3 . C o n s ti tu ti o n o f s ta rs

(1)

3–1

3 . C o n s ti tu ti o n o f s ta rs

3–2 Constitutionofstars1

C o n s ti tu ti o n o f s ta rs

Inordertosolvethestructureequationsthephysicalpropertiesofthestellar materialandinteractionwiththeradiationfieldneedtobeknown. 1.Equationofstate:

ρ ( P ,T ,X

i

)

⇒

c

P,

δ

,∇ad 2.Opacities:

κ ( P ,T ,X

i

)

3.nuclearreactionratesandenergyproduction(incl.neutrinolosses):

r

ij

( P ,T ,X

i

)

,

ǫ ( P ,T ,X

i

)

,

ǫ

ν

( P ,T ,X

i

)

3–3 Equationofstate1

E q u a ti o n o f s ta te : T h e id e a l g a s

Idealgas:

P = n k T

with

ρ = n µ m

Hand

m

H

=

1

.

6605·10−24

g

P=ρkT µmHρ=µmHP kT lnρ=lnP−lnkT+lnµmH dlnρ=αdlnP−δdlnkT+(ϕdlnµmH) α=∂lnρ ∂lnP=1δ=−∂lnρ ∂lnT=1(ϕ=∂lnρ ∂lnµ=1) Deviationsfromidealgascausedbyradiationpressureandionisation canbedescribedviaα,δ(andϕ),whichwilldeviatefromunity. 3–4 Equationofstate2

M e a n a to m ic w e ig h t

fullyionisedplasma:Element

i

withcharge

Z

i,massfraction

X

i,atomicweight

µ

iandpartialdensity

ρ

i,Partialnumberfraction: ni=ρi µimH=ρXi µimH Totalpressure: P=Pe+X iPi=ne+X ini! kT =X iZini+X ini! kT=X i(1+Zi)nikT=X i

ρ µimHXi(1+Zi)kT meanatomicweight(completelyionised): 1 µ=X i

Xi(1+Zi) µi neutralplasma: 1 µ=X i

Xi µi

(2)

M e a n a to m ic w e ig h t

Meanatomicweightpernucleon 1 µ0=X i

Xi µi meanatomicweightperelectron(completeionisation:

Z

ielectronsperion) 1 µe=X i

XiZi µi hydrogenµi Zi

=

1helium

=

4 2

=

2metals≈2 1 µe=X+1 2Y+1 2Z=X+1 2Y+1 2(1−X−Y)=X+1 2 µe=2 X+1completeionisation(partialionisation:later) 3–6 Equationofstate4

N o n -i d e a l e ff e c ts : ra d ia ti o n p re s s u re

Prad=1 3u=a 3T4 →P=Pgas+Prad=ρkT µmH+a 3T4 definition:β=Pgas P⇒1−β=Prad P ρ=µmH kT

P−a 3T4 α=∂lnρ ∂lnP=P ρ∂ρ ∂P=P ρµmH kT=P Pgas=1 β δ=−∂lnρ ∂lnT=−T ρ∂ρ ∂T=−T ρ −µmH kT2 P−a 3T4 −µmH kT

4a 3T3 =T ρ ρ T+µmH kT

4a 3T3 =1+1 ρµmH kT

4a 3T4=1+4Prad P−Prad=P−Prad+4Prad P−Prad =P+3Prad P−Prad=1+3Prad P 1−Prad P=1+3(1−β) 1−(1−β)=4−3β β ϕ=∂lnρ ∂lnµ=1

N o n -i d e a l e ff e c ts : ra d ia ti o n p re s s u re

energydensitypermass(monoatomicgas=3degreesoffreedom) u=3 2kTn ρ+aT4 ρ Withρ=µmHn

∇

ad

=

1+(4+β)(1−β) β2 5 2+4(4+β)(1−β) β2 limitingcases: withoutradiation:

β

→1⇒∇ad→2 5 radiationpressure:

