Stellar Structure and Evolution

Stellar Structure and EvolutionAchim Weiss

Max-Planck-Institut fur Astrophysik

03/2002 Stellar Structure – p.1

Lecture Content


What will be included . . .

Basic types of observations and global stellar quantities

Main assumptions and fundamental parameters

Stellar structure equations

Micro- (or input) physics

Stellar model calculations

The Standard Solar Model

Overview about stellar evolution for different massranges (including relevant objects)


What will be excluded . . .

hydrodynamics

rotation

late/final stages

explosive nucleosynthesis

atmospheres

theory of mass loss


Reading suggestions

Kippenhahn R. & Weigert A.: Stellar structure andevolution, Springer, 1990

Hansen C.J. & Kawaler S.D.: Stellar interiors, Springer,1994

de Loore C.W.H. & Doom C.: Structure and evolution ofsingle and binary stars, Kluwer, 1992

Padmanabhan T.: Theoretical Astrophysics, vol. II: Starsand stellar systems, Cambridge University Press, 2001

Chiosi C.: The Hertzsprung–Russell-Diagramm, (for theBritish Encyclopedia of Science), preprint 1999


More reading suggestions

Clayton D.,: Principles of Stellar Evolution andNucleosynthesis, Chicago University Press, 1983

Cox J.P. & Giuli R.Th., Principles of Stellar Structure,vol. I, Gordon & Breach, 1968, and 2nd, enlargededition by Weiss, Hillebrandt, Thomas, Ritter, to bepublished in 2002/2003 (Taylor & Francis)

Christensen-Dalsgaard J., Lecture Notes on StellarOscillations, available fromhttp://astro.ifa.au.dk/˜jcd/oscilnotes

various reviews in ARAA, in particular in vols. 37 (1999)and 38 (2000).


Fundamental Quantities

The interior structure and the global properties of a star aredetermined by

the initial mass Mi

the initial composition X, Y , Z (and individual metalabundances), where ~X(t = 0, r) = ~X(t = 0)

the age, which determinesM(t) (mass loss)~X(t, r) (nuclear reactions, transport processes)

The global quantities of interest are L, Teff (or R), ~X(R), M ,

vrot, Πosc


Basic facts and relation toobservations


The Significance of Stars

Stars contribute . . .

information (photons, neutrinos)

energy (radiation, winds, explosions)

matter (nucleosynthesis, winds, ejecta)

Stars allow to . . .

trace gravitational potentials

follow the history of galaxies and the Universe

probe basic physics

Stars host . . .

planets

life


The Significance of Stars - Applications

structure formation

population synthesis

galactic history

plasma physics

nuclear physics

neutrino and particle physics


Stellar data

photometry → brightness, colour

spectroscopy → Teff , g, ~X, M

astrometry → distance

eclipsing binaries → mass ratios, orbital elements,geometric quantities

occultations & interferometry → angular diameter → R

seismology → interior structure, c, ρ, Γ1 =(

∂ ln P∂ ln ρ

)

S


Photometry: Basics

Photometry: Astronomical observations make use ofseveral broad-band filters to measure the flux of a star indifferent wavelength-bands.Best-known filter system: Johnson-Cousins-GlassUBV RIJHK; the V filter is similar to the human eye.

Illustration of the UBV-filter re-

sponse functions


Photometry: Basics

Stars are almost perfect Black Body radiators; spectrumcharacterized by temperature T

Flux in several bands → stellar temperature Teff

Iλ =2c2h

λ5exp

(

hc

λkT− 1

)−1( erg

cm3s

)

πF =

∫ ∞

0

dλIλ = σT 4( erg

cm2s

)

2π5k4

15c2h3=

ac

4= σ = 5.67 · 10−5 erg

cm2 s K4

Photometric observations of interest for this lecture yield

V ≡ Mv : brightness (luminosity)

B − V or V − I : colour (temperature)03/2002 Stellar Structure – p.13

Luminosity, magnitudes, etc.:

For the Sun,L = 4R2

(πF )

is the solar luminosity (integrated flux through solarsurface).Flux on earth: S = 1.36 · 106 erg

cm2s (solar constant):S = F (πR2

/d2) ⇒

L = 3.82 · 1033erg/s

withL = 4πσR2

T 4eff → Teff = 5770 K.

L, Teff , mass M (or g, R) are the basic global stellarparameters.


HRD & CMD

Hertzsprung-Russell-Diagram (HRD): log L – log Teff

Colour-Magnitude-Diagram (CMD): V – (B − V )

(more general: brightness vs. colour index)

Magnitudes measure stellar brightness in log L-system.

m2 −m1 = 2.5 log b1/b2

m: apparent magnitude; M : absolute magnitude:

M −m ≡ 5− 5 log d

d: distance in parsec (3.08 · 1018 cm);

M is m at 10 pc.


HRD & CMD

distance modulus m−M : 0.2(m−M) + 1 = log d.Indices to m and M denote filter, e.g. MV or mB.Bolometric luminosity or brightness isMbol = Mbol, − 2.5 log(L/L).Bolometric correction B.C.: Mbol −MV

For the sun, B.C. = −0.12 (Alonso et al. 1996) andMV, = 4.82± 0.02 → Mbol = 4.70 and therefore

Mbol = −2.5 logL

L+ 4.70

A further convention is that for a star of spectral class A0 V (e.g. α Lyr = Vega),

MV = MB = MU . This defines the bolometric corrections for U and B bands.


Colour-Magnitude-Diagrams

A globular cluster — mass : The HIPPARCOS solar neighbourhood —

mass, composition, age:

A dwarf spheroidal galaxy — mass, composition, age, distance


Spectroscopy: abundances and Teff

The Li- (Spite-) plateau

mass discrepancy in OB-stars


Astrometry→ distances

The Hyades main sequence

Distances allow brightness cal-ibrations, mass determinations,help spectroscopic analyses, . . .


Eclipsing binaries→masses and radii

Procyon:

evolutionary and astrometric mass

OGLE-17 in Ω Cen: age and mass


Seismology & oscillations→ ν & c

Power in solar oscillation modes

Solar sound speed and Γ1

and: hydrogen mass on white dwarfs, evolutionary state, . . .


The Structure Equations


Basic assumptions

Stars are self-gravitating objects of hot plasma;

emitting energy in the form of photons from the surface;

spherical symmetry (absence of rotation and magneticfields);

→ one-dimensional problem with radius r being the natural

coordinate (Euler description).


Mass and radius

Eulerian description: mass dm in a shell at r and ofthickness dr is

dm = 4πr2ρdr − 4πr2ρvdt

From the partial derivatives of this equation one can derive:

∂ρ

∂t= −r−2∂(ρr2v)

∂r

→ ∂ρ

∂t= −∇(ρv)

(continuity equation in 1-dimensional form and Eulerian de-

scription)


Lagrangian description

mass elements m (mass in a concentric shell). ⇒ r = r(m, t)

Variable change (r, t) → (m, t):∂

∂m= ∂

∂r∂r∂m

and(

∂∂t

)

m= ∂

∂r

(

∂r∂t

)

m+(

∂∂t

)

r

⇒∂r

∂m=

1

4πr2ρ(1)

This is the first structure equation (mass equation or massconservation).Contains transformation Euler- → Lagrange-description

∂

∂m=

1

4πr2ρ

∂

∂r


Gravity

Gravitational field ∇2Φ = 4πGρ(G = 6.673 · 10−8 dyn cm2 g−2).

In spherical symmetry:

1

r2

∂

∂r

(

r2∂Φ

∂r

)

= 4πGρ

g = ∂Φ∂r → g = Gm

r2 is solution of Poisson’s equation.Potential Φ vanishes for r →∞.

−∫∞0 Φdr is the energy required to disperse the complete

star to infinity.


Hydrostatic equilibrium

On layer of thickness dr two forces:gravity gρdr and pressure 4P = ∂P

∂r4r.

If shell is at rest (hydrostatic equilibrium):

∂P

∂r= −Gm

r2ρ

in Lagrangian coordinates:

∂P

∂m= − Gm

4πr4(2)

Second structure equation (hydrostatic equilibrium).


A simple estimate

. . . for central values of P and T :Replace derivatives in the hydrostatic equation bydifferences between center (Pc) and surface (P0 ≈ 0) →

Pc ≈2GM2

πR4

(M/2 and R/2 were used for mean mass and radius)Sun: Pc = 7 · 1015 (cgs units).With ρ = µP

RT and ρ = (3M)/(4πR3) ⇒

Tc =8

3

µ

RGM

R

ρ

ρc<

µ

RGM

R≈ 3 · 107 K


Motion:

If forces are not balanced → ∂P∂m + Gm

4πr4 = − 14πr2

∂2r∂t2

1. P = 0 → free-fall Gm/r2 = r; τff =√

R/|r| ≈√

R/g.

2. G = 0, τexpl ≈ R√

ρ/P (isothermal sound speed)

3. hydrostatic timescale τhydro ≈ 12(Gρ)−1/2

4. Examples: τhydro =

27 minutes for the Sun18 days for a Red Giant (R = 100R)4.5 seconds for a White Dwarf (R = R/50)

Conclusion: stars return to hydrostatic equilibrium within anextremely short time.


The Virial Theorem

From hydrostatic equilibrium equation by integration:

∫ M

0

Gm

rdm = 3

∫ M

0

P

ρdm

where the l.h.s. = −Eg (gravitational energy).Monatomic ideal gas: P/ρ = (2/3)u = (2/3)cvT (u: specificinternal energy) ⇒Ei :=

∫

udm.

⇒ Eg = −2Ei (3)

General gas, 3(P/ρ) = ζu (ζ = 3(γ − 1)); for monatomic gasζ = 2 (γ = 5/3) and for photon gas 1. →

ζEi + Eg = 0.


The Virial Theorem

Total energy: W = Ei + Eg = (1− ζ)Ei = ζ−1ζ Eg

L + dWdt = 0 ⇒ L = (ζ − 1)dEi

dt .

monatomic gas:

ζ = 2 ⇒ L = −Eg

2= Ei, (4)

On contraction, half of the energy liberated is radiated, the

other half is used to increase the internal energy (star heats

up) — or vice versa: stars lose energy from surface → must

contract (overall) — negative specific heat!


