The Relativistic Lighthouse · AI Generator #GEN-2FBC3664
Physics
Problem
A neutron star rotates with period T=1.337s and emits a narrow beam of electromagnetic radiation from a hot spot on its surface. This beam sweeps across space like a lighthouse, creating the pulsar phenomenon observed by radio telescopes on Earth. The neutron star has radius R=12km and the hot spot is located at the star's equator.
Due to the star's rotation, the hot spot moves at relativistic speeds. An observer on Earth receives pulses of radiation each time the beam sweeps past. However, the frequency of the electromagnetic waves within each pulse is Doppler-shifted due to the relativistic motion of the hot spot.
When the hot spot is moving directly toward Earth (at the point in its circular orbit closest to our line of sight), the emitted radiation has a rest frequency of f0=1.420GHz (the 21-cm hydrogen line).
(a) What is the speed of the hot spot as a fraction of the speed of light?
(b) What frequency does an Earth-based radio telescope observe for this radiation when the hot spot is moving directly toward Earth?