On May 6, 9:40 pm, Eric Gisse wrote:
On May 6, 12:14 pm, Albertito wrote:



On May 6, 8:56 pm, PD wrote:

On May 6, 2:40 pm, Albertito wrote:

On May 6, 3:06 pm, PD wrote:

On May 1, 7:47 am, Albertito wrote:

This the correct addition of velocities

w = u + v,

Well, there's a problem here. This formula appears to be incorrect,
according to experimental evidence.
Have you done a literature search of the experimental evidence?

Well, there's a problem for you. This formula,
w = u + v, appears to be incorrect, according
to experimental evidence, when used with the
relativistic Doppler furmula f' = sqrt((1 - v/c)/(1 +v/c))f,
but it is still waiting for a test when used with this
Doppler formula, f' = Exp(-v/c) f

No, that's not the problem. It appears to be incorrect when u, v, and
w are independently and directly measured, well outside the parochial
application of Doppler frequency shifting. Did you imagine that the
only test of this formula was Doppler frequency shifting?

and this is the correct Doppler formula for all moving
bodies,

f' = Exp(-v/c) f

Well, there's a problem here. This formula appears to be incorrect,
according to experimental evidence.
Have you done a literature search of the experimental evidence?

Well, there's a problem for you. This formula,
f' = Exp(-v/c) f , appears to be incorrect, according
to experimental evidence, when used with the
relativistic addition of velocities, but it is still
waiting for a test when used with this addition formula
w = u + v.

No, it appears to be incorrect according to experimental evidence,
when f, f' and v are all independently and directly measured.

Again I ask you, have you done a literature search of the experimental
evidence?

PD

Yes, I've done a literature search of the experimental evidence.
And no experimental tests have been found for that specific
set of equations. Nothing is said about that, neither that's wrong
nor right. OTOH, how do you intend to accomplish f, f' and v are
all independently and directly measured? Please, provide me a
reference/paper describing experiments where those three
parameters are all independently and directly measured, without
the assumption that the speed of light is invariant.

Why don't you do your own research since you are interested enough to
spend hours a week posting about relativity?

Why don't you do that research since you are interested
enough to spend hours a week reading all I post about
relativity?

On May 6, 12:40 pm, Albertito wrote:
On May 6, 3:06 pm, PD wrote:

On May 1, 7:47 am, Albertito wrote:

This the correct addition of velocities

w = u + v,


CRETIN. INCURABLE.
When you grow up you will grow into a Paco Velev,Koobee Wublee, Marcel
Luttgents, Juan****o Gonzalez. Your mother should have had an
abortion.

On May 6, 9:44 pm, PD wrote:
On May 6, 3:14 pm, Albertito wrote:



On May 6, 8:56 pm, PD wrote:

On May 6, 2:40 pm, Albertito wrote:

On May 6, 3:06 pm, PD wrote:

On May 1, 7:47 am, Albertito wrote:

This the correct addition of velocities

w = u + v,

Well, there's a problem here. This formula appears to be incorrect,
according to experimental evidence.
Have you done a literature search of the experimental evidence?

Well, there's a problem for you. This formula,
w = u + v, appears to be incorrect, according
to experimental evidence, when used with the
relativistic Doppler furmula f' = sqrt((1 - v/c)/(1 +v/c))f,
but it is still waiting for a test when used with this
Doppler formula, f' = Exp(-v/c) f

No, that's not the problem. It appears to be incorrect when u, v, and
w are independently and directly measured, well outside the parochial
application of Doppler frequency shifting. Did you imagine that the
only test of this formula was Doppler frequency shifting?

and this is the correct Doppler formula for all moving
bodies,

f' = Exp(-v/c) f

Well, there's a problem here. This formula appears to be incorrect,
according to experimental evidence.
Have you done a literature search of the experimental evidence?

Well, there's a problem for you. This formula,
f' = Exp(-v/c) f , appears to be incorrect, according
to experimental evidence, when used with the
relativistic addition of velocities, but it is still
waiting for a test when used with this addition formula
w = u + v.

No, it appears to be incorrect according to experimental evidence,
when f, f' and v are all independently and directly measured.

Again I ask you, have you done a literature search of the experimental
evidence?

PD

Yes, I've done a literature search of the experimental evidence.
And no experimental tests have been found for that specific
set of equations. Nothing is said about that, neither that's wrong
nor right. OTOH, how do you intend to accomplish f, f' and v are
all independently and directly measured? Please, provide me a
reference/paper describing experiments where those three
parameters are all independently and directly measured, without
the assumption that the speed of light is invariant.

