In article <
61b83d2c....@news.aioe.org>,
er...@flesch.org (Eric
Flesch) writes:
> On Mon, 13 Dec 2021 21:48:52 PST,
er...@flesch.org (Eric Flesch) wrote:
> >z = v / (c-v) Simple & easy. Should work quite adequately for z<0.1...
> >[[Mod. note -- (I suspect the author knows this, but others may not.)
> >
http://adsabs.harvard.edu/abs/1993ApJ...403...28H
>
> Thanks for that, it is a great discussion. The part relevant to me
> comes at the very end where Harrison describes the "habit of
> converting redshifts into radial velocities by means of the Doppler
> approximation V=cz" as being "convenient astronomically".
>
> Is *that* all that is used to produce the velocity figure!?
Yep, that's it!
In the old days, when 0.1 was a huge redshift, it sort of made sense: at
low redshift, the Doppler formula does give the recession velocity (in
the limit of 0 redshift), and differences of hundreds of km/s are easier
to visualize than the difference between 0.001 and 0.0025 or whatever.
Of course, although theoretically predicted by de Sitter and Lema=EEtre,
Hubble (whether or not he knew about their work) was very empirically
minded and used the standard astronomer conversion of redshift into
velocity (familiar from motions of double stars or whatever).
When redshift became larger, most (but not all---as Harrison points out,
even some professional astronomers at least seemed confused) realized
that it was just a placeholder for redshift, which also aided comparison
with older data.
> I avoided
> that as too simple, not to mention grossly wrong at z=1.
Right; it's a low-redshift approximation. But your formula, as far as I
know, has no justification.
> Well, if
> that's what they do, then my reverse equation z=v/(c-v) will show a
> 10% discrepancy at z=0.1, so I'd better go back and fix those.
The literature you have almost certainly has v=cz, so just convert back.
Where it is more difficult is where people observe something like the
distribution in redshift and flux, i.e. the luminosity-dependent
redshift distribution or, equivalently, the redshift-dependent
luminosity function. (Some astronomers call those relations "Hubble
diagrams"---redshift plotted against apparent magnitude or vice
versa---even for objects (QSOs, say) which are not standard candles and
hence no (approximately) limited relationship is even expected; again,
are you an empiricist or concerned with interpretation?) The
observational data, apparent magnitudes and redshifts, are clear. But
sometimes results are presented in terms of absolute magnitude (or
luminosity) and (co-moving) volume, which necessarily implies conversion
via some cosmological model, which they might have failed to specify.
Even if known, converting back to the original data is non-trivial and,
if binning is involved, impossible. Best for observers to (at least
also) report their data (not necessarily all the raw data) in terms of
observable quantities.