Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Similarities Between Electric and Gravitational Forces • Coulomb’s force: F12 = • q1 q2 r2 Felectric ee/r 2 e2 42 = = 4.17 × 10 = Fgravity Gme me /r 2 Gm2e 1 (1) (2) Gravitational Instability • The equations modeling the hydrodynamic effects governing the systems we consider: Hydrodynamic equation, Continuity equation, and Poisson equation • A homogeneous gas, without rotation and magnetic field, an analysis due to J. H. Jeans (1902, 1928) • Hydrodynamic equation, or conservation of momentum equation: ∂v ρ + ρ(v·∇)v = −∇P − ρ∇Φ (3) ∂t where ρ is the density, v is the macroscopic velocity of the “gas,” P is the pressure, and Φ is the gravitational potential 2 • Between ρ and v there exists also the continuity equation ∂ρ + ∇·(ρv) = 0 ∂t (4) • The Poisson equation: connects Φ and ρ ∆Φ = −4πGρ (5) with the boundary condition Φ → 0 for |r| → ∞ • In our modeling three properties are of immediate interest: mass (density), velocity, and temperature (energy). Here velocity is a three dimensional vector and hence we need at least five independent equations to determine the properties of the system 3 • Additional equation of state, or energy conservation equation: P = 2 cs ρ (6) where cs = const is the sound speed. That is, we consider an isotermic “gas” • So, we have five independent equations (3)–(5) for unknowns ρ, v, and Φ (and additional Eq. (6)) 4 • An equilibrium state: ρ = ρ0 = constant, P = P0 = constant, v = v0 = constant, and Φ = Φ0 = constant • Consider small perturbations ρ = ρ0 + ρ1 , P = P0 + P1 , v = v0 + v1 , and Φ = Φ0 + Φ1 with |ρ1 | ≪ ρ0 , . . . , |Φ1 | ≪ |Φ0 | • Linearised equations (3)–(6) are (under the additional assumption v0 = 0) 1 ∂v1 = − P1 − ∇Φ1 ∂t ρ0 ∂ρ1 = −ρ0 ∇·v1 ∂t P1 = c2s ρ1 ∆Φ1 = −4πGρ1 5 (7) (8) (9) (10) • If the functions ρ1 , P1 , v1 , Φ1 representing the solution of this system of equations increase with time the medium is called unstable; otherwise it is called stable • We form the divergence of (8), with P1 from (10), and the time derivative of (9) ∂v1 2 ρ1 = −∆ cs + Φ1 (11) ∇· ∂t ρ0 ∂ 2 ρ1 ∂v1 = −ρ0 ∇· (12) 2 ∂t ∂t • Eliminating of ρ0 ∇·(∂v1 /∂t) from these equations, and using (10), leads to the basic wave equation ∂ 2 ρ1 2 = 4πGρ ρ + c 0 1 s ∆ρ1 2 ∂t 6 (13) • Of the well known solutions of (13) we consider the plane wave propagating in the x-direction ρ1 = A exp[i(kx−ωt)]+A exp[−i(kx−ωt)] (14) where A = const is the amplitude, k = 2π/λ = const is the wavenumber, λ is the wavelength, and ω is the wavefrequency • Substituting (14) in (13), since ρ̈1 = −ω 2 ρ1 and ∆ρ1 = −k 2 ρ1 , we obtain the dispersion equation ω 2 = k 2 c2s − 4πGρ0 (15) • For ρ1 to increase (exponentially) with time at fixed x it is necessary to have a real coefficient of t in (14), i.e. ω 2 < 0 7 Gravitational (Jeans) Instability • According to (15), instability can only occur for k 2 < 4πGρ0 /c2s or for wavelengths s 4c2s λ > λjeans = (16) Gρ0 where λjeans is called the Jeans length • Examples of Jeans-unstable systems: Schematic view (a) (a) (b) (b) (c) 8 Trigonometric Parallax 9 Trigonometric Parallax 10 Trigonometric Parallax • Parsec (pc) 1′′ 360◦ ×60′ ×60′′ 1.5×1013 = 2πdpc 18 → 1 pc ≡ dpc = 3.1 × 10 cm • Light year (ly) = c × 1 yr = 3 × 1010 × 3.16 × 107 = 9.5 × 1017 cm • 1 pc = 3.1 × 10−1 ly 11 Trigonometric parallax – Example • Parallax of α Cen is 0′′ .751 0′′ .751 360◦ ×60′ ×60′′ • Distance to α Cen → → dα = 4.1 × 1018 cm = 4.1×1018 1′′ 3.1×1018 = 0′′ .751 = 1.3 pc • Distance to α Cen → 4.1×1018 9.5×1017 12 = 1.5×1013 2πdα = 4.3 ly Trigonometric Parallax 13 The Hertzsprung–Russell Diagram 14 The Hertzsprung–Russell Diagram • The pattern of lines depend on the temperatures (and pressures) • The spectral type of a star yields an estimate of its temperature: spectral type = function of Te 15 H-R Diagram – Spectra of Stars • The pattern of line depends of temperature (and pressures) 16 H-R Diagram – Spectra of Stars 17 The Hertzsprung–Russell Diagram • The spectral type of a star yields an estimate of its temperature: spectral type = function of Te 18 The Hertzsprung–Russell Diagram • O B A F G K M (RNS) • I: supergiants II: bright giants III: giants IV: subgiants V: dwarfs 19 The Hertzsprung–Russell Diagram 20 The Hertzsprung–Russell Diagram • Stellar luminosity classes 21 The Hertzsprung–Russell Diagram • Distances calculated durectly (“trigonometric parallax”) • Giants, main-sequence and white dwarfs 22 The Hertzsprung–Russell Diagram • All nearby stars 23 The Hertzsprung–Russell Diagram • Bright stars 24 The Hertzsprung–Russell Diagram • Evolution tracs of protostars 25 The Hertzsprung–Russell Diagram • T Tauri stars’ evolution tracs 26 The Hertzsprung–Russell Diagram • Main sequence stars 27 The End States of Stars • Ordinary stars sustain the thermal pressure • Heat flows from the star to the universe • Without nuclear-energy sources: the star will contract and get hotter • But violent or not, death is as inevitable • Astronomers belive four are possible: • 1. Nothing may be left: explosion • 2. White dwarf: mass ∼ 0.7M⊙ , radius ∼ 109 cm • 3. Neutron star: mass ∼ 1.4M⊙ , radius ∼ 106 cm • 4. Black hole: mass > 2M⊙ , radius > 1016 cm (Earth–Sun separation ≈ 1013 cm) 28