8.2: Doppler Effect
- Page ID
- 132324
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Measuring Radial Movement
Astronomers try to get the most information that they can from their data. This means understanding the properties of light. One property of light is that it behaves like a wave. When an object that gives off waves moves, it has an effect on those waves that can be used to measure the object's movement.
In 1842, Christian Doppler first measured the effect of motion on waves by hiring a group of musicians to play on an open railroad car as it was moving along the track. What was heard was probably similar to something you have experienced when a fire truck or ambulance has driven past you while using its siren. As the vehicle approaches, the sound from the siren becomes both louder, and higher pitched. After the vehicle passes and moves away, the siren decreases in volume and the pitch drops. The volume makes sense, but why does the pitch change? As the vehicle approaches, the sound waves coming from it pile up, which causes the increase in pitch. As the vehicle moves away the sound waves are spread out, lowering the pitch of the siren. After his experiment, Doppler applied what he learned to all waves, including light, and pointed out that if a light source is approaching or receding from the observer, the light waves will be, respectively, crowded more closely together or spread out.
The general principle, now known as the Doppler effect, is illustrated in Figure \(\PageIndex{1}\). In part (a) of the figure, the light source (S) is at rest with respect to the observer. The source gives off a series of waves, whose crests we have labeled 1, 2, 3, and 4. The light waves spread out evenly in all directions, like the ripples from a splash in a pond. The crests are separated by a distance, λ, where λ is the wavelength. The observer, who happens to be located in the direction of the bottom of the image, sees the light waves coming nice and evenly, one wavelength apart. Observers located anywhere else would see the same thing.
On the other hand, if the source of light is moving with respect to the observer, as seen in part (b), the situation is more complicated. Between the time one crest is emitted and the next one is ready to come out, the source has moved a bit, toward the bottom of the page. From the point of view of observer A, this motion of the source has decreased the distance between crests—it's squeezing the crests together, this observer might say.
In part (b), we show the situation from the perspective of three observers. The source is seen in four positions, S1, S2, S3, and S4, each corresponding to the emission of one wave crest. To observer A, the waves seem to follow one another more closely, at a decreased wavelength and thus increased frequency. (Remember, all light waves travel at the speed of light through empty space, no matter what. This means that motion cannot affect the speed, but only the wavelength and the frequency. As the wavelength decreases, the frequency must increase. If the waves are shorter, more will be able to move by during each second.)
The situation is not the same for other observers. Let's look at the situation from the point of view of observer C, located opposite observer A in the figure. For her, the source is moving away from her location. As a result, the waves are not squeezed together but instead are spread out by the motion of the source. The crests arrive with an increased wavelength and decreased frequency. To observer B, in a direction at right angles to the motion of the source, no effect is observed. The wavelength and frequency remain the same as they were in part (a) of the figure.
The illustration demonstrates that the Doppler effect is produced only by a motion toward or away from the observer, a motion called radial velocity. Sideways motion does not produce such an effect. Observers between A and B would observe some shortening of the light waves for that part of the motion of the source that is along their line of sight. Observers between B and C would observe lengthening of the light waves that are along their line of sight.
Color Shifts
When the source of waves moves toward you, the wavelength decreases a bit. If the waves involved are visible light, then the colors of the light change slightly. As wavelength decreases, they shift toward the blue end of the spectrum: astronomers call this a blueshift since the end of the spectrum is really violet, the term should probably be violetshift, but blue is a more common color. When the source moves away from you and the wavelength gets longer, we call the change in colors a redshift. Because the Doppler effect was first used with visible light in astronomy, the terms “blueshift” and “redshift” became well established. Today, astronomers use these words to describe changes in the wavelengths of radio waves or X-rays as comfortably as they use them to describe changes in visible light.
The greater the motion toward or away from us, the greater the Doppler shift. If the relative motion is entirely along the line of sight, the formula for the Doppler shift of light is
\[ \frac{\Delta \lambda}{\lambda} = \frac{v}{c} \nonumber\]
where λ is the wavelength emitted by the source, Δλ is the difference between λ and the wavelength measured by the observer, c is the speed of light, and v is the relative speed of the observer and the source in the line of sight. The variable v is counted as positive if the velocity is one of recession, and negative if it is one of approach. Solving this equation for the velocity, we find v = c × Δλ/λ.
If a star approaches or recedes from us, the wavelengths of light in its continuous spectrum appear shortened or lengthened, respectively, as do those of the dark lines. However, unless its speed is tens of thousands of kilometers per second, the star does not appear noticeably bluer or redder than normal. The Doppler shift is thus not easily detected in a continuous spectrum and cannot be measured accurately in such a spectrum. The wavelengths of the absorption lines can be measured accurately, however, and their Doppler shift is relatively simple to detect.
Examples and Exercises
We can use the Doppler effect equation to calculate the radial velocity of an object if we know three things: the speed of light, the original (unshifted) wavelength of the light emitted, and the difference between the wavelength of the emitted light and the wavelength we observe.
If the light from a gas cloud should be 656.3 nm but we observe the wavelength of 656.6 nm, how fast is this gas cloud moving toward or away from Earth?
Solution
Because the light is shifted to a longer wavelength (redshifted), we know this gas cloud is moving away from us. The speed can be calculated using the Doppler shift formula:
\[ \nu = c \times \frac{\Delta \lambda}{\lambda} = (3.0 \times 10^8 \text{ m/s}) \left( \frac{0.3 \text{ nm}}{656.3 \text{ nm}} \right) = (3.0 \times 10^8 \text{ m/s}) \left( \frac{0.3 \times 10^{-9} \text{ m}}{656.3 \times 10^{-9} \text{ m}} \right) = 140,000 \text{ m/s} = 140 \text{ km/s} \nonumber\]
Suppose the light of a star should be 500 nm, but is observed to be at 500.1 nm. How fast is the star moving toward or away from Earth?
- Answer
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Because the light is shifted to a longer wavelength, the star is moving away from us:
\( \nu = c \times \frac{\Delta \lambda}{\lambda} = (3.0 \times 10^8 \text{ m/s}) \left( \frac{0.1 \text{ nm}}{500 \text{ nm}} \right) = (3.0 \times 10^8 \text{ m/s}) \left( \frac{0.1 \times 10^{-9} \text{ m}}{500 \times 10^{-9} \text{ m}} \right) = 60,000 \text{ m/s} \) Its speed is 60,000 m/s.
- Use this Doppler Effect simulator to experiment with different objects, speeds, and distances.