The Kelvin-Helmholtz timescale

L ≈∣

∣

∣

∣

dEg

dt

∣

∣

∣

∣

⇒ τKH :=|Eg|L

≈ Ei

L

|Eg| ≈GM2

2R⇒ τKH ≈

GM2

2RL.

Sun: τKH = 1.6 · 107 yrs.

⇒ Sun could shine only for about 10 million years, if the

reactions at the center would be switched off.


The constant density model – I

specific internal energy of the ideal monatomic gas:

u =3

2

nkBT

ρ=

3

2

NAkB

µT =

3

2

Rµ

T =3

2

P

ρ,

exploit the Virial Theorem; assume ρ = ρ = const.

Ei = uM =3

2MRµ

T = −1

2Eg = −1

2

(

−3

5

GM2

R

)

using ρ = const.

⇒ T = 4.09 · 106µ(

MM

)2/3ρ1/3 (Sun: T ≈ 3 · 106)


The constant density model – II

hydrostatic equilibrium equation + r = (m/M)1/3R ⇒

∂P

∂m= − GM

4πR4

(

M

m

)1/3

.

Integration over M (P (M) = 0) →P (m) = Pc

[

1− (m/M)2/3]

= Pc

[

1− (r/R)2]

;

Pc = 38

GM2

R4 = 1.34 · 1015(

MM

)2 (RR

)4

and

T (m) = Tc

[

1− (m/M)2/3]

; Tc = 1.15 · 107µ(

MM

)(

RR

)


Local energy conservation

L(r) = Lr energy passing through sphere r per unit time;includes energy transported by radiation, convection,conduction, but not neutrino energies (lost!)Energy gained (or lost) by a shell of dm is

dLr = 4πr2ρεdm,

ε (erg/gs): specific energy generation rateSources for ε:

in a stationary mass shell: ε = εn(ρ, T, ~X) : nuclearenergy generation;

in non-stationary mass shell: interaction withsurrounding via Pdv = dq



(

εn −∂Lr

∂m

)

dt = dq

∂Lr

∂m= εn − T

∂s

∂t= εn + εg

εg: gravothermal energy

εg = −cP∂T

∂t+

δ

ρ

∂P

∂t= −cPT

(

1

T

∂T

∂t− ∇ad

P

∂P

∂t

)

(5)

(see thermodynamic relations)



Energy loss due to plasma neutrinos: −εν. energyconservation equation reads

∂Lr

∂m= εn + εg − εν . (6)

Conventions: εν > 0 for the neutrinos, therefore subtracted(loss);εg > 0 for contraction, liberating energy → Lr(m) raised.

Simplification: entropy changes due to composition changes

(mixing, burning) are ignored; usually of order 10−4 ∂Lr

∂m .


Global energy conservation

W =d

dt(Ekin + Eg + Ei + En) = −(L + Lν)

Def.: L =∫

∂Lr

∂m dm, Lν =∫

ενdm,∫

εndm = −dEn

dt .

Integrating εg over dm: from

εg = −∂u

∂t+

P

ρ2

∂ρ

∂t.

the first term gives −dEi/dt.



Second term. Use

Eg = −3

∫ M

0

P

ρdm,

ddt of integrated hydrostatic equilibrium →

∫ M

04πr3 ∂P

∂mdm = 4

∫ M

0

Gm

r

r

rdm = 4Eg

From l.h.s.: integration by parts, P (M) = 0, and ∂m∂r = 4πr2ρ

−3

∫ M

0

P

ρdm = 4Eg;



and finally, use this for dEg

dt ⇒

Eg = −∫ M

0

P

ρ2ρdm,

which corresponds to second term in εg;therefore we have shown that

d

dt

∫

εgdm = − d

dt(Eg + Ei)

and W is recovered, without the Ekin-term, however.

This is recovered when using the full equation of motion,

instead.


The nuclear timescale

τn := En/L

Nuclear energy reservoir:mass of fuel times q, the specific energy release in erg/g.Sun (H-burning): q = 6.3 · 1018 erg g−1 → 8.75 · 1051 erg.

→ τn = 7 · 1010 yrs

τn τKH τhydr

τn most important timescale → ∂Lr

∂m ≈ εn ↔ εg ≈ 0

(thermal equilibrium).

With mechanical equilibrium → complete equilibrium.


Energy transport

T -gradient in Sun: 4T/4r ≈ 107/1011 = 10−4 (K/cm).energy tranport by radiation, convection and conduction

∂T

∂m= −T

P

Gm

4πr4

(

∂ ln T

∂ ln P

)

= −T

P

Gm

4πr4∇

Radiative transport

cross section for scattering/absorption of radiation: σ

mean free path lph = 1/(nσ), n: absorber density;(lph ≈ 2 cm in Sun)

opacity definition ρκ = nσ; κ(T, ρ, ~X) (cm2/g)


Radiation diffusion

In analogy to particle diffusion equation:Diffusive flux ~j of particles (per unit area and time) is

~j = −D~∇n = −1

3vlph

~∇n

(D is called the diffusion constant; v the diffusion velocity; lpis the particle free path length and n the particle density).We now use U := aT 4 for the radiation density in place ofparticle density and c instead of v. In a 1-dimensionalproblem, we get for ~∇U

∂U

∂r= 4aT 3∂T

∂r


and for the radiative flux F (replacing ~j)

F = −4ac

3

T 3

κρ

∂T

∂r,

or F = −Krad∇T . Krad = 4ac3

T 3

κρ is the radiative conductivity.

With Lr = 4πr2F , we obtain

∂T

∂m= − 3

64acπ2

κLr

r4T 3(8)


The Rosseland mean opacity

κ is actually an appropriate mean of κν over frequency ν:

1

κ:=

∫∞0

1κν

∂Bν

∂T dν∫∞0

∂Bν

∂T dν

where

Bν(T ) =2hν3

c2

(

exp

(

hν

kT

)

− 1

)−1

is the Planck-function for the energy density flux of a blackbody. (U = aT 4 = (4π/c)

∫

Bνdν).

Note that the Rosseland mean is dominated by those fre-

quency intervals, where matter is almost transparent to radi-

ation!03/2002 Stellar Structure – p.45

Electron conduction

If energy is transported efficiently by conduction:

F = Frad + Fcond = −(Krad + Kcond)∇T

introduce formally κcond:

Kcond =4ac

3

T 3

κcondρ.

Replace κ in (8) by 1κ = 1

κrad+ 1

κcond

Effective ∇ in transport equation:

∇ = ∇rad =3

16πacG

κLrP

mT 4


Perturbations and stability: convection

Schematic picture: thetemperature excess DT ispositive, if the element ishotter than its surround-ing. DP = 0 due to hydro-static equilibrium. If Dρ <0, the element is lighterand will move upwards.

Take an element and lift it by 4r:

Dρ =

[(

∂ρ

∂r

)

e

−(

∂ρ

∂r

)

s

]

4r


Stability condition

The stability condition thus is(

∂ρ∂r

)

e−(

∂ρ∂r

)

s> 0.

EOS: d ln ρ = αd ln P − δd ln T − ϕdµ → the stabilitycondition changes into

(

δ

T

dT

dr

)

s

−(

δ

T

dT

dr

)

e

−(

ϕ

µ

dµ

dr

)

s

> 0

multiply the stability condition by the pressure scale height

HP := − dr

d ln P= −P

dr

dP=

P

ρg> 0 ⇒


Stability condition . . .

(

d ln T

d ln P

)

s

<

(

d ln T

d ln P

)

e

+ϕ

δ

(

d ln µ

d ln P

)

s

∇s < ∇ad +ϕ

δ∇µ

∇rad < ∇ad +ϕ

δ∇µ

The last equation holds in general cases and is called the

Ledoux-criterion for dynamical stability. If ∇µ = 0, the

Schwarzschild-criterion holds. Note: ∇µ > 0 and will sta-

bilize.


The four∇

In an unstable layer,the following rela-tions hold:

∇rad > ∇ > ∇e > ∇ad

The task of convection theory is to calculate ∇!


Convection in stars

highly turbulent (Reynolds number Re := vρlmη ≈ 1010; η

viscosity; lm = 109 cm; lab.: turbulence for Re > 100);)

3-dimensional and non-local

motion in compressible medium on dynamicaltimescales (v speed of blobs ≈ 103 cm/s = 10−5vsound)

3-d hydro simulations limited to illustrative cases

2-d hydro: larger stellar parameter range; shallowconvective layers

dynamical models: averages, simplifications, toocomplicated for stellar evolution

⇒ ∇ from simple approaches with additional extensions


The Mixing Length Theory

F =Lr

4πr2= Fconv + Frad

F =:4acG

3

T 4m

κPr2∇rad

Frad =4acG

3

T 4m

κPr2∇

Fconv = ρvcP (DT )

A blob starts somewhere with DT > 0 and loses identityafter a typical mixing length distance lm. On average

DT

T=

1

T

∂(DT )

∂r

lm2

= (∇−∇e)lm2

1

HP


DT → Dρ; buoyancy force: fb = −gDρ/ρ = gδDT/T .

The work done by it is (assume 50%)

1

2fb

lm2

= gδ(∇−∇e)l2m

8HP

50% → kinetic energy of the element, 50% → “pushingaway” the surrounding. Then,

v2 = gδ(∇−∇e)l2m

8HP

v2, DT/T in Fconv → Fconv = ρcP T√

gδ l2m4√

2H−3/2p (∇−∇e)

3/2


Te changes due to adiabatic cooling and radiation losses,(

dT

dr

)

e

=

(

dT

dr

)

ad

− λ

ρV cP v,

or ∇e −∇ad = λHP

8V vTcP; λ describes the radiation loss relative

to the blob’s energy; λ = Sf ≈ 8acT 3

3κρ DT Sd

At this point, geometric factors and other assumptions come

in (for the details see Kippenhahn & Weigert).


multiplication by HP/T →

∇e −∇ad =λHP

ρV cP vT

use definition of λ with average DT , and replace “formfactor” lmS/V d by some geometrical measure, e.g. ≈ 9/2

lm(6

instead of 4.5 for sphere). Above equ. then becomes

∇e −∇ad

∇−∇e≈ 6acT 3

κρ2cP lmv

Finally, one obtains 5 equations for v, Fconv, Frad, ∇ and ∇e

in terms of P, T, ρ, lm,∇ad,∇rad, cP , g.


