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Home , Asymptotic giant branch, Hayashi track, Luminosity, Main sequence

(2) low- : ideal-gas law, Kramer’s 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 • mass–radius relationship • main-sequence phase: core burning phase . central determined by characteristic . zero-age (ZAMS): homogeneous nuclear-burning temperature (hydrogen fusion: composition 7 8 Tc ∼ 10 K; fusion: Tc ∼ 10 K) Scaling relations for main-sequence stars . from (3) ⇒ R ∝ M (in reality R ∝ M0.6−0.8) • use dimensional analysis to derive scaling relations (3) very massive stars: pressure, electron (relations of the form L ∝ M ) scattering opacity, i.e. 1/2 • replace differential 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 • power-law index in mass– relationship de- • → P ∼ (1) R4 creases from ∼ 5 (low-mass) to 3 (massive) and 1 (very R4T4 massive) • radiative transfer → L ∝ (2) ¢ M M 4 • to derive luminosity–mass relationship, specify • near a : L ' L¯    M¯  equation of state and opacity law • main-sequence lifetime: TMS ∝ M/L −3 (1) massive stars: ideal-gas law, electron scattering 10 M typically: TMS = 10 yr   opacity, i.e.  M¯ 

¡ kT M • pressure is inverse proportional to the mean molecular ¢ ∼   ¢ ' . P = kT 3 and Th = constant £ £ mH mH R  weight £ kT GM

⇒ ∼ (3) . higher £ (fewer particles) implies higher tempera-

£ mH R ture to produce the same pressure, but T is fixed 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 

• opacity at low depends strongly on    

  § 

¢ ¢  ∝ ¡ £

(for bound-free opacity: Z) ¤¦¥ 

¨ ©  

. low-metallicity stars are much more luminous at a  given mass and have proportionately shorter - . 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 transport zone by radiation just below surface

N.B. Tc increases with Ms; ¡ c decreases with Ms.

• Hydrogen-burning limit: Ms ' 0.08 M¯ . low-mass objects (brown dwarfs) do not burn hy- drogen; they are supported by electron degeneracy

• maximum mass of stars: 100 – 150 M¯  • Giants, supergiants and 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 • observationally new-born stars appear as embedded the core before becoming degenerate) /T Tauri stars near the 6 • 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 → becomes a • 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 • energy source: hydrogen burning (4 H→ 4He) envelope → few stars observed in transition region ()

→ mean molecular weight £ increases in core from 0.6 to 1.3 → R, L and Tc increase (from

Tc ∝ £ (GM/R)) −3 10 M • lifetime: TMS ' 10 yr    M¯  after hydrogen exhaustion: • formation of isothermal core • hydrogen burning in shell around inert core (shell- burning phase)

→ growth of core until Mcore/M ∼ 0.1 (Sch¨onberg-) . 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 . ) • energy source: hydrogen burning in thin shell above the degenerate core giant envelope (convective) (R= 10 - 500 R O. ) H-burning shell • core mass grows → temperature in shell increases → luminosity increases → star ascends red-giant branch

• : 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

• when the core mass reaches Mc ' 0.48 M¯ → ignition of helium → helium flash 6.2.5 THE (HB) 6.2.4 log L asymptotic • 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 ()] 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

• runaway stops when matter • 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 • H burning in shell (usually the dominant energy • disruption of star? source) . energy generated in • lifetime: ∼ 10 % of main-sequence lifetime runaway: (lower efficiency of He burning, higher luminosity) Mburned . E = kTFermi • 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 • after exhaustion of central He 42 Mburned ¡

   

→ E ∼ 2 × 10 J ( £ ' 2) 0.1 M 109 kg m−3  ¯    → core contraction (as before) → degenerate core

. compare E to the binding energy of the core → 2 43 −2 Ebind ' GMc/Rc ∼ 10 J (Mc = 0.5 M¯; Rc = 10 R¯)

→ expect significant 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. • H burning and He burning (in thin shells) • 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

• low-/intermediate-mass stars (M ∼< 8 M¯) do not ex- perience nuclear burning beyond helium burning • 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 infinity without requiring energy source . complication: radiative losses

• after ejection: log L rapid transition hot CO core is exposed planetary and ionizes the ejected shell ejection → phase (lifetime AGB ∼ 104 yr) • 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

• P ∼ Pe ∝ ( ¡ / £ emH) ∼ GM /R

1 5/3 −1/3

→ R ∝ ( £ emH) M me

Mass ( M¯) Radius ( R¯) • note the negative exponent 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 0.50 § 0.05 0.0115 § 0.0012

Relativistic degeneracy (when pFe ∼ mec) • first white dwarf discovered: Sirius B (companion of 4/3 2 4 • 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 ¡ Teff ) R ∼ 10 R¯ ∼ R⊕ 9 −3

→ ¡ ∼ 10 kg m • Chandrasekhar (Cambridge 1930) . white dwarfs are supported by electron degeneracy pressure

. white dwarfs have a maximum mass of 1.4 M¯ • most white dwarfs have a mass very close to M ∼ 0.6 M¯: MWD = 0.58 § 0.02 M¯ • most are made of and (CO white dwarfs) • some are made of He or O-Ne-Mg THE CHANDRASEKHAR MASS

• consider a star of radius R containing N Fermions (electrons or neutrons) of mass mf

• the mass per Fermion is £ f mH ( £ f = mean molecular weight per Fermion) → number density n ∼ N/R3 → volume/Fermion 1/n • Heisenberg uncertainty principle 3 1/3 [ x p ∼ h¯] → typical momentum: p ∼ h¯ n → Fermi energy of relativistic particle (E = pc) hc¯ N1/3 E ∼ h¯ n1/3 c ∼ f R • gravitational energy per Fermion

GM( £ f mH)

E ∼ − , where M = N £ m g R f H → total energy (per particle) 1/3 2

hcN¯ GN( £ m ) E = E + E = − f H f g R R • stable configuration has minimum of total energy • if E < 0, E can be decreased without bound by de- creasing R → no equilibrium → • maximum N, if E = 0 hc¯ 3/2 → ∼   ∼ × 57 Nmax 2 2 10

G( £ mH)   f 

Mmax ∼ Nmax ( £ emH) ∼ 1.5 M¯ Chandrasekhar mass for white dwarfs

2 2 MCh = 1.457   M¯

 £ e 