How do astronomers convert z (when < 0.1) to km/s (cz vs relativistic formula)

What is the RESEARCH convention for z0.1, cz or actual (relativistic
velocity)?

When, say, a highly accurate 21cm redshift is obtained, the researcher
generally publishes it (if z 0.1) in km/s. My question is this:

QUESTION: Is there a generally accepted standard for converting z to
km/s (when z 0.1)?

For example, if z = 0.02, then the corresponding ACCURATE velocity
is 5936km/s, but the cz approximation is 5996 km/s, a difference of 60
km/s (in a measurement that sometimes has an rms error of ONLY 4
km/s).

[above computed with the usual formulas of special relativity [ z+1 =
sqrt(1-(beta**2), where beta is v/c; or v =
c*(((1+z)**2)-1)/(((1+z)**2)+1) ]] In case the size of the error
seems "off" to you: The reason why the relativistic correction for
redshift velocity is so much more than, say, the relativistic
correction for mass or time, is because of that (1+z) term: at v =
6000 km/s, the relativistic correction for mass or time or (1+z) is
only 0.02%, but changing (1+z) by 0.02% changes z by almost 50 times
that amount, about 1%.

Most articles that give these km/s z measurements (e.g., Theureau et
al A&A Sup.130, 333(1998)) do not mention which of the two methods
they use. Common sense would tell you that they wouldn't want to
introduce such large errors in their data, but most catalogs (e.g.,
NED) that give both forms use the inaccurate cz formula. I'm assuming
that the astronomers who actually measure these things always do it
one way or the other (otherwise these redshifts couldn't be compared
meaningfully), but which one?

Any help is greatly appreciated.

(rhmd) wrote in message ...
What is the RESEARCH convention for z0.1, cz or actual (relativistic
velocity)?

When, say, a highly accurate 21cm redshift is obtained, the researcher
generally publishes it (if z 0.1) in km/s. My question is this:

QUESTION: Is there a generally accepted standard for converting z to
km/s (when z 0.1)?


Hubble's constant

Regards

Carsten Nielsen
Denmark

[Mod. note: that's not what the original poster was asking. -- mjh]

rhmd wrote:
What is the RESEARCH convention for z0.1, cz or actual (relativistic
velocity)?

When, say, a highly accurate 21cm redshift is obtained, the researcher
generally publishes it (if z 0.1) in km/s. My question is this:

QUESTION: Is there a generally accepted standard for converting z to
km/s (when z 0.1)?

For example, if z = 0.02, then the corresponding ACCURATE velocity
is 5936km/s, but the cz approximation is 5996 km/s, a difference of 60
km/s (in a measurement that sometimes has an rms error of ONLY 4
km/s).

[above computed with the usual formulas of special relativity [ z+1 =
sqrt(1-(beta**2), where beta is v/c; or v =
c*(((1+z)**2)-1)/(((1+z)**2)+1) ]] In case the size of the error
seems "off" to you: The reason why the relativistic correction for
redshift velocity is so much more than, say, the relativistic
correction for mass or time, is because of that (1+z) term: at v =
6000 km/s, the relativistic correction for mass or time or (1+z) is
only 0.02%, but changing (1+z) by 0.02% changes z by almost 50 times
that amount, about 1%.

Most articles that give these km/s z measurements (e.g., Theureau et
al A&A Sup.130, 333(1998)) do not mention which of the two methods
they use. Common sense would tell you that they wouldn't want to
introduce such large errors in their data, but most catalogs (e.g.,
NED) that give both forms use the inaccurate cz formula. I'm assuming
that the astronomers who actually measure these things always do it
one way or the other (otherwise these redshifts couldn't be compared
meaningfully), but which one?

Any help is greatly appreciated.

The actual measurement is z. cz is often tabulated for history and
convenience; what comes into distance for nontrivial cz is just
z, while the special-relativistic convention is useful only
in retrsicted cases (such as comparising the velocity dispersions
of objects at different redshifts and avoiding the "stretch" introduced
by cz). There was an issue at one point of different scales from
optical and radio lines, in which radio astronomers took
z = delta nu / nu while optical astronomers would use delta lambda/lambda
(frequency versus wavelength conventions).

I think there is an explication o this somewhere on Ned Wright's
WWW site (travelling now and can't cut-and-paste as usual).

