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

[xox]

How about this aberration formula:


Velocity of light according to a moving observer = Velocity of that light according to a static observer "+" velocity of the moving observer


"+" means relativistic velocity addition

Velocity of light means the speed and the direction of light, velocity vector.

The above is false, why do you keep posting fringe stuff?

 

[xox]

I thought I understood how stellar aberration conformed to Special Relativity. The CHANGE in that angle comes from the CHANGE in our orbital velocity direction about the sun over six months. And it is the same for all stars. That is fine if there are no significant changes in a star’s state of motion relative to our sun over short periods of time. “Significant” is relative to our orbital velocity; and “short” is relative to six months.

Correct.

Now I read that there are binary stars out there that DO have significant velocity component changes parallel to the plane of our orbit during short periods of time. Yet we observe no corresponding changes in the aberration angles from those stars.

Do you have a reference?

 

[jartsa]

Velocity of light according to a moving observer = Velocity of that light according to a static observer

Now you are thinking about composition of parallel velocities, I guess.

When light's and observer's velocities are parallel or anti-parallel, then the combined velocity is the light's velocity. So no change of velocity in those cases, so no aberration.

In other cases there is a change of velocity, and aberration.

 

[mfb]

The above is false, why do you keep posting fringe stuff?

It is not false, it follows from the definition of relativistic velocity addition.

Now I read that there are binary stars out there that DO have significant velocity component changes parallel to the plane of our orbit during short periods of time. Yet we observe no corresponding changes in the aberration angles from those stars.

Do you have a reference?

See any introductory relativity book or astronomy book. It is so basic that you won't find it mentioned in publications any more.
There are binaries orbiting each other with a significant fraction of the speed of light. If this would lead to abberation, their apparent position would move wildly through the sky and it would be impossible to observe them properly (or even give their positions in the sky).

The link I sent you says exactly the opposite, the formula is valid if and only if the change in position of the observed star is much smaller than the distance star-observer (Earth).

This is true for all observations of objects outside the solar system humans ever made.

 

[xox]

Now you are thinking about composition of parallel velocities, I guess.

When light's and observer's velocities are parallel or anti-parallel, then the combined velocity is the light's velocity. So no change of velocity in those cases, so no aberration.

In other cases there is a change of velocity,

The above is a collection of misconceptions.

$$u'_x=\frac{u_x+V}{1+u_xV/c^2}$$
$$u'_y=\frac{u_y}{\gamma(V)(1+u_xV/c^2)}$$


Making $(u_x,u_y)=(c,0)$ produces $(u'_x,u'_y)=(c,0)$

and aberration.

Aberration, yes. "Change in velocity", not so much.

 

[xox]

It is not false,

So, according to you "Velocity of light according to a moving observer = Velocity of that light according to a static observer "+" velocity of the moving observer " is not false?

You seem to agree with jartsa that the second SR postulate is false. Interesting.

it follows from the definition of relativistic velocity addition.

The formula (not definition) of relativistic velocity is derived from the assumption that light speed is invariant, so, it does not support jartsa's claim "Velocity of light according to a moving observer = Velocity of that light according to a static observer "+" velocity of the moving observer ".
Actually, (and unsurprisingly) it supports exactly the opposite, see post 40.

 

[jartsa]

The above is a collection of misconceptions.

$$u'_x=\frac{u_x+V}{1+u_xV/c^2}$$
$$u'_y=\frac{u_y}{\gamma(V)(1+u_xV/c^2)}$$Making $(u_x,u_y)=(c,0)$ produces $(u'_x,u'_y)=(c,0)$

Aberration, yes. "Change in velocity", not so much.

The math is wrong because:

When you see aberration happening, you see light rays turning. (When you are accelerating)

Light ray turninig = light ray's velocity changing.

 

[PAllen]

So, according to you "Velocity of light according to a moving observer = Velocity of that light according to a static observer "+" velocity of the moving observer " is not false?

You seem to agree with jartsa that the second SR postulate is false. Interesting

Jarts said '+' meant relativistic velocity addition. For light, this means the speed stays the same, but for all but parallel/antiparallel motion, the direction changes. Direction is part of velocity. Point is, the full relativistic velocity addition formula works for light as well as for material bodies.