Clock signals sent from Voyager. f is set by the oscillator prior to
launch and whose stability is engineered, v is known by time-separated
surveying to landmarks (like planets) by the satellite, and f' is
measured on the ground. Resolution is 18% of v/c, certainly capable of
distinguishing between your expression and the correct one.


Houston, we have a problem, the voyager probe seems
not to be where it should! Is it the Pioneer anomaly, or
what? How is it that knowing f and f' with enough accuracy
the beta v/c is still within a 18% in solar system gravitational
field?

No, in order to distinguishing between my expression and
the relativistic one you need at least a third-order term of v/c.
Even with a second-order term of v/c both expression still
remain undistinguished.

On May 6, 4:06*pm, Albertito wrote:
On May 6, 9:44 pm, PD wrote:



On May 6, 3:14 pm, Albertito wrote:

On May 6, 8:56 pm, PD wrote:

On May 6, 2:40 pm, Albertito wrote:

On May 6, 3:06 pm, PD wrote:

On May 1, 7:47 am, Albertito wrote:

This the correct addition of velocities

* * * * *w = u + v,

Well, there's a problem here. This formula appears to be incorrect,
according to experimental evidence.
Have you done a literature search of the experimental evidence?

Well, there's a problem for you. This formula,
w = u + v, appears to be incorrect, according
to experimental evidence, when used with the
relativistic Doppler furmula *f' = sqrt((1 - v/c)/(1 +v/c))f,
but it is still waiting for a test when used with this
Doppler formula, f' = Exp(-v/c) f

No, that's not the problem. It appears to be incorrect when u, v, and
w are independently and directly measured, well outside the parochial
application of Doppler frequency shifting. Did you imagine that the
only test of this formula was Doppler frequency shifting?

and this is the correct Doppler formula for all moving
bodies,

* * * * f' = Exp(-v/c) f

Well, there's a problem here. This formula appears to be incorrect,
according to experimental evidence.
Have you done a literature search of the experimental evidence?

Well, there's a problem for you. This formula,
f' = Exp(-v/c) f , appears to be incorrect, according
to experimental evidence, when used with the
relativistic addition of velocities, but it is still
waiting for a test when used with this addition formula
w = u + v.

No, it appears to be incorrect according to experimental evidence,
when f, f' and v are all independently and directly measured.

Again I ask you, have you done a literature search of the experimental
evidence?

PD

Yes, I've done a literature search of the experimental evidence.
And no experimental tests have been found for that specific
set of equations. Nothing is said about that, neither that's wrong
nor right. OTOH, how do you intend to accomplish f, f' and v are
all independently and directly measured? Please, provide me a
reference/paper describing experiments where those three
parameters are all independently and directly measured, without
the assumption that the speed of light is invariant.

Clock signals sent from Voyager. f is set by the oscillator prior to
launch and whose stability is engineered, v is known by time-separated
surveying to landmarks (like planets) by the satellite, and f' is
measured on the ground. Resolution is 18% of v/c, certainly capable of
distinguishing between your expression and the correct one.

Houston, we have a problem, the voyager probe seems
not to be where it should!

The difference is still MUCH smaller than the difference your formula
would suggest. While there is a SMALL anomaly seen in the Voyager
data, we know for certain that your formula is not supported by those
data.

You did say you know how to do a literature search, right? This means
more than just hearing that something was funny about Voyager.

Is it the Pioneer anomaly, or
what? How is it that knowing f and f' with enough accuracy
the beta v/c is still within a 18% in solar system gravitational
field?

No, in order to distinguishing between my expression and
the relativistic one you need at least a third-order term of v/c.
Even with a second-order term of v/c both expression still
remain undistinguished.

Nope, because your expansion gets the relative size of the first and
second order terms wrong. You might want to look again. Then again,
you might not.


PD

On May 6, 2:38*pm, Albertito wrote:
On May 6, 10:16 pm, PD wrote:



On May 6, 4:06 pm, Albertito wrote:

On May 6, 9:44 pm, PD wrote:

On May 6, 3:14 pm, Albertito wrote:

On May 6, 8:56 pm, PD wrote:

On May 6, 2:40 pm, Albertito wrote:

On May 6, 3:06 pm, PD wrote:

On May 1, 7:47 am, Albertito wrote:

This the correct addition of velocities

* * * * *w = u + v,

Well, there's a problem here. This formula appears to be incorrect,
according to experimental evidence.
Have you done a literature search of the experimental evidence?

