AstroProjects
Cosmological redshift
Understanding cosmological redshift
The expanding universe
As you probably know, the whole universe is
expanding, so distant galaxies are being carried
away from us as space itself expands. A good
way of visualising this is to imagine blowing up
a balloon on to which bits of cotton wool have
been stuck to represent galaxies. The surface
of the balloon represents the whole universe.
The analogy is a useful one, but you have to
remember that in reality space is three
dimensional, whereas the surface of the balloon
is only two dimensional. Just as space is
believed to have a finite volume, yet no 'edge'
to it, so the balloon has a finite area, but no
edge.
In the model, the speed at which two cotton wool
'galaxies' move apart will be proportional to their
distance apart. Similarly, in the real universe, the
speed at which a distant galaxy is receding from
our own galaxy, the Milky Way, will be
proportional to its distance from us. In essence,
this is Hubble's Law.
In symbols, the recession velocity of a galaxy v is
at distance r is
v = H0 r
where H0 is called the Hubble constant. It is
usual to measure v in km s-1 and r in Mpc.
Hubble's constant has the value
H0 = 71 km s-1 Mpc-1 according to the most recent
measurements.
[Note: Astronomers normally measure distances
in megaparsec (Mpc) rather than millions of light
years (Mly). 1 Mpc = 3.26 Mly = 3.09 x 1013 km.]
Using cotton wool 'galaxies' stuck to the
two dimensional surface of a balloon to
illustrate why the recession velocity of a
galaxy is proportional to its distance from
us (Hubble's Law). The expansion of
space is represented by the expansion of
the balloon as it is inflated.
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Richard Beare, 14th December, 2007
Version 1.00
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AstroProjects
Cosmological redshift
Cosmological redshift
This expansion of space also causes light waves from these distant galaxies to be stretched as
they travel towards us. This means that their wavelengths increase so that they (generally)
appear redder to us. This effect is called the 'cosmological redshift'. It is measured by the
fractional increase in the wavelength  of a light wave as it travels towards us: z =  / 
It can be shown that the redshift is given by:
z=v/c
where v is the recession velocity of the distant galaxy and c is the velocity of light.
This is the same formula as for the Doppler effect =  /  = v / c. However, it is important to
realise that the cosmological redshift is due to the expansion of space itself, whereas a Doppler
shift occurs when a source of light moves through space relative to the observer.
Substituting v = H0 r, into z = v / c gives:
z = (H0 /c) r
Redshift and distance
Measuring the redshift z of a cluster of galaxies (say) should therefore enable us to find the
distance d to it in Mpc, using the known values of Hubble's constant H0 = 71 km s-1 Mpc-1 and
the velocity of light c = 3.00 x 105 km s-1.
There is, however, something else to be taken into account. Galaxies in clusters are not
stationary, but orbit the centre of mass of the cluster with speeds of up to 1500 km s-1 or so.
This additional velocity gives rise to an additional red (or blue) shift of up to 0.005 due to the
Doppler effect.
If we based our distance estimate for the whole cluster on just one galaxy, this could result in an
error of up to 20 Mpc (70 Mly). To avoid this, it is best to average the redshifts for a few galaxies
in the cluster in order to estimate the cluster distance.
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Richard Beare, 12th December, 2007
Version 1.22
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AstroProjects
Cosmological redshift
Spectra and redshifts in SDSS
If SDSS has obtained spectra for some of the galaxies in a cluster (say), it will have automatically
calculated their redshifts and you can use these to find out how far away the cluster is. The
following instructions explain how to do this.
1) Which galaxies have spectra?
To find out which galaxies SDSS has obtained spectra for, click the Objects with spectra
box in the Navigate Tool. This marks objects with spectra by small red squares (light blue
squares if using a white background).
Objects for which spectra have been obtained in galaxy cluster ACO 2151. (No spectra
have been obtained outside the plate position indicated by the circular arc.)
2) Find an average value for the redshift of your cluster Select a number of galaxies for which
spectra have been obtained. Click on each of these in turn to select it, and then click the
Explore button at the bottom right of the Navigate window. If the selected object has a
spectrum, you will find the redshift z given immediately above the spectrum in the Navigate
window. If you click on the spectrum you will get an enlarged image of it, and also find the
redshift z just above the x-axis of the graph.
Beware: very confusingly, the symbol z is also used for the object's magnitude in the infrared. This is the z given higher up in the Explore window as well in the Navigate window.
3) Find the distance to your cluster Knowing the cluster's redshift, you can now work out its
distance r using Hubble's law. Rearranging the formula given above, z = (H0 /c) r, you can
use the following formula:
r = cz /H0
-1
-1
where H0 = 71 km s Mpc and c = 3.00 x 105 km s-1.
Note about SDSS spectra At the time of writing (July, 2007) SDSS had not finished obtaining spectra
for all the objects that will eventually be included, so there are some areas of the sky where there are
images but no spectra as yet. To obtain spectra of individual objects, SDSS uses a circular plate drilled
with holes that align exactly with the images of the objects. Optical fibres then feed the light coming
through the 600 or so holes to separate spectrographs that measure the spectra. The plates with their
optical fibres replace the camera when spectra are being obtained. To see the positions of plates in
images, click the Plates box in the Navigate Tool when using a fairly low 'magnification'. This will show
you where spectra have been obtained and where they have not.
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Richard Beare, 12th December, 2007
Version 1.22
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