OK, now that I understand the barycentre, The centre of mass of the solar system would lie either very close to the sun, or within the sun.
On top of this, the barycentre is not a fixed position but rather changes as the planets revolve around the sun. So, our spacecraft would have difficulty maintaining this position as well as resisting the heat generated by the sun.
I can understand how the motion of the earth's contribution to the doppler effect can be determined. The earth's velocity in its orbit around the sun is about 28km/s so this contribution to the doppler effect is small.
If we aimed our spectrograph at a specific star, say Arcturus, then the redshift would be measured when the earth moves away from Arcturus and the blue shift when earth moves toward Arcturus. Any difference between the red and blue component could be construed as the motion of Arcturus itself.
Good enough (don't forget that unless the observatory is at one of the Earth's poles there is also the motion of the observatory around the Earth's rotational axis, and that for instruments aboard spacecraft there is the motion of the spacecraft wrt the solar system barycentre).
WRT to laboratory spectroscopy, once the at rest spectra of the elements have been determined, the next step would be to compare these known rest spectra with the spectra of moving galactic objects. This would enable a determination to be made of the motion of the extragalactic body. ie if it is redshifted the extragalactic body is moving away and if blue shifted , moving towards us. By subtracting the results obtained in the previous paragraph would leave the redshift of the extragalactic body.
Fair enough, at a very high level.
However, there are some points that are very important to your, and PS', question about alternative explanations of the redshift; I'll go over some of these in a later post.
WRT to the gravitational aspect, I am not quite sure as to how this is removed from the equation. Initially, I thought, as we had a reference to the barycentre, we could measure the gravitational effect on redshift much the same way as Eddington did when he measured the bending of light due to the sun during a solar eclipse. Perhaps I need some clarification here.
The first thing to note about the gravitational redshift is that it is predicted by GR.
The second thing is that Pound and Rebka (and others) measured it, in labs here on Earth, in controlled conditions, in a series of experiments (starting in 1959?). The experiment has been repeated many times and in many different ways; the results are consistent with the GR predictions.
With some simplification, the gravitational redshift of a compact, spherical object* of mass M and radius r, observed at a great distance (zero relative velocity) is z = GM/rc^2 (G is the gravitational constant; c is the usual suspect).
If the roles of source and observer are reversed, then there's a gravitational blueshift.
Would you care to take a crack at estimating the gravitational blueshift that observers on the surface of the Earth must factor out of any spectrum of a distant source?
As to the What Now, I think observation and more observation and comparisons to what we see in the lab and what we observe in the sky.
For readers who have not been keeping up: we've done a thorough literature search and have made a list of all the things that scientists have found, in the lab, that can result in redshifts (Doppler effect, Compton effect, ...).
Our quest is to outline how a diligent scientist might go about investigating the possibility of alternative explanations for the redshifts conventionally ascribed to 'cosmological expansion'.
Certainly nothing wrong with "
observation and more observation and comparisons to what we see in the lab and what we observe in the sky"!
But perhaps that's a little too terse?
For example, wouldn't it be a very good idea to understand - in considerable detail - the actual, physical mechanisms at work in each of the (empirically founded) causes of redshift? IOW, to see if you can account for them - quantitatively - within the framework of the physics found in standard textbooks? And if you can't, then maybe some more research would be needed, to nail down just what's going on, to as many decimal places (significant figures) as the errors and uncertainties in your lab results allow?
*
more precisely, this is the redshift from a source on the surface of such an object; we also assume that the object is not rotating (etc)