Well, there's a problem for you. This formula,
w = u + v, appears to be incorrect, according
to experimental evidence, when used with the
relativistic Doppler furmula *f' = sqrt((1 - v/c)/(1 +v/c))f,
but it is still waiting for a test when used with this
Doppler formula, f' = Exp(-v/c) f

No, that's not the problem. It appears to be incorrect when u, v, and
w are independently and directly measured, well outside the parochial
application of Doppler frequency shifting. Did you imagine that the
only test of this formula was Doppler frequency shifting?

and this is the correct Doppler formula for all moving
bodies,

* * * * f' = Exp(-v/c) f

Well, there's a problem here. This formula appears to be incorrect,
according to experimental evidence.
Have you done a literature search of the experimental evidence?

Well, there's a problem for you. This formula,
f' = Exp(-v/c) f , appears to be incorrect, according
to experimental evidence, when used with the
relativistic addition of velocities, but it is still
waiting for a test when used with this addition formula
w = u + v.

No, it appears to be incorrect according to experimental evidence,
when f, f' and v are all independently and directly measured.

Again I ask you, have you done a literature search of the experimental
evidence?

PD

Yes, I've done a literature search of the experimental evidence.
And no experimental tests have been found for that specific
set of equations. Nothing is said about that, neither that's wrong
nor right. OTOH, how do you intend to accomplish f, f' and v are
all independently and directly measured? Please, provide me a
reference/paper describing experiments where those three
parameters are all independently and directly measured, without
the assumption that the speed of light is invariant.

Clock signals sent from Voyager. f is set by the oscillator prior to
launch and whose stability is engineered, v is known by time-separated
surveying to landmarks (like planets) by the satellite, and f' is
measured on the ground. Resolution is 18% of v/c, certainly capable of
distinguishing between your expression and the correct one.

Houston, we have a problem, the voyager probe seems
not to be where it should!

The difference is still MUCH smaller than the difference your formula
would suggest. While there is a SMALL anomaly seen in the Voyager
data, we know for certain that your formula is not supported by those
data.

You did say you know how to do a literature search, right? This means
more than just hearing that something was funny about Voyager.

*Is it the Pioneer anomaly, or
what? How is it that knowing f and f' with enough accuracy
the beta v/c is still within a 18% in solar system gravitational
field?

No, in order to distinguishing between my expression and
the relativistic one you need at least a third-order term of v/c.
Even with a second-order term of v/c both expression still
remain undistinguished.

Nope, because your expansion gets the relative size of the first and
second order terms wrong. You might want to look again. Then again,
you might not.

PD

How do you know my expansion gets the relative
size of the first and second order terms wrong?

Its' called the Taylor series.

Have you gone out of your way for me to see it?
Prove it! Show me those terms are wrong.

For what purpose? If you can't obtain the Taylor expansion of e^x,
then you are unqualified to discuss anything in physics anyway.

On May 7, 12:08 pm, Eric Gisse wrote:
On May 7, 1:45 am, Albertito wrote:



On May 7, 10:08 am, Eric Gisse wrote:

On May 7, 12:06 am, Albertito wrote:

On May 7, 2:52 am, Eric Gisse wrote:

On May 6, 2:38 pm, Albertito wrote:

On May 6, 10:16 pm, PD wrote:

On May 6, 4:06 pm, Albertito wrote:

On May 6, 9:44 pm, PD wrote:

On May 6, 3:14 pm, Albertito wrote:

On May 6, 8:56 pm, PD wrote:

On May 6, 2:40 pm, Albertito wrote:

On May 6, 3:06 pm, PD wrote:

On May 1, 7:47 am, Albertito wrote:

This the correct addition of velocities

w = u + v,

Well, there's a problem here. This formula appears to be incorrect,
according to experimental evidence.
Have you done a literature search of the experimental evidence?

Well, there's a problem for you. This formula,
w = u + v, appears to be incorrect, according
to experimental evidence, when used with the
relativistic Doppler furmula f' = sqrt((1 - v/c)/(1 +v/c))f,
but it is still waiting for a test when used with this
Doppler formula, f' = Exp(-v/c) f

No, that's not the problem. It appears to be incorrect when u, v, and
w are independently and directly measured, well outside the parochial
application of Doppler frequency shifting. Did you imagine that the
only test of this formula was Doppler frequency shifting?

and this is the correct Doppler formula for all moving
bodies,

f' = Exp(-v/c) f

Well, there's a problem here. This formula appears to be incorrect,
according to experimental evidence.
Have you done a literature search of the experimental evidence?

