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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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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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  • Observational Evidence for Stellar Mass Black Holes

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    J. Astrophys. Astr. (1999) 20, 197–210 Observational Evidence for Stellar Mass Black Holes Tariq Shahbaz, University of Oxford, Department of Astrophysics, Nuclear Physics Building, Keble Road, Oxford, 0X1 3RH, England. e-mail:[email protected] Abstract. I review the evidence for stellar mass black holes in the Galaxy. The unique properties of the soft X-ray transient (SXTs) have provided the first opportunity for detailed studies of the mass-losing star in low-mass X-ray binaries. The large mass functions of these systems imply that the compact object has a mass greater than the maximum mass of a neutron star, strengthening the case that they contain black holes. The results and techniques used are discussed. I also review the recent study of a comparison of the luminosities of black hole and neutron star systems which has yielded compelling evidence for the existence of event horizons. Key words. X-ray sources—black holes. 1. Black holes and the soft X-ray transients It is widely believed that black holes power AGN and some X-ray binaries, and that they inhabit the nuclei of many normal galaxies. The compulsion to believe that black holes are physical objects is driven by our faith in general relativity and the idea that a black hole is a logical end point of stellar and galactic evolution. The proof for the existence of stellar black holes has been a subject of considerable effort for the last three decades, since the first dynamical mass estimate for the compact object in Cyg X-l (Webster & Murdin 1972; Bolton 1972).
  • Millisecond Pulsars, Their Evolution and Applications

    Millisecond Pulsars, Their Evolution and Applications

    J. Astrophys. Astr. (September 2017) 38:42 © Indian Academy of Sciences DOI 10.1007/s12036-017-9469-2 Review Millisecond Pulsars, their Evolution and Applications R. N. MANCHESTER CSIRO Astronomy and Space Science, P.O. Box 76, Epping, NSW 1710, Australia. E-mail: [email protected] MS received 15 May 2017; accepted 28 July 2017; published online 7 September 2017 Abstract. Millisecond pulsars (MSPs) are short-period pulsars that are distinguished from “normal” pulsars, not only by their short period, but also by their very small spin-down rates and high probability of being in a binary system. These properties are consistent with MSPs having a different evolutionary history to normal pulsars, viz., neutron-star formation in an evolving binary system and spin-up due to accretion from the binary companion. Their very stable periods make MSPs nearly ideal probes of a wide variety of astrophysical phenomena. For example, they have been used to detect planets around pulsars, to test the accuracy of gravitational theories, to set limits on the low-frequency gravitational-wave background in the Universe, and to establish pulsar-based timescales that rival the best atomic-clock timescales in long-term stability. MSPs also provide a window into stellar and binary evolution, often suggesting exotic pathways to the observed systems. The X-ray accretion- powered MSPs, and especially those that transition between an accreting X-ray MSP and a non-accreting radio MSP, give important insight into the physics of accretion on to highly magnetized neutron stars. Keywords. Pulsars—general—stars: evolution—gravitation. 1. Introduction pulsar was found in September 1982 at Arecibo Obser- vatory in a very high time-resolution search of the The first pulsars discovered (Hewish et al.
  • The Mass-Ratio Distribution of Spectroscopic Binaries Henri M.J

    The Mass-Ratio Distribution of Spectroscopic Binaries Henri M.J

    The mass-ratio distribution of spectroscopic binaries Henri M.J. Boffin ESO, Vitacura, Santiago, Chile. [email protected] Introduction The distribution of orbital elements, and in particular the orbital period and the eccentricity, can reveal much about the formation mechanisms of binary systems as well as their subsequent evolution. In the same vein, the distributions of the masses of the two components, M1 and M2, or similarly, of M1 and the mass ratio, q=M2/M1, are clues to critical questions related to binaries: did the binaries form through random pairing (in which case the mass of the individual components would be drawn from the IMF)? Does the mass ratio distribution depend on the primary mass (as does apparently the multiplicity of stars)? How will the systems evolve (some systems being possible only if the mass ratio is close to one, i.e. the system is made of quasi-twins)? How do families of stars compare to each other? It is thus a reasonable thing to do to try to estimate for different samples of binary stars the distribution of their mass ratios and many attempts have been done in the literature. I will hereby review some of the pitfalls of such endeavours. 1. Spectroscopic binaries and exoplanets We can distinguish between single- and double-lined spectroscopic binary (SB1 and SB2), depending on whether we can see the radial velocity variations of one or both components. The distinction will be due to the magnitude difference between the components, as too large a difference impedes the detection of the secondary star.