16. Hydrogen Shell Burning

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16. Hydrogen Shell Burning a) Chandrasekhar-Shönberg Limit After ignition of H-burning in shell, entral He-ore is inert : T too low for ignition of He ( 17) no nulear energy generation in ore dt/dr ~

1 16. Hydrogen Shell Burning a) Chandrasekhar-Shönberg Limit After ignition of H-burning in shell, entral He-ore is inert : T too low for ignition of He ( 17) no nulear energy generation in ore dt/dr ~ 0 in ore (in TE) Properties of He ore ruial for post-main-sequene evolution Consider idealized situation Core: temperature T 0 mass M radius R Pressure P 0 at ore-surfae, follows from virial theorem: 3 4πRP 0 = Eg + 3( γ 1) Ei 2 GM TM 0 E P0 C1 C M g V = 3 2 R R R NkT 0 0 Ei M μ R,max U W This has a maximum at: C M P C T = 3, 0,max = 4 T μ M Envelope: pressure P env temperature T env at radius R Homology: 2 M U Penv 4 4 R T Penv C MV M Tenv = T = 0 5 μ μ 4 2 env 0 R W 2 4 1

2 Pressure must be ontinuous at ore/envelope boundary so that P 0 = P env this is only possible when P env P 0,max C 5 4 T M C T M q M M q = μ μ env Chandrasekhar & Shönberg (1942): If q ore q CS : isothermal ore is apable of supporting the weight of the envelope If q ore > q CS, then ore annot support the envelope it must ontrat release of gravitational energy temperature gradient no longer isothermal Quantitative: q CS ~ 0.10 for a He-ore and normal envelope See KW 30.5 for more details CS q CS μ 037. μ 2 2 env b) Evolution of the Core More realisti treatment of ore inludes (partial) degeneray of eletron gas for M > 1.4M two stable solution branhes: Core ~ non-degenerate and q ore < q CS Core ~ degenerate and q ore > q 1 Whih branh is seleted by a star depends on its evolutionary history 2

3 M > 6M q ore > q CS at ignition of H-burning shell ore annot beome isothermal ontinuing gravitational ontration on τ KH T rises until He ignites at ~10 8 K M < 6M q ore < q CS at ignition of H-burning shell isothermal ore develops with T ~ T(H-burning shell) As H is burned, q ore steadily inreases and R dereases slightly so that ρ inreases and partial degeneray inreases 2.5 M < M < 6 M q ore > q CS before ore beomes fully degenerate rapid ore ontration until seond stable branh is reahed, followed by slow evolution until He ignites M < 2.5M Core degenerates before q ore > q CS q CS does not apply Degeneray pressure allows q ore to beome very large and remain in thermal equilibrium As q ore inreases, ore ontrats slightly and T rises slowly M < 0.33 M T never exeeds 10 8 K H shell ontinues to burn outwards Result is a degenerate star omposed of He: He white dwarf 0.33 M < M < 2.5M Cores all evolve to about the same degenerate state When T exeeds 10 8 K, He ignites under degenerate onditions, leading to the helium flash ( 17) 3

4 Evolution of the ore in the log T versus log ρ plane ) Evolution of the Envelope Ignition of H-shell: R inreases rapidly T eff dereases deep onvetive envelope forms star approahes Hayashi line ( 12h) T eff annot derease further into Hayashi s forbidden region, as star would adjust on τ ff L must inrease as R inreases Star asends the Hayashi line the red giant branh (RGB) 4

5 Massive stars ross HRD rapidly: few stars in Hertzsprung Gap d) Evolution of low-mass stars Degenerate He-ore: mass M, radius R Hydrogen envelope: hemial abundane X H L &M = with E H the energy gain per unit mass of H XHEH Sine extended envelope is nearly weightless, properties of shell soure are mainly determined by M and R Saling relations for Lead to: α L = M R β a κ = κ PT b, ε = ε ρ λ T ν 0 0 with α, β funtions of a, b, λ,ν Typial ase: eletron sattering: a=b=0 CNO yle λ=1, ν~13 7 M M Then: T0 L 16/ 3 R R ( 13e; KW 32.2) 5

Degenerate ore is effetively a white dwarf ( 21), so that the mass-radius relation holds: R dereases as M inreases 13 / st In NR limit this gives: M R = (f 12; n=3/2 polytrope) L M T M 8 10 4/ 3 L

6 Degenerate ore is effetively a white dwarf ( 21), so that the mass-radius relation holds: R dereases as M inreases 13 / st In NR limit this gives: M R = (f 12; n=3/2 polytrope) L M T M / 3 L inreases strongly as ore mass grows; T inreases slowly T =10 8 K when M = 0.45M (independent of M) helium flash Max T ours off-enter, due to neutrino losses ( 21) Evolution of 1.3 M star Convetion zone beomes very deep during H-shell burning phase, and reahes into previously mixed ore enrihed material is transported to surfae This is alled the first dredge-up 6

