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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

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Example:Internal evolutionof 5 M star

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When XC sufficiently large: α+12C→16O importantC/O core develops

Blue loopExtent depends on mass MSensitive to details of convective overshootingStar traverses Cepheid instability strip more than once (§22)