β

→0⇒∇ad→1 4 3–8 Equationofstate6

N o n -i d e a l e ff e c ts : Io n is a ti o n B o lt z m a n n e q u a ti o n

occupationdensitiesoftwoboundenergylevelsofanioninthermodynamic equillibrium: ni nj=gi gje−Ei−Ej kT

g

i

,g

j

=

statisticalweights(numberofstatesofsameenergy;see:Pauli exclusionprincipal). Usingthepartitionfunction: U(T):=X jgje−Ej kT ni N=gi U(T)e−Ei kT

(3)

N o n -i d e a l e ff e c ts : Io n is a ti o n

generalizeBoltzmannformulatoionisation: upperstate=2-particles(Ion+freeelectron) •EnergyE=Eion+p2 2mep

=

electronmomentum •statisticalweightg

=

gup·Ge

(

p

)

; statisticalweight:ion:gup;electronwithmomentump:Ge(p) →insertintoBoltzmannformula: nup(p) nlow=gupGe(p) glowexp −Eup−Elow kT

exp

−p2/2me kT

|{z} equilibriumdistributionforfreeelectrons(Maxwelldistribution) thermalmeanvalueoverallmomenta: nup(p) nlow=gup glowexp −Eup−Elow kT

·

∞Z 0

Ge(p)exp

−p2/2me kT

dp 3–10 Equationofstate8

N o n -i d e a l e ff e c ts : Io n is a ti o n

StatisticalweightofthefreeelectronGe

(

p

)

continuousspectrumofstates:numberofstatesintheinterval[p,p

+

dp]? phasevolume:dΩ

=

∆x·∆y·∆z |{z} space·∆px·∆py·∆pz |{z} momentumspace

quantummechanics: Heisenberg’principleofuncertainty: ∆x·∆px≈h∆y·∆py≈h∆z·∆pz≈h HencethereisafinitenumberofphasecellsavailabledΩ

/

h3 eachcanbe occupiedwith2electronsofdifferentspin(Pauli’sexclusionprinciple). ⇒Ge(p)dp=2·dΩ h3 dΩ

=

4

π

p2 dpdV(sphericalcoordinates);electrondensityne

=

numberofelectronspervolume;ForoneelectronthereisdV

=

1

/

neavailable. dΩ=1 ne4πp2dpGe(p)=21 ne4π h3p2dp

N o n -i d e a l e ff e c ts : Io n is a ti o n

Sahaequation: nup nlow=1 ne2 2πmekT h2

3 2gup glowe−Eup−Elow kT Makinguseofthepartitionfunction

U

jforionisationstage

j

wecancomputethe densityratiooftwosubsequentstagesofionisation:

N

jand

N

j+1 Nj Nj+1=neUj Uj+1C1T−3 2e^χ^j kT=:neΦj(T)

C

1

=

1 22πmekT h2−3 2 ionisationfraction(

N =

PJ j=1

N

j): Nj N=

QJ−1 l=j[neΦl(T)] 1+PJ m=1Qj−1 l=m[neΦl(T)] 3–12 Equationofstate10

N o n -i d e a l e ff e c ts : Io n is a ti o n

Example:hydrogenplasma Ineutralhydrogen

U

I≈2 IIionisedhydrogen

U

II

=

1 NI NII=neUI UIIf(T)=2nef(T) degreeofionisation(

n

e

= N

II

,N

I

+ N

II

= N

): x=NII NI+NII=ne NI+NII (1−x)N xN=xN2f(T)→1−x 2x2N=f(T) (quadraticequationfor

x

)

(4)

N o n -i d e a l e ff e c ts : Io n is a ti o n

Example:hydrogenplasma electronpressure: Pe=nekT=(N+ne)kT |{z} Pgas

ne N+ne =PgasxN N(1+x)=Pgasx (1+x) Foranyarbitrarychemicalcomposition: Numberoffreeelectronsperatom(incl.ions): E:=ne N=x ρ=(N+ne)µmH=Nµ0mH=neµemH µ=ρ mH(N+ne)=ρ mH(1+E)N=µ0 1+E=µeE 1+E 3–14 Equationofstate12

N o n -i d e a l e ff e c ts : Io n is a ti o n

Example:HydrogenPlasma

ρ

∼

µ

P Tund

µ =

µ0 1+Emit

µ

0

= co n st

. lnρ=−ln(1+E)+lnP−lnT+const δ=−∂lnρ ∂lnT=1 1+E∂E ∂lnT+1 applicationtohydrogenplasma:(

E = x

).differentiateSahaequation:∂E ∂lnT: δ=1+1 2x(1−x) 5 2+χH kT +4(1−β) β (incaseofradiationpressure).

