Chapter 2 Stellar Structure Equations 1 Mass conservation equation

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Therefore near the centre T (r) = T c − 1 8ac k c ρ 2 cq c r 2 . (24) Tc 3 Note that these relations hold regardless of the functional dependence P (ρ, T ), q (ρ, T ), and k (ρ, T ). 3 Simple stellar models So far, we have 3 unknowns, namely, m(r), ρ(r) and P (r) but only two equations, namely, the mass conservation equation and equation of hydrostatic equilibrium. So, we need another equation to relate m, ρ and P . Astronomers have proposed a number of extremely simplified equations to fulfill this task. Since all these proposed equations are not derived from first principle, they are nothing more than crude approximations of the realistic situation. Yet, these approximations are sometimes quite useful to investigate the approximate structure of a star. The first such model is called constant density model, namely, we assume ρ is a constant inside the entire star. The second one is called the linear density model, namely, ρ(r) = ρ c (1 − r/R) where ρ c is a constant. You may solve m(r) and P (r) in the above two models yourself. 4 The polytrope model The last such model I am going to introduce, which is the most important model of this kind, is called the polytrope model. We assume that P = Kρ γ (25) for some constants K and γ. Note that this model is based on the observation that P and ρ follow the above equation for ideal gas in various situations such 6
as adiabatic expansion. With this in mind, we expect that γ varies from 4/3 to 5/3. Combining the mass conservation and hydrostatic equilibrium equations, we have ( ) 1 d r 2 dP = −4Gπρ . (26) r 2 dr ρ dr Write γ = 1 + 1/n, then the above equation can be written as [ ] (n + 1)K d r 2 (1−n)/n dρ ρ = −ρ . (27) 4πGnr 2 dr dr The boundary conditions for this differential equations are ρ(R) = 0 and dρ(0)/dr = 0. (Do you know why?) Let ρ = ρ c θ n where ρ c is the central density of the star, we have ( 1 d ξ 2 dθ ) = −θ n , (28) ξ 2 dξ dξ where r = [(n + 1)Kρ (1−n)/n c /4πG] 1/2 ξ. The corresponding boundary conditions are θ = 1 and dθ/dξ = 0 at ξ = 0. This equation is called Lane-Emden equation and can be solved numerically. The significance of the polytrope model is that Lane-Emden equation is independent of the mass M, radius R and central density ρ c of a star. So, once you have numerically solved the Lane-Emden equation for a given value of n, the numerical solution can be used to deduce the solution of any star with the same polytropic index n (or γ). For the special cases of n = 0, n = 1 and n = 5, Lane-Emden equation can be solved exactly. The case of n = 0 is easy. You may work out by yourself that θ(ξ) = 1 − ξ 2 /6; and hence ρ = ρ c whenever ξ ≠ √ 6 . (29) The case of n = 1, Lane-Emden equation can be rewritten as d 2 (ξθ) = −ξθ (30) dξ2 7
Page 1 and 2: Chapter 2 Stellar Structure Equatio
Page 3 and 4: Note that the R.H.S. above is Ω/3
Page 5: ( d 2 P dr 2 )c = − ( dρ Gm + ρ
Page 9 and 10: By substituting θ and ξ in the La
Page 11 and 12: ighter component: M = 8.30 × 10 33
Page 13 and 14: • The nuclear burning timescale:
Page 15 and 16: Note that we have assumed that the
Page 17 and 18: where Z s = ∑ i [ ] −(Es,i −
Page 19 and 20: The ion pressure is given by where
Page 21: Note that the first fraction in Eq.

Chapter 2 Stellar Structure Equations 1 Mass conservation equation

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