That is also what mfb obviously meant as well.

 

[PAllen]

The math is wrong because:

When you see aberration happening, you see light rays turning.

Light ray turninig = light ray's velocity changing.

The math is actually right. xox only plugged in the parallel case. If uy is not zero, you get the appropriate change in direction for the primed velocity.

 

[mfb]

xox: I think you confuse velocity (which depends on speed and direction) with speed.
The velocity of light can change while its speed is constant - this just means a change in direction.

 

[PAllen]

You must have a way of reading and interpreting nonsense as meaningful stuff.

It's called reading without assuming everyone is an idiot. Here is the direct quote from Jartsa's post:

"Velocity of light according to a moving observer = Velocity of that light according to a static observer "+" velocity of the moving observer


"+" means relativistic velocity addition

Velocity of light means the speed and the direction of light, velocity vector.
"

I see nothing incorrect in this.

 

[xox]

xox: I think you confuse velocity (which depends on speed and direction) with speed.

Clearly, I don't, I showed the transformation for both components.

The velocity of light can change while its speed is constant - this just means a change in direction.

I am fully familiar with that. PAllen managed to decipher jartsa's weirdly phrased claim as a case when $\vec{c}$ is not parallel with $\vec{V}$. I (and Meir Achuz) interpreted his claim as old ballistic theory of light.

 

[xox]

It's called reading without assuming everyone is an idiot. Here is the direct quote from Jartsa's post:

"Velocity of light according to a moving observer = Velocity of that light according to a static observer "+" velocity of the moving observer"+" means relativistic velocity addition

Velocity of light means the speed and the direction of light, velocity vector.
"

I see nothing incorrect in this.

Meir Achuz read exactly the same way I read it, as a support for ballistic theory.

 

[xox]

See any introductory relativity book or astronomy book. It is so basic that you won't find it mentioned in publications any more.

Please don't talk down.

There are binaries orbiting each other with a significant fraction of the speed of light. If this would lead to abberation, their apparent position would move wildly through the sky and it would be impossible to observe them properly (or even give their positions in the sky).

You are unclear, are you saying that there is NO aberration?
Besides, the exact claim that I questioned was that changes in speed do not lead to changes in the aberration angle. What is your exact take on this, you seem to be answering a different issue than the one being discussed.

 

[PAllen]

You are unclear, are you saying that there is NO aberration?
Besides, the exact claim that I questioned was that changes in speed do not lead to changes in the aberration angle. What is your exact take on this, you seem to be answering a different issue than the one being discussed.

If you see a formula that includes speed of the source, that formula is based on a reference angle in the rest frame of the source. Since the source is changing speed, the reference angle is changing as well. These effects balance such that speed of source has no direct contribution to observed aberration at all. For sources with changing motion [actually, in all cases IMO], it is much easier to analyze Earth's changing frame relative to a fixed inertial frame (e.g. sun's). In this analysis, there is no term for source speed at all, only source angular position (in the reference frame). The only speed terms are for Earth's orbital speed.

There is a secondary impact for an accelerating source. It's orbit has certain shape observed in the reference frame. The periodic change in the Earth's frame leads to a very small periodic change in the shape of the binary star orbit. That is, if the Earth observer plots the binary orbit against the background of the stellar COM position (with its periodic aberration), they see a very slightly different shape for the orbit than the solar observer does. I do not know if any of this is observable, in practice.

 

[PhilDSP]

I'd like to submit one particular geometric solution in as simple manner as possible:
$$\textrm{aberration} = \Theta = tan(\frac{\textrm{r}_{\textrm{T}} \times (\hat{\textrm{r}_{\textrm{H}}} \cdot \hat{\textrm{o}_{\textrm{N}}})}{\textrm{r}_{\textrm{L}}})$$
The vector $\textrm{r}_{\textrm{T}}$ is the transverse displacement of the observer from the point of photon emission.
The vector $\textrm{r}_{\textrm{L}}$ is the longitudinal displacement of the observer from the star in the same inertial system as the star.
The vector $\textrm{r}_{\textrm{H}}$ is the displacement of the observer from the point of photon emission. It represents the ray of the photon.

