A fully convective low-mass star of mass and radius is modelled as the polytrope…

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 Energy transport and the complete stellar model

Problem

A fully convective low-mass star of mass

M = 0.30\,M_\odot

and radius

R = 0.30\,R_\odot

is modelled as the

n = 3/2

polytrope built from the Lane-Emden equation. The standard

n = 3/2

Lane-Emden solution gives a central-to-mean density ratio

\rho_c/\bar\rho = 5.99

and a central-pressure coefficient

W_n = 0.770

such that

P_c = W_n\,GM^2/R^4

. Independent modelling of the same star gives a central temperature

T_c \approx 8\times10^{6}\,\mathrm{K}

, for a composition

X = 0.70

Z = 0.02

. ((a)) Compute the mean density

\bar\rho

and hence the central density

\rho_c

. ((b)) Compute the central pressure

P_c

. ((c)) Using the central series expansion

P(m) \approx P_c - \frac{3G}{8\pi}\left(\frac{4\pi\rho_c}{3}\right)^{4/3}m^{2/3}

, evaluate

P

m = 0.0010\,M

and state the fractional change from

P_c

. ((d)) Given

\rho_c

and

T_c

above, say in one sentence which opacity process dominates near the centre, and in a second sentence explain why the measured mass-luminosity relation for this star cannot be read off either homology relation derived for constant or for Kramers opacity.

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