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Stellar Structure and Evolution: Syllabus 3.3 the Virial Theorem and Its Implications (ZG: P5-2; CO: 2.4) Ph

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Stellar Structure and Evolution: Syllabus 3.3 the Virial Theorem and Its Implications (ZG: P5-2; CO: 2.4) Ph Page 2 Stellar Structure and Evolution: Syllabus 3.3 The Virial Theorem and its Implications (ZG: P5-2; CO: 2.4) Ph. Podsiadlowski (MT 2006) 3.4 The Energy Equation and Stellar Timescales (CO: 10.3) (DWB 702, (2)73343, [email protected]) (www-astro.physics.ox.ac.uk/˜podsi/lec mm03.html) 3.5 Energy Transport by Radiation (ZG: P5-10, 16-1) and Con- vection (ZG: 16-1; CO: 9.3, 10.4) Primary Textbooks 4. The Equations of Stellar Structure (ZG: 16; CO: 10) ZG: Zeilik & Gregory, “Introductory Astronomy & Astro- • 4.1 The Mathematical Problem (ZG: 16-2; CO: 10.5) physics” (4th edition) 4.1.1 The Vogt-Russell “Theorem” (CO: 10.5) CO: Carroll & Ostlie, “An Introduction to Modern Astro- • 4.1.2 Stellar Evolution physics” (Addison-Wesley) 4.1.3 Convective Regions (ZG: 16-1; CO: 10.4) also: Prialnik, “An Introduction to the Theory of Stellar Struc- • 4.2 The Equation of State ture and Evolution” 4.2.1 Perfect Gas and Radiation Pressure (ZG: 16-1: CO: 1. Observable Properties of Stars (ZG: Chapters 11, 12, 13; CO: 10.2) Chapters 3, 7, 8, 9) 4.2.2 Electron Degeneracy (ZG: 17-1; CO: 15.3) 1.1 Luminosity, Parallax (ZG: 11; CO: 3.1) 4.3 Opacity (ZG: 10-2; CO: 9.2) 1.2 The Magnitude System (ZG: 11; CO: 3.2, 3.6) 5. Nuclear Reactions (ZG: P5-7 to P5-9, P5-12, 16-1D; CO: 10.3) 1.3 Black-Body Temperature (ZG: 8-6; CO: 3.4) 5.1 Nuclear Reaction Rates (ZG: P5-7) 1.4 Spectral Classification, Luminosity Classes (ZG: 13-2/3; CO: 5.2 Hydrogen Burning 5.1, 8.1, 8.3) 5.2.1 The pp Chain (ZG: P5-7, 16-1D) 1.5 Stellar Atmospheres (ZG: 13-1; CO: 9.1, 9.4) 5.2.2 The CN Cycle (ZG: P5-9; 16-1D) 1.6 Stellar Masses (ZG: 12-2/3; CO: 7.2, 7.3) 5.3 Energy Generation from H Burning (CO: 10.3) 1.7 Stellar Radii (ZG: 12-4/5; CO: 7.3) 5.4 Other Reactions Involving Light Elements (Supplementary) 2. Correlations between Stellar Properties (ZG: Chapters 12, 13, 5.5 Helium Burning (ZG: P5-12; 16-1D) 14; CO: Chapters 7, 8, 13) 6. The Evolution of Stars 2.1 Mass-Luminosity Relations (ZG: 12-2; CO: 7.3) 6.1 The Structure of Main-Sequence Stars (ZG: 16-2; CO 10.6, 2.2 Hertzsprung-Russell diagrams and Colour-Magnitude Dia- 13.1) grams (ZG: 13-3; CO: 8.2) 6.2 The Evolution of Low-Mass Stars (ZG: 16-3; CO: 13.2) 2.3 Globular Clusters and Open (Galactic) Clusters (ZG:13-3, 14-2; OG: 13.4) 6.2.1 The Pre-Main Sequence Phase 6.2.2 The Core Hydrogen-Burning Phase 2.4 Chemical Composition (ZG: 13-3; CO: 9.4) 6.2.3 The Red-Giant Phase 2.5 Stellar Populations (ZG: 14-3; CO: 13.4) 6.2.4 The Helium Flash 3. The Physical State of the Stellar Interior (ZG: P5, 16; CO: 10) 6.2.5 The Horizontal Branch 3.1 The Equation of Hydrostatic Equilibrium (ZG: 16-1; CO: 6.2.6 The Asymptotic Giant Branch 10.1) 6.2.7 White Dwarfs and the Chandrasekhar Mass (ZG: 17-1; CO: 13.2) 3.2 The Dynamical Timescale (ZG: P5-4; CO: 10.4) Page 4 Useful Numbers Astronomical unit AU = 1.5 1011 m × Parsec