N o n -i d e a l e ff e c ts : Io n is a ti o n

Example:HydrogenPlasma specificheat

c

P

=

∂u ∂T P

+ P

∂v ∂T P

u

consistsof •kineticenergyofionsandelectrons •ionisationenergy u=3 2(n+ne)kT ρ+uion=

3 2(n+ne)kT (n+ne)µmH+uion =3 2kT µ0mH(1+E)+uion 3–16 Equationofstate14

N o n -i d e a l e ff e c ts : Io n is a ti o n

Example:HydrogenPlasma

E = x

forHydrogenplasma. cP= 5 2(1+x)+Φ2 H G(x) k µmH mit

Φ

H

=

5 2

+

χH kTand

G ( X ) =

2 x(1−x)

χ

Histheionisationenergyofhydrogen(13.6 eV). Forradiationpressure:5 2→5 2+4(1−β)(4+β) β2. ∇ad=Pδ TρcP=2+x(1−x)ΦH 5+x(1−x)Φ2 H (withoutradiationpressure) hydrogenconvectionzoneintheSun

(5)

N o n -i d e a l e ff e c ts : Io n is a ti o n

ingeneral:numberofelectronspernucleus E=ne N=X νi

ZiX r=0x^{r i} with

ν

i

=

ni N

=

Xiµ0 µi

x

^{r i}=degreeofionisationfor

r

-thstageofionisationofelement

i

. internalenergy: uion=X i

Xi µimH

ZiX r=0x^{r i}

r−1X s=0χ^{s i} ⇒

δ, c

P⇒∇ad needsnumericalsolution(

P

radtobeconsidered)

c

P:specificheatatconstant(total)pressure

P = P

gas

+ P

rad 3–18 Equationofstate16

L im it s fo r th e v a lid it y o f S a h a -B o lt z m a n n e q u a ti o n s

e.g.centreofthesun:

P

e

=

2

.

6·1017dyn cm2,

T =

17·106

K

⇒

n

H

=

1026

cm

−3 Sahaequationyields

x =

0

.

76,⇒24%of hydrogenshouldbeneutral! pressureionisation:decreaseofionisation potentials potentialseverelydisturbed,ifthedistance oftheatomsequaltheirdiameters.

L im it s fo r th e v a lid it y o f S a h a -B o lt z m a n n e q u a ti o n s

hydrogenatom: Bohr’sRadius

a

0, principlequantumnumber

n

, Radius:

a = a

0

n

2 distance:

d =

3 4πnH1 3 equate: a=d⇒n2=1 a0 3 4πnH1 3 InsertdensityofthecentreoftheSun

n

2

=

0

.

26, ⇒eventhegroundstateshouldnotexist. Sahaequationvalidfor

d >

10

a

0: ρmax=µ0mHnion<3µ0mH 4π(10a0)3 =2.66·10−3 µ0g cm3 3–20 Equationofstate18

N o n -i d e a l e ff e c ts

Degeneracy Athighdensity,Maxwell-distributionisnolongerapplicablebecauseofthePauli exclusionprincipleofquantummechanics⇒Fermidistribution later! Whicheffectdominates? Pgas=Prad⇔k µ0mHρT=a 3T4 ⇔T ρ1 3= 3k aµmH1 3 =3.2·107 µ1 3

(6)

3–21 Opacity1

O p a c it y

Whenwederivedtheequationofenergytransportbyradiationweassumedthat wecanuseanwavelengthindependentopacity.However,opacitycanbea strongfunctionofwavelength.Fourbasictypesofopacitysources: bound–bound Photonisabsorbedbyatom,whichisinan excitedstateafterwards→lineabsorption)

bound–free “Photo-ionisation”:anelectronisknockedfree fromanatom.Thresholdenergyforthis process→edges 3–22 Opacity2

O p a c it y

free–free Anfreeelectronpassingbyannucleus absorbsaphoton→continuousabsorption. (inverseprocessisknownasbremsstrahlung.)

