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a) Chandrasekhar-Schönberg Limit16. Hydrogen Shell Burning
After ignition of H-burning in shell, central He-core is ‘inert’:
Tc too low for ignition of He (§17)⇒ no nuclear energy generation in core⇒ dT/dr ~ 0 in core (in TE)
Properties of He core crucial for post-main-sequence evolution
Consider idealized situationCore: temperature T0
mass Mcradius Rc
4 3 130
2
0 0
0 10
3 2
2
4
π γ
μ
R P E E
E GMR
E N kT M
P C T MR
C MR
c g i
gc
c
ic
c
c
c
c
c
= + −
∝ −
∝
U
V|||
W|||
⇒ = −
( )
R C MT
P C TMc
c
c c,max ,max,= =3
00 4
04
4 2μ
Pressure P0 at core-surface, follows from virial theorem:
P MR
T T MR
P C TM
env
env
envenv
∝
= ∝
UV|
W|⇒ =
2
4
0
504
4 2μ μ
This has a maximum at:
Envelope: pressure Penvtemperature Tenvat radius Rc
Homology:
2
Pressure must be continuous at core/envelope boundaryso that P0 = Penv ⇒ this is only possible when Penv ≤ P0,max
⇒
Chandrasekhar & Schönberg (1942):
If qcore ≤ qCS: isothermal core is capable of supporting the weight of the envelope
If qcore > qCS, then core cannot support the envelope⇒ it must contract ⇒ release of gravitational energy⇒ temperature gradient ⇒ no longer isothermal
Quantitative: qCS ~ 0.10 for a He-core and normal envelope
See KW §30.5 for more details
C TM
C TM
q MM
qenv c c
cCS5
04
4 2 404
4 2 0μ μ≤ ⇔ = ≤
qCSc
env
≅ 0 372
2. μμ
b) Evolution of the CoreMore realistic treatment of core includes (partial) degeneracyof electron gas ⇒ for M > 1.4M two stable solution branches:
Core ~ non-degenerate and qcore < qCSCore ~ degenerate and qcore > q1
Which branch is ‘selected’by a star depends on itsevolutionary history
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M > 6Mqcore > qCS at ignition of H-burning shell ⇒ core cannot become isothermal⇒ continuing gravitational contraction on τKH
⇒ Tc rises until He ignites at ~108 K
M < 6Mqcore < qCS at ignition of H-burning shell ⇒ isothermal core develops with Tc ~ T(H-burning shell)As H is burned, qcore steadily increases and Rc decreases slightly
so that ρc increases and partial degeneracy increases
2.5 M < M < 6 Mqcore > qCS before core becomes fully degenerate⇒ rapid core contraction until second stable branch is reached,
followed by slow evolution until He ignites
M < 2.5MCore degenerates before qcore > qCS ⇒ qCS does not applyDegeneracy pressure allows qcore to become very largeand remain in thermal equilibriumAs qcore increases, core contracts slightly and Tc rises slowly
M < 0.33 MTc never exceeds 108 K H shell continues to burn outwardsResult is a degenerate star composed of He: He white dwarf
0.33 M < M < 2.5MCores all evolve to about the same degenerate stateWhen Tc exceeds 108K, He ignites under degenerate conditions, leading to the helium flash (§17)
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Evolution of the core in the log Tc versus log ρc plane
c) Evolution of the EnvelopeIgnition of H-shell: R increases rapidly ⇒ Teff decreases
⇒ deep convective envelope forms⇒ star approaches Hayashi line (§12h)
Teff cannot decrease further into Hayashi’s forbidden region, as star would adjust on τff ⇒ L must increase as R increases
Star ascends the Hayashi line ⇒ the red giant branch (RGB)
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Massive stars cross HRD rapidly: few stars in ‘Hertzsprung Gap’
d) Evolution of low-mass starsDegenerate He-core: mass Mc, radius Rc
Hydrogen envelope: chemical abundance XH
⇒ with EH the energy gain per unit mass of H
Since extended envelope is nearly weightless, properties ofshell source are mainly determined by Mc and Rc
Scaling relations for (§13e; KW§32.2)
Lead to: with α, β functions of a, b, λ,ν
Typical case: electron scattering: a=b=0CNO cycle λ=1, ν~13
Then:
&M LX Ec
H H
=
κ κ ε ε ρλ ν= =0 0P T Ta b ,
L M Rc c= α β
T MR
L MR
c
c
c
c0
7
16 3∝ ∝ /
6
Degenerate core is effectively a white dwarf (§21), so that themass-radius relation holds: Rc decreases as Mc increasesIn NR limit this gives: (cf §12; n=3/2 polytrope)
⇒
⇒ L increases strongly as core mass grows; Tc increases slowly
Tc=108K when Mc = 0.45M (independent of M) ⇒ helium flash
M R cc cst1 3/ =
L M T Mc c c∝ ∝−8 10 4 3/
Max T occursoff-center, dueto neutrino losses (§21)
Evolution of 1.3 M starConvection zone becomes very deep during H-shell burning phase, and reaches into previously mixed core ⇒
enriched materialis transported to surface
This is called the ‘first dredge-up’
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Homology: at fixed Mc, Rc: L ∝ μ7
⇒ when H-shell reaches deepest layer where surface convection penetratedμ drops suddenly
⇒ L drops (temporarily)
