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The time for the object to move the distance r is t = r/v (19.5)

However, a light or radio wave emitted from P has to travel a shorter distance than one emitted from O before reaching the observer. The path from P to the observer is shorter than the path from O to the observer by a distance x. This means that a light wave emitted from P takes x/c less time to reach us than one emitted at O. Therefore, from the point of view of the observer, the apparent time, from O to P is ta for the object to travel v (dt)

Observer

be detected as the proper motion. The distances x and y traveled along these two directions are

### Fig 19.12.

Superluminal expansion. (a) VLBA image of the radio jet 3C279, which shows apparent superluminal flows. Superluminal motion is shown in a 'movie' mosaic of five radio images made over seven years.The stationary core is the bright red spot to the left of each image.The observed location of the rightmost blue/green blob moved about 25 light years from 1991 to 1998, hence the changes appear to an observer to be faster than the speed of light, or 'superluminal'.The blue/green blob is part of a jet pointing within 2° to our line of sight, and moving at a speed of 0.997 times the speed of light. (b)The geometry of the problem.We measure a change in angular position d0 and relate that to a tangential velocity v by knowing the distance d. [(a) Ann Wehrle (Caltech/NASA/Wehrle, A. etal., Astrophys.J. Suppl., 133,297,2001]

Substituting from equations (19.4) and (19.5), (19.4) we have tapp = (r/v) - (r/c) cos 0 = (r/v) (1 - ß cos 0)

where we have set p = v/c. The apparent velocity across the sky, vapp, is then

Vapp t lapp r sin 0

Eliminating r gives v sin 0

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