The unit vector $\hat{\textrm{r}_{\textrm{H}}}$ is $\textrm{r}_{\textrm{H}}$ normalized to have length 1.
The unit vector $\hat{\textrm{o}_{\textrm{N}}}$ is normal to the plane containing the Earth's orbit.

Crossing $\textrm{r}_{\textrm{T}}$ with the other terms should give the correct adjustment for the angle of the star with respect to the orbital pole of the Earth (where aberration is maximum).

 

[PhilDSP]

Since an Earth-based observer will have a non-inertial relation to either the Sun or the star, the total of all motions needs to be summed during the time period that the photon was in flight. That will allow us to determine, geometrically, what the trajectory vector is for the impinging photon. Fortunately that's not terribly complicated, as I reported earlier

$$\Delta \textrm{r} = \int_{t_e}^{t_a} v(t) \ dt$$
$t_e$ is the time that the photon was emitted
$t_a$ is the time that the photon was absorbed by the observer
v(t) is the relative velocity between the point of emission and the observer at time t

Then

$\textrm{r}_\textrm{H} = \textrm{r}_\textrm{L} - \Delta \textrm{r} \ \ \ \ \ \$ $\textrm{r}_\textrm{T} = \textrm{r}_\textrm{L} \times (\textrm{r}_\textrm{L} + \Delta \textrm{r})$

 

[PAllen]

Since an Earth-based observer will have a non-inertial relation to either the Sun or the star, the total of all motions needs to be summed during the time period that the photon was in flight. That will allow us to determine, geometrically, what the trajectory vector is for the impinging photon. Fortunately that's not terribly complicated, as I reported earlier

$$\Delta \textrm{r} = \int_{t_e}^{t_a} v(t) \ dt$$
$t_e$ is the time that the photon was emitted
$t_a$ is the time that the photon was absorbed by the observer
v(t) is the relative velocity between the point of emission and the observer at time t

Then

$\textrm{r}_\textrm{H} = \textrm{r}_\textrm{L} - \Delta \textrm{r} \ \ \ \ \ \$ $\textrm{r}_\textrm{T} = \textrm{r}_\textrm{L} \times (\textrm{r}_\textrm{L} + \Delta \textrm{r})$

What theory of aberration is this? In SR, it is only based on the Lorentz transform. Instantaneously colocated inertial frames are all that is required for either an observer with acceleration or a source with acceleration (if you choose to involve the source frame at all - which is unnecessary and complicates things - but is not wrong). Past motion is irrelevant. There is no need for any integration.

 

[mfb]

Please don't talk down.

I don't do that, I give a reason why there won't be papers about it.

You are unclear, are you saying that there is NO aberration?
Besides, the exact claim that I questioned was that changes in speed do not lead to changes in the aberration angle. What is your exact take on this, you seem to be answering a different issue than the one being discussed.

There is no aberration from the motion of stars in binary systems.
Changes in speed of Earth change the aberration angle. Changes in speed of stars have no special effect.

 

[PAllen]

So far this has purely been classical - non-relativistic, as it hasn't invoked the Lorentz transform. I believe the OP is interested in the numeric differences between the two, I certainly am.

Past motion is irrelevant for any single observation. For 2 or more different observations it would also be irrelevant if the velocity was constant over the period that those observations were made. The integration is one means of making the different observations compatible.

Where do you get that? From the very first post, this as been about stellar aberration in the context of SR.

The references in Histspec's #5 cover all the issues in this thread and more. In some sense, all after that is redundant.

 

[xox]

If you see a formula that includes speed of the source,

All formulas are based on the relative speed between source and observer.

that formula is based on a reference angle in the rest frame of the source.

