A red giant hosts a transiting planet

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 hosts a transiting planet. The transit light curve alone (Chapter 2's method, which needs no stellar mass or radius input) gives a mean stellar density

\rho_{*,\rm transit} = 3.0\,\mathrm{kg\,m^{-3}}

. The same star shows solar-like oscillations with

\nu_{\max} = 50\,\mu\mathrm{Hz}

\Delta\nu = 6.0\,\mu\mathrm{Hz}

and

T_{\rm eff} = 4900\,\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

\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, and its mean density from these. ((b)) Compare this seismic density with the transit-based density as a percentage difference, and comment on what this says about the accuracy of the scaling relations. ((c)) The star's envelope has an adiabatic exponent

\Gamma_1 = 1.50

(reduced from

5/3

by partial ionisation). Using the one-zone formula and the seismic density, find the fundamental radial period, and confirm the star still satisfies the dynamical-stability condition. ((d)) Given

T_{\rm eff} = 4900\,\mathrm{K}

, state whether this giant lies in the classical instability strip and how any oscillations it shows are excited.

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