Five equations

Fconv + Frad =:4acG

3

T 4m

κPr2∇rad

Frad =4acG

3

T 4m

κPr2∇

v2 = gδ(∇−∇e)l2m

8HP

Fconv = ρcP T√

gδl2m

4√

2H−3/2p (∇−∇e)

3/2

∇e −∇ad

∇−∇e=

6acT 3

κρ2cP lmv


The MLT cubic equation

An analytical solution is possible and leads to a cubicequation:

(ξ − U)3 +8U

9(ξ2 − U2 −W ) = 0

where

ξ2 = ∇−∇ad + U2

U =3acT 3

cPρ2κl2m

√

8HP

gδ

W = ∇rad −∇ad

W and U can be calculated at any point (local!) and the cubic

equation can be solved.03/2002 Stellar Structure – p.57

The MLT cubic equation

Γ :=(∇−∇e)

1/2

2U=

(∇−∇e)

(∇e −∇ad)

is the ratio of energy transported by the blob over that lostfrom it, or the “efficiency” of the convection.

U small → Γ large: almost all flux transported byconvection→ ∇ ≈ ∇ad.

U large → Γ small; transport by convection but∇ ≈ ∇rad.

general case: convection superadiabatic:∇ = ∇ad + δ∇


The mixing length parameter

mixing length lm = αMLTHP ; αMLT: mixing-lengthparameter

αMLT of order 1

determined usually by solar models, αMLT ≈ 1.6 · · · 1.9or other comparisons with observations

NO calibration or meaning!

Examples for ∇: Sun: r = R/2, m = M/2, T = 107, ρ = 1,δ = µ = 1

→ U = 10−8 → ∇ = ∇ad + 10−5 = 0.4

(as long as ∇rad < 100 · ∇ad); at center, ∇ = ∇ad + 10−7.


Deficits of MLT

local theory → no overshooting from convectiveboundaries due to inertia of convective elements

time-independent → instantaneous adjustment; criticalif other short timescales (pulsations, nuclear burning)present

only one length scale, but spectrum of turbulent eddies→ improvements by Canuto & Mazzitelli

presence of chemical gradients ignored(semiconvection) → treatment of such layers unclear;probably slow mixing on diffusive timescale due tosecular g-modes; T -gradient radiative?


Chemical composition

The chemical composition of a star changes due to

nuclear burning

convection and other mixing

diffusion

Notation:relative element mass fraction: Xi := mini

ρ , with∑

i Xi = 1;hydrogen X, helium Y , “metals” Z = 1−X − Y .

mi and ni are particle masses and densities.


Changes due to nuclear reactions

∂Xi

∂t=

mi

ρ[∑

j

rji −∑

k

rik]

Energy released is εij = 1ρrijeij.

rij: number of reactions per secondeij: energy released per reactionqij = eij/mi energy released per particle massFor the conversion of hydrogen into helium, for example, weget

∂X

∂t= − εH

qH= −∂Y

∂t


Changes due to diffusion

Diffusion equation:

~vD = −1

cD(

~∇c + kT~∇ ln T + kP

~∇ ln P)

From Fick’s law~jD = c~vD = −D~∇c

with ~jD: diffusive particle flux, D: diffusion constant, c:concentration (relative number density).From the continuity equation:

∂c

∂t= −~∇~jD = ~∇(D~∇c) = D∇2c


Diffusion

Corresponding equations for the other two terms, with kT

and kP describing relative diffusion speeds w.r.t.concentration diffusion.

Sun: D ≈ 10 cm2 s−1; τdiff ≈ 1013 yrs; effect is measured:Y (t) = 0.246 ↔ Y (t = 0) = 0.275 (GARSOM4)

Low-mass stars (12–13 Gyr): possibly large effect, butnot yet measured

Mixing by convection can be described with highD ≈ 1015 cm2 s−1

Mixing due to rotation: D ≈ 1010 cm2 s−1


The overall problem

m = Mr: Lagrangian coordinate

r, P, T, Lr: independent variables

Xi: composition variables

ρ, κ, ε, . . .: dependent variables

initial value problem in time: r(m, t = 0), P (m, t = 0),T (m, t = 0), Lr(m, t = 0), ~X(m, t = 0) = ~X(t = 0) →integration with time

boundary value problem in space: r(m = 0, t) = 0,Lr(m = 0, t) = 0 and L = 4πσR2T 4

eff ,P (m = M) = P (τ = 2/3)


The equations

The four structure equations to be solved are:

∂r

∂m=

1

4πr2ρ

∂P

∂m= − Gm

4πr4− 1

4πr2

∂2r

∂t2

∂Lr

∂m= εn − εν − cP

∂T

∂t+

δ

ρ

∂P

∂t

∂T

∂t= − GmT

4πr4P∇


The equations

For energy transport, we have to find the appropriate ∇.In case of radiative transport, this is:

∇rad =3

16πacG

κLrP

mT 4

Finally, for the composition, we have

∂Xi

∂t=

mi

ρ

∑

j

rji −∑

k

rik

+ diff. or similar terms


Simple stellar models: Homology

Two stars (0, 1) are said to be homologous to each other, ifthe following homology assumption holds:

atm1

M1=

m0

M0→ r1

R1=

r0

R0

⇒ r(m

M

)

= Rfr

(m

M

)

P(m

M

)

= PcfP

(m

M

)

similarly for T , Lr, where the scaling functions fi are inde-

pendent of M , but constants are dependent on M and µ.


Homology

Assume power laws for the dependent variables:

ε = ε0ρλT ν

κ = κ0ρnT−s

P =Rµ

ρT

The equation of state can be written more general as

P ∼ ρχρTχT

n = 1, s = 3.5 for Kramer’s opacityλ = 1, ν = 7 for pp-chains

. . .


Homology

x := m/M ; from (radiative) structure equations

dP

dm=

P

m

d ln fP

d ln x= − Gm

4πr4→ P

m∼ m

r4

dr

dm=

r

m

d ln fr

d ln x=

1

4πr2ρ→ r

m∼ 1

r2ρ

dT

dm=

T

m

d ln fT

d ln x= − 3κ

64πac

Lr

r4T 3→ T

m∼ Lr

r4T 3

dLr

dm=

Lr

m

d ln fL

d ln x= ε → Lr

m∼ ε


Mass–Luminosity–Relation

From the relations for P and ρ and the EOS it follows that

P

ρ∼ m

r∼ T

µ

(or rT ∼ µm) and with that for Lr ⇒Lr ∼ µ4m3

in particular, for x = 1 or m = M , we have

L ∼ µ4M3,

which is the mass-luminosity relation for main sequence

stars. It does not depend on energy generation, but the pro-

portionality is determined mainly by opacity!


Mass–Luminosity–Relation

. . . and the observed counterpart


Mass-Radius–Relation

Since also Lr ∼ mε ∼ mρλT ν , we obtain (again for x = 1)

R ∼ µν−4

ν+3λ Mλ+ν−2ν+3λ

For λ = 1 and ν ≈ 5 (pp-cylce) ⇒

R ∼ µ0.125M0.5

For λ = 1 and ν ≈ 15 (CNO-cycle) ⇒

R ∼ µ0.61M0.78

These are mass-radius relations for the two main nuclear

cycles on the main sequence.


Mass-Radius–Relation

. . . and the observed counterpart


Main-Sequence–Relation

For R ∼ M3/4 (average exponent), L ∼ M3 and

L ∼ R2T 4eff

⇒ log L = 8 log Teff + const,

which is the equation for the main sequence; for R=const.one gets

log L = 4 log Teff + const,

which are shallower lines.Note also that since L ∼ M3 but τnuc ∼ M/L

→ τnuc ∼ M−2

more massive stars are brighter, but live for a shorter time!


Central values on the main sequence:

Set λ = 1 and µ=const.

Pc ∼ P ∼ P (x)fP (x) and Tc ∼ T ∼ T (x)

fT (x)

Also T ∼ MR , P ∼ M2

R4 , ρc ∼ Pc

Tc, and R ∼ M

ν−1ν+3 ;

Tc ∼ M4

ν+3

Pc ∼ M− 2(ν+5)ν+3

ρc ∼ M− 2(ν−3)ν+3

Tc ∼ ρ− 2

ν−3c


Central values on the main sequence:

Central values onmain-sequence (ho-mology results)


Numerical procedure


The complete problem

m is the Lagrangian coordinate;r, P, T, Lr are the independent variables;Xi are the composition variables.The four structure equations are:

∂r

∂m=

1

4πr2ρ

∂P

∂m= − Gm

4πr4− 1

4πr2

∂2r

∂t2

∂Lr

∂m= εn − εν − cP

∂T

∂t+

δ

ρ

∂P

∂t

∂T

∂t= − GmT

4πr4P∇


The complete problem. . .

For radiation, ∇ is

∇rad =3

16πacG

κLrP

mT 4

In case of convection

∇ = ∇c (= ∇ad)

For the composition we have

∂Xi

∂t=

mi

ρ

∑

j

rji −∑

k

rik

∑

i

Xi = 1


The complete problem. . .

In addition: ρ(P, T,Xi), κ(P, T,Xi), rjk(P, T,Xi), εn(P, T,Xi),εν(P, T,Xi), . . .

In space (mass 0 ≤ m ≤ M ): boundary value problem withboundary conditions:

at center: r(0) = 0, Lr(0) = 0

at r = R: P and T either from P = 0, T = 0 (“zero b.c.”)or from atmospheric lower boundary and:

Stefan-Boltzmann-law L = 4πσR2T 4eff


Central conditions

Series expansion in m around center:

r =

(

3

4πρc

)1/3

m1/3

P = Pc −3G

8π

(

4π

3ρc

)4/3

m2/3

Lr = (εg + εn − εν)cm

Stellar atmospheres (hydrostatic, grey):

optical depth: τ :=∫∞R κρdr = κ

∫∞R ρdr;

pressure P (R) :=∫∞R gρdr ≈ gR

∫∞R ρdr ⇒ P = GM

R223

1κ

In time, we have an initial value problem (zero-age model).


