame using Earth's velocity relative to the sun. Then convert angle in the solar frame to frame of Earth at t2.

3) Convert Earth observation to rest frame of star (symbolically, since you may not know it). Then convert to rest frame of Earth at t2.

The most direct, and pedagogical as to things like why there is no impact due to speed or acceleration of source is (1). The most practical, computationally, is (2). The historic derivation is closest in spirit to (3). However, (3) is needlessly complex in the case of an accelerating source (at minimum, you interpose the step of transforming from source rest frame at t1 to source rest frame at t2; then back to Earth frame at t2)

 

[micromass]

I think there has been enough discussion on this now. It's time to put this thread to rest.