pc = 3.26 ly =3.086 1016 m 6.3 The Evolution of Massive Stars (CO: 13.3) × Lightyear ly = 9.46 1015 m 6.4 Supernovae (ZG: 18-5B/C/D) × Mass of Sun M =1.99 1030 kg ⊙ × 6.4.1 Explosion Mechanisms Mass of Earth M =5.98 1024 kg ⊕ × 6 6.4.2 Supernova Classification =3 10− M × 3 ⊙ 6.4.3 SN 1987 A (ZG: 18-5E) Mass of Jupiter MJup =10− M ⊙ 6.4.4 Neutron Stars (ZG: 17-2; CO: 15.6) Radius of Sun R =6.96 108 m ⊙ × 6.4.5 Black Holes (ZG: 17-3; CO: 16) Radius of Earth R =6380km ⊕ 3 Radius of Jupiter RJup =10− R 7. Binary Stars (ZG: 12; CO: 7, 17) ⊙ Luminosity of Sun L =3.86 1026 W ⊙ × 7.1 Classification Effective temperature of Sun Teff =5780K 6 Central temperature of Sun Tc =15.6 10 K 7.2 The Binary Mass Function × Distance to the Galactic centre R0 =8.0kpc 7.3 The Roche Potential 1 Velocity of Sun about Galactic centre V0 =220kms− 7.4 Binary Mass Transfer Diameter of Galactic disc = 50 kpc 7.5 Interacting Binaries (Supplementary) Mass of Galaxy = 7 1011 M × ⊙ Appendices (Supplementary Material) A. Brown Dwarfs (ZG: 17-1E) B. Planets (ZG: 7-6; CO: 18.1) C. The Structure of the Sun and The Solar Neutrino Problem (ZG: P5-11, 10, 16-1D; CO: 11.1) D. Star Formation (ZG: 15.3; CO: 12) E. Gamma-Ray Bursts (ZG: 16-6; CO: 25.4) Page 6 Summary of Equations Opacity: Thomson (Electron) Scattering: Equation of Stellar Structure 2 1 κ =0.020 m kg− (1 + X)(page69) Equation of Hydrostatic Equilibrium: Kramer’s Opacity: dPr GMrρr 3.5 = (page 45) κ ρ T − (page 69) dr − r2 ∝ Low-Temperature Opacity: Equation of Mass Conservation: κ ρ1/2 T 4 (page 69) dMr 2 ∝ =4πr ρr (page 45) dr Energy Generation Rates (Rough!) Energy Conservation (no gravitational energy): PP Burning: 2 4 dL εPP ρ XH T (page 79) r =4πr2ρ ε (page 52) ∝ dr r r CNO Burning: 20 Energy Transport (Radiative Diffusion Equation): ε ρ XH XCNO T (page 79) CNO ∝ 4ac dT L = 4πr2 T 3 (page 55) Helium Burning (triple α): r − 3κρ dr 3 2 30 ε3 X ρ T (page 82) Energy Transport by Convection, Convective Stability: α ∝ He dT γ 1 T dP Stellar Timescales = − (page 57) dr γ P dr Dynamical Timescale: 1 Constitutive Relations tdyn (page 48) ≃ √4Gρ Equation of State, Ideal Gas: 3 1/2 30 min ρ/1000 kg m− − ρ ∼ P = NkT = kT (page 65) # $ µmH Thermal (Kelvin-Helmholtz) Timescale): Equation of State, Radiation Pressure: GM 2 tKH (page 51) ≃ 2RL 1 4 P = aT (page 66) 7 2 1 1 3 1.5 10 yr (M/M ) (R/R )− (L/L )− ∼ × ⊙ ⊙ ⊙ Equation of State, Electron Degeneracy (T =0K): Nuclear Timescale: 5/3 2 ρ tnuc Mc/M η (Mc )/L (page 52) P = K1 (page 66) ≃ !µemH " 10 3 10 yr (M/M )− (non-relativistic degeneracy) ∼ ⊙ (Radiative) Diffusion Timescale: ρ 4/3 P = K2 (page 67) l R2 ! " s µemH tdiff = N (page 53) × c ≃ lc (relativistic degeneracy) Notes: Notes: Page 8 Derived Relations Central Temperature Relation (for Ideal Gas): Miscellaneous Equations Distance Modulus: GMs µmH kTc (page 46) ≃ Rs (m M) =5log(D/10pc) (page 12) − V Virial Theorem: Absolute V Magnitude: 3(γ 1)U + Ω =0 (page50) − MV = 2.5logL/L +4.72 + B.C. + AV (page 12) Mass–Luminosity Relation (for stars 1 M ): − ⊙ ∼ ⊙ 4 Salpeter Initial Mass Function (IMF): M L L (page 85) ! " 2.35 ≃ ⊙ M f(M)dM M − dM (page 15) ⊙ ∝ Mass–MS Lifetime Relation (for stars 1 M ): ∼ ⊙ Black-Body Relation: 2 4 3 L =4π R σTeff (page 17) 10 M − s TMS 10 yr (page 85) ≃ ! M " Kepler’s Law: ⊙ 2 3 2π Mass–Radius Relation for White Dwarfs (non-relativistic): a = G(M1 + M2)(page25) ! P " 1 5/3 1/3 Notes: R (µemH) M − (page 98) ∝ me Chandrasekhar Mass for White Dwarfs: 2 2 MCh =1.457 M (page 99) !