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

[PAllen]

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

Sorry, I misread your formula which used beta where I use v. Substitute θ for β in what I said and it stands.

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.

Only if you include in aberration something no one else does: seeing an object move. A speed effect of aberration is the claim that two colocated objects with different speeds or changing speed (while remaining colocated) - are seen as not colocated from earth. That Earth sees a star that moves a lot relative to another star move, is obvious and is not aberration.

 

[xox]

Sorry, I misread your formula which used beta where I use v. Substitute θ for β in what I said and it stands.

This is the closest you'll come to admitting that you were wrong. Let me try to explain to you one more time: θ is not known and cannot be known. By contrast , β is known with very high accuracy (see , again, post 18). So, contrary to your claim, the two are not interchangeable.

Only if you include in aberration something no one else does: seeing an object move.

Has nothing to do with "seeing an object move". Has everything to do with angling the telescopes in such a way as to be able to "see" an object. Since I am getting tired of exchanging prose with you, let me put this in mathematical terms:

Say that in the frame of the emitter a ray of light propagates with velocity $\vec{c}$.
In the frame of the Earth, moving with velocity $V$ wrt the source, the velocity of the light ray is not $\vec{c}$ but:

$$\vec{c'}=\frac{1}{\gamma(V)(1-\beta cos \theta)}(\vec{c}+\frac{(\gamma(V)-1)cos \theta -\beta \gamma(V)}{\beta}\vec{V})$$

The departure of $\vec{c'}$ from $\vec{c}$ is the rigorous, mathematical definition of light aberration. The presence of $\gamma(V)$ is what makes it the definition of relativistic light aberration. In the above,

$$\beta=\frac{V}{c}$$
$$\gamma=\frac{1}{\sqrt{1-\beta^2}}$$
$$cos \theta =\frac{\vec{V} \cdot \vec{c}}{Vc}$$

A speed effect of aberration is the claim that two colocated objects with different speeds or changing speed (while remaining colocated) - are seen as not colocated from earth.

Definition by enumeration is a very poor way of doing physics, I gave you the exhaustive, rigorous definition.

That Earth sees a star that moves a lot relative to another star move, is obvious and is not aberration.

The Earth doesn't see" anything, the observers note that the position of the stars may be aberrated differently, as expressed by the formula given above.

 

[PAllen]

This is the closest you'll come to admitting that you were wrong. Let me try to explain to you one more time: θ is not known and cannot be known. By contrast , β is known with very high accuracy (see , again, post 18). So, contrary to your claim, the two are not interchangeable.

Has nothing to do with "seeing an object move". Has everything to do with angling the telescopes in such a way as to be able to "see" an object. Since I am getting tired of exchanging prose with you, let me put this in mathematical terms:

Say that in the frame of the emitter a ray of light propagates with velocity $\vec{c}$.
In the frame of the Earth, moving with velocity $V$ wrt the source, the velocity of the light ray is not $\vec{c}$ but:

$$\vec{c'}=\frac{1}{\gamma(V)(1-\beta cos \theta)}(\vec{c}+\frac{(\gamma(V)-1)cos \theta -\beta \gamma(V)}{\beta}\vec{V})$$

The departure of $\vec{c'}$ from $\vec{c}$ is the rigorous, mathematical definition of light aberration. The presence of $\gamma(V)$ is what makes it the definition of relativistic light aberration. In the above,

$$\beta=\frac{V}{c}$$
$$\gamma=\frac{1}{\sqrt{1-\beta^2}}$$
$$cos \theta =\frac{\vec{V} \cdot \vec{c}}{Vc}$$

Definition by enumeration is a very poor way of doing physics, I gave you the exhaustive, rigorous definition.

The Earth doesn't see" anything, the observers note that the position of the stars may be aberrated differently, as expressed by the formula given above.

