\thechapter Massive Stars and their Supernovae
\mainmatter\chapauthor

Friedrich-Karl Thielemann111University of Basel, 4056 Basel, Switzerland (f-k.thielemann@unibas.ch), Raphael Hirschi222University of Keele, Keele, ST5 5BG, United Kingdom, Matthias Liebendörfer333University of Basel, 4056 Basel, Switzerland, and Roland Diehl444Max Planck Institut für extraterrestrische Physik, 85748 Garching, Germany

Chapter \thechapter Massive Stars and their Supernovae

1 Cosmic Significance of Massive Stars

Our understanding of stellar evolution and the final explosive endpoints such as supernovae or hypernovae or gamma-ray bursts relies on the combination of

  • (magneto-)hydrodynamics

  • engergy generation due to nuclear reactions accomanying composition changes

  • radiation transport

  • thermodynamic properties (such as the equation of state of stellar matter).

Hydrodynamics is essentially embedded within the numerical schemes which implement the physics of processes (b) to (d). In early phases of stellar evolution, hydrodynamical processes can be approximated by a hydrostatic treatment. Nuclear energy production (b) includes all nuclear reactions triggered during stellar evolution and explosive end stages, also among unstable isotopes produced on the way. Radiation transport (c) covers atomic physics (e.g. opacities) for photon transport, but also nuclear physics and neutrino nucleon/nucleus interactions in late phases and core collapse. The thermodynamical treatment (d) addresses the mixture of ideal gases of photons, electrons/positrons and nuclei/ions. These are fermions and bosons, in dilute media or at high temperatures their energies can often be approximated by Maxwell-Boltzmann distributions. At very high densities, the nuclear equation of state is required to relate pressure and density. It exhibits a complex behavior, with transitions from individual nuclei to clusters of nucleons with a background neutron bath, homogeneous phases of nucleons, the emergence of hyperons and pions up to a possible hadron-quark phase transition.

The detailed treatment of all these ingredients and their combined application is discussed in more depth in textbooks (Kippenhahn and Weigert, 1994; Maeder, 2009; Arnett, 1996; Iliadis, 2007), and/or the preceding Chapter (3), where the evolution of low and intermediate mass stars is addressed. That chapter also includes the stellar structure equations in spherical symmetry and a discussion of opacities for photon transport. Ch. 8 and 9 (tools for modeling objects and their processes) go into more detail with regard to modeling hydrodynamics, (convective) instabilities and energy transport as well as the energy generation due to nuclear reactions and the determination of the latter. Here we want to focus on the astrophysical aspects, i.e. a description of the evolution of massive stars and their endpoints with a special emphasis on the composition of their ejecta (in form of stellar winds during the evolution or of explosive ejecta). Low and intermediate mass stars end their evolution as AGB stars, finally blowing off a planetary nebula via wind losses and leaving a white dwarf with an unburned C and O composition. Massive stars evolve beyond this point and experience all stellar burning stages from H over He, C, Ne, O and Si-burning up to core collapse and explosive endstages. In this chapter we want to discuss the nucleosynthesis processes involved and the production of radioactive nuclei555We focus especially on long-lived radioactivities which can be observed with gamma-ray satellites, and refractory isotopes which can be observed in dust condensations included in meteorites. in more detail. This includes all hydrostatic nuclear-burning stages experienced by massive stars, and explosive burning stages when a shock wave moves outward after a successful explosion was initiated, but also final wind ejecta from the hot proto-neutron star which emerged in the collapse and explosion phase. All these ejecta will enter the interstellar medium in galaxies, initially appearing as gas and dust in wind bubbles and supernova remnants, later determining the evolution of the larger-scale gas composition. The interstellar gas composition will evolve with time, and the composition of newly formed stars will witness this composition at the time of their formation.

Massive stars play an important role as contributors to the gas composition of the interstellar medium via wind losses or explosions. In astronomical terms they are the progenitors of blue supergiants (BSG), red supergiants (RSG), Wolf-Rayet (WR) and luminous blue variable (LBV) stars (Maeder and Meynet, 2010). At the end of their life, they explode as core collapse supernovae (ccSNe), observed as SNe of type II or Ib,c (Woosley and Bloom, 2006) and also as long soft gamma-ray bursts (GRBs Piran (2004)). After collapse, their cores become neutron stars or black holes. They are one of the main sites for nucleosynthesis, which takes place during both pre-SN (hydrostatic) burning stages and during explosive burning. A weak  process occurs during core He- (and C-) burning (The et al, 2007; El Eid et al, 2009) and the  process probably occurs during the explosion (Qian and Woosley, 1996). These s(low) and r(apid) neutron capture processes are mainly responsible for the heavy nuclei beyong the Fe-group. Radioactive isotopes like Al and Fe detected by the INTEGRAL satellite are produced by massive stars, plus many more radioactivities from the final explosive ejecta (like e.g. Ti, Ni, Co etc., see Sect. 4.4.2 and 4.5). Ch. 2 and 3 discussed also many long-lived heavy nuclei beyond Fe with half-lives larger than and up to years. As massive stars are probably not the origin of heavy -process nuclei (see Ch. 3), we will address here those nuclei which are clearly identified with the  process ( Th, y, U, y, U, y, U, y, Pu, y, Cm, y) and where especially Th and U, with half-lives comparable to the age of the Galaxy/Universe, can also serve as chronometers.

Massive stars, even though they are much less numerous than low mass stars, contribute significantly (about two thirds) to the integrated luminosity of galaxies. At high redshifts , or low metallicities , they are even more important drivers of characteristic phenomena and evolution. The first stars formed are thought to be all massive or even very massive, and to be the cause of the re-ionisation of the universe. As discussed above, if the final core collapse leads to a black hole, the endpoint of this evolution can be the origin of the subset of (long, soft) gamma ray bursts (GRBs). GRBs are the new standard candles for cosmology at high redshifts. They are visible from higher redshifts than usual SNe (of type I or II) are, and thus will impose tighter constraints on cosmological models. Massive stars with their large energy output can be seen out to significant (cosmological) distances – either directly through their thermal photospheric emission, or indirectly through the impact on their surroundings (ionization, or heated dust). In their collapsar and GRB extremes, emission is beamed into a jet, which makes them visible even at greater distances. This can also give us information on the star formation history at a very early age of the universe (10) beyond the reach of galaxy observations. Closer to home, recent surveys of metal poor halo stars provide a rich variety of constraints for the early chemical evolution of our Galaxy and thus the nucleosynthesis ejecta (astro-archeology).