Thomsonscattering Asingle,isolatedelectronisscatteringa photon→continuousabsorption,wavelength independent. FiguresfromM.Richmond/RIT(Rochester/USA)

3–23 Opacity3

O p a c it y

Wavelengthdependenceofopacitysources(schematic)⇒needforsomeclever averaging. 3–24 Opacity4

O p a c it y

Rosselandmean 1

κ = π a c T

Z∞ 0

1

κ

ν

∂ B

ν

∂ T dν

Physicalinterpretationwiththefrequencydependentdiffusionequation

F

ν

=

− 1

κ

ν

∂ B ∂ T

4

π

3

ρ

∇

T

•Contributiontotheintegralproportionaltonetflux. •Highestweightforfrequencyrangeswithhighestflux

(7)

3–25 Opacity5

O p a c it y

RelativecontributionstotheRosselandmeaninthestellarinteriorfortypical chemicalmixandconditions 3–26 Opacity6

O p a c it y

Opacitiesforsolarmix. Parameterforsolarinterior. dots=solarmodel majoreffortstocalculateopacitiesgoingon:OPAL(LosAlamos) OpacityProject(OP):Internationalcollaborations

3–27 Energyproduction1

E n e rg y p ro d u c ti o n in s ta rs

massdefect

∆ M

: thermonuclearfusionofatomicnuclei

X

tonuclei

Y

.

∆ M =

X X

M

X−

M

Y energyproduced:

E = ∆ M c

2 bindingenergypernucleon:

f = [( A

−

Z ) m

n

+ Z m

p−

M

nuc

]

c2 A 3–28 Energyproduction2

E n e rg y p ro d u c ti o n in s ta rs

netreaction:41

H

→4

H e

initialmassP

M

X=4×1

.

0081=4

.

0324

m

amu finalmass

M

Y=4

.

0039=4

.

0039

m

amu massdefect

∆ M

=0

.

0285

m

amu

∆ M

=0.7%ofinitialmass

E =

0

.

0285×1

.

6605·10−24 ×

(

3·1010

)

2

=

4

.

3×10−5

er g =

26

.

5

M eV

energyproductionpermass:

∆ E

P

M

x

m

amu

=

6

.

3×1018

er g g =

6

.

3×1014

J kg

(8)

N u c le a r re a c ti o n ra te s

Reactionrates

Coulombbarrier:

E

Coul

( r

0

)

≈

Z

1

Z

2

M eV

averagekineticenergyofprotonsinthe coreoftheSun

T

≈107

K

⇒

E = k T =

1

ke V E

kin≈1 1000

E

Coul

( r

0

)

numberofparticlesinthehighenergy tailoftheMaxwelldistribution:

f ( E )

∼

e

−E kT

f ( E

Coul

) = e

−1000

=

10−434 numberofnucleons intheSun1057 inthevisibleUniverse1080 3–30 Energyproduction4

N u c le a r re a c ti o n ra te s

Quantummechanicaltunnelling: Probability:

P

0

= p

0

E

−1 2

e

−2πη

η =

r

µ

2

Z

1

Z

2

e

2 4

π ǫ

0~√

E =

r

µ

2

α Z

1

Z

2

c

√

E µ =

m1m2 m1+m2isthereducedmass; makinguseofthefinestructurecon- stant

α =

e2 4πǫ0~c

=

1

/

137

p

0isaconstantthatdependson

µ

inthecoreoftheSun:(

T =

107

K

):

P

0

=

10−20 (1keVprotons)

N u c le a r re a c ti o n ra te s

Reactionsrates: Lab:numberofreactionsper timeinterval

dt

:(i:injectedparticles, t:targets) dN=σnids=σnivdt reactionspertime: dN dt=N τ=σ(v)vni reactionrate: r=dN dtdV=σ(v)vnint 3–32 Energyproduction6

N o n -r e s o n a n t re a c ti o n c ro s s s e c ti o n s

Fromquantummechanics:Maximumcrosssection

π λ

2 dB

λ

dB:deBroglie-wavelength

λ

dB

=

~

p =

~ √ 2

m E

crosssectionincludingtunneleffect:

σ ( E )

∼

π λ

2 dB

P

0

( E ) = S ( E )

1

E e

−2πη

η =

r

µ

2

α Z

1

Z

2

c

√

E S ( E )

:astrophysicalreactioncrosssection

S ( E )

mustbedeterminedinthelab. problem:crosssectionsareverysmall.Measurementsaremostlypossiblefor

E >

0

.