I claim this formula is the one that cannot (easily) be applied because it is relative to the emitter momentary rest frame. In the case of changing velocity of the emitter, you must deal with changing angle as expressed in emitter frames that change from one moment to the next. In your notation, the unprimed null vector is unknown, and the one that reaches Earth is changing moment to moment between changing emitter rest frames. Thus, for a tightly circling emitter, it is quite complicated to apply this formula in a way to predict the observed pattern of aberration for such a source.

Meanwhile, computing aberration relative to a solar frame, with only the velocity of Earth involved, does not have this problem, and does not (and has no place) for emitter speed (only emitter position in the solar frame). An observation on any night can be transformed to a solar frame with the very well known velocity of Earth relative to sun. From there, you can compute the effect of the Earth's motion at any future time. It is this mode of analysis which shows, in a very simple way, why nothing about observable aberration includes any effect due to the speed of the emitter.

 

[xox]

Meanwhile, computing aberration relative to a solar frame, with only the velocity of Earth involved, does not have this problem, and does not (and has no place) for emitter speed (only emitter position in the solar frame).

What you claim above works if and only if the star is stationary wrt. the Sun. Does not work for stars moving wrt. the Sun.

 

[PAllen]

What you claim above works if and only if the star is stationary wrt. the Sun. Does not work for stars moving wrt. the Sun.

Nonsense. Transforming an observed light angle from Earth frame to sun frame is not affected in any way by the speed of whatever emitted the light. It is a straight Lorentz transform, whose result is Einstein's formula (which applies between any two frames, and is true independent of the motion of the source in either frame).

 

[xox]

Nonsense. Transforming an observed light angle from Earth frame to sun frame is not affected in any way by the speed of whatever emitted the light.

This is not what I said. What I told you is that, contrary to your claim, a light ray coming from a star moving wrt the Sun IS aberrated in the frame of the Sun. Nothing to do with any "Transforming an observed light angle from Earth frame to sun frame", nothing to do with the "Earth frame". Everything to do with the relative motion star-Sun.

 

[PAllen]

This is not what I said. What I told you is that, contrary to your claim, a light ray coming from a star moving wrt the Sun IS aberrated in the frame of the Sun. Nothing to do with any "Transforming an observed light angle from Earth frame to sun frame", nothing to do with the Earth. Everything to do with the relative motion star-Sun.

That aberration, again, is between the sun's observation and an observation in a momentary rest frame of the star. Nobody can see this or cares about this. Given, over some viewing period, a description of the motion of moving, accelerating star (imagine a very large binary system) in the solar frame (computed by adjusting the raw observation for day by day changes in Earth's motion), you can then extrapolate the stars position in the solar inertial frame, and than compute where to observe it at any future date.

A priori, nobody knows the velocity of a star relative to Earth or to the sun. However, earth-sun velocity is very precisely known. You can then derive the pattern of its motion seen from a convenient inertial frame by adjusting raw observations for Earth motion.

 

[exmarine]

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.

Brilliant comment! I need to re-think with this in mind. Thanks!

 

[xox]

That aberration, again, is between the sun's observation and an observation in a momentary rest frame of the star. Nobody can see this or cares about this. Given, over some viewing period, a description of the motion of moving, accelerating star (imagine a very large binary system) in the solar frame (computed by adjusting the raw observation for day by day changes in Earth's motion), you can then extrapolate the stars position in the solar inertial frame, and than compute where to observe it at any future date.

The rigorous thing to do is to compose the instantaneous velocity of the star wrt. the Sun with the instantaneous velocity of the Sun wrt. the Earth. Use the resultant speed to calculate the aberration. This is what I am showing in post 18.
What you are doing is composing the instantaneous position of the star with respect to the Sun with the instantaneous velocity of the Earth wrt. the Sun. This isn't rigorous, no wonder that applying this method enables you to claim that "see, the speed of the source (the star) does not affect the aberration".

A priori, nobody knows the velocity of a star relative to Earth or to the sun.