Numerical methods

Spatial problem

1. Direct integration : e.g. Runge-Kutta

2. Difference method : difference equations replace differentialequations

3. Hybrid methods : direct integration between fixedmesh-points; multiple-fitting method, but the variation ofthe guesses at the fixed points is done via aNewton-method


The Henyey-method

Write the equations in a general form:

Aji :=

yj+1i − yj

i

mj+1i −mj

i

− fi(yj+1/21 , . . . , y

j+1/24 )

upper index: grid-point (j + 1/2 mean value); lower index:i-th variableOuter and inner boundary conditions:

Bi = 0 i = 1, 2 Ci = 0 i = 1, . . . , 4

where the inner ones are to be taken at grid-point N − 1 and

the expansions around m = 0 have been used already.


Henyey-method (contd.)

→ 2 + 4 + (N − 2) · 4 = 4N − 2 equs.for 4×N unknowns -2 b.c.Newton-approach for corrections δyi

Aji +

∑

i

∂Aji

∂yiδyi = 0

H

δy11

δy12

...δyN

3

δyN4

=

B1...

Aji

...C4


Henyey-scheme

Matrix H containsall derivatives andis called Henyey-matrix. It containsnon-vanishing ele-ments only in blocks.This leads to a par-ticular method forsolving it (Henyey-method).

Expresse some of corrections in terms of others, e.g.:δy1

1 = U1δy23 + V1δy

24 + W1 ⇒ matrix equations for Ui, Vi, Wi.


Henyey-scheme

First block-matrix j = 1, 2:

∂B1

∂y11

∂B1

∂y12

. . . 0∂B2

∂y11

∂B2

∂y12

. . . 0∂A1

1

∂y11

∂A11

∂y12

. . . ∂A11

∂y22

......

......

∂A41

∂y11

∂A41

∂y12

. . . ∂A41

∂y22

U1 V1 W1

U2 V2 W2

U3 V3 W3...

......

U6 V6 W6

=

0 0 −B1

0 0 −B2

−∂A11

∂y23−∂A1

1

∂y24−A1

1...

......

−∂A14

∂y23−∂A1

4

∂y24−A1

4


Henyey-scheme

In the next block, y21 and y2

2 are replaced as before, and the

other variables are expressed in terms of y33 and y3

4. This

reduces the block of 4 equations in 8 unknowns into one of

4 unknowns. Repeat this until the last block consisting of

4 equations for 6 unknowns (2 expressed by previous co-

efficients). → δyN1 , . . . , δyN

4 . Then, the coefficient equations

of the previous block allow to calculate the corrections of

the previous block, and so on, until the first one is reached

again. This done, the Aji can be computed again (with the

derivatives) and the next iteration starts.


Integration in time

Xi(t +4t) = Xi(t) +∂Xi

∂t(T (t), P (t), . . .)4t

Improvement:backward differencing (e.g. nuclear network)

Xi(t +4t) = Xi(t) +4t∑

i

rij(t)Xi(t +4t)Xj(t +4t)

done for chemical evolution, mixing, diffusion, etc.


Microphysics – Equation of state,opacities, energy generation, . . .


Pressure and energy

Pressure as momentum transfer ⊥ area= n(ε) · ~p(ε) · ~v(ε)Mean over incident angle (1/3 · p · v) and particle energy ε:

P =1

3

∫ ∞

0n(ε)p(ε)v(ε)dε

relativistic case:

γ :=(

1− v2

c2

)−1/2; p = γmv; ε = (γ − 1)mc2 ⇒

P =1

3

∫ ∞

0nε

(

1 +2mc2

ε

)(

1 +mc2

ε

)−1

dε


Pressure and energy

limiting cases

non-relativistic: mc2 ε →PNR = 2

3

∫

nεdε = 23〈nε〉 = 2

3UNR

extrem-relativistic: mc2 ε → PER = 13UER

⇒ general relation P ∼ U (energy density)

Radiation pressure:

Prad =1

3U =

a

3T 4(

a = 7.56 · 10−15 erg

cm3K4

)


Equation of State - Ideal gas

P = nkBT =Rµ

ρT

with ρ = nµmu; µ: molecular weight, mass of particle permu.Several components in gas with relative mass fractionsXi = ρi

ρ → ni = ρXi

muµi

electrons and ions:

P = Pe +∑

i

Pi = (ne +∑

i

ni)kT.


Ionization – limiting cases

Completely ionized atoms (of mass fraction Xi and chargeZi):

P = nkT = R∑

i

Xi(1 + Zi)

µiρT =

Rµ

ρT

µ :=(

∑

iXi(1+Zi)

µi

)−1: mean molecular weight

For a neutral gas, µ =(

∑

iXi

µi

)−1.

The mean molecular weight per free electron is

µe :=

(

∑

i

XiZi

µi

)−1

=2

(1 + X)


Ideal gas with radiation pressure

P = Pgas + Prad

β := Pgas

P ⇒(

∂β∂T

)

P= −4(1−β)

T and(

∂β∂P

)

T= (1−β)

T .

Furthermore

α :=

(

∂ ln ρ

∂ ln P

)

T

=1

β

δ := −(

∂ ln ρ

∂ ln T

)

P

=4− 3β

β

ϕ :=

(

∂ ln ρ

∂ ln µ

)

T,P

= 1


. . . more derivatives

cP :=

(

dq

dT

)

P

=

(

du

dT

)

P

+ P

(

dv

dT

)

P

=Rµ

[

3

2+

3(4 + β)(1− β)

β2+

4− 3β

β2

]

∇ad :=

(

∂ ln T

∂ ln P

)

ad

=Rδ

βµcP=:

Γ2 − 1

Γ2

γad :=

(

d ln P

d ln ρ

)

ad

=1

α− δ∇ad=: Γ1


. . . more relations

Γ3 :=

(

d ln T

d ln ρ

)

ad

+ 1

Γ1

Γ3 − 1=

Γ2

Γ2 − 1

Limiting cases:

For β → 0, cP →∞, ∇ad → 1/4 and γad → 4/3.

For β → 1, cP → 5R2µ , ∇ad → 2/5, and γad → 5/3.


Ionization

Saha-equation (see introductory course for derivation):

nr+1

nrPe =

ur+1

ur2(2πme)

3/2

h3(kT )5/2 exp(−χr/kT ),

with

nr: number density of atoms in ionization state r;

χr ionization energy;

ur partition function (statistical weight)

Pe = nekT the electron pressure (k = kB)


Hydrogen ionziation in Sun

n = n0 + n1; ne = n1; x := n1

n0+n1

Pe = Pgasne

ne+n = Pgasx

x+1

⇒ x2

1− x2=

u1

u0

2

Pgas

(2πme)3/2

h3(kT )5/2e−χ1/kT

u0 = 2, u1 = 1 are ground-state stat. weights; χ1 = 13.6 eV.

solar surface, T = 5700 K, Pgas = 6.8 · 104, → x ≈ 10−4

in envelope Pgas = 1012, T = 7 · 105, → x ≈ 0.99.

mean molecular weight: µ = µ0/(E+1), where E the number

of free electrons per all atoms.


Ionization profile

Illustration of ionization of hydrogen and he-

lium within a stellar envelope. In panel (b)

the corresponding run of ∇ad is shown. The

depression is due to the increase in cP due

to ionziation. Since ∇ad is getting smaller,

convection will set in.


Electron degeneracy

The distribution of electrons in momentum space(Boltzmann equation; p is momentum):

f(p)dpdV = ne4πp2

(2πmekT )3/2exp

(

− p2

2mekT

)

dpdV

Pauli-principle:

f(p)dpdV ≤ 8πp2

h3dpdV


Completely degenerate gas

f(p) =8πp2

h3for p ≤ pF ∝ n

1/3e non-rel.

= 0 for p > pF

EF = p2F

2me∝ n

2/3e is the Fermi-energy; for EF ≈ mec

2, ve ≈ c

→ relativistic complete degeneracy and EF ∝ n1/3e

pF mec (non-relativistic) →Pe = 1.0036 · 1013

(

ρµe

)5/3Pe = 2

3Ue

pF mec (relativistic) →Pe = 1.2435 · 1015

(

ρµe

)4/3Pe = 1

3Ue

Pi Pe


Partial degeneracy

Finite T → Fermi-Dirac statistics:

f(p)dpdV =8πp2

h3

1

1 + exp(

EkT −Ψ

)dpdV

Ψ = ne

T 3/2 is the degeneracy parameter.

At constant Ψ, T ∝ ρ2/3 for the non-relativistic case, and∝ ρ1/3 in the relativistic one.

For Fermions, the following relations are valid:

ne =8π

h3

∫ ∞

0

p2dp

1 + exp(

EkT −Ψ

)

Pe =8π

3h3

∫ ∞

0

p3v(p)dp

1 + exp(

EkT −Ψ

)


Partial degeneracy

. . . Ue = 8πh3

∫∞0

Ep2dp

1+exp( EkT−Ψ)

f(p) for partially degenerate gas

with ne = 1028 cm−3 and T =

1.9·107 K corresponding to Ψ =

10.

