# 2.7 The Doppler Effect 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. Applied to cosmic sources of electromagnetic radiation, it has become one of the most important observational tools in all of twentieth-century astronomy. Here’s how it works.

Imagine a wave moving from the place where it is generated toward an observer who is not moving with respect to the wave source (Figure 2.22a). By noting the distances between successive wave crests, the observer can determine the wavelength of the emitted wave. Now suppose that the wave source begins to move (Figure 2.22b). Because the source moves between the times of emission of one wave crest and the next, wave crests in the direction of motion of the source are seen to be closer together than normal, while crests behind the source are more widely spaced. Thus, an observer in front of the source measures a shorter wavelength than normal, while one behind sees a longer wavelength. Figure 2.22 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.

The greater the relative speed of source and observer, the greater the observed shift. 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) by  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, we can assume that the recession velocity is small compared to the speed of light. Only when we come to discuss the properties of black holes (Chapter 13) and the structure of the universe on the largest scales (Chapter 17) will we have to reconsider this formula.

Note that, in the figure, the source is shown in motion. However, the same general statements hold whenever there is any relative motion between source and observer. Note also that only motion along the line joining source and observer, known as radial motion, appears in the above equation. Motion transverse (perpendicular) to the line of sight has no significant effect.

In astronomical parlance, the 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 holds 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.

The Doppler effect is not normally noticeable in visible light on Earth—the speed of light is so large that the wavelength change is far too small to be noticeable for everyday terrestrial velocities. However, using spectroscopic techniques, astronomers routinely use the Doppler effect to measure the line-of-sight velocity of cosmic objects by determining the extent to which known spectral lines are shifted to longer or shorter wavelengths. For example, suppose that an astronomer observes the red line in the spectrum of a star to have a wavelength of 657 nm, instead of the 656.3 nm measured in the lab. (How does she know it is the same line? Because, as illustrated in Figure 2.23, she realizes that all the hydrogen lines are shifted by the same fractional amount—the characteristic pattern of lines identifies hydrogen as the source.) Using the above equation, she calculates that the star’s radial velocity is times the speed of light. In other words, the star is receding from Earth at a rate of 320 km/s. The Doppler Effect The Doppler Effect Figure 2.23 Doppler Shift Because of the Doppler effect, the entire spectrum of a moving object is shifted (in this case toward longer wavelengths, corresponding to motion away from the observer), allowing specific spectral lines to be recognized and their shifts measured. The spectrum at top shows the slight redshift of the hydrogen lines from an object moving at a speed of 320 km/s away from the observer. The spectrum at bottom shows the unshifted spectrum of the object at rest.

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. CONCEPT CHECK

How might the Doppler effect be used in determining the mass of a distant star? 