## Doppler shift

Moving Observer

Moving Source

Higher v (shorter X

Doppler shift for waves in an elastic medium, such as sound waves. (a) Moving observer. On the left the observer is moving toward the source, encountering wave crests more frequently than for a stationary observer.The frequency appears to increase (and the wavelength to decrease). On the right, the observer is moving away from the source, encountering waves at a lower frequency, corresponding to a lower frequency, corresponding to a longer wavelength. (b) Moving source.The motion of the source distorts the wave pattern, so the circles are no longer concen-tric.The observer on the left has the source approaching, producing a shorter wavelength (and a higher frequency). The observer on the right has the source receding, producing a longer wavelength (and a lower frequency).

Higher v (shorter X

shorter X Longer X) (Higher v)

shorter X Longer X) (Higher v)

Longer X (Lower v)

Doppler shift for waves in an elastic medium, such as sound waves. (a) Moving observer. On the left the observer is moving toward the source, encountering wave crests more frequently than for a stationary observer.The frequency appears to increase (and the wavelength to decrease). On the right, the observer is moving away from the source, encountering waves at a lower frequency, corresponding to a lower frequency, corresponding to a longer wavelength. (b) Moving source.The motion of the source distorts the wave pattern, so the circles are no longer concen-tric.The observer on the left has the source approaching, producing a shorter wavelength (and a higher frequency). The observer on the right has the source receding, producing a longer wavelength (and a lower frequency).

It is possible for both the source and the observer to be moving. If their combined motion brings them closer together, the wavelength will decrease and the frequency will increase. If their combined motion makes them move farther apart, the wavelength will increase, and the frequency will decrease. If there is no instantaneous change in their separation, there is no shift in wavelength or frequency.

The shift only depends on the component of the relative velocity along the line joining the source and observer, since this is the only component that can change the distance, r, between them. We call this component the radial velocity (Fig. 5.4). We refer to the line joining the source and observer as the line of sight. From our definition of radial velocity, vr, we can see that it is given by vr = dr/dt

Note that if the source and observer are moving apart, r is increasing, and vr > 0. If the source and observer are moving together, r is decreasing, and vr < 0.

Longer X (Lower v)

Suppose the source is moving with a speed vs in a direction that makes an angle 0 with the line of sight, and the observer is moving with a speed vo in a direction making an angle fi with the line of sight. Taking the components of the two velocities along the line of sight as vs cos 0 and vo cos fi, and subtracting the get the relative radial velocity, gives

In astronomy we are interested in the Doppler shift for electromagnetic waves. The underlying physics is a little different, because there is no mechanical medium for these waves to move through. They can travel even in a vacuum. (We will discuss this point further in Chapter 7.) For sound waves, the actual amount of Doppler shift depends on whether the source or observer (or vs cos 8

Source vo cos 9 Observer

Radial velocity.The horizontal line is the line of sight between the source and observer.The radial velocity is the difference between the line of sight components of the observer and source velocities.

vr = vs cos both) is moving. For electromagnetic waves, only the relative motion counts.

As long as the relative speed of the source and the observer is much less than the speed of light, the results for electromagnetic radiation are relatively simple. If A is the wavelength at which a signal is received, and A0 is the wavelength at which it was emitted, called the rest wavelength, the wavelength shift AA is defined by

The simple result is that the wavelength shift, expressed as a fraction of the original wavelength, is equal to the radial velocity, expressed as a fraction of the speed of light. That is

If vr > 0, then A A > 0. For a spectral line in the middle of the visible part of the spectrum, a shift to longer wavelength is a shift to the red, so this is called a redshift. The name applies even if we are in other parts of the spectrum. A positive radial velocity always produces a redshift. If vr < 0, then AA < 0 , and we have a blueshift.

We now look at what happens to the frequency. We remember that A = c/v, so dA/dv = —c/v2

This means that Av = (into equation (5.4) gives

v/A ) AA. Substituting

We now add this to the original wavelength to find the observed wavelength:

(b) If we take the negative of vr, we just get the negative of AA, so

This gives a wavelength of

If we observe two spectral lines, their wavelength shifts will be different, since each is shifted by an amount proportional to its own rest wavelength. Thus, the spacing between spectral lines will be shifted.

### 5.2.2 Circular orbits

We now look at Doppler shifts produced by a star in a circular orbit. The orbital speed is v, and the radius is r. The angular speed of the star in its orbit (in radians per second) is given by o> = v/r. The situation is shown in Fig. 5.5. Suppose the star is moving directly away from the observer at time t = 0. At that instant the radial velocity vr = v. As the stars moves, the component of its v(t = n/2œ) M

The shift in frequency, expressed as a fraction of the rest frequency, is the negative of the radial velocity, expressed as a fraction of the speed of light. (So Av/v0 = —AA/A0.)

Example 5.1 Doppler shift

The Ha line has a rest wavelength, A0 = 656.28 nm. What is the observed wavelength for a radial velocity (a) vr = 10 km/s, and (b) vr = —10 km/s?

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