µe " ⊙ Schwarzschild Radius (Event Horizon) for Black Holes: 2GM M RS = 3km (page 112) c2 ≃ !M " ⊙ Notes: Page 10 STELLAR STRUCTURE AND EVOLUTION 1.1 LUMINOSITY (ZG: 11; CO: 3.1) 1. OBSERVABLE PROPERTIES OF STARS (‘power’, [J/s=W]) Basic large-scale observable properties: ∞ 2 ∞ Ls = L d = 4 Rs F d Luminosity %0 %0 Surface temperature where F is the radiative flux at wavelength at the stellar surface, Radius R the stellar radius. Energy may also be lost in the form of Mass s neutrinos or by direct mass loss (generally unobservable). Further observable: Astronomers measure: Spectrum ... yields information about surface chemical composition and gravity 2 f =(Rs/D) F at Earth’s surface Evidence from: To obtain L we must know the star’s distance D and correct • Individual stars for: • Binary systems ◃ absorption in the Earth’s atmosphere (standard • methods) Star clusters....these reveal how stars evolve with time • ◃ absorption in interstellar space (negligible for nearby stars) Nuclear physics...energy source, synthesis of heavy elements • Measurements from the Hipparcos satellite (1989–1993) have • No direct information about physical conditions in stellar interiors yielded parallaxes accurate to 0.002 arcsec for about 100,000 (except from helioseismology and stars. The largest stellar parallax (Proxima Centauri) is 0.765 solar neutrinos) arcsec. No direct evidence for stellar evolution......typical timescale 106 109 − Notes: years.......(except for a few very unusual stars and supernovae) Notes: Page 12 THE UBV SYSTEM the UBV system (ultraviolet, blue, visual) which can be extended 1.2 STELLAR MAGNITUDES (ZG: 11; CO: 3.2, 3.6) • into the red, infrared (RI) measure stellar flux (i.e. f = L/4 D2, L: luminosity, D: distance) • 26 3 2 ◃ for Sun: L = 3.86 10 W, f = 1.360 10 Wm− (solar ⊙ × × constant) ◃ luminosity measurement requires distance determination (1A.U. = 1.50 1011 m) × define apparent magnitudes of two stars, m1,m2,by • m1 m2 = 2.5logf2/f1 − zero point: Vega (historical) m = 26.82 • → ⊙ − to measure luminosity define absolute magnitude M to be the • approximate notation for magnitudes apparent magnitude of the object if it were at a distance 10 pc region apparent absolute solar value (1 pc = 3.26 light years = 3.09 1016 m) × ultraviolet U or mU MU 5.61 define bolometric magnitude as the absolute magnitude corre- • sponding to the luminosity integrated over all wavebands; for blue B or mB MB 5.48 the Sun Mbol = 4.72 visual V or mV MV 4.83 ⊙ (near yellow) in practice, the total luminosity is difficult to measure because • of atmospheric absorption and limited detector response colours (colour indices): relative magnitudes in different wave- define magnitudes over limited wavelength bands • • length bands, most commonly used: B V, U B − − Notes: define bolometric correction: B.C.
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