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Main-Sequence Phase: Hydrogen Core Burning Phase

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(2) low-mass stars: ideal-gas law, Kramer's opacity law, ¡3:5 ¢ i.e. / ¡ T 6.1 THE STRUCTURE OF MAIN-SEQUENCE STARS 7:5 5:5 £ M (ZG: 16.2; CO 10.6, 13.1) ) L / R0:5 ² mass{radius relationship ² main-sequence phase: hydrogen core burning phase . central temperature determined by characteristic . zero-age main sequence (ZAMS): homogeneous nuclear-burning temperature (hydrogen fusion: composition 7 8 Tc » 10 K; helium fusion: Tc » 10 K) Scaling relations for main-sequence stars . from (3) ) R / M (in reality R / M0:6¡0:8) ² use dimensional analysis to derive scaling relations (3) very massive stars: radiation pressure, electron (relations of the form L / M ) scattering opacity, i.e. 1=2 ² replace di®erential equations by characteristic quanti- 1 4 M 3 . P = aT ! T » ) L / M ties (e.g. dP=dr » P=R, ¡ » M=R ) 3 R GM2 ² power-law index in mass{luminosity relationship de- ² hydrostatic equilibrium ! P » (1) R4 creases from » 5 (low-mass) to 3 (massive) and 1 (very R4T4 massive) ² radiative transfer ! L / (2) ¢ M M 4 ² to derive luminosity{mass relationship, specify ² near a solar mass: L ' L¯ 0 1 @ M¯ A equation of state and opacity law ² main-sequence lifetime: TMS / M=L ¡3 (1) massive stars: ideal-gas law, electron scattering 10 M typically: TMS = 10 yr 0 1 opacity, i.e. @ M¯ A ¡ kT M ² pressure is inverse proportional to the mean molecular ¢ » 0 1 ¢ ' . P = kT 3 and Th = constant £ £ mH mH @R A weight £ kT GM ) » (3) . higher £ (fewer particles) implies higher tempera- £ mH R ture to produce the same pressure, but T is ¯xed 4 3 c £ M 7 . substituting (3) into (2): L / (hydrogen burning (thermostat): Tc » 10 K) ¢ Th . during H-burning £ increases from » 0:62 to » 1:34 ! radius increases by a factor of » 2 (equation [3]) Turnoff Ages in Open Clusters ² opacity at low temperatures depends strongly on § ¢ ¢ / ¡ £ metallicity (for bound-free opacity: Z) ¤¦¥ ¨ © . low-metallicity stars are much more luminous at a given mass and have proportionately shorter life- times . mass-radius relationship only weakly dependent on metallicity blue stragglers ! low-metallicity stars are much hotter . subdwarfs: low-metallicity main-sequence stars ly- ing just below the main sequence General properties of homogeneous stars: Upper main sequence Lower main sequence (Ms > 1:5 M¯) (Ms < 1:5 M¯) core convective; well mixed radiative CNO cycle PP chain ¢ electron scattering Kramer's opacity ¡3:5 ¢ ¢ ' 3 ¡ T surface H fully ionized H/He neutral energy transport convection zone by radiation just below surface N.B. Tc increases with Ms; ¡ c decreases with Ms. ² Hydrogen-burning limit: Ms ' 0:08 M¯ . low-mass objects (brown dwarfs) do not burn hy- drogen; they are supported by electron degeneracy ² maximum mass of stars: 100 { 150 M¯ ² Giants, supergiants and white dwarfs cannot be chemically homogeneous stars supported by nuclear burning 6.2 THE EVOLUTION OF LOW-MASS STARS . contraction stops when the core density becomes (M »< 8 M¯) (ZG: 16.3; CO: 13.2) high enough that electron degeneracy pressure can support the core 6.2.1 Pre-main-sequence phase (stars more massive than » 2 M¯ ignite helium in ² observationally new-born stars appear as embedded the core before becoming degenerate) protostars/T Tauri stars near the stellar birthline 6 ² where they burn deuterium (Tc » 10 K), often still while the core contracts and becomes degenerate, the accreting from their birth clouds envelope expands dramatically ! star becomes a red giant ² after deuterium burning ! star contracts . the transition to the red-giant branch is not well ! Tc » ( £ mH=k) (GM=R) increases until hydrogen 7 burning starts (Tc » 10 K) ! main-sequence phase understood (in intuitive terms) . for stars with M »> 1:5 M¯, the transition occurs 6.2.2 Core hydrogen-burning phase very fast, i.e. on a