Are you talking about the formula $cos \theta_{obs}=\frac{cos \theta_{src}-v/c}{1 -v/ccos \theta_{src}}$? Because I am talking about something totally different: $\tan \theta_{obs}=\frac{v}{c}$, where $v=v(t)$ is the relative speed between the source and the observer. See here. Even in $cos \theta_{obs}=\frac{cos \theta_{src}-v/c}{1 -v/ccos \theta_{src}}$ $v=v(t)$ is the relative speed between the source and the observer.

Since the source is changing speed, the reference angle is changing as well.

Yes, obviously. But this is not what I am talking about.

These effects balance such that speed of source has no direct contribution to observed aberration at all.

The "speed of the source" (the star) in the baricentric reference frame of the Sun is a definite contributor to the aberration, just as much as the speed of the Earth , in the Sun reference frame. See here

For sources with changing motion [actually, in all cases IMO], it is much easier to analyze Earth's changing frame relative to a fixed inertial frame (e.g. sun's).

Yes, ..which is exactly what I posted earlier.

In this analysis, there is no term for source speed at all, only source angular position (in the reference frame). The only speed terms are for Earth's orbital speed.

Well, the source-angular position is nothing but a function of the source velocity in the Sun - anchored frame. So, to say that "aberration is not a function of the speed of the star wrt. the Sun-based frame" is just a misnomer. As an aside, could you put what you said in words into math, the way I did it? This would make things a lot clearer.

 

[xox]

I don't do that, I give a reason why there won't be papers about it.

The way you phrased it, sure sounded like it.

There is no aberration from the motion of stars in binary systems.

Could you please prove this?

Changes in speed of Earth change the aberration angle.

Yes, obviously. See my post here.

Changes in speed of stars have no special effect.

Aberration depends on the relative velocity between source (the star) and the observer. How can you claim that the "changes in the speeds of stars have no (special? what is special?) effects"? Can you prove your claim mathematically? How does your claim jibe with the definition of aberration (either relativistic or non-relativstic)?

 

[mfb]

The "speed of the source" (the star) in the baricentric reference frame of the Sun is a definite contributor to the aberration, just as much as the speed of the Earth , in the Sun reference frame. See here

It is not. A moving star (relative to our sun) won't appear where it currently is, but this is not called aberration - it is just the time delay, the star had some years (or much more) time to move forward between the emission of light and our detection.

There is no aberration from the motion of stars in binary systems.

Could you please prove this?

Sure.

HM Cancri is a binary system where the white dwarfs orbit each other with velocities above .1% c, but they always appear at the same position in the sky.

For even higher speeds, this picture for example. The source of a relativistic jet and the relativistic jet appear directly next to each other.

Aberration depends on the relative velocity between source (the star) and the observer. How can you claim that the "changes in the speeds of stars have no (special? what is special?) effects"? Can you prove your claim mathematically? How does your claim jibe with the definition of aberration (either relativistic or non-relativstic)?

See the first part of this post. The effects of special relativity get split in different effects, a constant relative velocity between star and sun is not included in the aberration.

 

[xox]

It is not. A moving star (relative to our sun) won't appear where it currently is, but this is not called aberration

So, the argument boils down to the fact that the aberration "is not called aberration"? A ray of light coming from a (distant) source is no longer aberrated as a function of the relative speed between the source and the receiver?

- it is just the time delay, the star had some years (or much more) time to move forward between the emission of light and our detection.

While the ray of light covers the distance star-Earth $ct$, the star has moved by $vt$ , where $v$ is the relative speed between the star and Earth. This results into an aberration of $\tan \theta'=\frac{vt}{ct}=\frac{v}{c}$. Is this no longer called aberration?
We are discussing:
- whether the ray of light has a different angle in the frame of the emitter vs. the frame of the receiver, i.e. whether the well known phenomenon known as aberration of light is present
-whether or not changes in the speed of the star can be perceived as changes in the aberration angle.
Can you put this prose in math form, please? I asked this before, I am really interested in seeing the mathematical explanation.

Sure.

HM Cancri is a binary system where the white dwarfs orbit each other with velocities above .1% c, but they always appear at the same position in the sky.For even higher speeds, this picture for example. The source of a relativistic jet and the relativistic jet appear directly next to each other.

