Chapter 2 Stellar Structure Equations 1 Mass conservation equation

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Note ∑ that G is minimized in chemical equilibrium. Since G = E − T S − s µ sN s , we have N s = − ∂G ∂µ s ∣ ∣∣∣T,V,{µi } i≠s = g sV λ 3 s ( ) µs − E 0,s exp . (63) kT By rearranging terms, the above equation becomes ( ) ( ) Ns λ 3 s ns λ 3 s µ s = kT ln + E 0,s = kT ln + E 0,s , (64) g s V g s where n s is the number density of species s. Now consider a specific atomic species and use the label s, 0 to denote its sth ionized state in its ground level. For example, the label 0, 0 of H refers to the state of hydrogen atom with its electron in the lowest energy level. And I use the label e to denote electron. Clearly, at non-zero temperature, reactions such as X s+ ⇋ X (s+1)+ + e − (65) may occur, where X denotes the atomic species. Hence, at thermodynamic equilibrium, µ s,0 = µ s+1,0 + µ e . (66) Since E 0,(s+1,0) + E 0,e − E 0,(s,0) = χ s+1 is the (s + 1)th ionization potential of the atomic species, substitute this and Eq. (64) into above equation, we have n s+1,0 n e λ 3 s+1,0λ 3 eg s,0 n s,0 λ 3 s,0g s+1,0 g e = e −χ s+1/kT . (67) From Eq. (57), g e = 2 and the approximation that the masses of the sth and the (s + 1)th ionized atom are about the same, the above equation becomes where n s+1,0 n s,0 f s+1 (T ) = 2(2πm ekT ) 3/2 h 3 = g s+1,0 g s,0 n e f s+1 (T ) , (68) ( exp − χ ) s+1 , (69) kT In general, not all the sth ionized atoms in a thermalized system are in the ground state. From Eq. (55), Eq. (68) can be re-written as n s+1 n s = Z s+1 Z s n e f s+1 (T ) , (70) 16
where Z s = ∑ i [ ] −(Es,i − E s,0 ) g s,i exp kT is the partition function of those sth ionized atoms in various energy levels. (71) Eqs. (68) and (70) are known as the Saha equation. Saha equation is useful in astronomy. For instance, if we know the electron partial pressure P e , then the electron number density n e can be well approximated by P e /kT in many cases. We now apply Saha equation to estimate the degree of ionization of our Sun. For simplicity, we consider only the form of Saha equation in Eq. (68). The mean density of our Sun is about 1.4 g/cm 3 . By virial theorem, the mean temperature of our Sun is about 6 × 10 6 K. We simplify our analysis by assuming that the Sun is made up of entirely hydrogen. Thus, χ 1 = 13.6 eV. Let us define the degree of ionization by n 1 /(n 0 + n 1 ) ≡ x. Then Saha equation demands that x 1 − x = f 1(T ) ≈ m Hf 1 (T ) n e ρx . (72) By solving above equation we get x ≈ 0.95. In other words, almost all hydrogen in our Sun is ionized. By the same token, it is straight-forward to check that most atoms in a main sequence star are ionized. On the other hand, in the solar atmosphere where T ≈ 6000 K and n e ≈ 10 13 cm −3 . Saha equation tells us that the degree of ionization of hydrogen is about 10 −4 . Note however that Saha equation has its limitations. It works for the most region of a star, where is in thermal equilibrium. On the other hand it may not be applicable to stellar atmosphere, where thermal equilibrium may not be applied. Also, it assumes classical non-relativistic ideal gas, so it may not work near the core of some stars, where degenerate may occur. 7.2 Simple Equation of State for Stars As most of the stellar material should be ionized, the dominant interaction between particles in stellar interior (besides gravity) is electrostatic in nature. The typical distance between particles in stellar interior d ≈ (Ām H /¯ρ) 1/3 where Ā is the average atomic weight of particles, m H is the mass of hydrogen atom, and ¯ρ is the average density of a star. Consequently, the typical 17
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 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: Note that we have assumed that the
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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