Nuclear Astrophysics: The origin of heavy elements Lecture 1: Evolution massive stars
Gabriel Martínez Pinedo
Isolde lectures on nuclear astrophysics May 9–11, 2017 Introduction Astrophysical reaction rates Hydrostatic Burning Phases Outline
1 Introduction
2 Astrophysical reaction rates
3 Hydrostatic Burning Phases Hydrogen Burning Advanced burning stages Introduction Astrophysical reaction rates Hydrostatic Burning Phases What is Nuclear Astrophysics?
Nuclear astrophysics aims at understanding the nuclear processes that take place in the universe. These nuclear processes generate energy in stars and contribute to the nucleosynthesis of the elements and the evolution of the galaxy.
Hydrogen mass fraction X = 0.71 3. The solar abundance distribution Helium mass fraction Y = 0.28 Metallicity (mass fraction of everything else) Z = 0.019 Heavy Elements (beyond Nickel) mass fraction 4E-6 a-nuclei Disk solar abundances: Gap 12C,16O,20Ne,24Mg, …. 40Ca general trend; less heavy elements B,Be,Li Elemental 0 10 Halo (and isotopic) 10-1 composition 10-2 + Sun + of Galaxy at 10-3 location of solar -4 n r-process peaks (nuclear shell closures) system at the time o 10 Bulge
i
t -5
c 10 of it’s formation
a -6
r
f
10 r -7 e 10 s-process peaks (nuclear shell closures) b -8
m 10
u -9
n 10 10-10 10-11 Fe peak U,Th -12 (width !) 10 Fe Au Pb 10-13 0 50 100 150 200 250 mass number
K. Lodders, Astrophys. J. 591, 1220-1247 (2003) Introduction Astrophysical reaction rates Hydrostatic Burning Phases Cosmic Cycle Introduction Astrophysical reaction rates Hydrostatic Burning Phases Nucleosynthesis processes
In 1957 Burbidge, Burbidge, Fowler and Hoyle and independently Cameron, suggested several nucleosynthesis processes to explain the origin of the elements.
νp-process “neutrino-proton process” s-process 184 rp-process “slow process” via chain of stable nuclei through “rapid proton process” 162 via unstable proton-rich nuclei neutron capture through proton capture Pb (Z=82)
Proton dripline 126 (edge nuclear stability) Number of protons
Sn (Z=50)
r-process “rapid process” via Ni (Z=28) 82 unstable neutron-rich nuclei
Neutron dripline 50 (edge nuclear stability) 28 Fusion up to iron 20 Big Bang Nucleosynthesis 2 8 Number of neutrons Introduction Astrophysical reaction rates Hydrostatic Burning Phases Composition of the Universe after Big Bang Ma tt e r C o m po sit io n 80 76 after Big Bang Nucleosynthesis 70.683 in Solar System
60 ) % (
on i
act 40 r F ass
M 27.431
24
20
1.886 8.00E-08 0 HHeMetals Stars are responsible of destroying Hydrogen and producing “metals”. This requires weak interaction processes to convert protons into neutrons. Introduction Astrophysical reaction rates Hydrostatic Burning Phases Nuclear Alchemy: How to make Gold in Nature?
Pieter Bruegel (The Elder): The Alchemist Introduction Astrophysical reaction rates Hydrostatic Burning Phases Nuclear Alchemy: How to make Gold in Nature?
Stars are the cauldrons where elements are synthesized Introduction Astrophysical reaction rates Hydrostatic Burning Phases Star formation
Stars are formed from the contraction of molecular clouds due to their own gravity. Contraction increases temperature and eventually nuclear fusion reactions begin. A star is born. Contraction time depends on mass: 10 millions years for a star with the mass of the Sun; 100,000 years for a star 11 times the mass of the Sun.
The evolution of a Star is governed by gravity Introduction Astrophysical reaction rates Hydrostatic Burning Phases What is a star?
A star is a self-luminous gaseous sphere. Stars produce energy by nuclear fusion reactions. A star is a self-regulated nuclear reactor. Gravitational collapse is balanced by pressure gradient: hydrostatic equilibrium. mdm dF = −G = [P(r + dr) − P(r)]dA = dF grav r2 pres dm = 4πr2ρdr mρ dP −G = r2 dr Further equations needed to describe the transport of energy from the core to the surface, and the change of composition (nuclear reactions). Supplemented by an EoS: P(ρ, T). Introduction Astrophysical reaction rates Hydrostatic Burning Phases Stellar Evolution
Star evolution, lifetime and death depends on mass?
