Mathematical Principles of Theoretical Physics

7.1. ASTROPHYSICAL FLUID DYNAMICS 407 whereAμis the electromagnetic potential, the time components ofgμ νare as g 00 =− ( 1 + 2 ...

408 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY ρ= mass + electromagnetism + potential + heat. 5.Equation of state: (7.1.68) p=f(ρ,T). ...

7.1. ASTROPHYSICAL FLUID DYNAMICS 409 whereP= (Pr,Pθ,Pφ)is the momentum density field,Tis the temperature,pis the pressure, ρis ...

410 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY The Universe and all stars are in the momentum-flow state, i.e. they are fluid spheres ...

7.1. ASTROPHYSICAL FLUID DYNAMICS 411 Then, in view of (7.1.28)-(7.1.31), the equation (7.1.76) can be expressed by (7.1.79) 1 α ...

412 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY supplemented with the boundary conditions: (7.1.86) ∂T ∂r = 0 , Pr= 0 , ∂Pθ ∂r = ∂Pφ ∂ ...

7.2. STARS 413 For the main-sequence stars, we have the following the relation (7.2.2) L/L⊙= (m/m⊙)α, whereαtakes different valu ...

414 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY 8.Red giants.Stars are held together by gravity. The gravitational forcepulling inward ...

7.2. STARS 415 total force consists of four parts: the viscous friction, the pressure gradient, the relativistic effect, and the ...

416 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY Hereα,ψ,Tsatisfy equations (7.1.79)-(7.1.80) with the boundary conditions (7.1.81)-(7. ...

7.2. STARS 417 3.Derivation of formula (7.2.14).To deduce (7.2.14) we have to derive the gravitational potentialα. The first equ ...

418 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY whereξis as in (7.2.12). By (7.1.82) and (7.2.16)-(7.2.17), we can deduce that (7.2.21 ...

7.2. STARS 419 Dynamic transitions of (7.2.24) take place at(u,λ) = ( 0 ,λ 1 )provided thatλ 1 satisfies the following principl ...

420 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY Based on the dynamic transition theory introduced early in this section, the stellar c ...

7.2. STARS 421 In particular,σchas the same order of magnitude as the first eigenvalueλ 1 of the the following equations in the ...

422 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY By (7.2.34), neutron stars will eventually stop convection. Due to the high rotationΩ ...

7.2. STARS 423 For stellar atmosphere circulations, the star radiusr 0 is much greater than the convection heighth=r 1 −r 0. Hen ...

424 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY It implies that the forceFin (7.2.40) is explosive asδ→1. In addition, by (7.2.38) the ...

7.2. STARS 425 For the stars with rotation~Ω, the corresponding eigenvalue equations with the Coriolis force are (7.2.43) Pr [ ...

426 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY the nonzero components of the Levi-civita connections are Γ^000 = 1 2 c ut, Γ^011 = 1 ...