Well, there's a problem for you. This formula,
f' = Exp(-v/c) f , appears to be incorrect, according
to experimental evidence, when used with the
relativistic addition of velocities, but it is still
waiting for a test when used with this addition formula
w = u + v.

No, it appears to be incorrect according to experimental evidence,
when f, f' and v are all independently and directly measured.

Again I ask you, have you done a literature search of the experimental
evidence?

PD

Yes, I've done a literature search of the experimental evidence.
And no experimental tests have been found for that specific
set of equations. Nothing is said about that, neither that's wrong
nor right. OTOH, how do you intend to accomplish f, f' and v are
all independently and directly measured? Please, provide me a
reference/paper describing experiments where those three
parameters are all independently and directly measured, without
the assumption that the speed of light is invariant.

Clock signals sent from Voyager. f is set by the oscillator prior to
launch and whose stability is engineered, v is known by time-separated
surveying to landmarks (like planets) by the satellite, and f' is
measured on the ground. Resolution is 18% of v/c, certainly capable of
distinguishing between your expression and the correct one.

Houston, we have a problem, the voyager probe seems
not to be where it should!

The difference is still MUCH smaller than the difference your formula
would suggest. While there is a SMALL anomaly seen in the Voyager
data, we know for certain that your formula is not supported by those
data.

You did say you know how to do a literature search, right? This means
more than just hearing that something was funny about Voyager.

Is it the Pioneer anomaly, or
what? How is it that knowing f and f' with enough accuracy
the beta v/c is still within a 18% in solar system gravitational
field?

No, in order to distinguishing between my expression and
the relativistic one you need at least a third-order term of v/c.
Even with a second-order term of v/c both expression still
remain undistinguished.

Nope, because your expansion gets the relative size of the first and
second order terms wrong. You might want to look again. Then again,
you might not.

PD

How do you know my expansion gets the relative
size of the first and second order terms wrong?

Its' called the Taylor series.

Let's expand both Doppler factors to 6th-order term.

Why? Sixth order effects aren't visible by any current experiment of
which I'm aware.

Let x=v/c be the beta,
D' is the Doppler factor in my model, and
D'' is the relativistic factor,

D' = Exp(-x)

Which we should _once again_ note that you do not derive your
equation, you pull it out of the air. Special relativity is more than
a specific case of the Doopler formula.

D'' = sqrt((1 - x)/(1 + x))

then

D' = 1 - x + x^2/2 - x^3/6 + x^4/24 - x^5/120 + x^6/720
D'' = 1 - x + x^2/2 - x^3/2 + 3x^4/8 - 3x^5/8 + 5x^6/6

Where in D' are the first and second order terms wrong?

http://img156.imageshack.us/img156/7604/taylorxh5.gif

They aren't, but the higher order terms are. Congratulations - you
matched the first and second order terms in the v c expansion of
the Doppler formula. Except the formulas are DIFFERENT.

http://math.ucr.edu/home/baez/physic...periments.html

Of course, they are different. I bet you bet the correct one
is the relativistic D'' = sqrt((1 - x)/(1 + x)) :-)
Since they are different and they can predict the same
observed Doppler shifts to second order, you can't simply
rule out my formula before a experimental test were performed
to distinguish them for higher order terms.

Yea, I can. Your formula is arbitrary and not derived while the
special relativistic Doppler formula is but one part of a diverse
theory that has been tested in countless times in different ways.

Your formula has no independent experimental backing, and you have
made no effort to do a literature search on the subject. Instead you
rely on folks like me to do /your/ work for /you/.





Go to the library and read.

Have you gone out of your way for me to see it?
Prove it! Show me those terms are wrong.

For what purpose? If you can't obtain the Taylor expansion of e^x,
then you are unqualified to discuss anything in physics anyway.

Maybe, the unqualified person to discuss anything in physics
is you? What kind of Taylor expansion has you performed to
see that the first and second order terms in D' are wrong?

Which one of us is pulling arbitrary equations out of our butt to
explain something he does not understand? Hint: Not me.