This may have a simple mathematical explanation, the variation in the velocity of the two stars may produce a variation in the aberration angle that is below the current measurement capabilities. Again, I would welcome a complete mathematical treatment, could you do this?

See the first part of this post. The effects of special relativity get split in different effects, a constant relative velocity between star and sun is not included in the aberration.

Can you explain this in mathematical terms? Stating it , even repeatedly, does not constitute a convincing argument. As a matter of fact, this is also the request of the thread originator.

 

[mfb]

So, the argument boils down to the fact that the aberration "is not called aberration"? A ray of light coming from a (distant) source is no longer aberrated as a function of the relative speed between the source and the receiver?

It boils down that you use a definition of aberration no one else does, I think. And one that fails as soon as the motion is not uniform any more.

This is obvious but we are not discussing the trivial fact that the star has moved between the time the ray of light was emitted and the time the ray arrived on Earth, we are discussing:

But that is exactly the (only) effect a relative velocity between star and sun has.

- whether the ray of light has a different angle in the frame of the emitter vs. the frame of the receiver, i.e. whether the well known phenomenon known as aberration of light is present

How do you compare angles and directions in frames with a relative motion?

-whether or not changes in the speed of the star can be perceived as changes in the aberration angle.

Certainly not, see the binary stars.

Can you put this prose in math form, please? I asked this before, I am really interested in seeing the mathematical explanation.

Just view everything in the frame of the sun and there is absolutely no reason to expect any aberration effect (and nothing to calculate). Light travels in a straight line and does not care about the velocity of the emitter.
For the moving star, the direction it has to emit light to hit our Earth will be different the point where the star sees our earth, this is the same effect of the time delay just seen from the other direction.

This may have a simple mathematical explanation, the variation in the velocity of the two stars may produce a variation in the aberration angle that is below the current measurement capabilities. Again, I would welcome a complete mathematical treatment, could you do this?

No. A .1% motion of the objects in the sky (corresponding to their .1% c velocity) would be plain obvious to every observer. See above, there is no mathematical treatment needed for a straight line.

Stating it , even repeatedly, does not constitute a convincing argument. As a matter of fact, this is also the request of the thread originator.

Reducing a physical problem to one that can be solved without any calculation is one of the most convincing arguments I know.

 

[xox]

It boils down that you use a definition of aberration no one else does, I think. And one that fails as soon as the motion is not uniform any more.

So, according to you, $tan \theta'=\frac{v}{c}$ is no longer a valid expression for aberration if $v=v(t)$?

How do you compare angles and directions in frames with a relative motion?

I am starting to see your problem, this is not about "comparing angles". This is not about "frameS" , it is about ONE frame, the Earth (lab) frame and ONE angle (variable in time), the angle $\theta'(t)=arctan \frac{v(t)}{c}$.

Certainly not, see the binary stars.

Can you provide the mathematical analysis? This is the third time I am asking for it.

Just view everything in the frame of the sun and there is absolutely no reason to expect any aberration effect (and nothing to calculate). Light travels in a straight line and does not care about the velocity of the emitter.

But the binary stars (and other stars as well) MOVE wrt. the Sun. Even worse, they MOVE wrt. the Earth observer.

 

[PhilDSP]

Where do you get that? From the very first post, this as been about stellar aberration in the context of SR.

The references in Histspec's #5 cover all the issues in this thread and more. In some sense, all after that is redundant.

Yes, you're probably right. And casting the classical formula into a distance traveled problem rather than a velocity problem is an unecessary complication. I deleted post 56 but can no longer delete 19, 21, 28, 51 and 52. If a moderator can do that, I agree they should be deleted.

 

[PAllen]

All formulas are based on the relative speed between source and observer.

This is not true. Einstein's derivation related change observed light angle between any two inertial frames. The only velocity term is that between the two frames. The observed light source need not be at rest in either frame, and its velocity in either frame does not enter the formula at all. All confusion on this, in the SR case, is related to the historic convention from the era of Bradley of using the rest frame of the source as one of the frames, and the rest frame of the observer as the other. But Einstein's derivation and formula have no such requirements.