Age Planetary Nebula
Red Giant White Dwarf Low-mass Star Mass
Intermediate -mass Star
ECSN
Supernova Red Supergiant Neutron Star Massive Star CCSN
Stellar Cloud Black Hole with Protostars
Why such distinct outcomes? Introduction Astrophysical reaction rates Hydrostatic Burning Phases Core evolution
Hydrostatic equilibrium together with equation of state determines the evolution of the star core. , 407 (2012). 62
Green: transition from relativistic to non-relativistic electrons. from H.-T. Janka, Annu. Rev. Nucl. Part. Sci. Introduction Astrophysical reaction rates Hydrostatic Burning Phases Where does the energy come from? Energy comes from nuclear reactions in the core. 1 4 + 4 H → He + 2e + 2νe + 26.7 MeV
E = mc2
The Sun converts 600 million tons of hydrogen into 596 million tons of helium every second. The difference in mass is converted into energy. The Sun will continue burning hydrogen during 5 billions years. Energy released by H-burning: 6.45 × 1018 erg g−1 = 6.7 MeV/nuc Solar Luminosity: 3.85 × 1033 erg s−1 Introduction Astrophysical reaction rates Hydrostatic Burning Phases Nuclear Binding Energy
Liberated energy is due to the gain in nuclear binding energy.
Rest mass energy is converted into kinetic energy of the particles. This translates into heat that raises the temperature unless energy transported away. During hydrostatic burning phases stars maintain an equilibrium: produced energy is transported and later emitted at the surface. Introduction Astrophysical reaction rates Hydrostatic Burning Phases Types of reactions
Nuclei in the astrophysical environment can suffer different reactions: Decay 56 56 + Ni → Co + e + νe 15O + γ → 14N + p dn a = −λ n dt a a In order to dissentangle changes in the density (hydrodynamics) from changes in the composition (nuclear dynamics), the abundance is introduced:
na ρ Ya = , n ≈ = Number density of nucleons (constant) n mu dY a = −λ Y dt a a Rate can depend on temperature and density Introduction Astrophysical reaction rates Hydrostatic Burning Phases Types of reactions
Nuclei in the astrophysical environment can suffer different reactions: Capture processes a + b → c + d dn a = −n n hσvi dt a b
dYa ρ = − YaYbhσvi dt mu
decay rate: λa = ρYbhσvi/mu 3-body reactions: 3 4He → 12C + γ dY ρ2 α − Y3hαααi = 2 α dt 2mu 2 2 2 decay rate: λα = Yαρ hαααi/(2mu) Introduction Astrophysical reaction rates Hydrostatic Burning Phases Reaction rates
Consider na and nb particles per cubic centimeter of species a and b. The rate of nuclear reactions a + b → c + d is given by: rab = nanbσ(v)v, v = relative velocity In stellar environment the velocity (energy) of particles follows a thermal distribution that depends of the type of particles. Nuclei (Maxwell-Boltzmann) ! m 3/2 mv2 N(v)dv = N4πv2 exp − dv = Nφ(v)dv 2πkT 2kT Electrons, Neutrinos (if thermal) (Fermi) g 4πp2 N(p)dp = dp (2π~)3 e(E(p)−µ)/kT + 1 photons (Bose) 2 4πp2 N(p)dp = dp (2π~)3 epc/kT − 1 Introduction Astrophysical reaction rates Hydrostatic Burning Phases Stellar reaction rate
The product σv has to be averaged over the velocity distribution φ(v) (Maxwell-Boltzmann) Z ∞ hσvi = φ(v)σ(v)vdv 0 that gives: ! m 3/2 Z ∞ mv2 m m hσvi = 4π v3σ(v) exp − dv, m = 1 2 2πkT 0 2kT m1 + m2
or using E = mv2/2
!1/2 8 1 Z ∞ E hσvi = σ(E)E exp − dE 3/2 πm (kT) 0 kT Introduction Astrophysical reaction rates Hydrostatic Burning Phases Inverse reactions
Let’s have the reaction
a + b → c + γ Q = ma + mb − mc
We are interested in the inverse reaction. One can use detailed-balance to determine the inverse rate. Simpler using the concept of chemical equilibrium. dN dN ab = c = 0 dt dt NaNb λc NaNbhσvi = Ncλc = Nc hσvi