2 Hydrostatic and Explosive Burning in Massive Stars

Following the motivation for studying massive stars in the previous section, we now discuss the ingredients for their modeling. Thermonuclear energy generation is one of the key aspects: It shapes the interior structure of the star, thus its evolutionary time scales, and the generation of new chemical elements and nuclei. Without understanding these, the feedback from massive stars as it determines the evolution of galaxies cannot be understood in astrophysical terms.666Empirical descriptions from observations of a multitude of galaxies are often utilized to substitute such astrophysical models in cosmological simulations. Thermonuclear burning, nuclear energy generation and resulting nuclear abundances are determined by thermonuclear reactions and weak interactions. The treatment of the required nuclear/plasma physics, and a detailed technical description of reaction rates, their determination and the essential features of composition changes and reaction networks is presented in Ch. 9. Here we want to discuss which types of reactions are involved specifically in the evolution of massive stars and their catastrophic end stages. Nuclear burning can in general be classified into two categories:

  • hydrostatic burning stages on timescales dictated by stellar energy loss

  • explosive burning due to hydrodynamics of the specific event.

Massive stars (as opposed to low and intermediate mass stars) are the ones which experience explosive burning (2) as a natural outcome at the end of their evolution and they undergo more extended hydrostatic burning stages (1) than their low- and intermediate-mass cousins. Therefore, we want to address some of these features here in a general way, before describing the evolution and explosion in more detail.

The important ingredients for describing nuclear burning and the resulting composition changes (i.e. nucleosynthesis) are (i) strong-interaction cross sections and photodisintegrations, (ii) weak interactions related to decay half-lives, electron or positron captures, and finally (iii) neutrino-induced reactions. They will now be discussed777A review of the sources for this microphysics input is given for (i) in Ch. 9 and for (iii) in Ch. 8. We will review some of the required weak interaction rates (ii) in the subsections on late phases of stellar evolution / core collapse and the description of the explosion..

2.1 Nuclear Burning During Hydrostatic Stellar Evolution

Hydrostatic burning stages are characterized by temperature thresholds, permitting thermal Maxwell-Boltzmann distributions of (charged) particles (nuclei) to penetrate increasingly larger Coulomb barriers of electrostatic repulsion. These are (two body) reactions as discussed in Equ. 9.6 and 9.9 of Ch. 9, representing terms of the type in the network equation (9.1). H-burning converts H into He via pp-chains or the CNO-cycles. The simplest pp-chain is initiated by H(p,)H(p,)He and completed by He(He,2p)He. The dominant CNO-cycle chain C(p,N(C(p,N(p,O(N(p,C is controlled by the slowest reaction N(p,O. The major reactions in He-burning are the - reaction He(2)C and C()O. The - reaction, being essentially a sequence of two two-body reactions with an extremely short-lived intermediate nucleus Be, is an example for the term in Equ. 9.1, which includes the product of three abundances. The H- and He-burning stages are also encountered in low and intermediate mass stars, leaving white dwarfs as central objects. They are discussed in much more detail with all minor reaction pathways in Ch. 3. Massive stars, the subject of the present Chapter, undergo further burning stages up to those involving the production of Fe-group nuclei. Table 1 lists these burning stages and their typical central densities and temperatures, their duration and the typical luminosity in photons (from Woosley and Weaver (1995)), which involve the reaction types given below. For further details see Sect. 3.

Fuel (g cm) ( K) (yr) (erg s)
Hydrogen 5.6(0) 0.04 1.0(7) 2.7(38)
Helium 9.4(2) 0.19 9.5(5) 5.3(38)
Carbon 2.7(5) 0.81 3.0(2) 4.3(38)
Neon 4.0(6) 1.70 3.8(-1) 4.4(38)
Oxygen 6.0(6) 2.10 5.0(-1) 4.4(38)
Silicon 4.9(7) 3.70 2 days 4.4(38)
Table 1: Burning stages of a 20M star
(a) basic energy generation
C(C,NeC(C,p)Na
Na(p,NeNa(p,MgC(O
(b) fluxes (a)
Ne(MgNa(,p)Mg(p,Al
Ne(n,Ne(p,Na (Ne(,n)Mg(n,Mg
Ne(,n)MgNe(p,NaMg(p,Al(Mg
(c) low temperature, high density burning
C(p,N(C(,n)O(Ne
Mg(p,Al(Mg
Ne(n,Ne(n,Ne(Na(n,Na(Mg +  processing
Table 2: Major Reactions in Carbon Burning
  • Heavy-ion fusion reactions: In C-burning the reaction C(C,)Ne dominates, in O-burning it is O(O,)Si. The corresponding reaction rates (after integrating over a Maxwell-Boltzmann distribution of targets and projectiles) have the form given in Equ. 9.9 of Ch. 9 and contribute to the second term in Equ. 9.1. Reactions going beyond these key reactions are provided in tables 2 and 3. Further features as well as the status of nuclear cross sections are discussed in recent reviews on hydrostatic burning stages (Haxton et al, 2006; Buchmann and Barnes, 2006; Costantini et al, 2009; Wiescher et al, 2010) and Ch. 9.

    (a) basic energy generation
    O(O,SiO(O,p)PO(O,n)S(P
    P(p,Si(S
    Si(Mg(,p)Al(,p)Si
    S(n,S(n,Si(S
    Si(n,Si(,n)S(,p)Cl
    Si(p,P(Si
    electron captures
    S(P(p,n)S
    Cl(S(p,n)Cl
    (b) high temperature burning
    S(Ar(,p)K
    Ar(n,Ar(Cl
    Cl(,p)S(Ar(p,K(p,Ca
    Cl(S(,p)S
    Ar(Ca(Ti
    Ca(,p)Sc(p,Ti
    (c) low temperature, high density burning
    P(SP(n,P
    S(P(p,n)S
    P(p,Si
    Table 3: Major Reactions in Oxygen Burning
  • Photo-disintegrations: The alternative to fusion reactions are photodisintegrations which start to play a role at sufficiently high temperatures when 30 (the Q-value or energy release of the inverse capture reaction). This ensures the existence of photons with energies in the Planck distribution and leads to Ne-Burning [Ne(O, Ne(Mg] at K (preceding O-burning) due to a small Q-value of 4 MeV and Si-burning at temperatures in excess of 310K [initiated like Ne-burning by photodisintegrations]. Such photodisintegrations (after integrating over a thermal (Planck) distribution of photons at temperature ) have the form given in equation (9.4) of Ch. 9 and act similar to decays with a temperature-dependent decay constant, contributing (like decays) to the first term in equation (9.1). In table 4 we provide some of the main reactions of Ne-burning, which is initiated by the photodisintegration of Ne.