thermal timescale of the ² energy source: hydrogen burning (4 H! 4He) envelope ! few stars observed in transition region (Hertzsprung gap) ! mean molecular weight £ increases in core from 0.6 to 1.3 ! R, L and Tc increase (from Tc / £ (GM=R)) ¡3 10 M ² lifetime: TMS ' 10 yr 0 1 @ M¯ A after hydrogen exhaustion: ² formation of isothermal core ² hydrogen burning in shell around inert core (shell- burning phase) ! growth of core until Mcore=M » 0:1 (SchÄonberg-Chandrasekhar limit) . core becomes too massive to be supported by thermal pressure ! core contraction ! energy source: gravitational energy ! core becomes denser and hotter Evolutionary Tracks (1 to 5M O. ) 6.2.3 THE RED-GIANT PHASE degenerate He core -2 (R core = 10 R O . ) ² energy source: hydrogen burning in thin shell above the degenerate core giant envelope (convective) (R= 10 - 500 R O . ) H-burning shell ² core mass grows ! temperature in shell increases ! luminosity increases ! star ascends red-giant branch ² Hayashi track: log L vertical track in H-R diagram Hayashi forbidden . no hydrostatic region solutions for very cool giants . Hayashi forbidden region ¡ 4000 K log T (due to H opacity) eff ² when the core mass reaches Mc ' 0:48 M¯ ! ignition of helium ! helium flash 6.2.5 THE HORIZONTAL BRANCH (HB) 6.2.4 HELIUM FLASH log L asymptotic ² ignition of He under degenerate conditions giant branch (for M »< 2 M¯; core mass » 0:48 M¯) He flash . i.e. P is independent of T ! no self-regulation RR Lyrae [in normal stars: increase in T ! decrease in ¡ (expansion) ! decrease in T (virial theorem)] horizontal branch red-giant branch . in degenerate case: nuclear burning ! increase in T ! more nuclear burning ! further increase in T ! thermonuclear runaway log T eff ² runaway stops when matter ² He burning in center: conversion of He to mainly C becomes non-degenerate 12 16 n(E) and O ( C + ¡ ! O) (i.e. T » TFermi) kT = E Fermi Fermi ² H burning in shell (usually the dominant energy ² disruption of star? source) . energy generated in ² lifetime: » 10 % of main-sequence lifetime runaway: (lower e±ciency of He burning, higher luminosity) Mburned . E = kTFermi ² RR Lyrae stars are pulsationally unstable £ mH characteristic E < number of | {z } Fermi E (L, B ¡ V change with periods » 1 d) | {z } energy particles easy to detect ! popular distance indicators 2=3 ² after exhaustion of central He 42 Mburned ¡ 0 1 0 1 ! E » 2 £ 10 J ( £ ' 2) 0:1 M 109 kg m¡3 @ ¯ A @ A ! core contraction (as before) ! degenerate core . compare E to the binding energy of the core ! asymptotic giant branch 2 43 ¡2 Ebind ' GMc=Rc » 10 J (Mc = 0:5 M¯; Rc = 10 R¯) ! expect signi¯cant dynamical expansion, but no disruption (tdyn » sec) ! core expands ! weakening of H shell source ! rapid decrease in luminosity ! star settles on horizontal branch 6.2.6 THE ASYMPTOTIC GIANT BRANCH (AGB) Planetary Nebulae with the HST degeneratedegenerate HeCO core core (R = 10 -2 R ) core O. ² H burning and He burning (in thin shells) ² H/He burning do not occur simultaneous, giant envelope (convective) but alternate ! (R= 100 - 1000 R O . ) He-burning shell thermal pulsations H-burning shell ² low-/intermediate-mass stars (M »< 8 M¯) do not ex- perience nuclear burning beyond helium burning ² evolution ends when the envelope has been lost by stellar winds . superwind phase: very rapid mass loss ¡4 ¡1 (M_ » 10 M¯ yr ) . probably because envelope attains positive binding energy (due to energy reservoir in ionization energy) ! envelopes can be dispersed to in¯nity without requiring energy source . complication: radiative losses ² after ejection: log L rapid transition hot CO core is exposed planetary and ionizes the ejected shell nebula ejection ! planetary nebula phase (lifetime AGB » 104 yr) white dwarf ² CO core cools, becomes cooling sequence log T eff degenerate ! white dwarf 50,000 K 3,000 K 6.2.7 WHITE DWARFS (ZG: 17.1; CO: 13.2) Mass-Radius Relations for White Dwarfs Non-relativistic degeneracy 5=3 2 4 ² P » Pe / ( ¡ = £ emH) » GM =R 1 5=3 ¡1=3 ! R / ( £ emH) M me Mass ( M¯) Radius ( R¯) ² note the negative exponent Sirius B 1:053 § 0:028 0:0074 § 0:0006 ! R decreases with increasing mass 40 Eri B 0:48 § 0:02 0:0124 § 0:0005 ! ¡ increases with M Stein 2051 0:50 § 0:05 0:0115 § 0:0012 Relativistic degeneracy (when pFe » mec) ² ¯rst white dwarf discovered: Sirius B (companion of 4=3 2 4 ² P » Pe / ( ¡ = £ emH) » GM =R Sirius A) ! M independent of R . mass (from orbit): M » 1 M¯ ! 