Mathematical Principles of Theoretical Physics

7.4. GALAXIES 447 galaxy nucleus Figure 7.7: A schematic diagram of elliptical galaxy. The metric is as in (7.4.3), and the corr ...

448 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY whereλ= (δ,Re), and theδ-factor and the Rayleigh number are defined by (7.4.12) δ= 2 M ...

7.4. GALAXIES 449 By the boundary conditions in (7.4.14), we obtain that (7.4.16) β 1 = rk 02 ζ 1 −rk 12 ζ 0 r 1 k^1 rk 02 −rk 0 ...

450 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY dictate the dynamic behaviors of spiral galaxies, which aredetermined by the physical ...

7.4. GALAXIES 451 The termF 1 corresponds to the Rayleigh-B ́enard convection with the Rayleigh number R=k 1 k 2 = βMrGr 0 r 1 γ ...

452 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY whereMbis the mass of the black hole core,Qis the heat source generated by the nuclear ...

7.4. GALAXIES 453 which is similar to the Taylor-Couette flow in a rotating cylinder. The rotating instability can generate a ci ...

454 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY It is the force (7.4.36) that not only causes the instability of the basic flow (7.4.2 ...

7.4. GALAXIES 455 Jet Accretion (a) (b) Accretion Jet Jet Figure 7.11: (a) A jet in the latitudinal circulation withk=1 cell, tw ...

456 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY Remark 7.18.The jets shown in Figures7.9and7.10are column-shaped. If the cell number k ...

7.5. THE UNIVERSE 457 Assume that ̇ris proportional tor, i.e. (7.5.5) r ̇=Hr, (His a constant). Then the Hubble law (7.5.1) foll ...

458 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY whereHis the Hubble constant. We infer then from (7.5.2) and (7.5.6) the ageTof the Un ...

7.5. THE UNIVERSE 459 whereM(r) = 4 πr^3 ρ/3, andρis the mass density. Thus, (7.5.10) can be rewritten as follows (7.5.11) r′′=− ...

460 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY whereR,ρ,pare the unknown functions. Equations (7.5.15)-(7.5.17) are called the Friedm ...

7.5. THE UNIVERSE 461 then the metric (7.5.8) satisfies the following equations R ̈=−^4 πG 3 ( ρ+ 3 p c^2 ) R+ Λc^2 3 (7.5.24) R ...

462 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY 7.5.2 Globular universe with boundary If the spatial geometry of a universe is open, t ...

7.5. THE UNIVERSE 463 Basic problems A static universe is confined in a ball with fixed radiusRsin (7.5.31), and the ball behave ...

464 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY y x r 1 y ̃ Figure 7.13: Due to the lensing effect, one atxcan also see the star atyas ...

7.5. THE UNIVERSE 465 Whenv/cis small, (7.5.38) can be approximatively expressed as (7.5.39) z≃v/c. In addition, Hubble discover ...

466 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY 7.5.3 Spherical Universe without boundary Bounded universe has finite energy and space ...