## Electromagnetic radiation, the Doppler effect, exoplanets and cosmology.

This page is a support page to the multimedia chapter The Doppler Effect in the volume Waves and Sound. It gives background information and further details.

#### The Doppler effect in electromagnetic radiation

In the multimedia tutorial and in the first supporting page, we concentrated on the Doppler effect in sound and also used water waves as an example. For electromagnetic radiation – light, radio, gamma rays etc, the same principles apply and we can often use the equations derived there for the frequency f ' observed when a source emits a frequency f:

where v is the velocity of the wave, vs is the velocity of the source, vo that of the observer, and where vs and vo are taken as positive for approaching and negative for receding.

For electromagnetic radiation, v = c is the speed of light, which is about 3 x 108 m.s−1. Consequently, the proportional changes in frequency and wavelength are very small for ordinary speeds vs and vo. However, frequency is one of the quantities that can be measured with very high precision, so the radar guns used by highway patrols (picture at right) are commonly capable of measuring frequency shifts of less than 1 part in 108, and so can accurately measure the speed of vehicles travelling at a few tens of  m.s−1.

#### Hunting exoplanets with the Doppler effect

Blue light has shorter wavelength (higher frequency) than red, so all features in the optical spectra of objects moving away from us are shifted towards the red end (a red shift). Objects moving towards us (much rarer in astronomy) have spectra that are blue shifted. In some but by no means all cases, the simple Doppler effect explains the shift.

The graph below was provided by Chris Tinney, who hunts exoplanets. It shows that the velocity of the star epsilon Reticuli has a component towards us that varies with a period of 439 earth days and a magnitude of about 40 m.s−1. In the animation (not to scale, of course) the red and blue shifts are indicated by red and blue waves. The motion of this star is due to the gravitational effect of a planet orbiting the star. Unlike the star, which shines, the planet is not visible: it is too far away to be seen in reflected light. However, we have made it visible in our animation. Although the planet's year is comparable with that of the earth, its mass is very much greater: for the moment, planets as small as ours have not been detected using this technique.

Most of the planets thus discovered are 'hot Jupiters': giant planets (big enough to move their star measurably) orbiting close to their star (which makes the effect both stronger and faster). Chris points out that most of the doppler shifts are much more asymmetrical than the one shown here, meaning that the orbits are usually highly elliptical.

#### Cosmological red shifts

Vesto Slipher observed in 1917 that the spectra of many of the distant objects in the universe are red-shifted. Edwin Hubble later interpreted this as being due to their receding from us, with speeds that increase with distance from us. With the exception of a few galaxies near to ours, such as the Clouds of Magellan, which are gravitationally bound, virtually all galaxies are receding from us and their light is redshifted (its wavelength has been increased) by a factor which astronomers and cosmologists call z.

The red shifts observed by astronomers come not only from the Doppler effect, but from at least three other effects. In the case of these galactic red shifts, the increased wavelength comes from a different effect. The universe and space itself are expanding. The further away a galaxy is, the longer its light has taken to reach us, and so the more its wavelength has been stretched over that time (the greater the z).

The animations above should be treated as cartoons. The one on the left shows a wave in a hypothetical, two dimensional universe that is spherical in three-space. As that universe expands, the length of the wave increases. The animation at right shows a few galaxies in the same unrealistic universe. The further apart they are, the longer the time light has taken to reach us, and so the more time the waves have had to expand, along with the space through which they have travelled.

If you have been wondering 'Why are all the galaxies going away from us? Is it something we said?', then the cartoon on the right answers your question. There is nothing special about our position: as space expands, everything in the universe recedes from everything else. If you inflate a balloon, then from any point on the balloon it appears that all other points are receding at a rate that increases with current distance of separation.

The spectra below, data kindly provided by Xiaohui Fan, show the same feature in a set of spectra from quasars with z values from 5.74 (bottom) to 6.42 (top). For all of these spectra, the wavelengths have been stretched by a factor of around six over the time it took their light to reach us. They are so distant that the time one calculates for the light to to reach us depends on which cosmological model one uses in the calculation. However, in all models it is of the order of 10 billion years.

There is a brief discussion of cosmological acoustics here on our acoustics FAQ.

#### Relativistic red shifts

There are two further causes of red shift in astronomical objects. First, objects that are moving with respect to an observer exhibit time dilation due to relative motion. Time passes more slowly on those objects: their clocks tick more slowly and their atoms emit radiation with lower frequency.

Consider for example an object whose motion is at right angles to the line separating us. It is neither approaching us nor receding from us. So there is no Doppler shift (in the ordinary sense of the word). It does, however, have relative motion and so exhibits time dilation. So it shows a relativistic red shift.

Another red shift comes from general relativity: Einstein's theory of gravitation. Objects at low gravitational potential exhibit time dilation, too. Near the surface of a star, the gravitational potential is lower than ours so, for massive stars with intense gravitational fields, there is a gravitational red shift. The light from the sun is gravitationally red-shifted (gravitational potential at the surface of the sun is lower than at the surface of the earth).

This discussion has taken us some way from our main topics. For readers wishing to go further, we provide an introduction to relativity called Einsteinlight.