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hwk8 - M ’ 1 M ± and R = 10 km Comment on the...

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AST371 - Introduction to Astrophysics Assignment 8 - Oct. 28, 2009 1. Consider an ideal gas that undergoes an adiabatic process. Starting with the relationship T ( r ) T 0 = P ( r ) P 0 ( γ - 1) where T 0 and P 0 are respectively the temperature and the pressure at the center of the distribution ( r = 0), show that the temperature distribution is given by dT ( r ) dr = T ( r ) P ( r ) γ - 1 γ dP ( r ) dr 2. Consider an ideal gas of H atoms in a stellar atmosphere, i.e., in a region just outside of the star, where the acceleration g = - GM r can be assumed to be constant; M is the mass of the star. Assuming hydrostatic equilibrium of the gas with the mass of the star, calculate the scale height of a gas of T = 10 6 K in the atmosphere of a neutron star, for which
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Unformatted text preview: M ’ 1 M ± and R = 10 km. Comment on the possibility of the existence of an atmosphere around a neutron star. 3. (a) Consider a polytropic gas of polytropic index n = 0 in hydrostatic equilibrium with its own gravitational mass, P ( r ) = Kρ ( r ) γ , γ = 1+1 /n . Show that a solution of the Lane-Emden equation is given by the function ξ ( x ) ξ ( x ) = 1-x 2 6 with the boundaries conditions ξ (0) = 1, dξ/dx | = 0. (b) Find the distribution of the normalized density and pressure as function of the dimensionless variable x , ρ ( x ) /ρ and P ( x ) /P , and plot them as function of x ....
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