A star of known mass and radius shows a periodic transit of period

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

Stars from the Inside, Ch Energy transport and the complete stellar model

Problem

A star of known mass

M_1 = 1.00\,M_\odot

and radius

R_1 = 1.00\,R_\odot

shows a periodic transit of period

P = 15.0\,\mathrm{days}

, fractional depth

\delta = 0.078

, and a radial-velocity semi-amplitude

K_1 = 18.6\,\mathrm{km\,s^{-1}}

. The orbit is circular and the inclination is

i = 90^\circ

(edge-on, as required for the transit to occur). ((a)) Using the single-lined mass function with

\sin i = 1

, find the mass

M_2

of the transiting body (assume solar mean molecular weight for both bodies). ((b)) Using the transit depth, find its radius

R_2

. ((c)) Assuming the same mean molecular weight as the Sun and the same opacity law, use both the electron-scattering homology relation

L \propto M^3

and the Kramers homology relation

L \propto M^{5.5}

(each normalised to

1\,L_\odot

1\,M_\odot

) to predict

L_2

in solar units. ((d)) State, with a one-sentence reason, where the true luminosity of this body is likely to fall relative to the two homology predictions, using the measured mass-luminosity relation of eclipsing binaries.

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