Using equilibrium condition for chemical potentials: µa + µb = µc
!3/2 n(Z, A) 2π~2 µ(Z, A) = m(Z, A)c2 + kT ln G(Z, A) m(Z, A)kT Introduction Astrophysical reaction rates Hydrostatic Burning Phases Inverse reactions
One obtains:
!3/2 !3/2 N N g g m m kT a b = a b a b e−Q/kT 2 Nc gc mc 2π~ Finally, we obtain:
!3/2 !3/2 g g m m kT λ = a b a b e−Q/kT hσvi c 2 gc mc 2π~
For a reaction a + b → c + d (Q = ma + mb − mc − md):
!3/2 gagb µab −Q/kT hσvicd = e hσviab gcgd µcd Introduction Astrophysical reaction rates Hydrostatic Burning Phases Rate Examples: 4He(αα, γ)
103 4 29 −3 He(αα,γ) nα = 10 cm 102 12C(γ,αα)4He
101 ) −1 100 Rate (s 10−1
10−2
10−3 0.1 1 10 Temperature (GK) Introduction Astrophysical reaction rates Hydrostatic Burning Phases Rate Examples: (p, γ)
1010 109 64 27 −3 Ga(p,γ) np = 10 cm 108 65Ge(γ,p) 107
) 106 −1 105 104 Rate (s 103 102 101 100 0.1 1 10 Temperature (GK) Introduction Astrophysical reaction rates Hydrostatic Burning Phases Rate Examples: (α, γ)
103 94 27 −3 102 Kr(α,γ) nα = 10 cm 98Sr(γ,α) 101
0
) 10 −1 10−1
Rate (s 10−2
10−3
10−4
10−5 0.1 1 10 Temperature (GK) Introduction Astrophysical reaction rates Hydrostatic Burning Phases Rate examples: (n, γ)
1010 109 130 22 −3 Cd(n,γ) nn = 10 cm 108 131Cd(γ,n) 107 130Cd beta decay
) 106 −1 105 104 Rate (s 103 102 101 100 0.1 1 10 Temperature (GK) Introduction Astrophysical reaction rates Hydrostatic Burning Phases Summary (reaction rates)
General features reaction rates: Reactions involving neutral particles, neutrons, are almost independent of the temperature of the environment. Charged particles reactions depend very strongly on temperature (tunneling coulomb barrier). At high temperatures inverse reactions (γ, n), (γ, p), (γ, α) become 3 important, nγ ∝ T We can distinguish two different regimes: Nuclear reactions are slower than dynamical time scales (expansion, contraction,...) of the system: Hydrodynamical burning phases. Nuclear reactions are faster than dynamical time scales: Explosive Nucleosynthesis. Introduction Astrophysical reaction rates Hydrostatic Burning Phases Hertzspung-Russell diagram Introduction Astrophysical reaction rates Hydrostatic Burning Phases Hydrogen burning: ppI-chain
Step 1: p + p → 2He (not possible) + p + p → d + e + νe Step 2: d + p → 3He d + d → 4He (d abundance too low)
Step 3: 3He + p → 4Li (4Li is unbound) 3He + d → 4He + n (d abundance too low) 3He + 3He → 4He + 2p
d + d not going because Yd is small and d + p leads to rapid destruction. 3 3 He + He goes because Y3He gets large as nothing destroys it. Introduction Astrophysical reaction rates Hydrostatic Burning Phases The relevant S-factors + 25 p(p, e νe)d: S 11(0) = (4.00 ± 0.05) × 10 MeV b calculated 3 −7 p(d, γ) He: S 12(0) = 2.5 × 10 MeV b measured at LUNA 3 3 4 He( He, 2p) He: S 33(0) = 5.4 MeV b measured at LUNA
Laboratory Underground for Nuclear Astrophysics (Gran Sasso). Introduction Astrophysical reaction rates Hydrostatic Burning Phases Reaction Network ppI-chain
dYp 2 ρ ρ 2 ρ = −Yp hσvipp − YdYp hσvipd + Y3 hσvi33 dt mu mu mu 2 dYd Yp ρ ρ = hσvipp − YdYp hσvipd dt 2 mu mu
dY3 ρ 2 ρ = YdYp hσvipd − Y3 hσvi33 dt mu mu 2 dY4 Y3 ρ = hσvi33 dt 2 mu Stiff system of coupled differential equations. Introduction Astrophysical reaction rates Hydrostatic Burning Phases pp chains
Once 4He is produced can act as catalyst initializing the ppII and ppIII chains. Introduction Astrophysical reaction rates Hydrostatic Burning Phases The other hydrogen burning: CNO bicycle
,?! . ,?! . %'. %(3 %+4 16.70 % 83.20 % 83.20 % 16.7083.20 % ,?! .
