A transit light curve (Chapter 2 method) of an old brown dwarf orbiting a Sun-like…

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 Pressure against gravity: the equation of state, polytropes and the Lane-Emden equation

Problem

A transit light curve (Chapter 2 method) of an old brown dwarf orbiting a Sun-like star of radius

R_* = 1.00\,R_\odot

gives a radius ratio

R_p/R_* = 0.115

; the radial-velocity mass function (Chapter 1) gives a companion mass

M_p = 45\,M_J

. Take

1\,R_\odot = 9.95\,R_J

. ((a)) An

n=1

polytrope of the same hydrogen-helium composition and the same constant

K

has a mass-independent radius; for a

1\,M_J

planet of this composition that radius is

1.00\,R_J

. What radius does the

n=1

polytrope predict for the

45\,M_J

companion, and how does this compare with the transit-derived radius? ((b)) The host star has mass

M_* = 0.85\,M_\odot

and

\mu = 0.61

. Using the Eddington standard-model relation

1-\beta = 0.003\,\mu^4\beta^4(M/M_\odot)^2

, estimate

1-\beta

for the host star (note

\beta \approx 1

makes this a one-step, not an iterated, estimate). ((c)) State the apsidal constant

k_2

that would be predicted for the host star if it were modelled as

n=3/2

(fully convective,

k_2 = 0.143

) versus

n=3

(more centrally concentrated,

k_2 = 0.0144

), and say which index is the more physically appropriate model for a star of

0.85\,M_\odot

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