The system of equations I've provided is not arbitrary,

Sum of velocities - w = u + v,
Doppler - f' = Exp(-x) f

Oh, yes it is quite arbitrary. Why don't you show us how you /derived/
your equations from first principles?



Under SR, you get

Sum of rapidities - r_w = r_u + r_v,
Doppler - f''= Exp(-r_w) f

Both systems of equation are isomorphic. The isomorphism

You have no idea what that word means.

is clear, under SR bodies move in a Minkowski spacetime,
in my model they move in a 3-d euclidean space plus time.

Neat! It is wrong - that's Galilean relativity.

Is the system of equation in SR arbitrary? IOW, is the
second postulate of SR arbitrary? Why is it necessarily
true that the speed of light is invariant in all inertial
frames of reference? I've showed to you there exists a
galilean relativity, equipped with a specific set of
rules to address relative anisotropy of light, that
can predict the same observable phenoma to second order
of v/c.

Do a literature search some time and stop wasting my time.

You have not answered my question:
Why is it necessarily true that the speed of light is
invariant in all inertial frames of reference?

Why isn't it arbitrary to assume the speed of
light is invariant in all inertial frames of reference?

Replace the second postulate of SR by this one:
"The speed of light is invariant, c, for all inertial
frames of reference, where the source of light
was at rest in the instant of emission"

Why is this new postulate arbitrary, but not the original
one of SR?

What does it mean? This means a photon always
propagates at c wrt source's rest frame in the instant
of emission, but it is not necessarily the same c wrt
the frame where the receiver is at rest. In ballistic
theories, you have c' = c - v, which has been proved
wrong. But, notice that's wrong, because c' = c(1 - v/c)
is only a first-order approximation, so we can easily
falsify it by experimental tests within our current
level of accuracy. Even under SR, we can interpret
there is a relative anisotropy of light, if we express

c' = c sqrt((1 - v/c)/(1 + v/c)).

But, from the spontaneous emission mechanism of all
atoms involved in the retransmission of that photon over
all possible paths, we get

c' = c Exp(-v/c)

On May 7, 3:52*am, Albertito wrote:
[snip]

I really couldn't be bothered to click "read more" when it became
apparent you don't understand why it is stupid to pluck equations out
of the air and pretend they are more fundamental.

On May 7, 12:55 pm, Eric Gisse wrote:
On May 7, 3:52 am, Albertito wrote:
[snip]

I really couldn't be bothered to click "read more" when it became
apparent you don't understand why it is stupid to pluck equations out
of the air and pretend they are more fundamental.

You still have not answered my question:
Why is it necessarily true that the speed of light is
invariant in all inertial frames of reference?

Why isn't it arbitrary to assume the speed of
light is invariant in all inertial frames of reference?

Read mo

Replace the second postulate of SR by this one:
"The speed of light is invariant, c, for all inertial
frames of reference, where the source of light
was at rest in the instant of emission"

Why is this new postulate arbitrary, but not the original
one of SR?

What does it mean? This means a photon always
propagates at c wrt source's rest frame in the instant
of emission, but it is not necessarily the same c wrt
the frame where the receiver is at rest. In ballistic theories,
you have c' = c - v, which has been proved wrong. But,
notice that's wrong, because c' = c(1 - v/c) is only a first-order
approximation, so we can easily falsify it by experimental
tests within our current level of accuracy. Even under SR,
we can interpret there is a relative anisotropy of light, if we
express

c' = c sqrt((1 - v/c)/(1 + v/c))

But, from the spontaneous emission mechanism of all
atoms involved in the retransmission of that photon over
all possible paths, we get

c' = c Exp(-v/c)

On May 7, 4:00*am, Albertito wrote:
On May 7, 12:55 pm, Eric Gisse wrote:

On May 7, 3:52 am, Albertito wrote:
[snip]

I really couldn't be bothered to click "read more" when it became
apparent you don't understand why it is stupid to pluck equations out
of the air and pretend they are more fundamental.

You still have not answered my question:
Why is it necessarily true that the speed of light is
invariant in all inertial frames of reference?

Why isn't it arbitrary to assume the speed of
light is invariant in all inertial frames of reference?

Read mo

Replace the second postulate of SR by this one:
* *"The speed of light is invariant, c, for all inertial
* * *frames of reference, where the source of light
* * *was at rest in the instant of emission"

Why is this new postulate arbitrary, but not the original
one of SR?