1

M eV

.Butastrophysicallyinterestingrange1...10keV! ⇒oftenrelyonextrapolation

(9)

N o n -r e s o n a n t re a c ti o n c ro s s s e c ti o n s

(lhs:totalcrosssection—rhs:astrophysicalfactorS(E)) 3–34 Energyproduction8

N u c le a r re a c ti o n ra te s

Velocitydistribution r=nintvσ(v)=nintZ∞ ovσ(E)f(E)dE Maxwell-Boltzmann-distributionforbothpartners⇒distributionofrelative velocitiesisalsoMaxwellian: f(E)dE=2 √ π

√ E (kT)3 2e−E kTdE mitσ(E)=S(E)1 Ee−2πηandE=1 2mv2 vσ(v)=Z∞ 0S(E)1 Ee−2πη |{z} σ(E)

r 2 µ√ E |{z} v

2 √ π

√ E (kT)3 2e−E kT |{z} f(E)

dE vσ(v)=2³ 2 √ πµ

1 (kT)3 2

Z∞ 0S(E)e−E kT−η√ EdE

G a m o v p e a k

mit η=2πη√ E=√ 2mπαZiZtc

S ( E )

≈

S

0

= co n st

fortherelevantenergyrange r∼vσ∼J=Z∞ 0eg(E)dE with

g ( E ) =

−E kT−η√ E Crosssectionforfusion: σ(E)=S(E) Eexp−r EG E! Gamovenergy:

E

G

= η

2

=

2

m c

2

( π α Z

1

Z

2

)

2

E

0isthepeakenergy 3–36 Energyproduction10

R e s o n a n t n u c le a r re a c ti o n s

Nuclearreaction

a + X

→

Y + b

,or

X ( a ,b ) Y

Excitedintermediatenucleus

C

∗ formswhichcandecayviadifferentchannels (

τ

≈10−21

s

): a+X→C∗ →X+a →Y1+b1 →Y2+b2 →... →C+γ

b

1

,b

2

,. ..

neutrons,protons,α-particles,noelectrons(

τ ( β

−

de ca y )

≈1

s

). Energylevelsoftheexitednucleus

C

∗ :

(10)

R e s o n a n t n u c le a r re a c ti o n s E < E

minstationarylevels⇒de-excitationtothegroundlevelbyγ-radiation, Widthofenergylevels:small

E > E

minquasistationarylevels,decayviatunnelling⇒smalllifetime

τ

⇒ largerwidth:

∆ E

·

τ

≈~(Heisenberg’suncertaintyprinciple) 3–38 Energyproduction12

R e s o n a n t n u c le a r re a c ti o n s

Oneresonanceatanappropriateenergymaydominatethereactionratetotally!

M e a s u ri n g c ro s s s e c ti o n s : L U N A

LUNA=LaboratoryforUndergroundNuclearAstrophysics(GranSasso,Italy) LUNAI-50 kV LUNAII-400 kV L’Aquila

Teramo Figure3.FloorplanoftheLaboratoriNazionalidelGranSasso (LNGS).TheplacesallocatedforthetwoLUNAaccelerator facilitiesaremarked. rhs:Rep.Prog.Phys72,086301(2009) 3–40 Energyproduction14

M e a s u ri n g c ro s s s e c ti o n s : L U N A

400KVacceleratorofLUNA Figure2 TheLUNAsetup.Thetwobeamlinesareintheforeground,andtheacceleratorisinthebackground.The beamlineontheleftisdedicatedtothemeasurementswithsolidtargets,whereastheoneontherighthosts thewindowlessgastarget.Thesetupforthestudyof3He(4He,γ)7Beisshownduringinstallation;theshield isonlypartiallymounted. Annu.Rev.Nucl.Part.Sci.60,53(2010)

