• velocity -> redshift

    From Eric Flesch@21:1/5 to All on Mon Dec 13 21:48:52 2021
    The occasional paper reports AGN redshift as a velocity, e.g., 5000
    km/sec. Since the original measurement was that of a redshift, i.e., wavelength displacement of spectral lines, the authors thus used some
    equation to convert that to velocity. But the equation is not given.

    So, treating this generically, I convert the velocity back to a
    redshift. But I'm not interested to use a cosmological model with
    various parameter values, instead I use a simple cosmology-free
    equation, to wit:

    z = v / (c-v)

    Simple & easy. Should work quite adequately for z<0.1 which is where
    one encounters such given velocities.

    In principle, it could work all the way to z=infinity, not that I want
    to. But I wonder if anyone has any thoughts on this.

    [[Mod. note -- (I suspect the author knows this, but others may not.)
    There is a superb discussion of this & many related issues in

    Edward R Harrison
    "The Redshift-Distance and Velocity-Distance Laws"
    Astrophysical Journal 403(1), 28-31 (Jan 1993)
    http://adsabs.harvard.edu/abs/1993ApJ...403...28H

    -- jt]]

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  • From Eric Flesch@21:1/5 to Eric Flesch on Tue Dec 14 13:46:23 2021
    On Mon, 13 Dec 2021 21:48:52 PST, eric@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!? I avoided
    that as too simple, not to mention grossly wrong at z=1. 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.

    Harrison did not mention my little pretty equation at all. :-)

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  • From Phillip Helbig (undress to reply@21:1/5 to Flesch on Tue Dec 14 13:47:01 2021
    In article <61b59704.168077578@news.aioe.org>, eric@flesch.org (Eric
    Flesch) writes:

    The occasional paper reports AGN redshift as a velocity, e.g., 5000
    km/sec. Since the original measurement was that of a redshift, i.e., wavelength displacement of spectral lines, the authors thus used some equation to convert that to velocity. But the equation is not given.

    They just multiply the speed of light by z.

    So, treating this generically, I convert the velocity back to a
    redshift. But I'm not interested to use a cosmological model with
    various parameter values, instead I use a simple cosmology-free
    equation, to wit:

    z = v / (c-v)

    Simple & easy.

    But where did it come from? The non-relativistic Doppler formula is
    v = cz, which holds for small v (i.e. v/c << 1). Your equation is
    equivalent to v = cz-vz. So, if v/c << 1 is small, then the second term
    on the r.h.s. is much smaller than the first, and thus your formula is approximately correct and the error is smaller than the error of using
    the non-relativistic Doppler formula in the first place (strictly
    speaking valid only in the limit of vanishing v). But is there any justification for your formula?

    Should work quite adequately for z<0.1 which is where
    one encounters such given velocities.

    Yes. Everything is linear to first order. :-) But that doesn't mean
    that all approximations are equally valid logically (even if the
    difference, mathematically, is negligible).

    In principle, it could work all the way to z=infinity, not that I want
    to. But I wonder if anyone has any thoughts on this.

    Define "work". Can you plug in a number and get another number? Yes.
    Does it mean anything useful? No.

    The equation for arbitrarily high redshift is very simple: v=HD, where H
    is the Hubble constant and D is the proper distance. That is the
    definition of the Hubble constant. However, the proper distance cannot
    be directly measured, but can be calculated given the cosmological model
    (which can be inferred from the dependence of other distances on
    redshift). Yes, v can become arbitrarily large. No, no conflict with
    special relativity.

    Not even wrong is using the relativistic Doppler formula for high
    redshift. Simple proof: it contains no cosmological parameters (not
    even the Hubble constant), so using it implies that the velocity at high redshift is independent of the cosmological model, which is absurd.

    [[Mod. note -- (I suspect the author knows this, but others may not.)
    There is a superb discussion of this & many related issues in

    Edward R Harrison
    "The Redshift-Distance and Velocity-Distance Laws"
    Astrophysical Journal 403(1), 28-31 (Jan 1993)
    http://adsabs.harvard.edu/abs/1993ApJ...403...28H

    -- jt]]

    Indeed. See also the corresponding chapter in his textbook Cosmology:
    The Science of the Universe. For that matter, everything Harrison wrote
    is worth reading.