Bill Keel

In article ,
(rhmd) writes:

What is the RESEARCH convention for z0.1, cz or actual (relativistic
velocity)?

Multiply z by the speed of light. Some folks might use 299792458 m/s,
some might use 300000000 m/s.

QUESTION: Is there a generally accepted standard for converting z to
km/s (when z 0.1)?

See above.

For example, if z = 0.02, then the corresponding ACCURATE velocity
is 5936km/s, but the cz approximation is 5996 km/s, a difference of 60
km/s (in a measurement that sometimes has an rms error of ONLY 4
km/s).

No, this is not the ACCURATE velocity...

[above computed with the usual formulas of special relativity [ z+1 =
sqrt(1-(beta**2), where beta is v/c; or v =
c*(((1+z)**2)-1)/(((1+z)**2)+1) ]]

....because the cosmological redshift is not a Doppler shift. If the
redshift is large enough to worry about relativistic corrections of the
type mentioned above, then it is large enough that it is not correct to
think of it as a Doppler shift.

The cz stuff is just a convention. Also, keep in mind that the measured
redshift is a combination of the cosmological redshift and the Doppler
shift due to peculiar motions. For low cosmological redshifts, the two
can be comparable in magnitude.

To properly calculate the recession velocity for larger redshifts, you
have to know the cosmological model.

For more information, see

@BOOK {EHarrison81a,
AUTHOR = "E. R. Harrison",
TITLE = "Cosmology, the science of the universe",
PUBLISHER = "Cambridge University Press",
YEAR = "1981",
ADDRESS = "Cambridge"
}

@ARTICLE {EHarrison93a,
AUTHOR = "E. R. Harrison",
TITLE = "The redshift-distance and velocity-distance laws
JOURNAL = "ApJ",
YEAR = "1993",
VOLUME = "403",
NUMBER = "1",
PAGES = "28"
}

On Fri, 30 Apr 2004 07:46:18 GMT, Phillip Helbig- wrote:
To properly calculate the recession velocity for larger redshifts, you
have to know the cosmological model.

I'd gotten the impression that this was model-independent.
Cosmological expansion dictates we can see only those parts of the
universe which are receding from us at sub-light speed. Therefore,
z=1 indicates a recession speed of .5c, z=3 shows .75c, etc. No?

Eric

(Phillip Helbig---remove CLOTHES to reply) wrote in message ...
For more information, see

@BOOK {EHarrison81a,
AUTHOR = "E. R. Harrison",
TITLE = "Cosmology, the science of the universe",
PUBLISHER = "Cambridge University Press",
YEAR = "1981",
ADDRESS = "Cambridge"
}

@ARTICLE {EHarrison93a,
AUTHOR = "E. R. Harrison",
TITLE = "The redshift-distance and velocity-distance laws
JOURNAL = "ApJ",
YEAR = "1993",
VOLUME = "403",
NUMBER = "1",
PAGES = "28"
}

A paper of mine, "A Cosmological Correspondence Principle", Pub. A. S.
P., Vol. 97, 697-699, Aug., 1985, discusses the Doppler and
cosmological redshifts, as well as some other effects, in some detail.

Donald Gudehus

In article ,
(Eric Flesch) writes:

On Fri, 30 Apr 2004 07:46:18 GMT, Phillip Helbig- wrote:
To properly calculate the recession velocity for larger redshifts, you
have to know the cosmological model.

I'd gotten the impression that this was model-independent.

Where?

Cosmological expansion dictates we can see only those parts of the
universe which are receding from us at sub-light speed.

Perhaps, depending on how you define the terms. Receding from us at
sub-light speed WHEN? Speed is distance/time, so what distance do you
mean? What time?

Therefore,
z=1 indicates a recession speed of .5c, z=3 shows .75c, etc. No?

This is definitely wrong.

See the references in my earlier post in this thread. In particular,
check out chapter 19 on cosmological horizons in Harrison's COSMOLOGY.

The cosmological redshift tells us the ratio of the scale factor now to
that at the time the light was emitted. This is the same in all
cosmological models. If you know the cosmological model, you can
calculate distances, velocities etc.

Imagine a cosmological model which is static, then expands, then is
static again. If light is emitted in the first static phase and
absorbed in the second, then there is a redshift, though at neither
emission or reception is the universe expanding. Also, the redshift
will be the same regardless of the distance (as long as the light is
emitted in one static phase and observed in the second).