Corresponding evolutionary track in HR Diagram
H-shell burning phase ends with He-flash at tip of giant branch
17. Core Helium Burninga) Nuclear Physics: the triple-alpha reactionObservationsAside from 1H and 4He, most abundant: 12C, 16O
Theory4He made in Big Bang nucleosynthesis, and in starsSince nucleus with A=5 is unstable, not possible to make nuclei heavier than 4He via proton capture8Be is unstable: cannot simply combine 4He and 4He
Salpeter (1952): ‘three-body encounter’ 3α→12C (cf Öpik 1951)
Scheme: 2 924 8
8 4 12 12
He keV BeBe He C C
+ ↔+ → → +
RST * γ(endothermic)
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Encounter lasts ~10-21 secReaction is resonant ⇒Lifetime of 8Be ~10-16 secTc~108 K ⇒ E0 ~100 keV
⇒ Sufficient for further α-capture to result in 12C, but only if this reaction is also resonant (Hoyle 1954)Subsequently confirmed in laboratory experiment (§9)
Energy generation rate
Further α-captures:
3α-reaction: very delicate process: small changes in strength of nuclear interaction ⇒ no elements heavier than 4He
UV||
W||
Small equilibrium concentration of 8Be: 10-9
ε ε ρ νν= = − + ≈ −0 43 2
8 0
3 43 2 40 20X TT
.
.
12 16
16 20
C O
O Ne
+ →
+ →
α
α
at slightly higher Tc
this step is very slow
b) Helium Flash (M < 2.5 M )Nuclear ignition in normal gasIncrease in T ⇒ increase in P ⇒ expansion ⇒ decrease in T ⇒ stable equilibrium is reached with ε equal to energy loss
Ignition of He in degenerate core of low mass starε > 0 ⇒ increase in T but ~ no effect on P (as this is provided
by degenerate electrons) ⇒ no expansion and cooling⇒ ε increases ⇒ T increases ⇒ ε increases ⇒ …
⇒ Thermonuclear Runaway: 40% of He core → 12C in few sec (!)εc > 1013 εc ~ 1014 erg/gm/sec
~ Lgalaxy
All the released energy is used for internal heating ⇒ lifts degeneracy in the entire core ⇒ runaway endsThis is the Helium Flashdiscovered by Schwarzschild
lc OL≈ 1011
KW §32; HKT §2.5
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Evolution of 1.3 M star through the Helium FlashNeutrino cooling ⇒off-center ignition
Example in KWIgnition slightly too far outwards(m/M~0.3), due to inaccuracies in early ν-cooling rates (§21d)Results qualitatively correct(but caption of KW fig 32.6 is wrong!)
He initially burns in a shell, which is convectively unstable This is separated from the convective envelope; in between, the now-extinct H-burning shell; this will re-ignite later onSubsequently, He burning also in coreMost likely, no sign of He-flash on surface of starEntire process difficult to follow numerically
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c) Zero Age Horizontal BranchAssume:
During He flash no mixing between He core and matter beyond the edge of the H-burning shellMc not changed during the flash; uniform He-abundance (?)
Then: star in TE consisting ofConvective core in which 3α→12CSurrounded by re-ignited H-burning shellThese lie on sequence in HRD at L~100L , range in TeffLocation of star on ZAHB influenced by Mc, M, XCNO
CommentsFor Pop I clusters indeed often a clump of stars is found atL~100L on the giant branch: the Red Clump Pop II clusters have horizontal branches that often extend to(very) high values of Teff: low XCNO!
Models with He cores in the HR-Diagram
To get good description of globular cluster HRD: need M ~ 0.7M on ZAHB: these stars should still be on MS⇒ mass loss as star ascends the giant branch (see §18)
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d) Horizontal Branch Evolution3α→12C in convective core: evolution away from ZAHB
There are differences in the details of the tracks, depending e.g. on XCNO, but general evolution is in the direction of theHayashi line: Asymptotic Giant Branch (AGB, see §19)
Convective core and envelope, and two energy sources ⇒details of semi-convection and convective overshoot important
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e) Evolution of stars with 2.5 M < M < 10 MHe ignition is nearly explosive, but no flash occursExpansion of core ⇒ ρ, T in H-burning shell decrease
⇒ εH decreases in shellContraction of envelope: ⇒ εH not too smallResult:
L decreases H-burning shell still produces bulk of energyHe-core convective, contains about 5% of total massTeff increases slowly, until energy transport in envelope goes from convection to radiation ⇒ envelope shrinks rapidlyuntil TE is reached againHere second phase of core He burning commencesImportance of He-core for energy generation increases slowlyCore continues to expand in radius and envelope contractsWhen YHe ~ 0.25 core contracts again, and envelope expands
KW §31
Example:Evolution of9 M star