(11)

M e a s u ri n g c ro s s s e c ti o n s : L U N A

400KVacceleratorofLUNA 4He+ beam 4×10–4 mbar 3He

0.7 mbar 3He

Water-cooled collimators Diameter: 15 mmDiameter: 7 mmLead shield LeadLead 135% HPGe detector

3He gas inlet ⁷Be catcher

Lead Calorimeter

Copper shield Roots pump 2,000 m3 h–1 Figure3 Thesetupforthestudyof3He(4He,γ)7Be.ThedevicetodetectRutherfordscattering,thecalorimeter,andthegermaniumdetector areindicated. Annu.Rev.Nucl.Part.Sci.60,53(2010) 3–42 Energyproduction16

M e a s u ri n g c ro s s s e c ti o n s : L U N A

S (eV b)

ECM (keV)

Solar Gamow peak

0.6 0.5 0.4 0.3 0.2 0.1 001020304050

Schmid et al. (54) LUNA 2002 (28)

Griffiths et al. (53) Figure5 The2H(p,γ)3HeastrophysicalfactorS(E)withthetotalerror. Annu.Rev.Nucl.Part.Sci.60,53(2010) theppIreaction:2

H ( p, γ )

3

H e

measuredatsolarenergies.

M e a s u ri n g c ro s s s e c ti o n s : L U N A

Solar Gamow peak ECM (keV)

Dwarakanath & Winkler (58) Krauss et al. (59) LUNA 1998–1999 (26, 27) Kudomi et al. (60)

Cross section (b)

10–5 10–6 10–7 10–8 10–9 10–10 10–11 10–12 10–13 10–14 10–15 10–16 1030100 Figure6 Crosssectionofthe3He(3He,2p)4Hereaction.DatafromLUNA(26,27)andfromothergroups(58–60). ThedashedgraylineistheextrapolationbasedonthemeasuredS(E)-factor(27). Annu.Rev.Nucl.Part.Sci.60,53(2010) theppIreaction3

H e (

3

H e,

2

p )

4

H e

measuredatsolarenergies. 3–44 Energyproduction18

M e a s u ri n g c ro s s s e c ti o n s : L U N A

Counts (keV h)

–1

1003He(4He,γ)7Be 10–1 10–2 10–3 00.5

Laboratory γ-ray background 1.0 Eγ (MeV)

1.52.0 Figure7 3He(4He,γ)7Bespectrumatabeamenergyof250keV(red)andthelaboratorybackground(blue). Annu.Rev.Nucl.Part.Sci.60,53(2010) theppIIreaction3

H e (

4

H e, γ )

7

B e

measuredatsolarenergies.

(12)

M e a s u ri n g c ro s s s e c ti o n s : L U N A

10340K207Bi 208TI 2H(p,γ)3He 14N(p,γ)15O 15N(p,γ)16O 11B(p,γ)12C

102 101 100 10–1 10–2 051015

10–3 10–4 10–5

Counts (keV h) –1

Eγ (MeV)

Laboratory γ-ray background Figure9 Bismuthgermanatespectrumtakenwitha100-keVprotonbeamonnitrogenofnaturalisotopicabundance. Red,peaksfrom14N(p,γ)15Oand15N(p,γ)16O;darkyellow,beam-inducedbackground;blue,laboratory background.Inspiteofthesmallisotopicabundanceof15N(0.366%only),thepeakarisingfrom 15N(p,γ)16Ocanbeeasilyseenthankstothemuch-reducedbackground. Annu.Rev.Nucl.Part.Sci.60,53(2010) protonreactionsonnitrogenisotopesimportantintheCNOcycle. 3–46 Energyproduction20

M e a s u ri n g c ro s s s e c ti o n s : L U N A

78 6 5 4 3 2 1 0050100150200

S (keV b)

Lamb & Hester (82) Schröder et al. (83) NACRE extrapolation 1999 (81) LUNA data 2004 (29) (solid target) LUNA extrapolation 2004 (30) TUNL data 2005 (93) LUNA data 2006 (31) (gas target) Solar Gamow peak ECM (keV) Figure10 AstrophysicalS(E)-factorofthe14N(p,γ)15Oreaction.Theerrorsarestatisticalonly(thesystematicones aresimilar). Annu.Rev.Nucl.Part.Sci.60,53(2010) the14

N ( p, γ )

15

O

reactionoftheCNcyclehasnotyetbeenmeasuredatsolar energies,butLUNAresultsarelowerbyafactorof2thanearlierones.Important forageofoldeststarclustersandsolarneutrinorates.