The equation of state for normal stellar matter:

P = Pion+Pe+Prad =Rµ0

ρT +8π

3h3

∫ ∞

0

p3v(p)dp

1 + exp(

EkT −Ψ

)+a

3T 4


Non-ideal effects

finite size of atoms → pressure ionizationimportant already in Sun and low-mass stars

Coulomb interaction – low density → pressure reductionimportant in many stars (envelopes, but also solar core)

Coulomb interaction – high density → crystallizationwhite dwarfs, neutron stars

configuration effects → van der Waals gas; quantumeffects (spin–spin–interactions)

neutronizationneutron stars


Pressure ionization

defect of Saha-equation: P ↑, T ≈ const. → ionizationdegree decreases

reason: assumption of point-like atoms

picture: atomic potentials overlap and decreasecontinuum → electrons are set free

effect important for rs ≡ de

a0< 1, where a0 classical

atomic radius and de =(

34π

)1/3n−1/3e mean electronic

distance

calculation either with simple recipes (hard-sphereapproach) or quantum-statistical means


Electrostatic corrections

interaction parameter: thermal energy of species irelative to Coulomb energy

Γ ≡ (Zie)2

dikT

Γ 1 → negligible

Γ < 1 → Debye-Hückel treatment

Γ 1 → Wigner-Seitz-approximation


Debye-Hückel approximation

Debye-radius rD =√

mukT4πe2ρξ = 8.9 · 10−9

√

T6/ρξ cm ;

(ξ := 〈Z/A〉)→ reduction of potentialV (r) = −Ze2

r exp(−r/rD) ≈ −Ze2

r + Ze2

rD

screening acts as additional energy Es = Ze2

rD

reduction of free energy → gas pressure reduction,P = −

(

∂F∂V

)

T,V

non-degenerate gas → Pg = nkT

[

1− 0.032 ρ1/2

T3/26

µξ3/2

]

Sun: µ ≈ 0.5, ξ ≈ 2 → Pg = nkT [1− 0.015]


Crystallization

at high ρ and low T (white dwarfs) ions form latticewhich minimizes (Coulomb) energy

critical condition: therm. energy ≈ Coulomb-energy perion of charge Ze

ΓC ≡ (Ze)2

rionkT = 2.7 · 10−3 Z2n1/3ion

T

ΓC ≈ 100 → crystallization

melting temperature

Tm ≈(Ze)2

Γck

(

4πρ

3µ0mu

)1/3

= 2.3 · 103Z2µ−1/30 ρ1/3


A phase diagram of hydrogen

limits: pressure ionization (rs), degeneracy θi, Coulomb interac-tion Γ, radiation pressure, ionization


EOS - practical use

analytical EOS (Saha,radiation pressure) withsome corrections(pressure ionization)

semi-analytical:Eggleton-Faulkner-Flannery and derivates

detailed tables (MHD for Sun; OPAL for Sun andgeneral purposes; Saumon-Chabrier for H/He-gas athigh densities)


Opacity

Electron scattering: Thomson-scattering:

κsc = 8π3

r2e

memu= 0.20(1 + X) cm2g−1

Compton-scattering: T > 108: momentum exchange →κ < κsc

free-free transitions:κff ∝ ρT−7/2 (Kramers formula)

bound-free transitions:κbf ∝ Z(1 + X)ρT−7/2

b-f for H−-ion below 104 K (major source)

bound-bound transitions: below 106 K. No simple formula.

e−-conduction: κc ∝ ρ−2T 2


Opacities – practical usecomplications: complex line structures, many elements,molecules, underlying EOS, transition probabilities . . .

no on-line calculationaccurate enough →treat separately

use of pre-calculatedtables for many mixtures

Opacity Project (Sun,atomic data); OPAL (Sunand stars); Alexander(low T ; molecules)


Nuclear reactions

conversion of mass to energy; E = mc2

Sun: Mc2 = 2 · 1054 erg

maximum lifetime → 1013 years at L

H→ He conversion: (4MH −MHe)c2 = 26.73 MeV

(6.4 · 1018 erg/gm)

rel. to rest mass: 26.73/3724 = 0.007 →1011 yr

He → C conversion: (3MHe −MC)c2 = 7.27 MeV

lifetime (L ≈ 10 · · · 102L) → 109 yr

fusion of heavier elements → shorter burning times


Fundamental terms – I

Reaction a + X → Y + b (Mi: rest mass)

energy conservation:EaX + (Ma + MX)c2 = EbY + (Mb + MY )c2

Eij: kinetic energy of i + j in center-of-mass system ij

atomic mass excess:4MAZ = (MAZ − Amu)c2

or = 931.48(MAZ/mu − A) MeV

→ EaX + (4Ma +4MX) = EbY + (4Mb +4MY )

Some mass excesses (in MeV):p: 7.288; d: 13.136; 3He: 14.931; 4He: 2.425; 12C: 0.0;16O: -4.727; 56Fe: -60.605


Fundamental terms – II

Cross section σ(cm2): “probability per pair of reactionpartners for reaction to occur”σ(cm2) = number of reactions/nucleus X/time

number of incident particles a/area/time

Reaction rate r(reactions/(cm3 s))NX : number density of X; Na: number density of a;v: velocity of a relative to X; σ(v): cross section (flux-,or velocity-dependent)

In a gas, relative velocities follow a velocity distributionΦ(v) (

∫

Φ(v)dv = 1) ⇒

raX = NaNX

∫ ∞

0vσ(v)Φ(v)dv ≡ NaNX〈σv〉


Fundamental terms – III

Correction for identical reaction partners:raX = (1 + δaX)−1NaNX〈σv〉λ ≡ 〈σv〉 (reaction rate per pair of particles)

lifetime of X against reactions with a:(

∂NX

∂t

)

a= − NX

τa(X) = −raX → τa(X) = (λa(X)Na)−1

velocity distribution Φ(v) (both nuclei are Maxwellian):N1(v1)d

3v1N2(v2)d3v2 =

N1N2(m1m2)

3/2

(2πkT )3 exp[

− (m1+m2)V2

2kT − µv2

2kT

]

d3v1d3v2

V : velocity of center of mass; v: velocity relative toc.o.m.µ: reduced mass (m1m2/(m1 + m2))

→ r = N1N2

( µ2πkT

)3/2 ∫vσ(v) exp

(

− µv2

2kT

)

4πv2dv


Non-resonant reaction rates

Coulomb barrier: V = 1.44Z1Z2

R(fm) MeV

thermal energy: kT = 8.62 · 10−8T keV

⇒ particles with average energy cannot penetrate barrier,but those with high enough energies are too few

Gamow: tunnel-effect; penetration probability∝ exp−2πZ1Z2e

2

v

→ cross-sections proportional to such a factor⇒ σ(E) ≡ S(E)

E exp−2πZ1Z2e2

v

this leaves S(E) as a slowly varying function to bedetermined


〈σv〉 for non-resonant reactions

〈σv〉 =

(

8

µπ

)12

(kT )−32

∫ ∞

0S(E) exp

(

− E

kT− b√

E

)

dE

where b ≡ 31.28Z1Z2A keV12 ; A =

(

A1A2

A1+A2

)12

= µMu

S(E): slowly for steller energies → replace it by average S0:

〈σv〉 =

(

8

µπ

)12 S0

(kT )32

∫ ∞

0exp

(

− E

kT− b√

E

)

dE

Integrand has maximum at the most effective energy

E0 =(

bkT2

)

23 = 1.220(Z1Z2AT 2

6 )13 keV > kT !


〈σv〉 for non-resonant reactions

S(E) lab. measurements for T ≥ 100 keV (but3He(3He, 2p)4He at 20 keV)

extrapolated to lower energy and characterized byS(E = 0) and dS

dE |E=0

exponential integrand in 〈σv〉 replaced by

exp

(

−E0

kT− b√

E0

)

exp

(

−E − E0

4/2

)2

with exp(

−E0

kT − b√E0

)

= exp(

−3E0

kT

)

and 4 = 4√3(E0kT )1/2 = 0.75(Z2

1Z22AT 5

6 )16 keV


The Gamow-peak

S(E0): ⇒ S(E0) +(

dSdE

)

E0(E − E0)

Gaussian shape for Gamow peak:correction factor F (τ) = 1 + 5

12τ + O(τ−2) (τ = 3E0/kT )


Resonant reactions

conditions:

a + X → W → Y + b: W is compound nucleus

quasistationary states of compound nucleus, Ei

Ei ≈ kT

angular momentum, isotopic spin and parity conserved

Consequences:

In resonances, cross section dramatically increased

Some resonances predicted(3α →8 Be + α →12 C)


Electron (Coulomb) shielding

effective potential: Utot(d12) = Z1Z2e2

d12+ U(d12)

radius Rsh of shielding cloud (e− and ions) larger thaninterparticle distance

turning radius R0 ≈ Z1Z2e2

E0 Rsh

U(d12) ≈ U0 (at nuclear radius)

penetration factor in star at E = that in laboratory atE − U0

reaction rate r′12 = r12e−U0/kT


Weak screening

kT Coulomb energy of ions (Debye-Hückel approx.)

potential around nucleus: V1 ≈ eZ1

d exp(

− drD

)

rD =(

kT4πe2ρN0ξ

)1/2; ξ =

∑

Z(Z2+Z)XZ

AZ

Z2 close to Z1 → −U0

kT = 0.188Z1Z2ρ12 ξ

12 T

− 32

6

enhancement f ≈ 1− U0

kT (weak screening; Salpeter1954)


Strong screening

Coulomb energy of ions kT

pure electron cloud around ion

electrostatic energy of cloud U = − 910

(Ze)2

RZ;

RZ =(

3Z4πne

)12

−U0

kT = 0.205[(Z1 + Z2)53 − Z

53

1 − Z53

2 ](

ρµe

)13

T−16

for stellar astrophysics mostly unimportant!