He + He ν %'3 0%)4 00 %+/
,?! . ,?! . ,?! . ,?! . %&. %)3 %*4 Introduction Astrophysical reaction rates Hydrostatic Burning Phases Neutrino spectrum (Sun)
This is the predicted neutrino spectrum ]
1 Gallium Chlorine SuperK − 1012 s 2 − pp 1010 ±1%
8 13 10 N 7 ) [for lines, cm 15 Be pep 1 O − ±10% ±1.5%
6 MeV 10 +20% 1 17 8 − F B –16% s 2 − 104
± hep ? 102
0.10.2 0.5 12 5 10 20 Flux at 1 AU (cm Neutrino energy (MeV) Introduction Astrophysical reaction rates Hydrostatic Burning Phases Energy generation: CNO cycle vs pp-chains Introduction Astrophysical reaction rates Hydrostatic Burning Phases Consequences
Stars slightly heavier than the Sun burn hydrogen via CNO cycle. CNO cycle goes significantly faster. Such stars have much shorter lifetimes
Mass (M ) lifetime (yr) 0.8 1.4 × 1010 10 )'*+",-..% 1.0 1 × 10 .(- .%/0#10/2% 1.7 2.7 × 109 #!(#%(+/%*'-!( 3 -!4+ 3.0 2.2 × 108 5.0 6 × 107 !"#$$%'!( 7 9.0 2 × 10 5#6",-..% 16.0 1 × 107 .(- .%.('11%#!% 6 (+/%,-'!% 25.0 7 × 10 ./7/!4/ 40.0 1 × 106 Introduction Astrophysical reaction rates Hydrostatic Burning Phases Helium Burning
Once hydrogen is exhausted the stellar core is made mainly of helium. Hydrogen burning continues in a shell surrounding the core. 4He + p produces 5Li that decays in 10−22 s. Helium survives in the core till the temperature become large enough (T ≈ 108 K) to overcome the coulomb barrier for 4He + 4He. The produced 8Be decays in 10−16 s. However, the lifetime is large enough to allow the capture of another 4He: 3 4He → 12C + γ Hoyle suggested that in order to account for the large abundance of Carbon and Oxygen, there should be a resonance in 12C that speeds up the production. 12C can react with another 4He producing 16O 12C + α → 16O + γ These are the two main reactions during helium burning. Introduction Astrophysical reaction rates Hydrostatic Burning Phases Hoyle State and tripple α reaction
Red giant structure Introduction Astrophysical reaction rates Hydrostatic Burning Phases End of helium burning
Nucleosynthesis yields from stars depend on mass:
stars with M . 8 M These stars eject their envelopes during helium shell burning producing planetary nebula and white dwarfs. Constitute the site for the s process.
stars with M & 12 M These stars will ignite carbon burning under non-degenerate conditions. The subsequent evolution proceeds in most cases to core collapse. These stars make the bulk of newly processed matter that is returned to the interstellar medium.
stars with 8 M . M . 12 M The end products of these stars is still under discussion and may depend on metallicity. Potentially they can produce ONe white-dwarfs (planetary nebula), thermonuclear supernova, and/or electron capture supernova. Introduction Astrophysical reaction rates Hydrostatic Burning Phases Stellar life
Nuclear burning stages (e.g., 20 solar mass star)
Main T Time Main Fuel Secondary Product Product (109 K) (yr) Reaction
14 7 CNO H He N 0.02 10 4 H Æ 4He
18 22 4 12 O, Ne 6 3 He Æ C He O, C s-process 0.2 10 12C DJ)16O Ne, 3 12C + 12C C Mg Na 0.8 10 20Ne JD)16O Ne O, Mg Al, P 1.5 3 20Ne DJ)24Mg Cl, Ar, 16O + 16O O Si, S K, Ca 2.0 0.8 Ti, V, Cr, 28Si JD)… Si Fe Mn, Co, Ni 3.5 0.02 Introduction Astrophysical reaction rates Hydrostatic Burning Phases Energy generation vs energy loss
net nuclear energy loss (burning + neutrino losses; in erg / g / s)
<•1014<•1013<•1012<•1011<•1010 <•109 <•108 <•107 <•106 <•105 <•104 <•103 <•102 <•101 <•100 <•10•1 0
net nuclear energygeneration (burning + neutrino losses; in erg / g / s)
0 >10•1 >100 >101 >102 >103 >104 >105 >106 >107 >108 >109>10 10 >1011 >1012 >1013 >1014
(blue giant) Total mass of the star convection semiconvection (reduces by mass loss) radiative envelope
convective envelope (red supergiant)
H shell burning He shell burning
C shell burning
O O O shell burning C shell burning burning O Si (raditive) He burning H burning C burning Ne O Si
22 M evolution, from Heger and Woosley 2001. Introduction Astrophysical reaction rates Hydrostatic Burning Phases Photon vs neutrino luminosities
Neutrino luminosity is larger than photon luminosity. Weak interaction processes start to determine the evolution.