    (a) basic energy generation
    Ne(ONe(Mg(Si
    (b) fluxes (a)
     Na(p,NeNa(,p)Mg(,n)Si
    Ne(n,Ne(,n)Mg(n,Mg(,n)Si
    Si(n,Si(n,Si
    Mg(,p)Al(,p)Si
    Mg(p,Al(n,Al(Si
    (c) low temperature, high density burning
    Ne(,n)Mg(n,Mg(n,Mg(Al
    Ne left from prior neutron-rich carbon burning
    Table 4: Major Reactions in Neon Burning
  • Electron capture reactions: Massive stellar cores eventually lead to electron-gas degeneracy, i.e. the Pauli exclusion principle for fermions determines the population of energy states rather than the Boltzmann statistics, valid only for low densities / high temperatures. The Fermi energy of electrons is

    (1)

    Here is the density of the electron gas , denotes the matter density and Avogadro’s number. In late stages of O-burning, in Si-burning (and during the later collapse stage) this Fermi energy of (degenerate) electrons, increases to the level of nuclear energies (MeV). In a neutral, completely ionized plasma, the electron abundance is equal to the total proton abundance (summing over all abundances of nuclei, including protons/hydrogen) and limited by the extreme values 0 (only neutrons) and 1 (only protons) with typical values during stellar evolution close to 0.5 or slightly below. Such conditions permit electron captures on protons and nuclei, if the negative Q-value of the reaction can be overcome by the electron (Fermi) energy. The general features for typical conditions are presented in table 5, example reactions were already given in table 3.

       or    p(n
       or    Z(Z-1
    gcm)=0.75 MeV
    gcm)=4.70 MeV
    Table 5: Electron Capture

    Thus, at sufficiently high densities, electron captures - which are energetically prohibited - can become possible and lead to an enhanced neutronization of the astrophysical plasma, in addition to the role of beta-decays and electron captures with positive Q-values (Nomoto and Hashimoto, 1988). In degenerate Ne-O-Mg cores (after core C-burning of stars with M), electron captures on Ne and Mg cause a loss of degeneracy pressure support and introduce a collapse rather than only a contraction, which combines all further burning stages on a short collapse time scale (Nomoto, 1987). In Si-burning of more massive stars, electron capture on intermediate mass and Fe-group nuclei becomes highly important and determines the neutronization () of the central core. As discussed in Ch. 9, such rates contribute to the one-body reaction terms in Equ. 9.1 with the effective decay constants in Equ. 9.5 being a function of and , the electron number density.

       or    n()p
       or    p((n
       or    Z(Z+1
       or    Z(Z-1
    Table 6: Neutrino Reactions

    Neutrino cross section on nucleons, nuclei and electrons are minute, by comparison to above reactions. It therefore requires high densities of the order g cm that also the inverse process to electron/positron capture (neutrino capture) can occur on relevant timescales. The same is true for other processes such as e.g. inelastic scattering, leaving a nucleus in an excited state which can emit nucleons and alpha particles. Such neutrino-induced reactions can be expressed in a similar way as photon and electron captures, integrating now over the corresponding neutrino distribution. The latter is, however, not necessarily in thermal equilibrium and not just a function of temperature and neutrino densities. Neutrino distributions are rather determined by (neutrino) radiation transport calculations (see Ch. 8, where also other neutrino scattering processes are discussed).

All the reactions presented above and occurring at different times in the sequence of burning stages, contribute to the three types of terms in the reaction network equation (Equ.9.1 in Ch. 9). If one is interested to show how nuclear abundances enter in this set of equations, it can also be written in the form888The formal difference to Equ. 9.1 is that one does not sum here over the reactions but rather over all reaction partners (see also the equation following Table 3.2 in Ch. 3). However, in total, all the terms which appear are identical. Due to the different summation indices, the P’s have a slightly different notation, ’s denote decay rates called in Ch. 9, and correspond to of reactions between nuclei and , while includes a similar expression for three-body reactions (Nomoto et al, 1985). A survey of computational methods to solve nuclear networks is given in Hix and Thielemann (1999a); Timmes (1999). (Like for electron abundances , the abundances in Eq.(2) are related to number densities and mass fractions of the corresponding nuclei via , where is the mass number of nucleus and .)

(2)

Core Si-burning, the final burning stage during stellar evolution, which is initiated by the photodisintegration Si(Mg close to  K - and followed by a large number of fusion and photodisintegration reactions - ends with nuclear reactions in a complete chemical equilibrium999all strong (thermonuclear) and photodisintegration reactions are equilibrized, while weak interaction reactions, changing , may occur on longer timescales. (nuclear statistical equilibrium, NSE) and an abundance distribution centered around Fe (as discussed in Ch. 9, Equ. 9.14 and 9.15). These temperatures permit photodisintegrations with typical Q-values of 8-10 MeV as well as the penetration of Coulomb barriers in capture reaction. In such an NSE the abundance of each nucleus Y is only dependent on temperature , density , its nuclear binding energy , and via charge conservation on . is altered by weak interactions on longer timescales. A quasi-equilibrium (QSE) can occur, if localized nuclear mass regions are in equilibrium with the background of free neutrons, protons and  particles, but offset from other regions of nuclei and thus their NSE values (Hix and Thielemann, 1996, 1999b; Hix et al, 2007). Different quasi-equilibrium regions are usually separated from each other by slow reactions with typically small Q-values. Such boundaries between QSE groups, which are due to slow reactions, can be related to neutron or proton shell closures, like e.g. , separating the Si- and Fe-groups in early phases of Si-burning.