Reaction types

1. standard reactions:(p, γ), (γ, p) (photo-disintegration)(n, γ), (γ, n)

(α, γ), (γ, α)

(p, n), (n, p)

(α, n), (n, α)

(α, p), (p, α)

e−- (νe!) and β+ (νe) -decay and -capture

2. multiple-particle reactions: 3α-process

3. ion-ion reaction: 12C(12C, γ)24Mg or 12C(12C, p)23Ne

4. URCA-process:e− + (Z,A) → (Z − 1, A) + νe → (Z,A) + e− + νe (A odd)


Reaction rate sources

Caughlan & Fowler (ADNDT 40, 283, 1988): Z ≤ 14;NA〈σv〉; Q-value and factor for inverse reaction

NA〈σv〉ρN1: interaction rate/second (reciprocal oflifetime)NA〈σv〉ρNAN1N2/(1 + δ12): reactions/(g sec)

NA〈σv〉ρNAN1N2/(1 + δ12) 1.6022 · 10−6Q6: energygeneration in erg/(g sec)fit-formulae like (p(p, β+)d):NA〈σv〉 =4.01E-15/T9ˆ(2/3)*DEXP(-3.380/T9ˆ(1/3))*(1.+0.123*T9ˆ(1/3)+1.09*T9ˆ(2/3)+0.938*T9)


Reaction rate sources

Adelberger et al. (Rev. Modern Physics, 70, 1265,1998): update on solar reaction rates (i.e. hydrogenburning) – attempt to define recommendable standard

NACRE (Angulo et al., Nuc. Phys. A 656, 3, 1999)

individual sources for selected reactions (e.g.22Ne(p, γ)23Na from El Eid & Champagne 1995)


Hydrogen burning

pp-chains:

Energy per completion: 26.20 (ppI), 25.67 (ppII), 19.20 MeV

(ppIII).03/2002 Stellar Structure – p.129

pp-chains reactions - I

1. p(p, β+νe)d

1.442 MeV; Eν = 0.263 MeVrpp ∼ ρ2X2

H ; τp(H) ≈ 1010 yrs

pep-reaction p(pe−, νe)d:

rpep = 5.51 · 10−5ρ(1 + X)T−1/26 (1 + 0.02T6)rpp

2. d(p, γ)3He

d(p, γ)3He favored over d(d, γ)4He

dDdt = λpp

H2

2 − λpdHD

D equilibrium abundance:(

DH

)

e= λpp

2λpd


pp-chains reactions - II

3. 3He(3He, 2p)4He

equation for 3He-abundance:

d3He

dt= λpdHD − 2λ33

(3He)2

2

d3He

dt= λpp

H2

2− 2λ33

(3He)2

23He achieves equilibrium value:

(3He

H

)

e

=

(

λpp

2λ33

)1/2


ppI-chain in equilibrium

equation for 4He-abundance:

d4He

dt= λ33

(3He)2

2= r33

first step: 3H → 3He + ν at rate rpp liberates 6.936 MeV(-0.262 for ν)ε(3H →3 He) = 1.069 · 10−5rpp erg g−1 sec−1

second step: 23He →4 He liberates 12.858 MeV

total energy is sum: εppI = 1.069 · 10−5rpp + 2.060 · 10−5r33

or (3He in equilibrium) d4Hedt = rpp

2 ; εppI = 2.099 · 10−5rpp


pp-chains reactions - III

4. 3He(4He, γ)7Be competitive reaction to ppI-termination

5. 7Be(e−, ν)7Li7Be-neutrinos (Homestake experiment);Eν = 0.80 MeVfollowed (immediately) by 7Li(p,4 He)4He (ppII-chain)

6. 7Be(p, γ)8B

about 10−3 as frequent as e−-capturefollowed by decay 8B →8 Be + β+ν → 2 4He (0.8 sec)ppIII-chain; B-ν (Kamiokande); Eν = 7.2 MeV


pp-chains reactions - IV

7. ppI vs. ppII and ppIII4He is only catalyst to allow 3He and p to produceanother 4He

energy/reaction therefore the same in ppII andppIII except for ν-losses7Li and 7Be in few years in eqilibrium4He and H equations are then:

d4He

dt= λ33

(3He)2

2+ λ34

3He4He

dH

dt= −3λpp

H2

2+ 2λ33

(3He)2

2− λ34

3He4He

branching between subcycles T -dependent03/2002 Stellar Structure – p.134

CNO-cycle

14N(p, γ)15O slowest reac-tion;qCNO ≈ 25MeV ;C & O ⇒ 14N .


CNO-cycle – remarks

12C/13C ≈ 3 . . . 6 (solar: 90)

at low T : C → N

Equivalent cycles involving Na and Mg exist andoperate partially at higher temperature (≈ 5 · 107 K).

26Mg

27Si(p,γ)

24Mg

26Al25Mg

22Ne21Na

23Na20Ne

22Na21Ne

27Al

(p,γ) (p,γ)

(p,γ)

(p,γ) (p,γ)

(p,α)

(p,γ)

(β+) (β+)

(β+)

(β+)

MgAlNeNa 25Al

(p,γ) (p,α)28Si

(p,γ)

(β+)


Hydrogen-shell abundance profile


pp or CNO?

εpp ∼ ρX2H ↔ εCN ∼

ρXHXCN

With increasing tem-perature, the CNO-cycle becomes moredominant.


Helium burning – 3α-process

No stable isotopes with A = 5, 8

3 particle process unlikely

Salpeter: 3α is two-step process8Be breakup lifetime of 10−16 sec

equilibrium 4He +4 He 8 Be;4He8Be ≈ 109

8B +4 He →12 C + γ

resonance in 12C! (predicted: Hoyle, measured: Fowler)

T -separation: no simultaneous p- and α-processes

at T ≈ 108, ε3α ∼ T 408 ! (pp ∼ 3 · · · 7, CNO ∼ 15)


Helium burning – further reactions

12C(α, γ)16O: uncertain by factors 3; C-production!

Q(12C →16 O) = 7.161 MeV16O(α, γ)20Ne

Q(16O →20 Ne) = 4.73 MeV

at higher temperatures further α-captures(→ α-elements: Mg, Si, S, Ca, Sc, Ti; Supernovae II)Q(20Ne →24 Mg) = 9.31 MeV


Carbon burning – I

1. 12C +12 C (at 6− 7 · 108 K)many resonances in compound nucleus 24Mg

many energetically interesting reaction channels:→24 Mg + γ (13.93 MeV) !→23 Na + p (2.238 MeV) !!→20 Ne + α (4.616 MeV) !!→23 Mg + n (-2.605 MeV) –→16 O + 2α (-0.114 MeV) !liberated particles → buildup of 16O, α-elements, etc.complicated nucleosynthesis begins


Carbon burning – II

2. 12C +16 O: Coulomb-barrier too high

3. 16O +16 O: (at ≈ 1 · 109 K)similar channels as for 12C +12 C:→32 S + γ (16.539 MeV)→31 P + p (7.676 MeV)→28 Si + α (9.593 MeV)→31 S + n (1.459 MeV)→24 Mg + 2α (-0.393 MeV)high ν energy losses (plasma) dominate overnuclear energy production!


Beyond carbon burning

Photo-disintegration (T ≥ 8 · 108 K)similar to ionization (but now nuclear instead ofatomic forces to be overcome by radiation bath)20Ne + γ →16 O + α

implies that C-burning products might be destroyedlater at higher T!

. . . afterwards rearrangement of nuclei abundances atincreasing temperatures by emission and capture ofparticles until 56Fe (max. in binding energy/nucleon);controlled by photodisintegration of 28Si


Burning- and life-times

burning times:

H : 1010 (yrs)

He : 108

C : 104

... :...

Si : hrs

lifetimes for Z = 0.004, Y = 0.240:mass (M) H-burn. He-burn.0.6 5.3 · 1010 1.3 · 108

1.0 7.0 · 109 1.1 · 108

1.6 1.5 · 109 1.2 · 108

2.0 8.3 · 108 2.5 · 108

3.0 3.1 · 108 6.1 · 107

5.0 1.0 · 108 9.4 · 106

9.0 3.3 · 107 2.0 · 107

15.0 1.4 · 107 8.9 · 105

20.0 9.8 · 106 6.6 · 105

40.0 5.0 · 106 4.3 · 105

100. 3.1 · 106 2.7 · 105


Plasma neutrino emission

Stellar plasma emits neutrinos, which leave star withoutinteraction and lead to energy loss Lν.Processes are:

1. Pair annihilation: e− + e+ → ν + ν at T > 109 K.

2. Photoneutrinos: γ + e− → e− + ν + ν (as Comptonscattering, but with ν-pair instead of γ).

3. Plasmaneutrinos: γpl → ν + ν; decay of plasma state γpl

4. Bremsstrahlung: inelastic nucleus–e− scattering, butemitted photon replaced by a ν-pair.

5. Synchroton neutrinos: as synchroton radiation, butagain a photon replaced by a ν-pair.


Regions of plasma-neutrino processes




A low-mass main-sequence star

Sun in main-sequence phase

hydrogen fusion in stellar center

standard test case for theory

solar neutrino problem

helioseismology


Solar quantities

Quantity value accuracy source

d 1.4959 · 1013 cm 10−8 triangulation;

radar & laser

M 1.989 · 1033 g 10−3 Kepler’s 3rd law

R 6.955 · 1010 cm 3 · 10−5 angular diameter

g 2.74 · 104 cm/s2

L 3.846 · 1033 erg/s 10−4 solar constant

Teff 5779 K ±4.5 K Stefan-Boltzmann~X Z/X = 0.0245 ±0.0010 photosph.;meteor.

Z/X = 0.0230 (new measurements)

Z O = 0.49, C = 0.30

N = 0.05, Fe = 0.07

X&Y X = 0.713 Y = 0.270 solar models

t 4.57 · 109 yrs ±0.03 · 109 meteorites


Interlude on star formation

Proto-stars contract along Hayashi-line

fully convective → homogeneous

heat up (virial theorem)

energy lost taken from contraction(εg)∂Lr

∂m = −T ∂S∂t

finally hydrogen burning starts

M < 0.08M – brown dwarfs

stellar structure theory when star contracts on thermaltimescale

→ Tc ≈ 105 K and T (R) ≈ 3000 K; lg L/L ≈ 3


Pre-MS evolution

During the contraction:

106 yrs: Tc ≈ 6 · 105 K →deuterium burning

107 yrs: Tc ≈ 8 · 106 K→CN-equilibrium

3 · 107 yrs; pp-chain

pp-nuclei in equilibriumabundances (3He)

4 ·107 yrs: εg ≈ 0 and L =∫

εndm (ZAMS)


Evolution on the main-sequence

The Sun at its present age:

Xc(t) = 0.36 (50% offuel)

from t = 0:L = 0.68L &Teff = 5600 K

Tc & Pc increased by7% & 30%.