45 15Msun 44 Neutrino luminosity s/s)
g 43 42 41 40 minosity (er u 39
L Photon luminosity g 38 Lo
37 Fig. 9.2 A comparison between the neutrino losses and various nuclear energy generation reactions according to Hayashi et al. (1962). The rates are given in erg/g/s as a function of temperature in 0.01 0.1 1 10 units of 108 K. The density is 104 g/cm3. Note that several reactions shown in the figure are no longer considered relevant to massive stars. The neutrino rates have been updated since 1962, but Central Temperature in billion degrees K the general picture of neutrino losses taking over energy generation remains today
From G. Shaviv, The Synthesis of the Elements Introduction Astrophysical reaction rates Hydrostatic Burning Phases Carbon Burning
Burning conditions:
for stars > 8 Mo (solar masses) (ZAMS) T~ 600-700 Mio r ~ 105-106 g/cm3 Major reaction sequences:
dominates by far
of course p’s, n’s, and a’s are recaptured … 23Mg can b-decay into 23Na
Composition at the end of burning:
mainly 20Ne, 24Mg, with some 21,22Ne, 23Na, 24,25,26Mg, 26,27Al
16 20 of course O is still present in quantities comparable with Ne (not burning … yet) 21 Introduction Astrophysical reaction rates Hydrostatic Burning Phases Carbon fusion
As the α, proton and neutron (at T > 1.1 GK) channels are open in 12C + 12C fusion, the reaction proceeds dominantly by the strong interaction. Each fusion reaction produces a light nuclide and a heavier fragment. At the high temperatures light particles react much faster than the basic 12C + 12C fusion reaction. They produce other nuclides, eg. α particles react with 22Ne (produced by α reactions on 14N) via the 22Ne(α, n)25Mg. Neutrons react very fast producing nuclei with neutron excess like 23Na, 25,26Mg, 29,30Si. α particle reactions lead to an increase in the abundance of 20Ne, 24Mg, and a small extend of 28Si. Carbon gets depleted and finally is less abundant than 16O. Introduction Astrophysical reaction rates Hydrostatic Burning Phases Core C burning evolution (constant temperature)
Christian Iliadis, Nuclear Physics of Stars. Introduction Astrophysical reaction rates Hydrostatic Burning Phases Neon Burning Neon burning Burning conditions:
for stars > 12 Mo (solar masses) (ZAMS) T~ 1.3-1.7 Bio K r ~ 106 g/cm3
Why would neon burn before oxygen ??? Answer: Temperatures are sufficiently high to initiate photodisintegration of 20Ne
20Ne+g ‰ 16O + a equilibrium is established 16O+a ‰ 20Ne + g
this is followed by (using the liberated helium) 20Ne+a ‰ 24Mg + g
so net effect: 22 Introduction Astrophysical reaction rates Hydrostatic Burning Phases Core Ne burning evolution (constant temperature)
Christian Iliadis, Nuclear Physics of Stars. Introduction Astrophysical reaction rates Hydrostatic Burning Phases Oxygen Burning
Burning conditions:
T~ 2 Bio r ~ 107 g/cm3
Major reaction sequences:
(5%)
(56%)
(5%)
(34%)
plus recapture of n,p,d,a
Main products:
28Si,32S (90%) and some 33,34S,35,37Cl,36.38Ar, 39,41K, 40,42Ca 27 Introduction Astrophysical reaction rates Hydrostatic Burning Phases Oxygen burning and quasi-equilibrium
Oxygen (and Silicon burning) show quasi-equilibrium behavior. Some reactions are fast enough to keep abundances of nuclei in relative equilibrium. Specific reactions connecting the different regions are not fast enough to be in equilibrium. Nuclei not in equilibrium are 16O and 24Mg:
16O + 16O → 28Si + α 24Mg + α → 28Si + γ
Two 16O and one 24Mg are converted into two 28Si; additional α capture on 28Si produces 32S, 36Ar, 38Ar, and 40Ca (quasi-equilibrium region) Introduction Astrophysical reaction rates Hydrostatic Burning Phases Core O burning evolution (constant temperature)
Christian Iliadis, Nuclear Physics of Stars. Introduction Astrophysical reaction rates Hydrostatic Burning Phases Silicon Burning
Burning conditions:
T~ 3-4 Bio r ~ 109 g/cm3
Reaction sequences:
• Silicon burning is fundamentally different to all other burning stages.
