A thin pulsating shell sits at radius around a point mass

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Astrophysics & Cosmology · Stellar structure, energy generation and evolution

Stars from the Inside, Ch Pulsating stars: the kappa mechanism, the Leavitt law and asteroseismology

Problem

A thin pulsating shell sits at radius R=50 R⊙R = 50\,R_\odot around a point mass M=5 M⊙M = 5\,M_\odot, obeying the one-zone radial oscillation frequency ω2=(3Γ1−4) GM/R3\omega^2 = (3\Gamma_1 - 4)\,GM/R^3. ((a)) Compute the oscillation period Π=2π/ω\Pi = 2\pi/\omega for Γ1=5/3\Gamma_1 = 5/3, Γ1=3/2\Gamma_1 = 3/2 and Γ1=7/5\Gamma_1 = 7/5. ((b)) Compute the period for Γ1=1.34\Gamma_1 = 1.34, close to 4/34/3, and describe how Π\Pi behaves as Γ1→4/3+\Gamma_1 \to 4/3^+. ((c)) For Γ1=1.30\Gamma_1 = 1.30, below 4/34/3, ω2\omega^2 is negative. State what this means physically and compute the associated e-folding timescale τ=1/∣ω∣\tau = 1/|\omega|.

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