From: Robert A. Knop Jr. (robert.a.knop@vanderbilt.edu)
Date: Tue Apr 15 2003 - 08:01:03 PDT
From Kolb & Turner p. 41:
d_L^2 = R^2(t_0) r_1^2 (1+z)^2
where R(t_0) is the scale factor at the time of detection, r_1 is the
coordinate distance to the object, and z is the redshift. In this case,
r_1 can be figured out as r_1(z). Equivalently, work out the proper
distance to the object at time of detection, and that is R(t_0)r_1(z) ;
this gives us (most of) our standard luminosity distance integral
(missing one factor of (1_z)).
r_1(z) should clearly just use that z that comes from cosmological
redshift, since this is giving you the radius of the sphere surrounding
the emitting object, and as such you want the real distance.
The other z, in the (1+z)^2 above, however, should use your observed
(geocentric) redshift, as those terms are to take care of (1) the
redshifting of the photons (and corresponding energy loss) and (2) time
dilation. Energy loss and time dilation will happen if it's a doppler
shift or a cosmological redshift, so the total redshift is appropriate
here.
Probably what this means is that to do it *right*, we need to use *both*
heliocentric and CMB-based redshifts, putting the right one in the right
place.
Does anybody agree with this, or can anybody point out a flaw in my
reasoning?
-Rob
-- --Prof. Robert Knop Department of Physics & Astronomy, Vanderbilt University robert.a.knop@vanderbilt.edu
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