All you need to know is the angular velocity of one star about the other star, as Brown does in his derivation:

" The coordinates of the smaller star revolving at a radius R and angular speed w around the larger star in a plane perpendicular to the Earth are x2(t) = -vt + Rcos(q), y2(t) = Rsin(q), z2(t) = L, and t2(t) = t, where q = wt + f is the angular position of the smaller star in its orbit. Again, since light travels along null paths, a pulse of light arriving on Earth at time t = 0 was emitted at time t = T satisfying the relation
Dividing through by L2 and re-arranging terms, we have
Consequently, for L sufficiently great compared to R, the second two terms on the right side are negligible, so we have again T = , and hence the tangents of the angles of incidence in the x and y directions are
The leading terms in these tangents represent just the inherent "static" angular separation between the two stars viewed from the Earth, and these angles are negligibly small for sufficiently large L. Thus the tangent of the aberration angle is (again) essentially just , and so, as before, we have sin(a) = v, which of course is the same as for the central star. Incidentally, recall that Bradley's original formula for aberration was tan(a) = v, whereas the corresponding relativistic equation is sin(a) = v. The actual aberration angles for stars seen from Earth are small enough that the sine and tangent are virtually indistinguishable."

A collection of approximations, indeed, but he does use the angular velocity of one star as it orbits the "stationary" one. He does use a combination of the angular velocity of the "orbiting star" with the position of the "central star" wrt. the Sun.

 

[PhilDSP]

The actual aberration angles for stars seen from Earth are small enough that the sine and tangent are virtually indistinguishable."

My appreciation for Brown's explanations is growing, but something inside does shutter a bit on encountering a crude approximation without a rigorous analysis of what the implied error entails. Is there any chance that the approximation error can be further analyzed? What does it mean with respect to other parameters? (A simple answer may suffice)

Then again, this may be a digression from the crucial issue...

 

[TrickyDicky]

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?

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.

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.
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?

 

[exmarine]

FWIW, the light went on for me with the excellent comment by PAllen, #?.

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.

He is eggzackly right. Any number of passing inertial observers must see a perfect sphere of outgoing light from the very same flash. The Lorentz transform applies between any two, though obviously neither share the inertial reference frame of the light source. I, and perhaps some others, had a sort of semi-ballistic idea about the source without realizing it. All the examples and proofs, from Brown on down were quite unconvincing for me. In my opinion, this is the key to understanding stellar aberration: the state of motion of the light source has no bearing on anything.

 

[TrickyDicky]

FWIW, the light went on for me with the excellent comment by 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?
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.

 

[exmarine]

...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 ...

I am probably in over my head here, but I don't think the correct aberration is ONLY obtained with the relative velocity between us and the sun. It is just the most convenient. I think any other "non-accelerating" frame would work. Or is that not precisely your concern?

 

[TrickyDicky]

I am probably in over my head here, but I don't think the correct aberration is ONLY obtained with the relative velocity between us and the sun. It is just the most convenient. I think any other "non-accelerating" frame would work. Or is that not precisely your concern?

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 ?

 

[TrickyDicky]

From the point of view of an observer riding on the binary star, the position of Earth in the sky varies back and forth. If he had to aim his photons, he would need to take this motion into account. But he does not - he sprays photons in all directions. The photon that winds up hitting Earth goes straight to Earth. His motion changes only which photon this is.

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.

 

[TrickyDicky]

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.

 

[exmarine]

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 ?

More carefully, isn't it the annual CHANGES in our relative velocity with the sun that gives the correct observed changes in the aberration angle? So wouldn't our annual CHANGES be the same with any other inertial frame?

 

[TrickyDicky]

More carefully, isn't it the annual CHANGES in our relative velocity with the sun that gives the correct observed changes in the aberration angle?

Yes.

So wouldn't our annual CHANGES be the same with any other inertial frame?

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.

 

[exmarine]

??

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

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

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

 

[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.