# 16.5 Relating Redshift and Distance

Determining the Hubble constant using supernova type 1a. Click here for original source URL

Hubble showed that the red shift of a galaxy is correlated with its distance from the Milky Way. Let us look at the implications of the Hubble relation in a bit more detail. We start with the way that red shift is defined. Astronomers observe the light from almost every galaxy to be red shifted. The only exceptions are a few very nearby galaxies that are bound by gravity to the Local Group. Suppose we observe a galaxy and label the wavelength of any spectral feature as λ (lambda). The spectral feature will typically be an absorption line of hydrogen, calcium, or magnesium. The wavelength of that same spectral feature observed in a gas in the laboratory is λ_{0}. The difference in the two wavelengths (λ-λ_{0}), which is a positive number, represents how much the wavelengths of the galaxy’s light have been shifted to longer wavelengths (Δλ). The red shift of a galaxy is defined as:

z = Δλ/ λ_{0}

Since Δλ = λ-λ_{0}, we get z = (λ-λ_{0}) / λ_{0}. Now we can use the Doppler effect (Δλ/ λ_{0} = v/c) to define the red shift in terms of the recession velocity of the galaxy (v) and the speed of light (c):

z = v / c

This equation is actually an approximation that is only valid to describe the red shift of galaxies when the recession velocity v is much smaller than the velocity of light c. In general, cosmological red shifts are not the same as Doppler shifts. Red shifts can be measured very accurately. A typical velocity measurement for a galaxy might be 12,540 ± 120 km/s, which corresponds to a red shift of z = v / c = 12,540 / 3x 10^{5} = 0.0418. The uncertainty in velocity corresponds to a red shift uncertainty of 120 / 3 x 10^{5} = 0.0004, so we would quote the complete measurement as z = 0.0418 ± 0.0004.

Remember that the conceptual basis for galaxy red shifts is quite distinct from the Doppler effect. The Doppler effect is due to the relative motions of objects traveling through space or another material medium. The cosmological red shift is due to the expansion of space itself (or more correctly, in terms of the theory of general relativity, the expansion of space-time). Measuring the galaxy’s red shift is one step towards determining its distance. The next step is using the Hubble relation. The Hubble relation is a (locally) linear correlation between the redshift of a galaxy and its distance from the Milky Way. If you graph this relation, the slope of the line is the Hubble constant, or a measure of the expansion rate of the universe. Mathematically, the Hubble relation can be expressed as:

v = H_{0 }d

In this equation, v is the velocity of the galaxy in km/s, d is the distance in Mpc, and H_{0} is the Hubble constant in km/s/Mpc. If this relationship is expanded out to the most distant observed supernovae, astronomers find that the relationship curves. The curvature indicates a change in the expansion rate in the distant past. This curve indicates that the universe isn't just expanding, it's accelerating.

Determining this curve and fitting it from data is difficult. Real galaxies do not follow a perfect Hubble relation. There is always a scatter around the line caused by the fact that galaxies interact with each other by gravity, which gives them a component of velocity that is not due to the expansion of the universe, called a peculiar velocity. The amount of this extra velocity (which may be a red shift or a blue shift) is 100 to 200 km/s. Galaxies tug each other around in space, so there is no Hubble relation for the very nearest galaxies — velocity and distance are not correlated. For a galaxy with a recession velocity of 1000 km/s, the peculiar velocity is significant fraction of the recession velocity, 10 to 20 percent. For a much more distance galaxy with a recession velocity of 10,000 km/s, the peculiar velocity is only 1 or 2 percent of the recession velocity. In other words, the peculiar velocity can be ignored for sufficiently distant galaxies and the Hubble expansion is smooth and well determined. Peculiar velocities have no systematic effect on the slope of the Hubble relation. They just add a scatter to the correlation between distance and velocity. The largest uncertainty in the slope of the Hubble relation — and therefore in the value of the Hubble constant — is caused by systematic errors which alter the distance scale.

We can combine the Hubble relation and the definition of red shift to relate the distance of a galaxy to its red shift. First, rearrange the terms of the Hubble relation to calculate distance as d = v / H_{0}. Then rearrange the terms of the red shift equation to get v = z c. Combining the two results gives:

d = z c / H_{0}

Again, this formula is only appropriate if the recession velocity is much less than the speed of light, or if z << 1. Knowing the value of the Hubble constant, red shift can be used as a distance indicator. This technique is a very powerful tool, since astronomers can measure red shifts of galaxies so faint that it is impossible to get more detailed information about them by other means.

While it is not unusual to measure a red shift to a precision of four significant figures or an accuracy of a few percent, astronomers cannot calculate distance with that level of accuracy. In late 2010, NASA combined existing data from the Hubble Space Telescope, Chandra Observatory, Wilkinson Microwave Anisotropy Probe, and a variety of other sources to derive a best estimate of Hubble's Constant of 70.8 ± 1.6 km/s/Mpc if space is assumed to be flat, or 70.8 ± 4.0 km/s/Mpc otherwise. This means the current expansion rate of the universe is known with an impressive precision of just a few percent.