Figure 1: Burning timescales in () seconds for fuel exhaustion of He-, C-, and O-burning (top) and Ne- and Si-burning (buttom), as a function of temperature. Density-dependent timescales are labeled with a chosen typical density (in g cm). They scale with for C- and O-burning and for He-burning. Ne- and Si-burning, initiated by photodisintegrations, are not density-dependent. The almost constant He-burning timescale beyond =K=1 permits efficient destruction on explosive timescales only for high densities.

All reactions discussed above, occurring during all stellar burning stages, are essentially related to nuclei from H to the Fe-group, and not much beyond.

  • Neutron capture processes: Through neutron capture reactions, also during regular stellar evolution, there is a chance to produce heavier nuclei. During core and shell He-burning specific -induced reactions can liberate neutrons which are responsible for the slow neutron capture process ( process). A major neutron source is the reaction Ne(Mg, with Ne being produced via succesive -captures on the H-burning CNO product N(F(O(Ne. If occurring, the mixing of C into H-burning shells can produce an even stronger neutron source C(O via C(N(C. In massive, rotating, low metallicity stars, mixing can lead to the production of primary N and Ne, i.e. a neutron source which does not reflect the initial metallicity of N in the CNO-cycle, and can thus be much stronger. Ch. 3 discusses in full detail the strong  process via a combination of C and Ne in He-shell flashes of low and intermediate mass stars. In a similar way mixing processes can also occur in massive stars due to rotation or convective instabilities. Without such mixing processes only secondary (metallicity-dependent) Ne is available for Ne(Mg and core He-burning as well as shell C-burning lead to a weak  process (The et al, 2007). The  process can in principle form elements up to Pb and Bi through a series of neutron captures and -decays, starting on existing heavy nuclei around Fe (Käppeler and Mengoni, 2006). Weak  processing, based on secondary Ne, does not proceed beyond mass numbers of . The production of heavier nuclei is possible in massive stars if primary N and Ne are available.

2.2 Explosive Burning

Figure 2: Final results of explosive Si-burning as a function of maximum temperatures and densities attained in explosions before adiabatic expansion. For temperatures in excess of 5K any fuel previously existing is photodisintegrated into nucleons and  particles before re-assembling in the expansion. For high densities this is described by a full NSE with an Fe-group composition, favoring nuclei with maximum binding energies and proton/nucleon ratios equal to . For lower densities the NSE breaks into local equilibrium groups (quasi-equilibrium, QSE) with group boundaries determined by reactions with an insufficiently fast reaction stream. Alpha-rich freeze-out (insufficient conversion of  particles into nuclei beyond carbon) is such a QSE-behavior. Lines with 1% and 10% remaining -particle mass fraction are indicated as well as typical conditions for mass zones in type Ia and core-collapse supernovae.

Many of the hydrostatic nuclear-burning processes occur also under explosive conditions at higher temperatures and on shorter timescales (see Fig.1), when often the -decay half-lives are longer than the explosive timescales, producing significant abundances of unstable isotopes as burning proceeds. This requires in general the additional knowledge of nuclear reactions for and among unstable nuclei. The fuels for explosive nucleosynthesis consist mainly of = nuclei like C, O, Ne, Mg, or Si (the ashes of prior hydrostatic burning), resulting in heavier nuclei, again with . At high densities also electron captures on nuclei - can occur at substantial rates due to energetic, degenerate electrons when Fermi energies are high, as already discussed for late hydrostatic burning stages.

Explosive Si-burning differs strongly from its hydrostatic counterpart and can be divided into three different regimes: (i) incomplete Si-burning and complete Si-burning with either (ii) a normal (high density, low entropy) or (iii) an -rich (low density, high entropy) freeze-out of charged-particle reactions during cooling from NSE. At initially-high temperatures or during a normal freeze-out, the abundances remain in a full NSE. The full NSE can break up in smaller equilibrium clusters (quasi-equilibrium, QSE), for a detailed discussion see Hix and Thielemann (1996, 1999b); Hix et al (2007). An example for such QSE-bevavior is an -rich freeze-out, caused by the inability of the - reaction He(2C, and the He(n,Be reaction to keep light nuclei like n, p, and He, and nuclei beyond =12 in an NSE during declining temperatures, when densities are low. This causes a large -particle abundance after freeze-out of nuclear reactions. This effect, most pronounced for core collapse supernovae, depends on the entropy of the reaction environment, being proportional to in a radiation dominated plasma (see Fig. 2).

-process nucleosynthesis (rapid neutron capture) relates to environments of explosive Si-burning, either with low or high entropies, where matter experiences a normal or -rich freeze-out. The requirement of a neutron to seed-nuclei ratio of 10 to 150 after freeze-out of charged particle reactions101010Such neutron/seed ratio is required in order to produce all, including the heaviest, -process nuclei via neutron capture from seed nuclei at their abundances before freeze-out. translates into =0.12-0.3 for a normal freeze-out. For a moderate 0.40 an extremely -rich freeze-out is needed (see the disussion in Section 4). Under these conditions the large mass fraction in He (with ) permits ratios of remaining free neutrons to (small) abundances of heavier seed nuclei, which are sufficiently high to attain -process conditions. In many cases QSE-groups of neutron captures and photodisintegrations are formed in the isotopic chains of heavy elements during the operation of the  process.

3 Evolution of Massive Stars up to Core Collapse

In Section 2 we have discussed nuclear burning processes in detail, including also individual reactions which are of relevance during the evolution of massive stars. This relates to the main focus of this book, the production of (radioactive) nuclei in astrophysical environments. In the present section we will discuss the physics of stellar evolution and major related observational features; but we refer to review articles or textbooks for technical descriptions of treatments of mass, energy, and momentum conservations equations as well as energy transport (via radiation or convective motions) (Maeder, 2009; Maeder and Meynet, 2010; Heger et al, 2003; Limongi et al, 2000; Limongi and Chieffi, 2003, 2006; Ohkubo et al, 2008; El Eid et al, 2009) (but see also the hydrostatic stellar struture / evolution equations in spherical symmetry, as presented in Ch. 3). Stellar-evolution calculations as discussed here are based on the Geneva code of A. Maeder and G. Meynet and their students (Maeder, 2009; Maeder and Meynet, 2010). This numerical implementation of stellar evolution includes (i) an adaptative reaction network for the advanced burning stages, which is capable to follow the detailed evolution of and a large set of nuclei; (ii) a discretization of the stellar-structure equations, modified in order to damp instabilities occurring during the advanced stages of evolution; (iii) the treatment of dynamical shear in addition to the other mixing processes (such as, e.g., horizontal turbulence, secular shear and meridional circulation); and (iv) the treatment of convection as a diffusive process from O-burning onwards. This allows to follow the evolution of massive stars from their birth until the stage of Si-burning, including all nuclear burning stages discussed in Sect. 2, for a wide range of initial masses, metallicities and stellar rotation. Here the treatment of rotation and mixing effects still utilizes methods based on spherical symmetry. Full multi-dimensional calculations of mixing processes during stellar evolution have recently been established (Meakin and Arnett, 2007; Arnett et al, 2009) and might open up a new era for our understanding of the evolution of stars.

