The Relativistic Lighthouse · AI Generator #GEN-533AA8CD

Physics

Problem

A neutron star rotates with period

T = 0.033\;\text{s}

, emitting a narrow beam of radio waves from a hot spot on its surface. The beam sweeps across space like a lighthouse, and we observe it as a pulsar. The neutron star has radius

R = 12\;\text{km}

and we detect the pulses when the beam points directly toward Earth. Due to the neutron star's rapid rotation, the hot spot moves at relativistic speeds. When the beam points toward us, the hot spot is moving with velocity

v

perpendicular to our line of sight. (a) Calculate the speed

v

of the hot spot. (b) The neutron star emits radio waves at frequency

f_0 = 1.4\;\text{GHz}

in its rest frame. Due to relativistic effects, we observe a Doppler-shifted frequency when the beam points toward us. Calculate the observed frequency

f_{\text{obs}}

. Take

c = 3.0 \times 10^8\;\text{m/s}

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