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27.1 Quasars  (Page 3/9)

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Recession speed of a quasar

The formula for the Doppler shift, which astronomers denote by the letters z , is

z = Δ λ λ = v c

where λ is the wavelength emitted by a source of radiation that is not moving, Δλ is the difference between that wavelength and the wavelength we measure, v is the speed with which the source moves away, and c (as usual) is the speed of light.

A line in the spectrum of a galaxy is at a wavelength of 393 nanometers (nm, or 10 –9 m) when the source is at rest. Let’s say the line is measured to be longer than this value (redshifted) by 7.86 nm. Then its redshift z = 7.86 nm 393 nm = 0.02 , so its speed away from us is 2% of the speed of light ( v c = 0.02 ) .

This formula is fine for galaxies that are relatively nearby and are moving away from us slowly in the expansion of the universe. But the quasars and distant galaxies we discuss in this chapter are moving away at speeds close to the speed of light. In that case, converting a Doppler shift (redshift) to a distance must include the effects of the special theory of relativity, which explains how measurements of space and time change when we see things moving at high speeds. The details of how this is done are way beyond the level of this text, but we can share with you the relativistic formula for the Doppler shift:

v c = ( z + 1 ) 2 − 1 ( z + 1 ) 2 + 1

Let’s do an example. Suppose a distant quasar has a redshift of 5. At what fraction of the speed of light is the quasar moving away?

Solution

We calculate the following:

v c = ( 5 + 1 ) 2 − 1 ( 5 + 1 ) 2 + 1 = 36 − 1 36 + 1 = 35 37 = 0.946

The quasar is thus receding from us at about 95% the speed of light.

Check your learning

Several lines of hydrogen absorption in the visible spectrum have rest wavelengths of 410 nm, 434 nm, 486 nm, and 656 nm. In a spectrum of a distant galaxy, these same lines are observed to have wavelengths of 492 nm, 521 nm, 583 nm, and 787 nm respectively. What is the redshift of this galaxy? What is the recession speed of this galaxy?

Answer:

Because this is the same galaxy, we could pick any one of the four wavelengths and calculate how much it has shifted. If we use a rest wavelength of 410 nm and compare it to the shifted wavelength of 492 nm, we see that

z = Δ λ λ = ( 492 nm − 410 nm ) 410 nm = 82 nm 410 nm = 0.20

In the classical view, this galaxy is receding at 20% of the speed of light; however, at 20% of the speed of light, relativistic effects are starting to become important. So, using the relativistic Doppler equation, we compute the true recession rate as

v c = ( z + 1 ) 2 − 1 ( z + 1 ) 2 + 1 = ( 0.2 + 1 ) 2 − 1 ( 0.2 + 1 ) 2 + 1 = 1.44 − 1 1.44 + 1 = 0.44 2.44 = 0.18

Therefore, the actual recession speed is only 18% of the speed of light. While this may not initially seem like a big difference from the classical measurement, there is already an 11% deviation between the classical and the relativistic solutions; and at greater recession speeds, the divergence between the classical and relativistic speeds increases rapidly!

Quasars obey the hubble law

The first question astronomers asked was whether quasars obeyed the Hubble law and were really at the large distances implied by their redshifts. If they did not obey the rule that large redshift means large distance, then they could be much closer, and their luminosity could be a lot less. One straightforward way to show that quasars had to obey the Hubble law was to demonstrate that they were actually part of galaxies, and that their redshift was the same as the galaxy that hosted them. Since ordinary galaxies do obey the Hubble law, anything within them would be subject to the same rules.
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Source:  OpenStax, Astronomy. OpenStax CNX. Apr 12, 2017 Download for free at http://cnx.org/content/col11992/1.13
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