H y d ro g e n b u rn in g

41

H

→4

H e ∆ M c

2

=

26

.

731

M eV

Twoprotonsareconvertedintoneutrons(β+ -decay).Twoneutrinosare produced,whichcarryaway2...30%oftheenergy. Tworeactionchainsforhydrogenburning: proton-protonchain, CNOcycle 3–48 Energyproduction22

H y d ro g e n b u rn in g : p p c h a in s

Threevariantsoftheppchain pp1pp2pp3

energyproductionrate: ǫpp∼ρX2 T5 (approximationvalidfor T≈107 K).

(13)

H y d ro g e n b u rn in g : p p c h a in s

timescalesforppreactions: (I)1

H +

1

H

→2

D + e

+

+ ν

1010 years (II)2

D +

1

H

→3

H e

10s (III)3

H e +

3

H e

→4

H e +

21

H

106 years Changeofabundancesof2

D

and3

H e

inrelationto1

H

(I)und(II)havetooccurtwice,before(III)canstart (I)creates2

D

,(II)destroys2

D

, (II)creates3

H e

,(III)destroys3

H e

. 3–50 Energyproduction24

H y d ro g e n b u rn in g : p p c h a in s

inanequilibriumstateabundanceof2

D

und3

H e

willbedeterminedby: rI=N1H τ1H=rII=N2D τ2D rII=N2D τ2D=2rIII=2N3He τ3He ⇒N2D N1H=τ2D τ1H=3·10−17 N3He N1H=τ3He τ1H=5·10−5 energyproductioncoefficient: ǫpp∼ρX2T5 (validforT≈107 K).FusionprocesswiththesmallestT-sensitivity.

H y d ro g e n b u rn in g : C N O c y c le

CNOcycle(s) •catalyticprocess •catalyst(e.g.C)mustbe available •sequenceofprotoncap- turesfollowedbyβdecay •energyproductionrate: ǫCNO∼ρXXCNOT16 (approximationvalidfor T≈107.5 K). 3–52 Energyproduction26

C N O -c y c le (B e th e -W e iz s ä c k e r- p ro c e s s )

timescalesforthereactionsintheCN-cycle: (I)12

C ( p, γ )

13

N

107 years (II)13

C ( p, γ )

14

N

106 years (III)14

N ( p, γ )

15

O

108 years (IV)15

N ( p, α )

12

C

105 years

β

+ -Zerfälle13

N ( e

+

ν )

13

C

,15

O ( e

+

ν )

15

N

: 1...10min

(14)

H y d ro g e n b u rn in g : C N O c y c le

TheCNO-cyclemodifiestherelativeabundancesoftheC,N,andOisotopes •thetotalnumberof CNOnucleiremains constant, •however,mostCand Onucleiconverted into14 Nbecause 14

N ( p, γ )

15

O

isthe slowestreactionof theCNOcycle(s)

EvolutionofCNO-isotopeswithtime: Equilibriumreachedafter≈108 years. 3–54 Energyproduction28

H y d ro g e n b u rn in g : C N O c y c le

TheCNO-cyclemodifiesrelativeabundancesoftheC,N,andOisotopes •totalnumberofCNO nucleiremainsconstant, •mostCandOnucleicon- vertedinto14 Nbecause 14

N ( p, γ )

15

O

istheslow- estreaction •theabundanceratio ofthecarbonisotopes 12

C /

13

C

decreasesto≈ 4. ”Smokinggun”forfor CNOcycle(solarvalue: 89).(lhs:absolutevalues—rhs:isotoperatios)

H y d ro g e n b u rn in g

Additionalcycles:NeNa-,MgAl-cycles 3–56 Energyproduction30

H e liu m b u rn in g

Combiningof4 He-nucleitoform12 C(and16 O). 4He+4He→8Be−92keV endothermreaction⇒8 Bedecaysto4 Hewithin

τ (

8

B e) =

2

.

