The Relativistic Lighthouse · AI Generator #GEN-6978DD72
Physics
Problem
A neutron star rotates with period T=1.337s and emits a narrow beam of radio waves from a hotspot on its surface. The beam sweeps across space like a lighthouse, creating a pulsar that astronomers observe on Earth. The neutron star has radius R=12.0km and the hotspot is located at the star's equator.
Due to the star's rapid rotation, the hotspot moves at a significant fraction of the speed of light. As the beam sweeps past Earth, astronomers measure the pulse duration (the time interval during which they receive the radio signal) to be Δtobs=2.50ms.
The pulse duration depends on two effects: (1) the geometric time it takes for the beam to sweep across the line of sight to Earth, and (2) relativistic time dilation of the emission process at the hotspot.
Assuming the beam has angular width Δϕ=0.1rad as measured in the neutron star's rest frame, and that Earth is located far from the neutron star such that the beam appears to sweep across Earth at approximately constant angular velocity, find the ratio of the observed pulse duration to what would be observed if there were no relativistic effects.