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

[Dale]

Another point in this thread that I think deserves to be better explained is that of the relativistic composition of velocities, a formula was given by jartsa that involved addition of the velocity of photons coming from the stellar source to observer's velocity wrt sun, I was not aware that relativistic addition of velocities could be performed with photons velocities, always saw it done with objects moving below c. Again this seems to go counter the restriction to use the unphysical photon's frame.

You can use the relativistic velocity addition formula with photons without implying a reference frame going at c.

So the usual way of writing the velocity addition formula is:
$$v'=\frac{v+u}{1+vu/c^2}$$
Where $v$ is the velocity of the object in the unprimed frame, $v'$ is the velocity of the object in the primed frame, and $u$ is the relative velocity between the two frames. As long as $u<c$ you are dealing with two standard inertial frames and not a mythical photon frame. $v$, on the other hand, can be whatever. It can be c or (in principle) even greater.

 

[PAllen]

What could seem puzzling if one considers aberration of sufficiently distant stars is why this particular change in velocity wrt the sun prevails over many other faster ones that depend on the Earth's motion, like its motion in the galaxy, or the local group..., and for closer ones any other relative motion wrt random objects orbiting near the earth.

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.

 

[PAllen]

I am too left wondering about the usual explanation given for annual stellar aberration and can imagine some questions many could come up with, so maybe somebody could try and address these points.
It is normally accepted that the relativistic explanation follows the same reasoning as the classical Newtonian one, just giving a better approximation since it uses the relativistic velocity addition.

No, not really. The Bradley derivation requires the assumption of corpuscular light theory, with speed affected by emitter. That is why there was no satisfactory explanation from the time when wave theory was accepted until SR. 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).

The first question would be, how can it be the same reasoning, classical Newtonian physics is based in an absolute space and absolute velocities, so it makes sense to attribute the aberration effect just to the observer's velocity, but it doesn't sound so right in the relativistic paradigm, more so if we consider the case of binaries mentioned by the OP.

No, the Bradley derivation was based on relative velocity of source and observer, and the observed seasonal pattern was considered due to variation of this relative velocity due to Earth's orbit. Bradley assumed all stars were 'stationary' in a frame of the 'firmament'.

Bradley's approach could most readily handle binaries by comparing the seasonally changing Earth frame to a binary COM frame that would be considered to be the same as the frame of other stars.

So if we are using relativity and consider just relative velocities between the source and the observer one may wonder why shoul the sun's frame be important in the relative motion between the Earth and the stellar source. Surely the Earth is in relative motion wrt many objects in a periodic way that we do not care for in order to determine aberration wtt distant stars. So what is special sbout the sun's frame?

The only reason to introduce the sun's frame is convenience. Especially, to handle parallax and substantial movement of a (closer) source, combined with aberration, it is convenient to transform daily observations to a common sun frame. Then, you have a model of motion of bodies in the sun frame. Once you have such a model, you just compute Earth observation frame via translation, rotation, and boost from solar model.

 

[PAllen]

That's a nice comment, but IMO it doesn't by itself clarify the specific questions I posed above: if indeed any two inertial frames(the Earth's and another) are valid, why the correct aberration is only obtained using the relative velocity of the Earth and the Sun frames, and not using the Earth and any other object's frame with motion periodically related to the Earth , what is special about the specific sun-earth velocity for the observed aberration of light coming from distants stars?

You can get the correct aberration using many different conventions. As I have said several times, you only need relative velocity between Earth at time t1 and time t2, to compute aberration (change in observed angles due to change of frame) between those two times. To account for parallax, you would also need change in position of earth. To account for detectable motion of star, you need a model of its motion in some inertial frame. If you had such a model in the frame of Earth at perihelion, then you could predict the where to look for any object using nothing but information in the earth-perihelion frame, plus knowledge of position and speed of Earth at any other time relative to this frame.

While this would work, it is simply much more convenient to use a solar frame to define reference position and motion of all bodies (including earth).

Note again that I'm sticking to relativistic reasoning to formulate these questions, they don't come up if one uses prerelativistic classical explanations(Bradley, Fresnel) as there the Earth's motion is absolute.

No, this is wrong. Bradley's derivation completely respected Galilean relativity. The difficulties with aberration after Bradley was that there was no theory of light that fully respected relativity once you abandoned Bradley's corpuscular model.

 

[PAllen]

For Bradley classical annual aberration only the earth-Sun relative velocity in the aberration formula gives the correct angle correction, have you ever found a different velocity used ?

This is just wrong. Bradley's approach used relative velocity of Earth and star. Then, he assumed that stars didn't move, and computed how aberration relative to stellar frame changed due change in earth/star relative velocity over the seasons. The only relevance of the sun was that Bradley simply assumed that the sun would be at rest relative to stars. This is obviously not strictly correct, but plenty good enough to explain observations of his day.

 

[PAllen]

I considered this approach at first, and even found a paper that tries to explain it using relative motion between photons sent from the source and the observer, instead of source-observer relative motion, but this clearly can't be correct, mostly because as we all know the photon's frame can't be employed like this in physics.

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.

 

[PAllen]

Yes.

No, unless this change would casually amount to 30 km/s too.

What could seem puzzling if one considers aberration of sufficiently distant stars is why this particular change in velocity wrt the sun prevails over many other faster ones that depend on the Earth's motion, like its motion in the galaxy, or the local group..., and for closer ones any other relative motion wrt random objects orbiting near the earth.

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

 

[PAllen]

??

V_{earth}=V_{sun}\pm{V_{orbit}}
\Delta{V_{earth}}=2V_{orbit}

V_{earth}=V_{any other inertial frame}\pm{V_{orbit}}
\Delta{V_{earth}}=2V_{orbit}

What am I missing? Relativistic addition wouldn't change this would it?

Well, it would change it a little (the factor of 2 would not be exact). However, the choice a frame with respect to which to express Earth's velocity would not change the net delta over 6 months. You are precisely correct in this.

 

[PhilDSP]

Can it be stated then, that stellar aberration is a different physical process than doppler shift? Aberration is caused by the wavenumber, vector $k$ being affected by the Earth's motion but not the angular frequency $\omega$ of the light wave. Doppler shift is caused by the angular frequency being affected by source-sink relative velocity $\omega$ but not the wavenumber $k$.

 

[PAllen]

Can it be stated then, that stellar aberration is a different physical process than doppler shift? Aberration is caused by the wavenumber, vector $k$ being affected by the Earth's motion but not the angular frequency $\omega$ of the light wave. Doppler shift is caused by the angular frequency being affected by source-sink relative velocity $\omega$ but not the wavenumber $k$.

I would express it as follows.

We directly observe 4-momenta of photons (their color and where they came from directly give us 4-momentum expressed in Earth frame). This is all we need to determine how this observation would look in a different frame (e.g. a different state of motion for the earth). Such a transform would slightly affect color, in principle.

Doppler is really the same phenomenon, but between different frames. The frames of interest are emitter rest frame at time of emission versus Earth frame now. Of course, a priori, we know nothing about stellar motion. To discover it, we need to detect Doppler by shift of identifiable spectral lines, and then observe change in angular position over some period corrected for aberration and parallax. Given these quantities, we can compute the emitter velocity at time of emission expressed in a convenient inertial frame. What we have effectively computed is what emitter frame and original 4-momentum transforms to our observed 4-momentum.

 

[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.