5-9 The wavelength of a spectral line is affected by the relative motion between the source and the observer
In addition to telling us about temperature and chemical composition, the spectrum of a planet, star, or galaxy can also reveal something about that object’s motion through space. This idea dates from 1842, when Christian Doppler, a professor of mathematics in Prague, pointed out that the observed wavelength of light must be affected by motion.
The Doppler Effect
In Figure 5-26 a light source is moving from right to left; the circles represent the crests of waves emitted from the moving source at various positions. Each successive wave crest is emitted from a position slightly closer to the observer on the left, so she sees a shorter wavelength—the distance from one crest to the next—than she would if the source were stationary. What is the effect on a full spectrum of light? All the lines in the spectrum of an approaching source are shifted toward the short-wavelength (blue) end of the spectrum. This phenomenon is called a blueshift.
The Doppler effect makes it possible to tell whether astronomical objects are moving toward us or away from us
The source is receding from the observer on the right in Figure 5-26. The wave crests that reach him are stretched apart, so that he sees a longer wavelength than he would if the source were stationary. All the lines in the spectrum of a receding source are shifted toward the longer-wavelength (red) end of the spectrum, producing a redshift. In general, the effect of relative motion on wavelength is called the Doppler effect. Police radar guns use the Doppler effect to check for cars exceeding the speed limit: The radar gun sends a radio wave toward the car, and measures the wavelength shift of the reflected wave. During reflection of the wave, the car acts like a moving source so the Doppler shift measures the speed of the car.
ANALOGY
You have probably noticed a similar Doppler effect for sound waves. When a police car is approaching, the sound waves from its siren have a shorter wavelength and higher frequency than if the siren were at rest, and hence you hear a higher pitch. After the police car passes you and is moving away, you hear a lower pitch from the siren because the sound waves have a longer wavelength and a lower frequency.
Suppose that λ0 is the wavelength of a particular spectral line from a light source that is not moving. It is the wavelength that you might look up in a textbook or determine in a laboratory experiment for this spectral line. If the source is moving, this particular spectral line is shifted to a different wavelength λ. The size of the wavelength shift is usually written as Δλ, where Δλ = λ − λ0. Thus, Δλ is the difference between the wavelength listed in textbooks and the wavelength that you actually observe in the spectrum of a moving star or galaxy.
Doppler proved that the wavelength shift (Δλ) is governed by the following simple equation:
Doppler shift equation
CAUTION!
The capital Greek letter Δ (delta) is commonly used as a symbol to denote change in the value of a quantity. Thus, Δλ is the change in the wavelength λ due to the Doppler effect. It is not equal to a quantity Δ multiplied by a second quantity λ.
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Interpreting the Doppler Effect
The velocity determined from the Doppler effect is called radial velocity, because v is the component of the star’s motion parallel to our line of sight, or along a “radius” drawn from Earth to the star. Of course, a sizable fraction of a star’s motion might be perpendicular to our line of sight, but this sideways or transverse movement across the sky does not cause Doppler shifts. Box 5-6 includes two examples of calculations with radial velocity using the Doppler formula.
CAUTION!
The redshifts and blueshifts of stars visible to the naked eye, or even through a small telescope, are only a small fraction of a nanometer. These tiny wavelength changes are far too small to detect visually. (Astronomers were able to detect the tiny Doppler shifts of starlight only after they had developed highly sensitive equipment for measuring wavelengths. This was done around 1890, a half-century after Doppler’s original proposal.) So, if you see a star with a red color, it means that the star really is red; it does not mean that it is moving rapidly away from us.
TOOLS OF THE ASTRONOMER’S TRADE
Applications of the Doppler Effect
Doppler’s formula relates the radial velocity of an astronomical object to the wavelength shift of its spectral lines. Here are two examples that show how to use this remarkably powerful formula.
EXAMPLE: As measured in the laboratory, the prominent Hα spectral line of hydrogen has a wavelength λ0 = 656.285 nm. But in the spectrum of the star Vega (Figure 5-21), this line has a wavelength λ = 656.255 nm. What can we conclude about the motion of Vega?
Situation: Our goal is to use the ideas of the Doppler effect to find the velocity of Vega toward or away from Earth.
Tools: We use the Doppler shift formula, Δλ/λ0 = v/c, to determine Vega’s velocity v.
Answer: The wavelength shift is
Δλ = λ − λ0 = 656.255 nm − 656.285 nm = −0.030 nm
The negative value means that we see the light from Vega shifted to shorter wavelengths—that is, there is a blueshift. (Note that the shift is very tiny and can be measured only using specialized equipment.) From the Doppler shift formula, the star’s radial velocity is
Review: The minus sign indicates that Vega is coming toward us at 14 km/s. The star might also have some motion perpendicular to the line from Earth to Vega, but such motion produces no Doppler shift.
By plotting the motions of stars such as Vega toward and away from us, astronomers have been able to learn how the Milky Way Galaxy (of which our Sun is a part) is rotating. From this knowledge, and aided by Newton’s universal law of gravitation (see Section 4-6), they have made the surprising discovery that the Milky Way contains roughly 10 times more matter than had once been thought! The nature of this unseen dark matter is still one of the great unsolved mysteries in astronomy.
EXAMPLE: In the radio region of the electromagnetic spectrum, hydrogen atoms emit and absorb photons with a wavelength of 21.12 cm, giving rise to a spectral feature commonly called the 21-centimeter line. The galaxy NGC 3840 in the constellation Leo (the Lion) is receding from us at a speed of 7370 km/s, or about 2.5% of the speed of light. At what wavelength do we expect to detect the 21-cm line from this galaxy?
Situation: Given the velocity of NGC 3840 away from us, our goal is to find the wavelength as measured on Earth of the 21-centimeter line from this galaxy.
Tools: We use the Doppler shift formula to calculate the wavelength shift Δλ, then use this to find the wavelength λ measured on Earth.
Answer: The wavelength shift is
Therefore, we will detect the 21-cm line of hydrogen from this galaxy at a wavelength of
λ = λ0 + Δλ = 21.12 cm + 0.52 cm = 21.64 cm
Review: The 21-cm line has been redshifted to a longer wavelength because the galaxy is receding from us. In fact, most galaxies are receding from us. This observation is one of the key pieces of evidence that the universe is expanding, and has been doing so since the Big Bang that took place almost 14 billion years ago.
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The Doppler effect is an important tool in astronomy because it uncovers basic information about the motion of planets, stars, and galaxies. For example, the rotation of the planet Venus was deduced from the Doppler shift of radar waves reflected from its surface. Small Doppler shifts in the spectrum of sunlight have shown that the entire Sun is vibrating like an immense gong. The back-and-forth Doppler shifting of the spectral lines of certain stars reveals that these stars are being orbited by unseen companions; from this astronomers have discovered planets around other stars and massive objects that may be black holes. Astronomers also use the Doppler effect along with Kepler’s third law to measure a galaxy’s mass. These are but a few examples of how Doppler’s discovery has empowered astronomers in their quest to understand the universe.
In this chapter we have glimpsed how much can be learned by analyzing light from the heavens. To analyze this light, however, it is first necessary to collect as much of it as possible, because most light sources in space are very dim. Collecting the faint light from distant objects is the key purpose of telescopes. In the next chapter we will describe both how telescopes work and how they are used.
CONCEPT CHECK 5-14
How is the spectrum changed when looking at the absorption spectrum from an approaching star as compared to how the star’s spectrum would look if the star were stationary?
CALCULATION CHECK 5-3
How fast and in what direction is a star moving if it has a line that shifts from 486.2 nm to 486.3 nm?