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For v V c, 3 is close to zero, and vapp = v sin 0, the expected result. However, for v close to c, vapp can be greater than v. In fact, vapp can be so much greater than v that it exceeds c.

To see this, we can find the angle that gives the maximum vapp for a given v. Taking equation (19.7), dividing by v, differentiating the result with respect to 0, and setting the result equal to zero, gives

This is just the quantity y that appears in the Lorentz transformations. We know that this quantity can become quite large as v approaches c.

Example 19.2 Superluminal expansion For an object moving away from the nucleus of a galaxy at v = 0.95 c, find the maximum value of vapp and the angle at which it must be moving to reach this maximum.

solution We have cos 0 = 3

so 0 = 18.2°. From equation (19.8) we have 0.95c v =-

There is a way to test this explanation. For a given speed and direction of motion, there should be a specific Doppler shift for radiation from the moving object. Unfortunately, these radio sources do not have any lines in their spectra. The Doppler shift alters the shape of the synchrotron spectrum, but the interpretation is difficult. Studies of the spectra of these sources are continuing.

19.3 I Seyfert galaxies

Seyfert galaxies are characterized by having nuclei that strongly dominate the total light from the galaxy. On a short exposure, they look like stars with a fuzzy patch around them and like a spiral galaxy on longer exposures (Fig. 19.14). For comparison, normal spirals never look like a star, even on a short exposure. This suggests that we are not resolving the nuclei of Seyferts in most of our images of them. (There are a few Seyferts for which the source of narrow emission lines has been resolved.) When we can study the fuzzy patch around the nucleus, it seems like the max