6 . P o s t- M a in -S e q u e n c e E v o lu ti o n

(1)

6–1

6 . P o s t- M a in -S e q u e n c e E v o lu ti o n

6–2 Post-Main-SequenceEvolution1

H R d ia g ra m o f th e g lo b u la r c lu s te r M 3

•Starsnotdistributeduni- formlyinHRdiagram.Ma- jorgroupingsalongmain sequenceandredgiant branch •Majorgroupingsindicate slowevolutionaryphases, i.e.stablephasesofstellar evolution •Obviously,certainconfigu- rationsofstellarmaterialare morestablethanothers

6–3 Post-Main-SequenceEvolution2

T h e O v e ra ll P ic tu re o f th e e v o lu ti o n o f a s o la r- lik e s ta r

Summary:Evolutionofasolar-likestar: 1.Mainsequence(MS)=coreHburning 2.Redgiantbranch(RGB) =Hshellburning 3.TipofRGB:coreheliumflash =heliumignitionindegenerateelectrongas 4.Horizontalbranch(HB) =coreHeburning 5.Asymptoticgiantbranch(AGB)=2shell burning(H+He)TipofAGB:envelopeejection throughdustformation&pulsations 6.Planetarynebula(PN):hotstarexcitesthe ejectedenvelopetoshine 7.whitedwarf=degenerateC/Oremnant 6–4 Post-Main-SequenceEvolution:theredgiantbranch1

P o s t- M a in -S e q u e n c e E v o lu ti o n : th e re d g ia n t b ra n c h

•noenergyproductionbyH-burning (orothernuclearprocesses)inthe core •corecontracts,envelopeexpands •thestarformsanextendedcon- vectiveenvelope(keywordsionisa- tionandopacity) •thestarevolvestotheHayashi lineintheHRdiagram •thestarbecomesaredgiant(first redgiantbranch–RGBorFGB)

(2)

6–5 Post-Main-SequenceEvolution:theredgiantbranch2

S tr u c tu re o f a fi rs t re d g ia n t b ra n c h s ta r

M ir ro r p ri n c ip le

•Hburninginthecoreceasesattheendofthemainsequencephase •aisothermalHecoreisformed,surroundedbyaHburningshell •theHecorecontracts •Mirrorprinciple:shellburningzonesactas“mirrors”,whichreversethe radialmovementsinsideandoutside(semi-empiricalrule)

H a y a s h i lin e (s ) in th e H R d ia g ra m

MeaningoftheHayashiline •TheHayashilinesarethepositionoffully convectivestarsintheHRdiagram •Hayashitheory •Redgiantsarenotfullyconvective,butthe completeenvelopeis.Thecorecontain- ingalargefractionofthemassistiny,the convectiveenvelopemakesupalmostthe wholestarintermsofvolume/radius •⇒veryclosetoHayashiline. 6–8 Post-Main-SequenceEvolution:theredgiantbranch5

H a y a s h i lin e : fu lly c o n v e c ti v e s ta rs

Fullyionised,noradiationpressure,efficientconvection (=adiabatictemperategradient)

dT dP = γ

ad−1

γ

ad

T P =

2 5

T P

⇒

T

∼

P

2 5 withtheadiabaticindexforanidealgas

γ

ad

=

5 3 Equationofstateforanidealgas(

µ = co n st

):

T

∼

P ρ

⇒

P ρ

∼

P

² 5⇒

P

³ 5∼

ρ

⇒

P

∼

ρ

⁵ 3 Polytropicrelation

P = K ρ

γ

= K ρ

1+1 nwith

γ =

5 3and

n =

3 2. Note:theconstant

K

isnotfixed–differentfromequationofstatefor degenerategas

(3)

H a y a s h i th e o ry fo r fu lly c o n v e c ti v e s ta rs

IdeasforthesolutionoftheLane-Emdenequationforfullyconvective stars(detailsinthePrialniktextbook) •Applythemass-radiusrelationforpolytropicstars(Lane–Emdenequation)

G M M

nn−1

R R

n3−n

= [( n +

1

) K ]

n 4

π G

•Complication:

K

and

R

areunknown! •

K

isapropertyofthestarandtakesdifferentvaluesfordifferentstars (differentfromwhitedwarfs). •Thevalueof

R

canbedeterminedbyjoiningtheconvectiveinteriorofthestar witharadiativephotosphereat

r = R

. 6–10 Post-Main-SequenceEvolution:theredgiantbranch7

H a y a s h i th e o ry fo r fu lly c o n v e c ti v e s ta rs

Fittingaphotosphereontop: •Radiationcanescapefromtheoutermostlayersofthestar(thephotosphere). •Thismakesenergytransportbyradiationveryefficientforregionswithan opticaldepth

τ <

1.Theseregionsarestableagainstconvection. •Thevalueof

R

canbedeterminedbyjoiningtheconvectiveinteriorofthestar witharadiativephotosphereat

r = R

with

M

r

= M ,r = R ,P = P

0

,T = T

eff. •Matchatopticaldepth

τ

≈1–convectiveinside,radiativeoutside. •Photosphericopacityisapproximatedbyasimplepowerlaw:

κ = κ

0

ρ

a

T

b “Fully”convectivestarshavelowtemperature:

T

eff

=

3000

K

. →

a =

1and

b =

4

H a y a s h i th e o ry fo r fu lly c o n v e c ti v e s ta rs

Thisresultsinasetoffourlinearequations,whichcanbesolvedtogivea relationbetweenluminosity,temperatureandmass:

lo g L = A lo g T

eff

+ B lo g M + co n st an t

with

A = (

7−

n )( a +

1

)

−4−

a + b

0

.

