The Stellar Forge: Neutron Star Crystallization · AI Generator #GEN-23BDE453

Physics · thermodynamics

Problem

Deep within a neutron star, matter exists in extreme conditions where nuclear pasta phases form crystal-like structures. Consider a simplified model where neutrons behave as a degenerate Fermi gas in a crystalline lattice. A small region of neutron matter with volume V=1.0×10−15  m3V = 1.0 \times 10^{-15}\;\text{m}^3 contains N=6.0×1020N = 6.0 \times 10^{20} neutrons at temperature T=108  KT = 10^8\;\text{K}. Due to the extreme density, the neutrons form a degenerate gas where quantum effects dominate. Suddenly, a gravitational wave passes through this region, causing an adiabatic compression that reduces the volume by exactly half. During this compression, the system remains in thermal equilibrium locally, but no heat can be exchanged with the surroundings due to the rapid timescale. (a) Calculate the Fermi energy EFE_F of the neutrons before compression. (b) Find the final temperature TfT_f after the adiabatic compression. (c) Determine the work done by the gravitational wave on this neutron matter during compression. Given: neutron mass mn=1.67×10−27  kgm_n = 1.67 \times 10^{-27}\;\text{kg}, Planck's constant ℏ=1.055×10−34  J⋅s\hbar = 1.055 \times 10^{-34}\;\text{J·s}, Boltzmann constant kB=1.38×10−23  J/Kk_B = 1.38 \times 10^{-23}\;\text{J/K}. For a degenerate Fermi gas: EF=ℏ22m(3π2n)2/3E_F = \frac{\hbar^2}{2m}(3\pi^2 n)^{2/3} where nn is number density, and the adiabatic relation is TV2/5=constantTV^{2/5} = \text{constant}.

Related theory

Browse the problem library