3.1 Stellar Evolution with Rotation

The evolution of all stars (including massive stars discussed here) is initiated by core H-burning, during which the star is found on the so-called main sequence (MS) in the Hertzsprung-Russell (HR) diagram, which relates the stellar luminosity to the stellar surface temperature (color). The observational appearance of a star after the completion of core H-burning is affected by the fact that the H-burning region continues to move outward as a burning shell. The He-core contracts and ignites core He-burning in the center, which produces mainly C and O. The star’s trajectory in the HR diagram leaves the main sequence, and its radius increases due to the increased radiation pressure. Depending on the resulting surface temperature it becomes a blue or red supergiant (BSG or RSG). Radiation pressure can rise to such extreme values that stars (more massive than 20-30 M) blow off their outer parts through strong stellar winds of velocities up to 2000 km s, exposing the more-interior parts of the star, the helium (or in some cases, the carbon) shell. Such a Wolf-Rayet (WR) star loses between and a few times  M per year, in comparison to our Sun losing  of its M per year through its solar wind. For non-rotating stars, the transition to the WR phase appears through the so-called Luminous Blue Variable stars (LBVs). LBVs are massive, intrinsically bright stars which display different scales of light and color variability, ranging from rapid microvariations to rare outbreaks of catastrophic mass loss. They represent a very short-lived (perhaps as little as 40,000 years) strongly mass-losing phase in the evolution of massive stars, during which they undergo deep erosion of the outer layers before they enter the Wolf-Rayet phase. For rotating stars, the WR phase may start before the star ends its main sequence, since rotation enhances mass loss and rotation-induced mixing further reduces the hydrogen content at the surface (Meynet and Maeder, 2003, 2005). In the following we discuss how these evolutionary phases depend on the initial properties of a star. Late burning phases progress much more rapidly than the H burning of the main sequence state. This is because the carriers of the star’s energy loss, which drives the evolution of a star, change from photons to neutrinos, which escape immediately at the densities discussed here, while photons undergo a multitude of scattering processes until they finally escape at the photosphere111111It takes a photon about 10 years to reach the surface, after it has been launched in the hot core of, e.g., our Sun.. The characteristics of late-burning stages are essentially identified by the size of a star’s C+O-core after core He-burning.

Figure 3: Stellar structure (Kippenhahn) diagrams, which show the evolution of the structure as a function of the time left until the core collapses after the completion of core Si-burning. The initial parameters of the models are given on the top of each plot. Coloring (shading) marks convective zones, and the burning stages are denoted below the time axis. Non-rotating and moderately rotating 20 M star models are shown, for different metallicities . indicates the rotation velocity at the surface of the star.

The evolution of stars is governed mainly by three initial parameters: (1) its mass , (2) its metallicity (, i.e. the mass fraction of pre-existing elements heavier than He from earlier stellar generations), and (3) the rotation rate or surface rotation velocity . Solar metallicity corresponds to121212The current value of solar metallicity is believed to be Z=0.014, see Ch. 1; the value of Z=0.02, which had been established before and was in common use till2005, remains a reference for comparisons, though. . The evolution can also be influenced by interior magnetic fields, and by a close binary companion. Rotation significantly affects the pre-supernova state, through the impact it has on the H and He-burning evolution. Two mass groups are distinguishable: Either rotationally induced mixing dominates (for 30 M), or rotationally increased mass loss dominates (for 30 M). For massive stars around solar metallicity, mass loss plays a crucial role, in some cases removing more than half of the initial mass. Internal mixing, induced mainly by convection and rotation, also has a significant effect on the evolution of stars. An important result is the production of primary N (via the CNO-cycle) and Ne (via -captures in He-burning), due to mixing of burning products (such as C) with hydrogen or ’s, respectively (see the discusssion in Sect. 2).

The general impact of metallicity can be summarized in the following way: Lower metallicity implies a (slightly) lower luminosity due to the lack of CNO-cycling in hydrogen burning, which leads to slightly smaller convective cores. A lower metallicity also implies a lower opacity due to the lack of heavier elements with their many spectral lines, reducing therefore also radiation pressure and hence mass loss (as long as the chemical composition has not been changed by burning or mixing in the part of the star under consideration). This results in lower metallicity stars being more compact and experiencing less mass loss. Prescriptions for mass loss as used in the Geneva stellar evolution code are described in detail in Meynet and Maeder (2005). Mass loss rates depend on metallicity as , where is the mass fraction of heavy elements at the surface of the star. The effects can be seen in Fig.3 which shows the interior structure of stars through so-called Kippenhahn diagrams of 20 M models for different metallicities and rotation velocities of the stars. These diagrams indicate regions (in radial mass coordinates) where matter is unstable against convection; here the energy transport is dominated by transporting hot matter rather than through the propagation of photons. The implications of such a behavior have already been described in Ch. 3, the evolution of low and intermediate mass stars, and the physical origin and treatment of these effects are addressed in Ch. 8.

With the exception of the outer convection zone, convective regions in most cases indicate burning zones, such as core H-burning, core He-burning, core C-burning etc.. They testify also the ignition of outward moving burning shells of the same nuclear burning stages. When comparing models for decreasing metallicities (without rotation, left column of Fig.3) one notices only minute reductions of the core sizes, but it is clearly seen that the outer (H-)burning shell moves further in towards smaller radial mass zones. In the third figure in this column we see merging of the H- and He-burning shells due to this effect, which leads to a largely-increased energy generation and extension of these combined burning zones.