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  • From Phillip Helbig (undress to reply@21:1/5 to Flesch on Tue Dec 14 14:31:23 2021
    In article <61b83d2c.341686578@news.aioe.org>, eric@flesch.org (Eric
    Flesch) writes:

    On Mon, 13 Dec 2021 21:48:52 PST, eric@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.

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  • From Eric Flesch@21:1/5 to Phillip Helbig on Tue Dec 14 16:42:32 2021
    On Tue, 14 Dec 2021 13:47:01 PST, Phillip Helbig wrote:
    eric@flesch.org (Eric Flesch) writes:
    z = v / (c-v)
    Simple & easy.

    But where did it come from?

    It's just an isomorphic mapping of non-relativistic cosmological
    recession inferred from the spectral line shifts. So non-relativistic recessional velocity could be written as

    V = c * z/(1+z)

    Where did it come from? As far as I know, this was how redshift was
    originally quantified as a measure (using spectral displacement as a placeholder for velocity), but I have no citation.

    But is there any justification for your formula?

    Only that it's accurate and trivial, e.g., at v=c/2, z=1 and light
    frequencies are halved. Tell me that's wrong.

    Define "work". Can you plug in a number and get another number? Yes.
    Does it mean anything useful? No.

    It has only so much meaning as "non-relativistic cosmological
    recession" has a meaning. No less, no more. Unless I'm missing
    something basic.

    However, the proper distance cannot
    be directly measured, but can be calculated given the cosmological model

    I was specifically avoiding cosmological models and cosmological
    distances. I was only looking for a conversion between redshift and cosmological recession velocity (which I understand to be
    non-relativistic). As it turns out, all I needed was z=V/c, silly as
    it may be, because V=cz is what is used to calculate the recessional
    velocities presented in some papers. They could have used
    V = cz/(1+z) , but they did not. I'm surprised that they did not, and
    that's the end of it, I guess.

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  • From Phillip Helbig (undress to reply@21:1/5 to Flesch on Thu Dec 16 05:09:07 2021
    In article <61b92469.400883390@news.aioe.org>, eric@flesch.org (Eric
    Flesch) writes:

    On Tue, 14 Dec 2021 13:47:01 PST, Phillip Helbig wrote:
    eric@flesch.org (Eric Flesch) writes:
    z = v / (c-v)
    Simple & easy.

    But where did it come from?

    It's just an isomorphic mapping of non-relativistic cosmological
    recession inferred from the spectral line shifts. So non-relativistic recessional velocity could be written as

    V = c * z/(1+z)

    Where did it come from? As far as I know, this was how redshift was originally quantified as a measure (using spectral displacement as a placeholder for velocity), but I have no citation.

    But is there any justification for your formula?

    Only that it's accurate and trivial, e.g., at v=c/2, z=1 and light frequencies are halved. Tell me that's wrong.

    It's wrong in the sense that at v=c/2 one can't say anything about the
    velocity without knowing the cosmological model. It is right in the
    sense that at z=1 frequencies are halved. But the frequency is the
    rest-frame frequency divided by (1+z). For z=1 both formulae give the
    same answer, but not in general.

    Interestingly, some people (such as Zel'dovich) used to use Delta rather
    than z to characterize redshifts, with Delta = 1 - 1/(1+z) = z/(1+z),
    which is similar to your formula. That has the advantage that it ranges between 0 and 1 rather than 0 and infinity. But nothing to do with
    velocity.

    It has only so much meaning as "non-relativistic cosmological
    recession" has a meaning. No less, no more. Unless I'm missing
    something basic.

    That has a meaning only at low redshift, in which case it would be
    easier to use v=cz.

    I was specifically avoiding cosmological models and cosmological
    distances. I was only looking for a conversion between redshift and cosmological recession velocity (which I understand to be
    non-relativistic). As it turns out, all I needed was z=V/c, silly as
    it may be, because V=cz is what is used to calculate the recessional velocities presented in some papers.

    Right.

    I'm not sure what you mean by cosmological recession velocity being non-relativistic. It's true that one can't calculate it with the
    relativistic Doppler formula, but that doesn't mean that the
    non-relativistic Doppler formula (or your formula) is correct, except in
    the limit of low redshift (in which case the relativistic Doppler
    formula is also correct).

    They could have used
    V = cz/(1+z) , but they did not. I'm surprised that they did not, and
    that's the end of it, I guess.

    What could have been the motivation to use V = cz/(1+z)?

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