3.5 The Doppler Effect

The Doppler Effect

Imagine a rocket ship launched from Earth with enough fuel to allow it to accelerate to speeds approaching that of light. As the ship’s speed increased, a remarkable thing would happen (Figure 3.14). Passengers would notice that the light from the star system toward which they were traveling seemed to be getting bluer. In fact, all stars in front of the ship would appear bluer than normal, and the greater the ship’s speed, the greater the color change would be. Furthermore, stars behind the vessel would seem redder than normal, while stars to either side would be unchanged in appearance. As the spacecraft slowed down and came to rest relative to Earth, all stars would resume their usual appearance. The travelers would have to conclude that the stars had changed their colors not because of any real change in their physical properties but because of the spacecraft’s own motion.

Figure 3.13 Astronomical Thermometer Comparison of blackbody curves for four cosmic objects. The frequencies and wavelengths corresponding to peak emission are marked. (a) A cool, invisible galactic gas cloud called Rho Ophiuchi. At a temperature of 60 K, it emits mostly low-frequency radio radiation. (b) A dim, young star (shown red in the inset photograph) near the center of the Orion Nebula. The star’s atmosphere, at 600 K, radiates primarily in the infrared. (c) The Sun’s surface, at approximately 6000 K, is brightest in the visible region of the electromagnetic spectrum. (d) Some very bright stars in a cluster called Omega Centauri, as observed by a telescope aboard a space shuttle. At a temperature of 60,000 K, these stars radiate strongly in the ultraviolet. (Harvard College Observatory; J. Moran; AURA; NASA)

This phenomenon is not restricted to electromagnetic radiation and fast-moving spacecraft. Waiting at a railroad crossing for an express train to pass, most of us have had the experience of hearing the pitch of a train whistle change from high shrill (high frequency, short wavelength) to low blare (low frequency, long wavelength) as the train approaches and then recedes. This motion-induced change in the observed frequency of a wave is known as the Doppler effect, in honor of Christian Doppler, the nineteenth-century Austrian physicist who first explained it in 1842. Applied to cosmic sources of electromagnetic radiation, it has become one of the most important measurement techniques in all of modern astronomy. Here’s how it works:

Figure 3.14 High-Speed Observers Observers in a fast-moving spacecraft will see the stars ahead of them seem bluer than normal, while those behind are reddened. The stars have not changed their properties—the color changes are the result of the observers’ motion relative to the stars.

Imagine a wave moving from the place where it is created toward an observer who is not moving with respect to the wave source, as shown in Figure 3.15(a). By noting the distances between successive wave crests, the observer can determine the wavelength of the emitted wave. Now suppose that the wave source is moving. As illustrated in Figure 3.15(b), because the source moves between the times of emission of one wave crest and the next, successive wave crests in the direction of motion of the source will be seen to be closer together than normal, whereas crests behind the source will be more widely spaced. An observer in front of the source will therefore measure a shorter wavelength than normal, while one behind will see a longer wavelength. (The numbers indicate successive wave crests emitted by the source and the location of the source at the instant each wave crest was emitted.)

Figure 3.15 Doppler Effect (a) Wave motion from a source toward an observer at rest with respect to the source. The four numbered circles represent successive wave crests emitted by the source. At the instant shown, the fifth wave crest is just about to be emitted. As seen by the observer, the source is not moving, so the wave crests are just concentric spheres (shown here as circles). (b) Waves from a moving source tend to "pile up" in the direction of motion and be "stretched out" on the other side. (The numbered points indicate the location of the source at the instant each wave crest was emitted.) As a result, an observer situated in front of the source measures a shorter-than-normal wavelength—a blueshift—while an observer behind the source sees a redshift. In this diagram the source is shown in motion. However, the same general statements hold whenever there is any relative motion between source and observer.

The greater the relative speed of source and the observer, the greater the observed shift. If the other velocities involved are not too large compared to the wave speed—less than a few percent, say—we can write down a particularly simple formula for what the observer sees. In terms of the net velocity of recession between source and observer, the apparent wavelength and frequency (measured by the observer) are related to the true quantities (emitted by the source) as follows:

A positive recession velocity means that the source and the observer are moving apart; a negative value means that they are approaching. The wave speed is the speed of light c in the case of electromagnetic radiation. For most of this text, the assumption that the recession velocity is small compared to the speed of light will be a good one. Only when we discuss the properties of black holes (Chapter 22) and the structure of the universe on the largest scales (Chapters 25 and 26) will we have to reconsider this formula.

Note that in Figure 3.15 the source is shown in motion (as in our train analogy), whereas in our earlier spaceship example (Figure 3.14) the observers were in motion. For electromagnetic radiation, the result is the same in either case—only the relative motion of source and observer matters. Note also that only motion along the line joining source and observer—known as radialmotion—appears in the above equation. Motion transverse (perpendicular) to the line of sight has no significant effect.*

A wave measured by an observer situated in front of a moving source is said to be blueshifted,because blue light has a shorter wavelength than red light. Similarly, an observer situated behind the source will measure a longer-than-normal wavelength—the radiation is said to be redshifted. This terminology is used even for invisible radiation, for which "red" and "blue" have no meaning. Any shift toward shorter wavelengths is called a blueshift, and any shift toward longer wavelengths is called a redshift. For example, ultraviolet radiation might be blueshifted into the X-ray part of the spectrum or redshifted into the visible; infrared radiation could be redshifted into the microwave range, and so on.

Because c is so large—300,000 km/s—the Doppler effect is extremely small for everyday terrestrial velocities. For example, consider a source receding from the observer at Earth’s orbital speed of 30 km/s, a velocity much greater than any encountered in day-to-day life. A beam of blue light would be shifted by only 30 km/s/300,000 km/s = 0.01 percent, from 400 nm to 400.04 nm—a very small change indeed, and one that the human eye cannot distinguish. (It is easily detectable with modern instruments, though.)

The importance of the Doppler effect to astronomers is that it allows them to determine the speed of any cosmic object along the line of sight simply by determining the extent to which its light is redshifted or blueshifted. Suppose that the beam of blue light just mentioned is observed to have a wavelength of 399 nm instead of the 400 nm with which it was emitted. (Let’s defer to the next chapter the question of how an observer might know the wavelength of the emitted light.) Using the above equation, the observer could calculate the source’s radial velocity to be 399/400 – 1 = – 0.0025 times the speed of light. In other words, the source is approaching the observer at a speed of 0.0025 c, or 750 km/s. The basic reasoning is simple but very powerful. The motions of nearby stars and distant galaxies—even the expansion of the universe itself—have all been measured in this way.

Motorists stopped for speeding on the highway have experienced another, much more down-to-earth, application. Police radar measures speed by means of the Doppler effect, as do the radar guns used to clock the velocity of a pitcher’s fastball or a tennis player’s serve. Notice, incidentally, that the Doppler effect depends only on the relative motion of source and observer; it does not depend on distance in any way.

In practice, it is hard to measure the Doppler shift of an entire blackbody curve, simply because it is spread over many wavelengths, making small shifts hard to determine with any accuracy. However, if the radiation were more narrowly defined and took up just a narrow "sliver" of the spectrum, then precise measurements of Doppler effect could be made. We will see in the next chapter that in many circumstances this is precisely what does happen, making the Doppler effect one of the observational astronomer’s most powerful tools.

*In fact, Einstein’s theory of relativity (see Chapter 22) implies that when the transverse velocity is comparable to the speed of light, a wavelength change, called the transverse Doppler shift, does occur. For most terrestrial and astronomical applications, however, this shift is negligibly small, and we will ignore it here.

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