5

(

3−

n )( a +

1

)

−1

B =

−

( n

−1

)( a +

1

) +

1 0

.

5

(

3−

n )( a +

1

)

−1 Polytropicindexforthefullyconvectivestar(adiabaticgradient):

n =

3 2

A =

4

.

5

a + b +

1

.

5 0

.

75

a

−0

.

25

B =

−0

.

5

a +

1

.

5 0

.

75

a

−0

.

25

a =

1

b =

4→

A =

20

B =

−4 6–12 Post-Main-SequenceEvolution:theredgiantbranch9

H a y a s h i lin e (s ) in th e H R d ia g ra m

MeaningoftheHayashiline •Partialconvectivestarshavezoneswithtem- peraturegradientsshallowerthantheadia- baticgradient∇

<

∇ad.Thesearehotter thancorrespondingfullyconvectivestarsand arefoundontheleftoftheHayashiline. •StarsontherightoftheHayashilinemust haveregionswithtemperaturegradients steeperthenadiabatic∇

>

∇ad.However, thiswouldcausestrongconvection(⇒crite- rionforconvection)andquickadjustmentto anadiabatictemperaturestructure. •Stablestarscannotexisttotherightofthe Hayashiline!

(4)

P o s t m a in -s e q u e n c e e v o lu ti o n

post-mainsequenceevolution forstarsofdifferentmass 6–14 Post-Main-SequenceEvolution:theredgiantbranch11

N e u tr in o p ro d u c ti o n re v is it e d

Productionofneutrinosfromnon-nuclearprocesses: temperaturesabove108 K,densitiesabove104 g/cm3 Photo-neutrinos:

γ + e

− →

e

−

+ ν + ν

scatteringofphotonsatelectrons,cf.Comptonscattering. NeutrinosfromBremstrahlungInelasticscatteringofelectronsbynuclei usuallyproducesBremsstrahlungsphotons.Athighenergiesoccasionally

ν /ν

-pairsareformed. Plasmaneutrinos

γ

Plasmon→

ν ν γ

Plasmon:QuantumofPlasmaoscillations. Plasmafrequency

ω

2 0

=

4πe2ne me CollectivemovementofElectrons.Electromagneticwavescanpropagate,if

ω > ω

0.

N e u tr in o p ro d u c ti o n re v is it e d

Neutrinosfrompairannihilation:attemperaturesabove4108 K:

e

−

+ e

+ →

ν + ν e

−

/e

+-pairsareformedviapairproductionfrom:

γ + γ

↔

e

−

+ e

+ probabilitytocreatea

ν /ν

-pair:10−19 ⇒manyhighenergyphotonsrequired ⇒becomesimportantat

T >

109

K

.Ratescaleswithdensityat∼

ρ

−1 . Specialcase:Neutrinosfromsynchrotronradiation:Inthepresenceof strongmagneticfieldsneutrinopairscanbeproducedoccasionallyinsteadof Synchrotronphotons. 6–16 Post-Main-SequenceEvolution:theredgiantbranch13

N e u tr in o p ro d u c ti o n re v is it e d

Suchneutrinosareimportantforlatestagesofevolution,butnotforthe mainsequence

(5)

N e u tr in o p ro d u c ti o n re v is it e d

TheUrcaprocess:AspecialcaseAtextremelyhighdensitieselectroncapture bynucleibecomepossible:

e

−

+ ( Z ,A )

→

( Z

−1

,A ) + ν

Urca-Prozeß(Gamov,Schönberg) forsomenucleiacyclicprocesscanoccur:

e

−

+ ( Z ,A )

→

( Z

−1

,A ) + ν ( Z

−1

,A )

→

( Z ,A ) + e

−

+ ν

neutrinoenergyislostwithoutanychangeinchemicalcomposition.Energy lossincreaseswithincreasing

T

and

ρ

. 6–18 Post-Main-SequenceEvolution:theredgiantbranch15

D if fe re n t e v o lu ti o n fo r d if fe re n t m a s s e s

Evolutionofcentraltemperatureanddensityforstarsofdifferentmasses

D if fe re n t e v o lu ti o n fo r d if fe re n t m a s s e s

initialmass

<

0

.

5

M

⊙verylowmassHeburningnotignited,nosignificant evolutionduringlifetimeofUniverse 0

.

5

.. .

2

.

3

M

⊙lowmassHeburningignitedindegeneratecore (heliumflash),noCburning 2

.

3

.. .

8

M

⊙intermediateHeburningignitedinnon-degenerate core,noCburning

>

8

M

⊙highmassHe,Cburning,ignitedinnon- degeneratecore→supernova