A distant galactic nucleus ejects a jet of material at right angles (left(90^{circ} ight)) to our

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A distant galactic nucleus ejects a jet of material at right angles \(\left(90^{\circ} \right)\) to our line of sight. We know the distance of the galaxy from the redshift of its spectral lines, so we can calculate how far the jet has traveled in a given time using the very small but growing angle between the galactic nucleus and jet as observed through our telescope. From this information we can find the velocity of the jet. Note that for such transverse motion it takes essentially the same time for light from the jet to reach us from the end of its journey as it does from its beginning, because it stays essentially the same distance from us throughout. But now suppose the jet is ejected at some angle \(\theta\) relative to our line of sight, so the jet's transverse velocity component is \(v_{\perp}=v \sin \theta\) and its velocity component towards us is \(v_{\|}=v \cos \theta\). And because the jet is getting closer to us, the time it takes light to reach us from it becomes smaller and smaller.

(a) In this case find an equation for the jet's apparent transverse velocity in the sky, defined as the transverse distance moved divided by the time interval as observed on earth, and show that this apparent velocity \(v_{\text {app }}\) can exceed the speed of light.

(b) For a given actual velocity \(v\), find the angle \(\theta\) that maximizes \(v_{\text {app }}\), and then find the magnitude of \(v_{\text {app }}\) in this case.

(c) Evaluate such a maximal \(v_{\text {app }}\) for the case \(v=0.99 c\). Such apparent superluminal velocities have often been observed by astronomers, even though no matter actually travels faster than light.

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NEETA SINGH

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