(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.