Astronomical Redshift

 

By Mike Luciuk

 

Studying astronomical redshift via spectral and photometric techniques has been an intensive astrophysical activity since 1929 when Hubble announced that galaxies are moving away from each other, i.e. the universe is expanding. Although Doppler redshift is well known to most amateur astronomers, this tutorial will also cover two less familiar causes of this phenomenon. 

 

Astronomical redshift is the lengthening of emitted radiation as detected by observers. The amount of lengthening is commonly denoted by z

	(1)
Where                is the wavelength of the observed light                            is the emitted wavelength

If z is positive, the observer sees a redshift. Negative z is a blueshift (wavelengths are shortened)

 

There are three causes of astronomical redshift:

1.   Doppler redshift is a result of relative radial motion between emitter and observer.

2.   Gravitational redshift occurs when the emitting body’s photons lose energy overcoming a gravitational field.

3.   Cosmological redshift is caused by the relativistic expansion of the universe, as quantified by the Hubble constant, H₀.

 

Total redshift effects can be represented by:

(2)

 

 

 

FIGURE 1. Wavelengths of spectral lines increase as z increases

 

The emitted spectrum in nanometers is at the bottom of the figure. The upper spectra have z values of 0.02, 0.06, and 0.25 respectively. Note that wavelengths move into the red as z increases.

 

DOPPLER REDSHIFT

In 1842, Christian Doppler pointed out that the observed wavelength of light is affected by the motion between the emitting source and the observer. We’ve all witnessed the acoustic corollary. An ambulance siren sounds at a higher pitch as it approaches us, then the pitch sounds lower as it speeds past us. In the same way, an approaching light source is seen as blueshifted as it approaches, and is redshifted when it moves away. The equation quantifying this effect when the velocity, v, is not too relativistic (less than 10% of light speed) is

 

(3)

 

Where          v is the radial velocity between the source and observer

                    c is the velocity of light, 300,000 km/s

 

For example, the hydrogen alpha line has a laboratory wavelength of 656.285 nm. If this spectral line in the light from a nearby star is 656.315 nm, we can calculate its receding speed as follows:

 

However, life gets more complicated at velocities where relativistic effects are important. Redshifts of 0.1 or greater must be handled with the following equations.

 

(4)

 

and

 

(5)

 

 

 

 

FIGURE 2. As radial velocity approaches light speed, z increases rapidly.

 

An example of equation (5) will be given later in this tutorial.

 

It should be noted that all electromagnetic radiation is affected by redshifting. Large values of z can lengthen ultraviolet light into the infrared spectrum and infrared into the radio spectrum.

 

 

GRAVITATIONAL REDSHIFT

Recall that one of the spectacular predictions of Einstein’s general relativity was that stellar light passing the edge of the Sun would be deflected 1.75 arc seconds. In addition, his theory predicted that electromagnetic radiation passing through a gravitational field would be redshifted. General relativity quantifies how mass-energy warps space time.

 

 

FIGURE 3. A rather simplistic view of how Earth’s mass might distort surrounding space time, forming a “gravitational hollow” that creates its gravitational field.

 

 

 As photons pass through a gravitational field, they lose energy, decreasing frequency and increasing wavelength. The gravitational redshift equation is

 

(6)

 

Where                   G is the gravitational constant

                              M is the mass of the body

                              c is the velocity of light

                              r is the distance from the body

 

For example, the redshift of starlight passing by the edge of the Sun would be

 

 

Massive as the Sun is, its gravitational redshift effect is quite small. More massive objects like black holes have a major redshift impact. The effect has been accurately measured on Earth (reference 1). In 1965, when gamma rays were directed upwards to a height of 22.5 meters, a gravitational redshift of 2.56x10^-15 was detected.

 

 

COSMOLOGICAL REDSHIFT

Cosmological redshifts become dominant for objects outside our Local Group of galaxies. These redshifts are a result of the stretching of space time, as postulated by general relativity, and not by radial motion. Light moving in that space will also be stretched. Its wavelength increases, and it becomes redshifted.

 

It’s important to note that stretching space time does not alter the size of stars, galaxies or even galaxy clusters. Their gravitational fields keep their dimensions unaltered.

 

Cosmological expansion occurred after the Big Bang creation of the Universe. The rapid initial expansion has slowed over time, and the current amount, the Hubble constant H₀  is 71 kilometers per second per megaparsec as determined by results from WMAP (Wilkinson Microwave Anisotropy Probe). When Hubble announced his results in 1929, he calculated a constant of almost 500 km/s/Mpc because his estimates for  24 galaxy distances were too small.

 

 

 

FIGURE 4. Hubble’s original graph.

 

In the seven decades since Hubble’s discovery, estimates for H₀ varied from 50 to 100 km/s/Mpc. However, WMAP’s 71 km/s/Mpc  seem to be universally accepted as the best estimate.

 

WMAP carried out extensive measurements of the 2.725⁰ K CMB (Cosmic Microwave Background) (reference (2)). WMAP astrophysicists estimate the Big Bang occurred 13.7 billion years ago, which ties in nicely with estimates of oldest known stars in globular clusters. They also estimate that the CMB’s redshift is 1089. We can use equation (5) to estimate the CMB’s velocity:

 

 

Or, the CMB is receding from us at 0.9999983 the speed of light.

 

 

SOME COMMENTS

Astronomical redshifts can have complex mechanisms ranging from the familiar Doppler to the general relativity-based gravitational and cosmological redshifts. In general, the higher z redshifts are due to the cosmological space expansion effect. The most distant galaxy yet detected, in March 2004, has a z of 10. This implies a light travel time of 13.18 billion years (reference (3).

 

From equation (1), wavelengths increase by a factor of z+1. Since photon energy varies as 1/wavelength, redshifted radiation has reduced energy by a factor of z+1. So distant stellar objects are not only faint due to their distance, but their observed light loses energy by a z+1 factor making the task of recording their spectra very difficult.

 

 

REFERENCES

 

(1) Spacetime Physics  p.272 by Taylor & Wheeler

 

(2) http://lambda.gsfc.nasa.gov/product/map/pub_papers/firstyear/basic/wmap_basic_results.pdf

 

(3) http://www.astro.ucla.edu/~wright/cosmolog.htm