7 Corresponding evolutionary trak in HR Diagram Homology: at fixed M, R : L μ 7 when H-shell reahes deepest layer where surfae onvetion penetrated μ drops suddenly L drops (temporarily) H-shell burning phase ends with He-flash at tip of giant branh 17. Core Helium Burning a) Nulear Physis: the triple-alpha reation Observations Aside from 1 H and 4 He, most abundant: 12 C, 16 O Theory 4 He made in Big Bang nuleosynthesis, and in stars Sine nuleus with A=5 is unstable, not possible to make nulei heavier than 4 He via proton apture 8 Be is unstable: annot simply ombine 4 He and 4 He Salpeter (1952): three-body enounter 3α 12 C (f Öpik 1951) Sheme: R S He + 92keV Be Be+ He C C + γ * 12 T (endothermi) 7

8 Enounter lasts ~10-21 se Reation is resonant Lifetime of 8 Be ~10-16 se T ~10 8 K E 0 ~100 kev U V W Small equilibrium onentration of 8 Be: 10-9 Suffiient for further α-apture to result in 12 C, but only if this reation is also resonant (Hoyle 1954) Subsequently onfirmed in laboratory experiment ( 9) Energy generation rate 3 2 ν ε = ε0 4ρ ν = X T T80. Further α-aptures: C+ α O at slightly higher T O+ α Ne this step is very slow 3α-reation: very deliate proess: small hanges in strength of nulear interation no elements heavier than 4 He b) Helium Flash (M < 2.5 M ) Nulear ignition in normal gas Inrease in T inrease in P expansion derease in T stable equilibrium is reahed with ε equal to energy loss Ignition of He in degenerate ore of low mass star KW 32; HKT 2.5 ε > 0 inrease in T but ~ no effet on P (as this is provided by degenerate eletrons) no expansion and ooling εinreases T inreases εinreases Thermonulear Runaway: 40% of He ore 12 C in few se (!) ε > ε ~ erg/gm/se l L O ~ L galaxy All the released energy is used for internal heating lifts degeneray in the entire ore runaway ends This is the Helium Flash disovered by Shwarzshild 8

9 Evolution of 1.3 M star through the Helium Flash Neutrino ooling off-enter ignition Example in KW Ignition slightly too far outwards (m/m~0.3), due to inauraies in early ν-ooling rates ( 21d) Results qualitatively orret (but aption of KW fig 32.6 is wrong!) He initially burns in a shell, whih is onvetively unstable This is separated from the onvetive envelope; in between, the now-extint H-burning shell; this will re-ignite later on Subsequently, He burning also in ore Most likely, no sign of He-flash on surfae of star Entire proess diffiult to follow numerially 9

10 ) Zero Age Horizontal Branh Assume: During He flash no mixing between He ore and matter beyond the edge of the H-burning shell M not hanged during the flash; uniform He-abundane (?) Then: star in TE onsisting of Convetive ore in whih 3α 12 C Surrounded by re-ignited H-burning shell These lie on sequene in HRD at L~100L, range in T eff Loation of star on ZAHB influened by M, M, X CNO Comments For Pop I lusters indeed often a lump of stars is found at L~100L on the giant branh: the Red Clump Pop II lusters have horizontal branhes that often extend to (very) high values of T eff : low X CNO! Models with He ores in the HR-Diagram To get good desription of globular luster HRD: need M ~ 0.7M on ZAHB: these stars should still be on MS mass loss as star asends the giant branh (see 18) 10

11 d) Horizontal Branh Evolution 3α 12 C in onvetive ore: evolution away from ZAHB There are differenes in the details of the traks, depending e.g. on X CNO, but general evolution is in the diretion of the Hayashi line: Asymptoti Giant Branh (AGB, see 19) Convetive ore and envelope, and two energy soures details of semi-onvetion and onvetive overshoot important 11

12 e) Evolution of stars with 2.5 M < M < 10 M KW 31 He ignition is nearly explosive, but no flash ours Expansion of ore ρ, T in H-burning shell derease ε H dereases in shell Contration of envelope: ε H not too small Result: L dereases H-burning shell still produes bulk of energy He-ore onvetive, ontains about 5% of total mass T eff inreases slowly, until energy transport in envelope goes from onvetion to radiation envelope shrinks rapidly until TE is reahed again Here seond phase of ore He burning ommenes Importane of He-ore for energy generation inreases slowly Core ontinues to expand in radius and envelope ontrats When Y He ~ 0.25 ore ontrats again, and envelope expands Example: Evolution of 9 M star 12

13 Example: Internal evolution of 5 M star 13

14 When X C suffiiently large: α+ 12 C 16 O important C/O ore develops Blue loop Extent depends on mass M Sensitive to details of onvetive overshooting Star traverses Cepheid instability strip more than one ( 22) 14

Stellar Physics lecture 6: stellar evolution

Stellar Physics lecture 6: stellar evolution Stellar Physis leture 6: stellar evolution 辜品高 gu@asiaa.sinia.edu.tw 01//16, 18 1 pre-main-sequene stars Calvet 00 vertial: Hayashi trak H- opaity ats as a thermostat horizontal: Henyey trak, radiative