A red giant observed by a photometric mission shows solar-like oscillations with…

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

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

Problem

A red giant observed by a photometric mission shows solar-like oscillations with frequency of maximum power

\nu_{\max} = 25\,\mu\mathrm{Hz}

and large frequency separation

\Delta\nu = 3.5\,\mu\mathrm{Hz}

, and has effective temperature

T_{\rm eff} = 4700\,\mathrm{K}

. Take solar reference values

\nu_{\max,\odot} = 3090\,\mu\mathrm{Hz}

\Delta\nu_\odot = 135.1\,\mu\mathrm{Hz}

T_{{\rm eff},\odot} = 5772\,\mathrm{K}

and mean solar density

\bar\rho_\odot = 1408\,\mathrm{kg\,m^{-3}}

. ((a)) Use the asteroseismic scaling relations to find the star's mass and radius in solar units. ((b)) Find its mean density in

\mathrm{kg\,m^{-3}}

. ((c)) Treat the envelope as an ideal monatomic gas with

\Gamma_1 = 5/3

(the adiabatic exponent of an ideal gas established from the equation of state). Using the one-zone radial-oscillation formula, find the angular frequency and period of the fundamental radial mode, and state whether the star satisfies the dynamical-stability condition on

\Gamma_1

. ((d)) The classical instability strip lies between about

6000\,\mathrm{K}

and

7500\,\mathrm{K}

. State whether the kappa mechanism is expected to drive self-excited pulsation in this star, and how its observed oscillations are excited instead.

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