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# 10.9 The Doppler Effect

In 1842, Austrian physicist Christian Doppler realized that we measure waves to have a wave length that depends on the motion of the source of waves. When waves leave a moving source they get "bunched up" in the direction of motion and "stretched out" in the direction away from the motion. Imagine stones being dropped into a pond at regular intervals. The ripples will spread out in concentric circles and the wavelength — the distance between successive crests or troughs — will be the same in every direction. If the source of waves is moving, the stones drop in at a different position each time and the waves are off-centered as they travel out.

Christian Doppler. Click here for original source URL

Plot of a sine wave, showing three pairs of corresponding points between which wavelength (lambda) can be measured. Click here for original source URL

If a source of waves is approaching you, there is a smaller distance between successive ripples; the wavelength is shorter. If the source is moving away from you, the wavelength is longer. Viewed from any direction between these special cases, the wavelength varies smoothly between the maximum compression of waves and the maximum stretching out of waves. Another special situation occurs when the view is at 90° (or transverse) to the direction of source motion. The wavelength is unaltered, regardless of how fast the source of waves is moving. This wavelength shift is called the Doppler effect.

Diagram of the Doppler effect. Click here for original source URL.

The Doppler effect applies to any source of waves. The waves can be ripples on the surface of a pond or sound waves traveling through the air. Since light is also a wave, a moving light source will experience the Doppler effect. Light emitted from a source moving toward us experiences a blue shift. Light emitted from a source moving away from us experiences a red shift. As a mnemonic aid, remember that recession produces red shifts.

The size of the Doppler effect is given by the simple formula

Δ λ / λ = v / c

Here, Δ λ is the small change or shift in wavelength and λ is the normal wavelength when the source is not moving. The fractional (or percentage) wavelength shift is equal to the velocity of the source as a fraction of the velocity of light.

In Doppler’s time, this equation was derived using a sound experiment. Scientists hired a brass band and got the trumpeters to belt out a single sustained note from an open railroad car as it passed by at a known speed. The scientists then measured the change in pitch of the note as the train approached and receded, compared with the note of a stationary musician. By repeating the experiment with different train speeds, they deduced the relationship given above.

Here is an example of the Doppler effect using sound waves. In this case, c is not the speed of light but the speed of sound — 330 meters per second. How fast would a police car have to be moving toward you for the pitch of its siren to increase by 50%? (This is called a "fifth" on the musical scale, or a change in the note from C to G, for example.) Increasing the pitch by a factor of 1.5 means decreasing the wavelength by a factor of 1/1.5 = 0.67, so Δ λ / λ = 1/3. To make this happen a police car would have to travel at 1/3 the speed of sound, or 110 meters per second (225 mph)!

The Doppler effect for light is generally very small. Imagine a supersonic jet fighter was heading toward us at night at 1000 mph. By how much would its wing lights be blue shifted? Since 1000 mph is 0.46 kilometers per second and the speed of light is 300,000 kilometers per second, the blue shift would only be 0.46 / 300,000 = 1.5 x 10-6 or about one part in a million. You would never see such a small effect.

Doppler Shift. Click here for original source URL.

How big is the Doppler effect in the case of planet detection. As seen from a distant vantage point, the reflex motion of the Sun caused by Jupiter is 13 meters per second — slightly faster than a sprinter can run. The Doppler shift is 0.013 / 300,000 = 4.3 × 10-8 or 4 parts in 100 million. Visible light has a wavelength of about 5 × 10-5 meters or 500 nanometers (nm). If a Sun-like star were being tugged by a Jupiter-like planet, the star’s light would be shifted from 500 nm to 500.00002 nm when the star wobbled toward us, and shifted from 500 nm to 499.99998 nm when the star wobbled away from us. Planet detection is difficult because it requires a fantastically precise measurement.

The nearly circular motion of the planet creates a nearly circular reflex motion of the star. This circular motion gives a sine-wave variation of the Doppler effect. We would have to make measurements for 12 years to see a complete cycle of variation. And to be safe we would want to observe for longer than 12 years to make sure that the phenomenon repeated and was cyclic.

If we are lucky enough that the plane of the planet orbit is parallel to our line of sight, then the planet is moving directly toward us and directly away from us as it orbits the star. We see the full Doppler effect. In general this will not be true. Suppose that the plane of the planet orbit was inclined at 45º to the line of sight. By simple geometry, the component of reflex motion in our direction is cos (45º) or 0.7 times smaller at every point in the orbit. The Doppler effect that we would observe is reduced. If a planet (of any size) orbits a star perpendicular to the line of sight, we would observe no Doppler effect at all.