How does rotation change this picture, and how do rotation-induced processes vary with metallicity? At all metallicities, rotation usually increases the core sizes via rotational mixing. The supply of more H-fuel leads to more energy generation and therefore a higher luminosity. The higher luminosity increases the radition pressure and stellar mass loss. The effect of increased core sizes (and smaller density gradients) can be viewed in all models with =300 km s in the second column of Fig.3. Clearly the convective core sizes are increased and the shell burning zones have moved outward. In the lowest metallicity case, the H/He-layers are separated again. In the intermediate metallicity case , the outer convection zone reaches the surface, and the star becomes a red supergiant. For metallicities (top row), the increased luminosity causes a sufficient increase in radiation pressure so that the mass loss is substantially enhanced (see the decrease of the stellar mass indicated by the top line). Mass loss becomes gradually unimportant for decreasing metallicities. For the rotating 20 M models the stellar fraction lost is more than 50% for solar metallicities, 13% at , less than 3% for , and less than 0.3% for .

Figure 4: The same as Fig. 3. Stellar structure diagrams for rapidly rotating stars of metallicity , over a mass range from 9 to 85 M. We see the drastically increasing amount of mass loss with increasing mass (enhancing mixing of burning products to the surface, and increasing opacities, i.e. acting like increased metallicities, plus some mass loss from critical rotation for the most massive stars). The two metallicity cases shown for the 20 M star show again that stars are less compact and show more and enhanced mass loss for higher metallicities (becoming a RSG which leads to the appearance of a large convective envelope).

This can be different for more massive stars (Meynet et al, 2006). In Fig.4, we show results for low metallicity stars with and fast rotation (500-800 km s) from 9 to 85 M. The surface layers of massive stars usually accelerate due to internal transport of angular momentum from the core to the envelope. Since at low , stellar winds are weak, this angular momentum dredged up by meridional circulation remains inside the star, and the star reaches critical rotation more easily. At the critical limit, matter can be launched into a Keplerian disk which probably dissipates under the action of the strong radiation pressure of the star. Such an effect can be seen for the 85 M star, which loses in total more than 75% of its initial mass, and initially about 10% due to critical rotation. The remaining mass loss occurs during the red supergiant phase after rotation and convection have enriched the surface in primary CNO elements. We can also see that this effect becomes vanishingly small for stars with masses M. The two 20 M models with varying metallicities and degrees of rotation again indicate the influence of metallicity and rotation on the compactness and mass loss of stars. In both cases the mass loss is negligible.

We have not shown here the evolution of extremely low metallicity stars. Below a metallicity of about , the CNO cycle cannot operate when H-burning starts after the star has been formed. The star therefore contracts until He-burning ignites, because the energy generation rate of H burning through the pp-chains cannot balance the effect of the gravitational force. Once enough C and O is produced, the CNO cycle can operate, and the star behaves like stars with for the rest of the main sequence. Metal-free stellar evolution models are presented in Chieffi and Limongi (2004); Heger and Woosley (2002); Umeda and Nomoto (2005); Ekström et al (2008).

Including the effects of both mass loss and rotation, massive star models reproduce many observables of stars with metallicities around solar . For example, models with rotation allow chemical surface enrichments already on the main sequence of core hydrogen burning (MS), whereas without the inclusion of rotation, self-enrichment is only possible during advanced burning evolution such as the red supergiant RSG stage (Heger and Langer, 2000; Meynet and Maeder, 2000). Rotating star models also better reproduce the ratio of star types, for the ones which retain their hydrogen surface layer (O stars), which lose the hydrogen layer completely (WR stars), and which even lose their helium layer. The latter affects also the appearance of later core collapse supernova explosions of massive stars. Indeed, rotation changes the supernova type due to the mass loss of the hydrogen envelope (turning such an event in optical observations from a type II supernova with a strong plateau phase to a IIb event with a smaller plateau, or even a Ib event for the case of complete loss of the hydrogen envelope, and a Ic event with the additional loss of the He-envelope). This is discussed in more detail in Sect. 4. Both aspects, the chemical surface enrichment in MS stars as well as the ratio of type Ib+Ic to type II supernovae, as a function of metallicity, are drastically changed compared to non-rotating models, which underestimate these ratios (Georgy et al, 2009; Meynet and Maeder, 2005). The value of 300 km s, used as the initial rotation velocity at solar metallicity, corresponds to an average velocity of about 220 km s on the main sequence (MS), which is close to the average observed value (Fukuda, 1982; Meynet et al, 2008). Observed ratios of stars of different types in the Magellanic clouds, as compared to our Galaxy (Maeder et al, 1999; Martayan et al, 2007), point to stars rotating faster at lower metallicities. Fast initial rotation velocities in the range of 600 – 800 km s (Hirschi et al, 2005) are supported by observations of very low- stars (Chiappini et al, 2006).

Figure 5: Abundance profiles for the 40 (top), 60 (middle) and 85 (bottom) models. The pre–SN and wind (yellow shaded area) chemical compositions are separated by a vertical dashed line located at the pre–SN total mass (), given below each plot.

Rotation affects all burning stages and the resulting Fe-core (we will discuss this issue further in the next subsection, see also Fig. 6). The size of the Fe-core in turn determines the final fate, whether a supernova explosion with neutron star formation or the collapse to a black hole occurs. The effects of rotation on pre-supernova models are most spectacular for stars between 15 and 25 . It changes the total size/radius of progenitors (leading to blue instead of red supergiants) and the helium and CO core (bigger by a factor of in rotating models). The history of convective zones (in particular the convective zones associated with shell H-burning and core He-burning) is strongly affected by rotation induced mixing (Hirschi et al, 2005). The most important rotation induced mixing takes place at low while He is burning inside a convective core. Primary C and O are mixed from the convective core into the H-burning shell. Once the enrichment is strong enough, the H-burning shell is boosted (the CNO cycle depends strongly on the C and O mixing at such low initial metallicities). The shell becomes convective and leads to an important primary N production while the convective core mass decreases, leading to a less massive CO-core after He-burning than in non-rotating models. Convective and rotational mixing brings the primary CNO to the surface with interesting consequences for the stellar yields. The yield of O, being closely correlated with the mass of the CO-core, is reduced. At the same time the C yield is slightly increased (Hirschi et al, 2005), both due to the slightly lower temperatures in core He-burning. This is one possible explanation for the high [C/O] ratio observed in the most metal-poor halo stars (see Fig. 14 in Spite et al (2005) and Fabbian et al (2009)) and in damped Lyman- systems DLAs (Pettini et al, 2008).