Are you talking about the formula $cos \theta_{obs}=\frac{cos \theta_{src}-v/c}{1 -v/ccos \theta_{src}}$? Because I am talking about something totally different: $\tan \theta_{obs}=\frac{v}{c}$, where $v=v(t)$ is the relative speed between the source and the observer. See here. Even in $cos \theta_{obs}=\frac{cos \theta_{src}-v/c}{1 -v/ccos \theta_{src}}$ $v=v(t)$ is the relative speed between the source and the observer.

That formula is Bradley' and it is derived from the ballistic theory of light. It is approximately correct to within observational limits, when both are valid.

Yes, obviously. But this is not what I am talking about.

You can't avoid it.

The "speed of the source" (the star) in the baricentric reference frame of the Sun is a definite contributor to the aberration, just as much as the speed of the Earth , in the Sun reference frame. See here

The formula you give there is not valid unless you factor in that the the θ to which θ' is related is ever changing for an accelerating source, in just such a way as to cancel the effect of the source motion on observed angular position.

Yes, ..which is exactly what I posted earlier.




Well, the source-angular position is nothing but a function of the source velocity in the Sun - anchored frame. So, to say that "aberration is not a function of the speed of the star wrt. the Sun-based frame" is just a misnomer. As an aside, could you put what you said in words into math, the way I did it? This would make things a lot clearer.

What I said is true, and very well known.

Rather than repeat what has been derived elsewhere, very clearly, I will simply point you to references already provided in this thread:

For the relativistic proof that a binary star's motion has no affect on aberrations see last sections of:

http://www.mathpages.com/rr/s2-05/2-05.htm

Note also, the link you provided: http://en.wikipedia.org/wiki/Stellar_aberration_%28derivation_from_Lorentz_transformation%29#Application:_Aberration_in_astronomy

makes no mention of source velocity.

Finally, Histspec also provided a link to Herschel's proof in 1844 using pre-relativistic aberration, than no effect from binary stars would be expected.

http://articles.adsabs.harvard.edu/full/1844AN...22..249H

 

[xox]

This is not true. Einstein's derivation related change observed light angle between any two inertial frames.

I made it quite clear, several times, that this is not the formula I am talking about, I am talking about $tan \theta'=\frac{v}{c}$

The only velocity term is that between the two frames.

I know that very well. This is precisely why the formula is totally useless in orienting the telescopes, I pointed this out several times as well.

But Einstein's derivation and formula have no such requirements.

I am not talking about the Einstein aberration formula.

For the relativistic proof that a binary star's motion has no affect on aberrations see last sections of:

http://www.mathpages.com/rr/s2-05/2-05.htm

Let's concentrate on the above because we can all see the complete derivation. First off, the derivation is an ugly mess. The mess aside, what Brown is saying is that under certain circumstances, the aberration of the "circling star" is close enough to the aberration of the "central star". The aberration formulas are clearly not the same, they become the same only after assuming R<<L (see his notation) AND neglecting some of the higher powers of $\frac{v}{c}$.

 

[PAllen]

I made it quite clear, several times, that this is not the formula I am talking about, I am talking about $tan \theta'=\frac{v}{c}$
I know that very well. This is precisely why the formula is totally useless in orienting the telescopes, I pointed this out several times as well.

This is false. Derivations based on Einstein's formula are the modern foundation for all aberration theory, and underlie orienting telescopes.

I am not talking about the Einstein aberration formula.

But the only accepted derivations today are based on Einstein's formula. We accept SR don't we?

[edit: Why don't you actually read the links, then come back with questions. They all concern aberration for the purpose of orienting telescopes.]

 

[xox]

This is false. Derivations based on Einstein's formula are the modern foundation for all aberration theory, and underlie orienting telescopes.

The $\theta$ in $cos \theta'=\frac{cos \theta +\beta}{1+ \beta cos \theta}$ is unknown , so the formula used is the one I posted in post 18. The discussion is not about the validity (it is valid), it is about its practical use (it isn't).

But the only accepted derivations today are based on Einstein's formula. We accept SR don't we?

