ast403_stellar structure notes

ast403_stellar structure notes - EQUATIONS OF STELLAR...

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EQUATIONS OF STELLAR STRUCTURE General Equations We shall consider a spherically symmetric, self-gravitating star. All the physical quantities will depend on two independent variables: radius and time , ( r, t ). First, we shall derive all the equations of stellar structure in a general, non spherical case, but very quickly we shall restrict ourselves to the spherically symmetric case. The microscopic properties of matter at any given point may be described with density ρ , tem- perature T , and chemical composition, i.e. the abundances of various elements X i , with i = 1 , 2 , 3 .... having as many values as there are elements. All thermodynamic properties and transport coeffi- cients are functions of ( ρ, T, X i ). In particular we have: pressure P ( ρ, T, X i ), internal energy per unit volume U ( ρ, T, X i ), entropy per unit mass S ( ρ, T, X i ), coefficient of thermal conductivity per unit volume λ ( ρ, T, X i ), and heat source or heat sink per unit mass ǫ ( ρ, T, X i ). All the partial derivatives of P , U , λ , and ǫ are also functions of ρ , T , and X i . Using these quantities the first law of thermodynamics may be written as T dS = d parenleftbigg U ρ parenrightbigg P ρ 2 dρ, (1 . 1) If there are sources of heat, ǫ , and a non-vanishing heat flux vector F , then the heat balance equation may be written as ρT dS dt = ρǫ div vector F. (1 . 2) The heat flux is directly proportional to the temperature gradient: vector F = λ T. (1 . 3) The equation of motion (the Navier-Stockes equation of hydrodynamics) may be written as d 2 vector r dt 2 + 1 ρ P + V = 0 , (1 . 4) where the gravitational potential satisfies the Poisson equation 2 V = 4 πGρ, (1 . 5) with V −→ 0 when r −→ ∞ . In spherical symmetry these equations may be written as 1 ρ ∂P ∂r + ∂V ∂r + d 2 r dt 2 = 0 , (1 . 6a) 1 r 2 ∂r parenleftbigg r 2 ∂V ∂r parenrightbigg = 4 πGρ, (1 . 6b) F = λ ∂T ∂r , (1 . 6c) 1 — 1
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1 ρr 2 ( r 2 F ) ∂r = ǫ T dS dt , (1 . 6d) It is convenient to introduce a new variable, M r : M r r integraldisplay 0 4 πr 2 ρdr , (1 . 7) which is the total mass within the radius r , and another variable, L r : L r 4 πr 2 F, (1 . 8) which is the luminosity, i.e. the total heat flux flowing through a spherical shell with the radius r , and also κ = 4 acT 3 3 ρ 1 λ , (1 . 9) where κ is the coefficient of radiative opacity (per unit mass) , c is the speed of light, and a is the radiation constant. The last equation is valid if the heat transport is due to radiation. Using the definitions and relations (1.7-1.9) we may write the set of equations (1.6) in a more standard form: 1 ρ ∂P ∂r + GM r r 2 + d 2 r dt 2 = 0 , (1 . 10a) ∂M r ∂r = 4 πr 2 ρ, (1 . 10b) ∂T ∂r = 3 κρL r 16 πacT 3 r 2 , (1 . 10c) ∂L r ∂r = 4 πr 2 ρ parenleftbigg ǫ T dS dt parenrightbigg , (1 . 10d) This system of equations is written in a somewhat inconvenient way, as all the space derivatives ( ∂/∂r ) are taken at a fixed value of time, while all the time derivatives ( d/dt ) are at the fixed mass zones. For this reason, and also because of the way the boundary conditions are specified (we shall
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