The fate of rotating stars at very low is therefore probably the following: M : Mass loss is insignificant and matter is only ejected into the ISM during the SN explosion. 30-40 MM : Mass loss (at critical rotation and in the RSG stage) removes 10-20% of the initial mass of the star. The star probably dies as a black hole without a SN explosion and therefore the feedback into the ISM is only due to stellar winds. M: A strong mass loss removes a significant amount of mass and the stars enter the WR phase. These stars therefore end as type Ib/c SNe and possibly as GRBs. This behavior is displayed in Fig. 5. At a metallicity , corresponding to an Fe/H ratio [Fe/H], C and O are shown in models to be mixed into the H-burning shell during He-burning. This raises the importance of the shell, and leads to a reduction of the CO-core size. Later in the evolution, the H-shell deepens and produces large amounts of primary nitrogen. For the most massive stars ( ), significant mass loss occurs during the red supergiant stage, caused by the surface enrichment in CNO elements from rotational and convective mixing.

The properties of non-rotating low- stars are presented in Heger et al (2003); Hirschi et al (2008), and several groups have calculated their stellar yields (Heger and Woosley, 2002; Chieffi and Limongi, 2004; Tominaga et al, 2007). All results for the non-rotating stars (whether at solar metallicity or for low- models) are consistent among these calculations, differences are understood from the treatments of convection and the rates used for CO. The combined contributions to stellar yields by the wind and the later supernova explosion (see Sect. 4) will be provided separately. The results for stellar models with metallicities close to solar can be described as follows: Rotating stars have larger yields in their stellar winds than the non-rotating ones, because of the extra mass loss and mixing due to rotation, for masses below . The C and O yields are increased by a factor 1.5–2.5 by rotation. At high mass loss rates (above  M), the rotating and non-rotating models show similar yield values. When the wind and explosive contributions are added, the total metal production of rotating stars is larger by a factor 1.5-2.5 (see Sect. 4). For very massive stars, the situation varies due to the extreme mass loss, as shown in Fig.5.

/M /M /M /M
11 10.56  3.47  1.75
15 13.49  5.29  2.72
20 16.31  7.64  4.35
30 12.91 12.68  8.01
40 12.52 16.49  8.98
60 17.08 25.17 12.62
80 22.62 34.71 17.41
Table 7: Stellar Properties (Limongi & Chieffi 2006)
/M    /M /M /M
 9 500   9.00   1.90   1.34
20 300   8.76   8.66   6.59
20   0 19.56   6.58   4.39
20 300 17.19   8.32   6.24
20 300 19.93   7.90   5.68
20 500 19.57   7.85   5.91
20 300 20.00   6.17   5.18
20 600 19.59   4.83   4.36
40 700 35.80 13.50 12.80
60 800 48.97 25.60 24.00
85 800 19.87 19.90 18.80
Table 8: Stellar Properties (Hirschi et al. 2007)

In order to give a quantitative impression of the influence of initial mass, metallicity and rotation on the evolution of stars, we present in Tables 7 and 8 results for (a) non-rotating solar metallicity stars (Limongi and Chieffi, 2006) and (b) rotating stars for varying metallicities (Hirschi, 2007). Table 8 corresponds to the models shown in Figs. 3, 4, and 5. Given are the initial and final mass (in order to give an impression of the mass loss), as well as the core size after central H-burning (the He-core) and after central He-burning (the CO-core), and in table 8 also the metallicity and initial rotational surface velocity in km s. As all burning stages after He-burning occur on significantly shorter timescales than the earlier burning phases, the CO-core size is the important quantity in order to determine the final outcome/fate of the star.

After this general discussion of stellar evolution, as it varies with initial mass, metallicity and rotation, we now focus on two long-lived isotopes Al and Fe, which have important contributions from the earlier burning stages in explosion ejecta.

Al

Long-lived Al is produced in core and shell H-burning via the NaMgAl-cycle (see Ch. 3) in the Mg(Al reaction and will be eventually ejected in the stellar wind during the WR-phase. Gamma-ray observations of the 1.809 MeV decay line of Al in systems like the Wolf-Rayet binary system 2 Vel, being the closest known Wolf-Rayet (WR) star, serve as a constraint to nucleosynthesis in Wolf-Rayet stars. From observations of the 2 Vel binary system including a WR star, Oberlack et al (2000) claimed that such WR stars must emit of the order M of Al by stellar winds. The amount of Al ejected into the interstellar medium is very sensitive to metallicity, initial stellar mass, rotation and mass loss rate, related to one or more of the physical effects discussed above. Results of detailed calculations can be found in Langer et al (1995); Meynet et al (1997); Palacios et al (2005); Limongi and Chieffi (2006); Tur et al (2009). Limongi and Chieffi (2006) provide an extended overview for the contribution from 11 to 120 M stars. The dominant source for the Al production during stellar evolution is the Mg(Al reaction. Therefore the resulting abundance depends (i) on this reaction rate converting Mg into Al, (ii) on the amount of Mg available, i.e. the total amount of matter in the NeNaMgAl-cycle (either in terms of the abundance/metallicity or in terms of the H-core size), and finally (iii) on the amount of Al distruction. In the part of the He-core (after H-burning) which undergoes He-burning, neutrons are produced via -reactions which destroy Al via Al(Mg and Al(Na. A further question is related to the amount of matter being ejected in winds (i.e. mass loss) during stellar evolution before Al can decay inside the star via -decay with a half-life of y.

He-burning, with its neutrons released, is destructive for Al, but shell C-burning is again a source of Al, also via Mg(Al, which is effective due to protons released in C(C,Na (see table 2 in Sect. 2). Convection in the C-burning shell brings in fresh C fuel and Mg which has been also produced in prior He-burning in the Ne(Mg reaction. Al production may be effective also in Ne-burning, based on Mg left over from C-burning and protons released via Na(Mg (see table 4). This Al only survives if rapidly convected outwards to lower temperature environments (Al may decay rapidly in hot regions due to thermal population of its short-lived isomeric state; see Fig. 1.3 in Ch. 1).