2 4 ¡2 existence of a maximum mass . radius (from L = 4 R ¡ Te® ) R » 10 R¯ » R© 9 ¡3 ! ¡ » 10 kg m ² Chandrasekhar (Cambridge 1930) . white dwarfs are supported by electron degeneracy pressure . white dwarfs have a maximum mass of 1:4 M¯ ² most white dwarfs have a mass very close to M » 0:6 M¯: MWD = 0:58 § 0:02 M¯ ² most are made of carbon and oxygen (CO white dwarfs) ² some are made of He or O-Ne-Mg THE CHANDRASEKHAR MASS ² consider a star of radius R containing N Fermions (electrons or neutrons) of mass mf ² the mass per Fermion is £ f mH ( £ f = mean molecular weight per Fermion) ! number density n » N=R3 ! volume/Fermion 1=n ² Heisenberg uncertainty principle 3 1=3 [ x p » h¹] ! typical momentum: p » h¹ n ! Fermi energy of relativistic particle (E = pc) hc¹ N1=3 E » h¹ n1=3 c » f R ² gravitational energy per Fermion GM( £ f mH) E » ¡ , where M = N £ m g R f H ! total energy (per particle) 1=3 2 hcN¹ GN( £ m ) E = E + E = ¡ f H f g R R ² stable con¯guration has minimum of total energy ² if E < 0, E can be decreased without bound by de- creasing R ! no equilibrium ! gravitational collapse ² maximum N, if E = 0 hc¹ 3=2 ! » 0 1 » £ 57 Nmax 2 2 10 BG( £ mH) C @ f A Mmax » Nmax ( £ emH) » 1:5 M¯ Chandrasekhar mass for white dwarfs 2 2 MCh = 1:457 0 1 M¯ @ £ e A.
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  • How Supernovae Became the Basis of Observational Cosmology

    How Supernovae Became the Basis of Observational Cosmology

    Journal of Astronomical History and Heritage, 19(2), 203–215 (2016). HOW SUPERNOVAE BECAME THE BASIS OF OBSERVATIONAL COSMOLOGY Maria Victorovna Pruzhinskaya Laboratoire de Physique Corpusculaire, Université Clermont Auvergne, Université Blaise Pascal, CNRS/IN2P3, Clermont-Ferrand, France; and Sternberg Astronomical Institute of Lomonosov Moscow State University, 119991, Moscow, Universitetsky prospect 13, Russia. Email: [email protected] and Sergey Mikhailovich Lisakov Laboratoire Lagrange, UMR7293, Université Nice Sophia-Antipolis, Observatoire de la Côte d’Azur, Boulevard de l'Observatoire, CS 34229, Nice, France. Email: [email protected] Abstract: This paper is dedicated to the discovery of one of the most important relationships in supernova cosmology—the relation between the peak luminosity of Type Ia supernovae and their luminosity decline rate after maximum light. The history of this relationship is quite long and interesting. The relationship was independently discovered by the American statistician and astronomer Bert Woodard Rust and the Soviet astronomer Yury Pavlovich Pskovskii in the 1970s. Using a limited sample of Type I supernovae they were able to show that the brighter the supernova is, the slower its luminosity declines after maximum. Only with the appearance of CCD cameras could Mark Phillips re-inspect this relationship on a new level of accuracy using a better sample of supernovae. His investigations confirmed the idea proposed earlier by Rust and Pskovskii. Keywords: supernovae, Pskovskii, Rust 1 INTRODUCTION However, from the moment that Albert Einstein (1879–1955; Whittaker, 1955) introduced into the In 1998–1999 astronomers discovered the accel- equations of the General Theory of Relativity a erating expansion of the Universe through the cosmological constant until the discovery of the observations of very far standard candles (for accelerating expansion of the Universe, nearly a review see Lipunov and Chernin, 2012).