Sure, we all accept SR. You are missing the point, the discussion is not about the validity of SR, it is about whether or not there are expected effects of the varying relative speed between the source and the receiver.

[edit: Why don't you actually read the links,

I actually read them, this is how I could detect all the typos and hacks in the mathpages link. I listed the conditions under which the aberrations of the two stars "become" similar (they are not identical), have you missed that?

then come back with questions.

I do not have questions, I am quite clear on the subject. Please stop talking down to me, I have a level of understanding that is equal to yours.

They all concern aberration for the purpose of orienting telescopes.

Correct. $cos \theta'=\frac{cos \theta +\beta}{1+ \beta cos \theta}$ while perfectly valid, is not the used formula , nor is it useful, for reasons explained above.

 

[PAllen]

[Edit: comments below fixed for confusing β and θ]

The $\theta$ in $cos \theta'=\frac{cos \theta +\beta}{1+ \beta cos \theta}$ is unknown , so the formula used is the one I posted in post 18. The discussion is not about the validity (it is valid), it is about its practical use (it isn't).

You don't need to know θ. If you apply this [Einstein] formula for the history of Earth frames relative to any given θ in the solar frame, using Earth velocity relative to the sun, you get the aberration pattern. You can allow for θ(t) for a star with significant movement. You don't need to incorporate its velocity. Given the aberration pattern, you know that if you found an object at one location in January, where to look for it in June.

Sure, we all accept SR. You are missing the point, the discussion is not about the validity of SR, it is about whether or not there are expected effects of the varying relative speed between the source and the receiver.

There aren't. Every reputable author says no. Do you have any reference that says yes?

I actually read them, this is how I could detect all the typos and hacks in the mathpages link. I listed the conditions under which the aberrations of the two stars "become" similar (they are not identical), have you missed that?

I did miss discussion of typos in the mathpages links. Can you indicate the posts in this now long thread?

Typos or not, the content of the mathpages are correct and accepted (except for stupid history debates about whether or not Einstein had a misunderstanding at some point).

 

[xox]

I thought I understood how stellar aberration conformed to Special Relativity. The CHANGE in that angle comes from the CHANGE in our orbital velocity direction about the sun over six months. And it is the same for all stars. That is fine if there are no significant changes in a star’s state of motion relative to our sun over short periods of time. “Significant” is relative to our orbital velocity; and “short” is relative to six months.

Now I read that there are binary stars out there that DO have significant velocity component changes parallel to the plane of our orbit during short periods of time. Yet we observe no corresponding changes in the aberration angles from those stars.

How can that be consistent with SRT? Only SOME changes in relative velocities between source and observer cause changes in aberration angles?

The above statement has a key qualifier, the effect effect is negligible if the distance between the binary stars and Earth , L, is much larger than the radius of their orbits, R. IN ADDITION, the terms in $\frac{v^2}{c^2}$ and larger NEED to be neglected as well. For a fairly muddled, typo-ladden proof, see here, at the bottom of the page.

 

[xox]

You don't need to know what you have renamed β.

I did not mention anything about $\beta$, my point was about not knowing the angle $\theta$.

Typos or not, the content of the mathpages are correct and accepted (except for stupid history debates about whether or not Einstein had a misunderstanding at some point).

I did not dispute the correctness of Brown's derivation, I simply pointed out that the aberrations for the two stars are NOT identical, the difference in their speeds DOES make a difference. That's all.

 

[PAllen]

The above statement has a key qualifier, the effect effect is negligible if the distance between the binary stars and Earth , L, is much larger than the radius of their orbits, R. IN ADDITION, the terms in $\frac{v^2}{c^2}$ and larger NEED to be neglected as well. For a fairly muddled, typo-ladden proof, see here, at the bottom of the page.

Those qualifiers simply rule out the case of a star that not only has some substantial velocity, it also changes position substantially (on scale of observation, e.g. a day or a year). The fact that you see it change position is not called aberration. Aberration would be the difference between how a platform stationary with respect to the sun would see it move versus how an Earth observer sees it move. It is just the seasonal ripple on such a substantial motion.