The Quantum Lighthouse · AI Generator #GEN-B968F511

Physics · modern-physics

Problem

A novel astronomical observation technique uses a rapidly rotating neutron star (pulsar) as a "quantum lighthouse." The pulsar rotates with period

T = 1.337\;\text{ms}

and emits X-ray photons of energy

E = 12.4\;\text{keV}

in a narrow beam that sweeps across Earth once per rotation. Due to the pulsar's rotation, the X-ray photons emitted tangentially from the neutron star's surface experience a relativistic Doppler shift. When the emission point moves toward Earth, we observe higher energy photons; when moving away, we observe lower energy photons. The neutron star has radius

R = 12\;\text{km}

and mass

M = 1.4 M_{\odot}

(where

M_{\odot} = 1.99 \times 10^{30}\;\text{kg}

). An X-ray telescope measures the energy difference between the maximum and minimum photon energies received during one pulse cycle to be

\Delta E = 82\;\text{eV}

. Using this "quantum lighthouse" technique, determine the distance

d

between Earth and the pulsar. Assume the pulsar's rotation axis is perpendicular to the line of sight to Earth, and that general relativistic effects can be neglected for this calculation. *Constants:*

c = 3.00 \times 10^8\;\text{m/s}

h = 4.14 \times 10^{-15}\;\text{eV·s}

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