• Complex network of fast (g,n), (g,p), (g,a), (n,g), (p,g), and (a,g) reactions
• The net effect of Si burning is: 2 28Si --> 56Ni,
need new concept to describe burning:
Nuclear Statistical Equilibrium (NSE)
Quasi Statistical Equilibrium (QSE)
28 Introduction Astrophysical reaction rates Hydrostatic Burning Phases Photodissociation reactions
Important role of photodissociation reactions
Fig. 5.51 Decay constants for the photodisintegrations of 28Si (solid lines) and 32S (dashed lines) versus temperature. The curves are cal- culated from the rates of the corresponding forward reactions.
Christian Iliadis, Nuclear Physics of Stars. Introduction Astrophysical reaction rates Hydrostatic Burning Phases Core Si burning evolution (constant temperature)
Christian Iliadis, Nuclear Physics of Stars. Introduction Astrophysical reaction rates Hydrostatic Burning Phases Nuclear Statistical Equilibrium
At high temperatures compositium can be aproximated by Nuclear Statistical Equilibrium. Composition is given by a minimum of the Free Energy: F = U − TS . Under the constrains of conservation of number of nucleons and charge neutrality. It is assumed that all nuclear reactions operate in a time scale much shorter than any other timescale in the system. Nuclear Statistical Equilibrium favors free nucleons at high temperatures and iron group nuclei at low temperatures. Production of nuclei heavier than iron requires that nuclear statistical equilibrium breaks at some point. Introduction Astrophysical reaction rates Hydrostatic Burning Phases Nuclear Statistical Equilibrium
The minimum of the free energy is obtained when:
µ(Z, A) = (A − Z)µn + Zµp
implies that there is an equilibrium between the processes responsible for the creation and destruction of nuclei:
A(Z, N) Zp + Nn + γ
Processes mediated by the strong and electromagnetic interactions proceed in a time scale much sorter than any other evolutionary time scale of the system. Introduction Astrophysical reaction rates Hydrostatic Burning Phases Nuclear abundances in NSE
Nuclei follow Boltzmann statistics:
2 !3/2 2 n(Z, A) 2π~ µ(Z, A) = m(Z, A)c + kT ln GZ,A(T) m(Z, A)kT
with GZ,A(T) the partition function:
X π G (T) = (2J + 1)e−Ei(Z,A)/kT ≈ exp(akT)(a ∼ A/9 MeV) Z,A i 6akT i Results in Saha equation:
3/2 !A−1 2 !3(A−1)/2 GZ,A(T)A ρ Z A−Z 2π~ B(Z,A)/kT Y(Z, A) = A Yp Yn e 2 mu mukT
Composition depends on two parameters: Yp, Yn Introduction Astrophysical reaction rates Hydrostatic Burning Phases Core evolution (Summary)
10 Fe
Si
25 Msun O 9 15 Msun C log Central T [K] He 8
H
0 2 4 6 8 10 log Central Density [g/cm**3]
From Woosley, Heger & Weaver, Rev. Mod. Phys. 74, 1015 (2002) Introduction Astrophysical reaction rates Hydrostatic Burning Phases Stellar Evolution (Summary)
0 1H
−2
−4 Stellar Evolution −6
232Th 4He −8 12 238 C 208Pb U 16O 56Fe 62Ni Binding Energy per nucleon (MeV) −10 0 30 60 90 120 150 180 210 240 Mass Number A
At the end of evolution the composition stellar core is given by Nuclear Statistical Equilibrium (NSE). Nuclear reactions operate in a time scale much shorter than the dynamical timescale in which the system evolves. Composition is given by a minimum of the Free Energy: F = U − TS . Under the constrains of conservation of number of nucleons and charge neutrality. Nuclear Statistical Equilibrium favors free nucleons at high temperatures and iron group nuclei at low temperatures for constant entropy.