εn: 98% pp-chain; 2%CNO-cycle


Solar neutrino production

The neutrino flux as produced within the solar core as a function of relative radius


The solar neutrino problem

Solar neutrinos - predicted and measured


Solutions that don’t work

Nuclear solution:reactions quite accurately known“no-solar-model”–argument: no combination ofneutrino source fluxes can match all

Astrophysical solution:solar structure from seismology accurately knownT as predictednon-standard models do not fit seismic structureno-solar-model argument

⇒ particle physics solution


Neutrino oscillations

extensions of standard model for particle physics allowmassive neutrinos

→ flavour changes (νe, νµ, ντ ) are possible

in vacuum and in presence of matter(Mikheyev–Smirnov–Wolfenstein–effect)

in nuclear reactions only νe are created (chargedcurrent)

depending on 4m2 and mixing angle sin2 2θ in detectorcertain fraction of other flavours arrive

radiochemical detectors: only νe measurable → deficit

electron scattering detectors: neutral charge crosssection much smaller → deficit


Neutrino oscillation solutions

10-4 10-3 10-2 10-1 100

sin22Θ

10-9

10-8

10-7

10-6

10-5

10-4

10-3

∆m2 [

eV2 ]

SMA

LMA

LOW

0.2 0.4 0.6 0.8 1.0sin22Θ

10-11

10-10

10-9

∆m2 [

eV2 ]

VO

Confidence regions in MSW-parameter space for 2-flavouroscillations


Helioseismology

Sun oscillates in > 105 eigenmodes

characterized by mode numbers n(radial), l (angular), and m (longitudinal)

measured by Doppler-observations oflines

frequencies of order mHz (5-min.oscillations)

differences of order µHz → days tomonths of observations

relative accuracy achieved 10−5!


p-modes

if restoring force is pressure (sound waves) → p-modes

propagation depends on sound speed profile, whichincreases inwards

outer turning point: steep ρ-gradient at photosphere

inner turning point: mode-dependent


Solar structure from p-modes

frequencies of standing waves depend on ρ, P , g, Γ1

with structure equations on ρ and Γ1 =(

∂ ln P∂ ln ρ

)

ad

inversion returns sound speed c(r) and density ρ

errors: measured frequencies, inversion method,reference model

⇒ seismic model ↔ comparison with theoretical model

forward comparison (model predictions for ν) alsopossible)


Sound speed comparison

(cSSM − cseis)/cseis) for three current Standard Solar Models and the uncertainty of the seis-

mic sound speed profile


Helioseismology: results

solar sound speed profile to 3 · 10−3 (0.2 ≤ r/R ≤ 0.8)

and < 1 · 10−2 everywhere

depth of convective envelope: rbcz = 0.713± 0.001R(SSM: 0.713R)

present surface helium content: Y = 0.246± 0.002(SSM: 0.245)

solar radius (seismic determination):R = (695.508± 0.026) · 106 m

central temperature known to better than 1%

exclusion of all non-standard models


Helioseismology: solar rotation

From SOHO/GOLF-data:



full evolutionary sequence from pre-MS or zero-age MSto solar age

unknown parameters: initial Y , Z, αMLT

fixed parameters: M, t

quantities to be matched: (Z/X), Teff(t), L

independently predicted quantities: rbcz, Y (t), c(r)

input physics: standard (OPAL κ & EOS; Adelbergerreaction rates; particle diffusion [sedimentation])

references: Bahcall & Pinsonneault (1998, 2001),Christensen-Dalsgaard et al. (1996), Schlattl & Weiss(1999)


The solar structure


Envelope structure

The temperature gradients in the outermost layers:


The Sun as a laboratory

Solar structure well known → deviations from SSM result inincompatibilities → use as test for non-standard oruncertain physics.

Examples:

determine the solar age

test overshooting from convective envelope

limit additional energy loss by axions

find deficits in EoS (relativistic electrons)

test non-Salpeter electron screening

limit uncertainties in nuclear reactions


Evolution of low-mass stars


ZAMS-star properties

The (zero-age) main sequence is the place of stars in corehydrogen burning with mass being the parameter along it.Radius increases with mass. Zero age = homogeneouscomposition (idealization)

Homology relations:

L ∝ M3µ4 and R ∝ µν−4ν+3 M

ν−1ν+3

pp-chain (for M . 1.5M), R ∝ µ0.15M0.5

CNO-cycle R ∝ µ0.6M0.8

central values

Tc ∝ M4

ν+3 Pc ∝ M−2(ν−5)

ν+3 ρc ∝ M−2(ν−3)

ν+3 Tc ∝ ρ−2

ν−3c

→ Central temperature ↑, but density ↓ with mass!03/2002 Stellar Structure – p.169

Brown dwarfs

stars that can never bestabilized by H-burning

estimate: H-burningtemperatureTc ≥ 5 · 106 K; fromhomology (ν ≈ 4):MBD < M(Tc,/Tc)

2 ≈0.1M

numerical result:MBD . 0.075M

first detection ongrounds of 7Li-presence


Interlude on 7Li

very fragile element; burnt via 7Li(p, α)4He atT & 2 · 106 K

primordial abundance [Li] = 2.2 (10−9.8 of H)

meteoritic (pre-solar) abundance [Li] = 3.3

solar photosphere abundance [Li] = 1.2 not explainedby SSM; T at rbcz not high enough

less massive stars → deeper convective zones → moreburning

at M . 0.4M completely convective → completeburning of 7Li

at M . 0.06M even Tc < 2 · 106 K → 7Li reappears!


Fully convective VLMS

VLMS: M/M . 0.7M

very cool LMS becomefully (almostadiabatically) convective

convective core due to3He +3 He-reactions

internal structurestrongly dependenton outer b.c. → atmo-spheres

radiative zone limits


VLMS - HRD

characteristicS-shape due toH2-dissociation (∇ad)and onset ofdegeneracy (constantradius track)

M–L–relation impor-tant for interpretationof N(MV ) → massbudget

HRD from M = 0.09 · · · 0.7 M


Comparison with observations

done both for fieldstars (solar Z) andclusters

important ingredients:EoS (non-ideal ef-fects), atmospheres,colour transforma-tions

VLMS and the CMD of NGC6397


Normal low-mass stars

M . 0.8M still on MS

M ≈ 0.8M (at age of universe) leave MS (turn-off) →found in Globular Clusters and Halo

metal-poor stars: relative overabundance of O andother α-elements w.r.t. Fe of [α/Fe] ≈ 0.3 · · · 0.4M . 1.5M (a few Gyr) → in disk and open clusters(around solar metallicity)

convective envelope below 1.2 . M1/M . 1.4

convective core above 1.1 . M2/M . 1.3(CNO-burning!)

end of MS: transition to thick → thin H-shell (alwaysCNO)


Low-mass stars: Evolution

MS and lower RGB evolution; influence of composition


Evolution up the Red Giant Branch

H-shell develops within former core (X < X(t = 0))

continuously deepening outer convective envelope,reaching into former core → first dredge-up

approaching H-shell finally pushes back convectiveenvelope

when reaching point of deepest convective extent →increased H-supply → shell lights up → L ↓ → verticalloop in HRD → bump in observed luminosity function

mass loss on upper RGB; according to Reimers formula

dM

dt= −4 · 10−13η

L

gR(M yr−1)


The first dredge-up

accompanied bysmall He-enrichmentof envelope

in field stars: larger ef-fects than predicted→additional deep mix-ing necessary


The RGB-bump

useful as distance indicator, check for theory


Core evolution on RGB

H-shell advances at Mc = LX0q

increasing He-core mass

L ∼ M7c (shell homology)

core degenerate → isothermal at Tc = Tsh

core contracting; shell and envelope expanding

small heating effect due to contracting material fromshell

high core density → plasma-ν-emission → T -inversion;hottest point slightly below shell

when Tmax ≈ 108 K → He-ignition


Luminosity Function

0.2 1.2 2.2 3.2 4.2 5.2 6.2M

0.3

0.8

1.3

1.8

2.3

2.8

3.3

Log

n

Z=0.0005 t=16 Gyr Z=0.0005 t=14 GyrZ=0.0002 t=14 Gyr

M30

v

luminosity functions test stellar evolution


Core Helium Flash

core develops independent of envelope

→ Mc at flash almost constant (Mc,fl ≈ 0.49M)

→ luminosity at flash Ltip ≈ constant → distance indicator

complex, almost dynamical behaviour; timescales downto hours

He-ignition off-center: if degenerate conditions →“low-mass star” (M . 2.3 · · · 2.5M)

→ nuclear energy into heating, not expansion → runaway

stops when T high enough for degeneracy lifted; thencore expansion

star settles on Horizontal branch/Red Giant Clump withnon-degenerate He-burning core and H-burning shell


Core Helium Flash

mixing of H into C-rich He-layers during flash possible!


Core Helium Flash


Horizontal Branch

after He-flash: same Mc →same L → horizontal

Teff determined by total (i.e.envelope) mass

distribution due to mass loss,but not understood

first HB parameter:composition (metal-poor →blue HB)

second parameter: unknown;age?


Past the HB. . .

He-shell burning (2shells) → AsymptoticGiant Branch (seeintermediate-massstars)

and/or extinction ofH-shell → crossing ofHRD at constant L

white dwarfThe solar evolution – from ZAMS to WD


Globular Clusters

Globular Clusters are excellent for test and use of stellarevolution theory:

very populous → well-defined CMD-branches

all stars at same distance

homogeneous composition (first order)

They can be used for

age determination

distance scale calibration (tip RGB; bump; HB)

calibration of theory (αMLT; colour transformations;diffusion; . . . )

detecting new effects (deep mixing; rotation)


GC Age determinations

Need:

metallicity [Fe/H]; [α/Fe]

distance (RR Lyr;MS-fitting to localsubdwarfs; . . . )

reddening

appropriate isochrones

→ TO-brightness ⇔ age

Or: age from relativechanges of CMD-branches


GC Ages: Comments

Problem: at high ages changes less sensitive to age(4VTO = 0.1mag ↔4t ≈ 1.5 Gyr)

Open clusters: younger; Y0 uncertain; fewer stars (noRGB)but some have Hipparcos-distances!

fitting of isochrones: quality is illusion of accuracy(colour transformation)

latest results: oldest galactic GC are ≈ 12 Gyr;α-enhancement, new EoS, new opacities (from solarmodel!) necessary; diffusion?


