Chapter 2 Stellar Structure Equations 1 Mass conservation equation

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(Remark : You can check above equation yourself and B n contains all n- dependent coefficients. In table 2 of Supplementary notes, you can see that B n varies very slowly with n. ) For a given M and P c we obtain ( ) 3/4 ρ c,1.5 B3 = . (42) ρ c,3 B 1.5 The central density of the polytropic star is given in terms of the mean density as M ρ c = D n ¯ρ = D n 4πR 3 /3 , (43) where [ ( ) ] −1 3 dθ D n = − , (44) ξ 1 dξ ξ 1 are some constants. For a given M we obtain the ratio of radii R(n) as R (1.5) R (3) = = ( D1.5 ( D1.5 ) 1/3 ) 1/3 ( ) 1/4 ρ c,3 B1.5 = D 3 ρ c,1.5 D 3 B 3 ( ) 1/3 ( ) 1/4 5.991 0.206 < 1, (45) 54.81 0.157 and therefore R(3) > R (1.5) . (46) This result makes sense because for a large polytropic index, the star is stiffer and hence its pressure can resist a bigger gravity so the star should be larger for the same mass. Example. Capella is a binary star discovered in 1899, with a known orbital period, which enables the determination of the mass and radius of the 10
ighter component: M = 8.30 × 10 33 g and R = 9.55 × 10 11 cm. Assuming that the star can be described by a polytrope of index 3, find the central pressure and the central density. Check wether the central pressure satisfies the inequality P c > GM 2 /8πR 4 . The central density is given by M ρ c = D n ¯ρ = D n 4πR 3 /3 , (47) and we obtain ρ c = 1.2 × 10 −1 g/cm 3 . In order to obtain the central pressure as a function of M and R we eliminate ρ c between the above equation and thus obtaining P c = (4π) 1/3 B n GM 2/3 ρ 4/3 c , (48) P c = GM 2 4πR 4 [ (3D n ) 4/3 B n ] . (49) The term in square brackets exceeds unity for all n and hence P c > GM 2 4πR 4 > GM 2 8πR 4 . (50) Thus the inequality is generally satisfied by polytropic models. For Capella, with n = 3, P c = 6.1 × 10 13 dyn/cm 2 . 5 Energy conservation and the energy production equation Now, we want to go further than simple stellar models. introducing a few more equations. We do so by 11
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 and 6: ( d 2 P dr 2 )c = − ( dρ Gm + ρ
Page 7 and 8: as adiabatic expansion. With this i
Page 9: By substituting θ and ξ in the La
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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