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I've done some search with the Nasa Extragalactic Database (NED) and I have a very basic question about the velocity/redshift conversion. For example, for the first object of this page, we have $v=19791km/s$ and $z=0.066016$.

If we use the simple formula $z=v/c$, we find the result of the database : $19791/299792=0.0660157709345$. But now if we use the formula : $z = \sqrt{\frac{c+v}{c-v}}-1$ we find $0.0683461749892$

Which is the correct one and why ?

Thank you very much.

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I suspect that the velocity formula must incorporate the expansion of the Universe. I'm not sure though: I wouldn't expect it to be a ~3% effect at that redshift. –  Warrick Jun 25 '12 at 11:29

2 Answers 2

up vote 2 down vote accepted

The simple formula is just the first-order expansion of the more complicated one about $v = 0$, the latter being exact for the Doppler effect of motion purely along the line of sight. The $v$ here refers to the peculiar motion of the galaxy.

Be aware that for all but the very nearest galaxies, the observed redshift comes almost entirely from the expansion of the universe, not from relative motion in the special-relativistic sense. Thus converting from redshift to velocity using either of the formulas mentioned, though a very common practice, can be misleading. For a thorough albeit technical discussion of subtleties related to this point, there is a paper by Davis and Lineweaver.

Edit: Since I have lately been using NED a lot, I came across this page in their documentation. Point 1 in particular notes that "no relativistic correction is applied" and so you may see "velocities in excess of the speed of light." (It also says $v = z/c$, but I hope that's just a typo.) There are two important points here. The first is that you can safely assume the values reported are redshift times the speed of light, possibly with a correction to a certain reference frame. The second is that even NASA is under the misconception that redshift of distant galaxies has something to do with Doppler shift, when this is just fundamentally false. The quantity $zc$ is really just a way of putting units to redshift, nothing more.

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Actually, now that you mention Davis & Lineweaver - there's another article (popular one) where Davis explains how Doppler and expansion redshifts are perfectly equivalent, it is really just a matter of one's choice of descriptive language. The real misconception is that superluminal speeds are not possible. Special Relativity only applies locally, and over the distances covered by a redshift of 1.4 (which, IIRC, is where expansion goes superluminal), this is no longer a good approximation. –  Thriveth Mar 24 '14 at 14:08

There are different formulas that give better approximations depending on the velocity distribution, and overall velocity magnitude. Suggest looking at :

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