A few words about redshifts, and what the published redshifts of galaxies mean (I'll focus on UV, optical/visual, and near-IR parts of the electromagnetic spectrum).
If you take a spectrum - split the light into colours - of an astronomical object, you send the light from the object into a spectrograph (sometimes called a spectroscope) and record the intensity of the light as a function of wavelength. The output may be a 1D spectrum (e.g. wavelength on the x-axis, intensity on the y-axis; all the light from what is sent into the spectrograph), a 2D spectrum (e.g. wavelength on the x-axis, angular distance on the y-axis, intensity coded as colour; the light sent into the spectrograph comes from a strip, perhaps a 'long slit'), or a data cube (each 2D pixel on the sky/object has its own spectrum; each could be displayed as a 1D spectrum).
The spectra of stars, planetary nebulae, HII regions, and galaxies (and many more objects besides) usually contain 'lines', which correspond to specific atomic transitions (the electron in an atom, or ion, 'jumps' from one allowed level to another; if the jump is 'down' - higher energy state to a lower one - the atom or ion emits light; if 'up, it absorbs light). The 'rest wavelengths' of the lines are very well known, either from high precision lab experiments or theory (e.g. many 'nebular lines' have never been observed in labs - we can't create vacuums hard enough for long enough). The difference between the observed wavelength (from the spectrum of the astronomical object) and the rest wavelength is called redshift, in the sense of (observed) - (rest). Note that negative redshifts are sometimes called blueshifts.
In this there is no assumption, or theory, concerning motion (Doppler or otherwise), gravity, distance, or anything else; the redshift - at this stage! - is simply the difference between what's observed and what's in a table.
To make the redshift of an object observed by one spectrograph - at a particular location and time - comparable (scientifically) with that observed (of the same object) by a different spectrograph (or the same one!) - at a different location and time (or different time, same location), a series of 'corrections' or 'transformations' are done. These convert what was actually observed to what would have been observed had the spectrograph been at the solar system's barycentre (or centre of mass/gravity); i.e. take out all the redshifts due to the relative motion of the spectrograph - at the time and location of the observation - with respect to the solar system barycentre. These transformations were exceedingly tedious to do in the days before astronomers had computers (actually, the people - usually women - who did all the calculations by hand were called, at the time, computers!
), but today they're just a simple routine (though not so simple for spectrographs aboard FUSE say, or the Hubble Space Telescope).
This works fine for stars, or objects which are essentially point sources; however, the redshift of a galaxy is rather more complicated.
Galaxies, obviously, are not points (well, some seem to be, but those in the M81 group are not). So, in principle at least, there could be a spectrum - and hence a redshift - for every pixel in the image of a galaxy (and as the number of pixels in a galaxy image depends - in part - on the resolution of the imaging system, spectrographs on the Hubble may be able to take far more spectra of an M81 group galaxy than one down here on the ground could).
And, in general, each part of a galaxy does have a different redshift! For spiral galaxies, how those redshifts vary across the galaxy is what becomes an estimate of the galaxy's 'rotation curve', when suitably analysed.
To get the redshift of the galaxy as a whole, two different methods are commonly used: in one, the redshift of the nucleus (if it can be obtained) is what's called the galaxy's redshift; in the other, the individual redshifts are averaged, using one of several weighting methods.
So a galaxy's redshift tells us whether the galaxy is heading away from us (a positive redshift), or towards us (a negative redshift; one day it will collide with us), right? Sorta; here's why:
* the redshift gives a measure of an object's line of sight motion only; if it's screaming across the sky, at right angles to the line of sight, you can't tell (from the redshift)
* the solar system is moving, relative to the nucleus of our own galaxy (the Milky Way), so the galaxy-to-galaxy redshift (that which you'd measure if you were at our galaxy's nucleus, and not moving relative to it) will nearly always be different than that which we observed out here in the galactic suburbs
There's nothing you can do about the former, using today's astronomical capabilities - at least not for galaxies in the M81 group - however, you can use estimates of the solar system's position and motion relative to the galaxy nucleus to transform the observed redshift to a 'galactocentric' redshift. There'll be some inevitable extra uncertainty of course (for example due to the viewing geometry), but for most purposes it won't be much more than that in the galaxy redshift estimate ('it' being the uncertainty, or 'error').
Finally, redshifts can be given a km/sec, or z; the former is easy to understand (the observed redshift is the same as that you'd see if the relative line of sight motion were so many km/sec); the latter is just a formula: z = (wavelength observed - rest wavelength) divided by rest wavelength.
I hope this helps some.