Salaris & Weiss results

Examples: M68 in (V − I); ages for the largest GC sample ever


Evolution of intermediate-mass stars


General features

2.5 ≤ M/M < 8: early evolution differs fromM ≤ 1.3M stars

mass range 1.3 < M/M < 2.5: properties of bothgroups

convective core and radiative envelope on the MS (forM > 1.3M); electron scattering opacity becomingimportant

hydrogen-burning via CNO-cycle; ε ≈ ρXZCNOT 18

rapid transition from MS to RGB (so-called Hertzsprunggap in HRD)

helium core remains non-degenerate

non-violent ignition of He at center

double-shell burning phase with degenerate C/O-core03/2002 Stellar Structure – p.192

Post-MS core evolution

Virial theorem applied to isothermal helium-core:

Eg = −∫ Mc

0

Gm

rdm = [4πr3P ]Mc

0 −∫ Mc

03P

ρdm

with (i.G.; T = Tc)

∫ Mc

03P

ρdm = 2Ei ≈ 3

Rµ

TcMc

⇒3Rµ

TcMc − CgGMc

Rc− 4πR3

cPc = 0

Here we have replaced Eg by an approximation with a“structure” constant Cg.


Post-MS core evolution

It follows that

⇒ Pc =3R4πµ

TcMc

R3c− CgG

4π

M2c

R4c

= C1TcMc

R3c− C2

M2c

R4c

, C1, C2 > 0

For fixed Mc, Tc, search the maximum, i.e.(

∂Pc

∂Rc

)

Mc,Tc

= 0.

This yields

Pc =2187R4

1024π2C3gG3µ4

T 4c

M2c∝ T 4

c

µ4M2c


Schönberg-Chandrasekhar mass

Envelope pressure Pe ∝ T 4c /M2 independent of Rc.

If maximum of Pc < Pe, no solution in equilibrium → corecontracts on a thermal timescale.⇒ critical Schönberg-Chandrasekhar mass q := Mc/M :

qSC = 0.37

(

µe

µc

)2

numerical value for solar-type stars is 0.08.

If q > qSC → degenerate (additional pressure source) or non-

isothermal (by contraction energy) core.


Hertzsprung-gap

Consequences:

1. after MS, cores of intermediate-mass stars contract onthermal timescale

2. envelope expands (why?) and gets cooler

3. fast crossing of HRD → gap

4. convection starts

5. further expansion via radius increase at constant Teff


Evolution of 5 M-star


Helium-burning phase

Helium ignites under non-degenerate conditions

M . 3 · · · 4M ignition during RGB ascent → clump

higher mass (M > 4M): excursion to higher Teff – loopthrough Cepheid-strip

loops depend on detailed structure

end of core helium burning: helium-burning shellaround C/O core and double-shell phase

asymptotic return to RGB: Asymptotic Giant Branch(AGB)

second dredge-up event


AGB-phase

Double-shell phase reached for M > 0.8M;intermediate-mass stars are prototype

special features: thermal pulses, nucleosynthesis ofrare elements (s-process); strong mass-loss

for 0.6 < Mc/M < 0.9: L/L = 5.92 · 104(Mc − 0.495)

Thermal pulsesrunaway events in helium shellduration: few hundred yearsinterpulse time: few thousand yearsstrong luminosity variations in shellvariable convective zonesdynamical phase possible


Thermal instability of helium shell

1. assume shell expands a little bit, e.g. due to a smallT -fluctuation or an increase in Y

2. if shell is very thin, the increase in thickness dDD is dρ

ρ

3. change of mean radius of shell, drr , however is almost 0,

because the shell is very thin

4. pressure in shell depends (hydrostatic equilibrium) onpressure exerted from the envelope

5. drr 1 → envelope does not “feel” any change in the

gravitational potential, and Pe stays constant

6. dρρ < 0 but dP = 0, for ideal gas → dT

T > 0 → heating

7. runaway until pressure is changing, too, or excessiveenergy carried away (convection)


A thermal pulse

The event of a thermal pulse ina 5M star (Pop. II). The firstpanel (a) shows the location ofthe two shells and the convec-tive regions; the second (b) thestellar luminosity, the third (c)the luminosity produced in H-and He-shell, (d) Teff and (e) thestellar radius. The time axis hasa varying scale.


Luminosity during TPs

805 806 8073

3.5

4

4.5

Thermal pulses in a 2.5M star over the whole AGB evolution


Mass loss on the AGB

M = const → evolution until Mc ≈ 1.4M

but: White Dwarf masses ≈ 0.6M

⇒ mass loss along AGB; up to 90% of mass lost!

M increase for high L; superwind: 10−4M/yr

Wind mechanism: radiative acceleration of dust formingin atmosphere

M = ηLα/(T βeffMγ)

ejected envelope expanding around star


The post-AGB HRD

AGB- and post-AGB evolution of a 2M star.


The post-AGB phase

Me < 10−2: envelope contracts

former core rapidly crosses HRD at constant L to highTeff , illuminating circumstellar shell with UV-photons

timescale: few ·104 yrs

at Teff ≈ 2 · 104 K UV-γ ionize circumstellar shell(Planetary Nebula)

later: H-shell extinguishes

WD cooling sequence reached

possibly late thermal pulse in post-AGB phase


Third dredge-up

AGB-stars show s-process elements (rare earths)enriched

process: thermal neutron captures (noCoulomb-barrier!); “s” for slow (compared to β-decays)

needs large n-flux

Two n-sources:

13C(α, n)16O and 22Ne(α, n)25Mg

Mechanism:TP: outer convective zone reaches He-rich layersbelow H-shellduring pulse (intershell convection) He-layersenriched in C


Nucleosynthesis on the AGB

→ formation of C-stars; observed even at low L

third dredge-up also mixes protons into the hot He-shell→ 13C source

12C(p, γ)13N(β+ν)13C → (p, γ)14N

(α, n)16O

16O(p, γ)17F(β+ν)17O → (p, α)14N

(α, n)20Ne


Nucleosynthesis on the AGB

or: H-shell converts C and O mostly into 14N

14N gets mixed into the He-shell, where

14N(α, γ)18F(β+ν)18O(α, γ)22Ne(α, n)25Mg

calculations: 3rd dredge-up, proton injection, ands-process more likely obtained, if

pulse number is highZ is lownon-standard convection (overshooting,semiconvection) is used

Example:110Cd + n →111 Cd + n . . .115 Cd

e−→ 115In + n → . . .


A different final evolution

. . . for M & 6M:

if mass high enough and M fails to remove envelope

Mc = MCH ≈ 1.44M reached

→ collapse and heating

C/O-core reaches C-burning ignition temperature(T ≥ 6 · 108 K)

core highly degenerate (similar to He-flash conditions,but more extreme)

→ thermal runaway →supernova explosion


Evolution of massive stars


General features

ignition of carbon in non-degenerate C/O-core →M & 8M

at M & 100M vibrational instability due toε-mechanism (positive feedback from nuclear reactionT -dependence) → disruption

extended convective cores on MS; overshooting

radiation pressure; electron scattering; relatively low ρ(20M: ρc ≈ 6.5)

mass loss strong and important (WR-stars: stellarwinds uncovering core during MS-phase)

nuclear burning stages from H- to Si-burning →Fe-cores; γ-disintegration → dynamical instabilityγad > 4/3) → core collapse → supernova


HRD of massive stars - I

Y = 0.285, Z = 0.02(Limongi et al. 2000): nomass loss, no overshoot-ing, semiconvection dur-ing He-burning


HRD of massive stars - II

Y = 0.28, Z = 0.02(Maeder, 1981): moder-ate mass loss, no over-shooting


HRD of massive stars - III

Y = 0.285, Z = 0.02(de Loore & Doom):parametrized mass loss &overshooting


Convective cores

Canonical onvectivecore sizes as func-tion of mass andmetallicity (Mowlaviet al., 1998); moder-ate overshooting α =0.2HP


Overshooting

large convective cores with high velocities and negativetemperature excess above Schwarzschild-boundary

extension of convective core due to overshootingchanges structure and increases fuel reservoir

→ extended lifetimes and broader main-sequence bands

from observations: canonical MS-bands too narrow →overshooting needed

inclusion: mostly parametric (fractions of HP – α);

mass mSch mα=0.25 mα=1.0

20 9 11 1440 24 26 3180 58 61 68


Semiconvection

after end of main-sequence phase; previous dyingconvective core and above → chemical gradients andconvection → semiconvection

crucial for understanding of SN1987A

20 M SN1987A progenitor evolution Left: Schwarschild-criterion; different physical aspects.

Right: Ledoux criterion; infinitely slow semiconvective mixing


Convection zones evolution

convectivezones untilbeginning ofcore carbonburning


Mass loss

Observational evidence:Wolf-Rayet stars – CNO-burning products at surfaceno red supergiants – envelope lost before redblue-to-red ratios – mass loss explainsspectroscopy

Influence:total masscore evolutionHRD-evolution

Inclusion: parametrized or ab-initio-calculations

Interaction with effect of overshooting


Mass loss on MS

Stellar mass at endof MS (Mowlavi et al.,1998), mass loss ac-cording to de Jager etal. + Kudritzki (solid) or2x the same (dashed)


Mass loss parametrization

1. function of L only: log M = m + n log L or M = NL/c2

2. of more stellar parmeters:log M = m + n log L + p log R− q log M

3. example:log(−M) = 1.24 log L + 0.16 log M + 0.81 log R− 14.02(de Jagers et al., 1989)

4. for WR-stars: log(−M) = α10−7M2.5 (α = 0.6 · · · 1.0)(Langer, 1989)

5. metallicity dependence (Kudritzki et al.): M ∼ Z1/2


Central evolution

Tc vs. ρc for massivestars during subse-quent nuclear burn-ing stages (Limongiet al.)


Documents

Stellar Structure and Evolution