What does it mean? This means a photon always
propagates at c wrt source's rest frame in the instant
of emission, but it is not necessarily *the same c wrt
the frame where the receiver is at rest. In ballistic theories,
you have c' = c - v, which has been proved wrong. But,
notice that's wrong, because c' = c(1 - v/c) is only a first-order
approximation, so we can easily falsify it by experimental
tests within our current level of accuracy. Even under SR,
we can interpret there is a relative anisotropy of light, if we
express

* * * * * *c' = c *sqrt((1 - v/c)/(1 + v/c))

No. The speed of light is invariant in SR.


But, from the spontaneous emission mechanism of all
atoms involved in the retransmission of that photon over
all possible paths, we get

* * * * * c' = c *Exp(-v/c)

On May 6, 5:38*pm, Albertito wrote:
On May 6, 10:16 pm, PD wrote:



On May 6, 4:06 pm, Albertito wrote:

On May 6, 9:44 pm, PD wrote:

On May 6, 3:14 pm, Albertito wrote:

On May 6, 8:56 pm, PD wrote:

On May 6, 2:40 pm, Albertito wrote:

On May 6, 3:06 pm, PD wrote:

On May 1, 7:47 am, Albertito wrote:

This the correct addition of velocities

* * * * *w = u + v,

Well, there's a problem here. This formula appears to be incorrect,
according to experimental evidence.
Have you done a literature search of the experimental evidence?

Well, there's a problem for you. This formula,
w = u + v, appears to be incorrect, according
to experimental evidence, when used with the
relativistic Doppler furmula *f' = sqrt((1 - v/c)/(1 +v/c))f,
but it is still waiting for a test when used with this
Doppler formula, f' = Exp(-v/c) f

No, that's not the problem. It appears to be incorrect when u, v, and
w are independently and directly measured, well outside the parochial
application of Doppler frequency shifting. Did you imagine that the
only test of this formula was Doppler frequency shifting?

and this is the correct Doppler formula for all moving
bodies,

* * * * f' = Exp(-v/c) f

Well, there's a problem here. This formula appears to be incorrect,
according to experimental evidence.
Have you done a literature search of the experimental evidence?

Well, there's a problem for you. This formula,
f' = Exp(-v/c) f , appears to be incorrect, according
to experimental evidence, when used with the
relativistic addition of velocities, but it is still
waiting for a test when used with this addition formula
w = u + v.

No, it appears to be incorrect according to experimental evidence,
when f, f' and v are all independently and directly measured.

Again I ask you, have you done a literature search of the experimental
evidence?

PD

Yes, I've done a literature search of the experimental evidence.
And no experimental tests have been found for that specific
set of equations. Nothing is said about that, neither that's wrong
nor right. OTOH, how do you intend to accomplish f, f' and v are
all independently and directly measured? Please, provide me a
reference/paper describing experiments where those three
parameters are all independently and directly measured, without
the assumption that the speed of light is invariant.

Clock signals sent from Voyager. f is set by the oscillator prior to
launch and whose stability is engineered, v is known by time-separated
surveying to landmarks (like planets) by the satellite, and f' is
measured on the ground. Resolution is 18% of v/c, certainly capable of
distinguishing between your expression and the correct one.

Houston, we have a problem, the voyager probe seems
not to be where it should!

The difference is still MUCH smaller than the difference your formula
would suggest. While there is a SMALL anomaly seen in the Voyager
data, we know for certain that your formula is not supported by those
data.

You did say you know how to do a literature search, right? This means
more than just hearing that something was funny about Voyager.

*Is it the Pioneer anomaly, or
what? How is it that knowing f and f' with enough accuracy
the beta v/c is still within a 18% in solar system gravitational
field?

No, in order to distinguishing between my expression and
the relativistic one you need at least a third-order term of v/c.
Even with a second-order term of v/c both expression still
remain undistinguished.

Nope, because your expansion gets the relative size of the first and
second order terms wrong. You might want to look again. Then again,
you might not.

PD

How do you know my expansion gets the relative
size of the first and second order terms wrong?
Have you gone out of your way for me to see it?
Prove it! Show me those terms are wrong.

No, you're right. I made an algebra error. To second order, the
expressions are identical and to higher order we don't yet have
experimental precision to distinguish, at least in the Doppler effect.

So tell me, do you have a physics motivation, a set of principles, by
which you can derive your proposed equations for frequency shift?

And referring to my earlier post, you are aware, aren't you, that
Galilean velocity addition has been directly ruled out in experiment
by direct measurement of the velocities?

PD