6·10−16

s

. τ(8 Be)stilllargecomparedtocollisionaltimescale:≈105 collisionsof4 He during

τ (

8

B e)

whichallowstoform12 C 8 Be+4 He→12 C+γ Highreactionratebecauseofaresonance. energyproductionrate:ǫ3α∼ρ2 Y3 T30

(15)

H e liu m b u rn in g

12

C

maycaptureanotherα-particle:12

C +

4

H e

→16

O + γ

resonantreaction(“subthreshhold”):

q =

7

.

16

M eV

Figure22.ThepresentstatusoftheSfactordatafor12C(α,γ)16O. ThetotalSfactormeasurementofERNA(filled-incircles)[12]is comparedwithrecentE1(opentriangles)andE2(opensquares) γ-raymeasurements[98]andtheEx=6.05MeVcascadedata (opencircles)[100].Thesolidlinerepresentsthesumofthesingle amplitudesofanR-matrixfit[99](thedottedanddashedlinesare theE1andE2amplitudes,respectively).Inaddition,theR-matrix fitof[100]totheircascadedata(dotted–dashedline)isshown.The lattercomponentisnotincludedinthesumandmightexplainthe highyieldintheERNAdatabetweentheresonances.

Furthercapturesarealsoendotherm 16

O ( α ,γ )

20

N e

,20

N e( α ,γ )

24

M g

,... But16

O ( α ,γ )

20

N e

isnon-resonant→ veryhightemperaturesneeded. noα-capturesbeyond16 Oduringhe- liumburningphasesofstellarevolu- tion. reactioncrosssectionisstill uncertainbyafactorof3. 3–58 Energyproduction32

H y d ro g e n a n d h e liu m b u rn in g

fractionsofenergyproductionfrompp-,CNO- and3α-processes: uppermainsequence(sp. typeAandearlier):CNO- cycle lowermainsequence(sp. typeGandlater)pp-chain 3α-process:T≥108K⇒ notduringmainsequence phase Moreonnuclearburningtoironlater!

N e u tr in o s

Theroleofneutrinos verysmallcross-section: σν≈ Eν mec22 ·10−44 cm−2 solarneutrinos:

E

ν≈1

M eV

⇒

σ

ν≈10−44

cm

−2 ,1018 timessmallerthan photonabsorptioncross-sections! meanfreepath: lν=1 nσν=µmH ρσν=2·1020 ρcm Sun

ρ =

1

g cm

−3 ⇒

l

ν

=

100

p c

whitedwarf

ρ =

106

g cm

−3 ⇒

l

ν

=

3000

R

⊙ ⇒neutrinosescapewithoutinteraction,theirenergyislostfromthestellar interior. 3–60 Energyproduction34

N e u tr in o s

neutrinoproductionduringhydrogenburning.5possiblereactions: 3He(3He,2p)4He 7Be(e–,ν)7Li 8Be*(4He)4He

8B(e+ν)8Be* * Be produced into an excited level

7Li(p,4He)4He 7Be(p,γ)8B

3He(4He,γ)7Be3He(p,e+ν)4He 14%0.02%

86%14%0.00003%

2H(p,γ)3He

1H(p,e+ν)2H 99.75%0.25%

1H(pe–,ν)2H Q/MeV 1

H +

1

H

→2

D + e

+

+ ν

(pp1)0.263 7

B e + e

−→7

L i+ ν

(pp2)0.80 8

B

→8

B e + e

+

+ ν

(pp3)7.2 13

N

→13

C + e

+

+ ν

(CNO)0.71 15

O

→15

N + e

+

+ ν

(CNO)1.0 Moreonstellarneutrinoslater: -non-nuclearneutrinoproduction -neutronisation -pairproduction:neutrino–anti-neutrino