A fraction of the Al produced during stellar evolution will again be destroyed, when a shock front is released in a supernova explosion and propagates through the stellar envelope; in particular, material from C and Ne-burning, being close to the Fe-core, will be affected. But there are also source processes for explosive Al production. The total yields, hydrostatic-evolution yields combined with the destruction and contribution from explosive burning, are given in Sect. 5.

Fe

Fe is produced by neutron captures on Fe, and destroyed again via Fe(Fe, i.e. during the  process. Generally, slow capture of neutrons released from the NeMg reaction in core He-burning leads to the so-called weak  process, producing nuclei up to nuclear mass numbers of around A=90. Fe is beta-unstable, thus in order for neutron capture to compete with this reaction branching (equating the neutron capture and beta-decay rates) requires a typcial neutron density of about cm. These are relatively high neutron densities for an  process, which also ensure that the destruction of Fe via neutron captures dominates over its decay with its half-life of y (Fig. 7.22 in Ch. 7, Rugel et al (2009)). Core He-burning will not provide sufficiently high-temperatures for the NeMg reaction to produce such high neutron densities. It requires the conditions in shell He-burning to do so. Apparently conditions are most favorable during shell He-burning at late evolutionary times when central O-burning has already active and a C-burning shell is existent as well (see Woosley and Weaver, 1995; Rauscher et al, 2002; Limongi and Chieffi, 2006; Tur et al, 2009). Fe yields are very sensitive to uncertainties in He-destruction reactions (such as the 3-rate and C(O) which compete with the neutron source reaction NeMg and neutron(-capture) poisons which compete with the production and destruction rates of Fe via neutron captures (Rauscher et al, 2002; Tur et al, 2009; Giron et al, 2010; Uberseder et al, 2009). Such uncertainties amount to factors of up to 5 from present rate uncertainties. Another possible effect which has not really been looked into, yet, is the amount of Ne available in He-burning. An important effect in low metallicity stars is the production of primary N (not enherited from CNO of previous stellar generations, but produced inside the star due to mixing of He-burning products with H). This causes the production of Ne in He-burning and can at low metallicities (with small seed abundances of Fe) permit sizable  processing, affecting again the abundance of Fe.

3.2 Late Burning Stages and the Onset of Core Collapse

Stars more massive than about 8 M will, after finishing core and shell H- and He-burning, lead to CO-cores which exceed the maximum stable mass of white dwarfs (the Chandrasekhar mass). For later burning stages, when the partial or full degeneracy of the electron gas is important, this critical limit decides upon further contraction and the central ignition of subsequent burning stages, i.e. C-, Ne-, O- and Si-burning. Dependent on the Fermi energy of the degenerate electron gas, electron capture on the C-burning products Ne and Mg can initiate a collapse, leading directly via nuclear statistical equilibrium to a central Fe-core. This evolution path occurs for stars in the range 8-10 M (Nomoto, 1987). More massise stars will proceed through all burning stages until Si-burning will finally produce an Fe-core. All burning stages after core H- and He-burning proceed on timescales which are shorter by orders of magnitude. The reason is that the energy carried away by freely escaping neutrinos dominates over radiation losses by photons which undergo a cascade of scattering processes before their final escape. Most of these neutrinos are created when central densities and temperatures permit neutrino production via new particle reactions, different from beta-decay or electron capture on nuclei. Following neutrino production reactions are relevant: (i) -pair annihilation (pair neutrinos), (ii) electron-photon scattering with neutrino-antineutrino pair creation (photo neutrinos), and (iii) neutrino-antineutrino pair creation from plasma oscillations (plasmon neutrinos). Neutrinos dominate the energy loss in stellar evolution from this point on, and lead to increasingly shorter burning timescales, although the photon radiation luminosity of the star remains roughly constant. The timescales for the individual burning stages are given in table 1 in section 2; these values refer to a 20 M star with solar metallicity and no mass loss (Weaver and Woosley, 1993). Effects of mass loss, rotation and metallicity can change these timescales somewhat (up to 20%). Due to the large difference in evolution timescales, the dominant mass loss by stellar winds occurs during H- and He-burning, and the final outcome of stellar evolution is determined by the CO-core size after He-burning. Therefore, given all dependencies of stellar evolution via initial metallicities and rotation, the initial main sequence mass of a star is less indicative for the final outcome than the size of its CO-core.

In the late phases of O- and Si-burning (discussed in Sect. 2), electrons are moderately to strongly degenerate, dependent on the initial stellar mass, and will be characterized by increasing Fermi energies. This will allow for electron captures on burning products, and will make matter more neutron-rich, i.e decrease , the electron or proton to nucleon (neutrons plus protons) ratio. In high density O-burning ( g cm) two electron capture reactions become important and lead to a decrease in , S(P and Cl((S. Such effects become more extensive at even higher densities in Si-burning and a large range of nuclei has been identified to be of major importance Co, Ni, Fe, Mn, Cu, Sc, V, Cr, Ti, Ga, Ge, Se, As, V, and Zn (Aufderheide et al, 1994). The amount of electron capture and the resulting has consequences for core sizes. (The core sizes of the late burning stages are shown in Figs. 3 and 4). The final size of the inner Fe-core represents the maximum mass which can be supported by the pressure of the degenerate electron gas. It is a function of , but also reflects temperature effects if the electron gas is not completely degenerate (Bethe, 1990), with being the entropy in electrons per baryon

(3)

Stars with masses exceeding roughly 10 M reach a point in their evolution where their Si-burning core (which will turn eventually into their Fe-core) exceeds this critical mass. At this point they collapse and bounce, if not too massive, to explode in spectacular core collapse events known as type II or Ib/c supernovae. These explosions create a neutron star or black hole at the end of the life of a star. They play a preeminent role in the nucleosynthesis and chemical evolution of a galaxy.

The collapse is initiated by the capture of degenerate electrons on nuclei, which reduces the dominant contribution of the pressure (i.e. the one from the degenerate electron gas). Alternatively, for lower densities and higher temperatures (in more massive stars), the pressure supporting the core is reduced by endoergic photodisintegrations of nuclei, reducing the thermal energy. The evolution in the core is determined by the competition of gravity (that causes the collapse of the core) and weak interaction (that determines the rate at which electrons are captured and the rate at which neutrinos are trapped during the collapse).

The early phases of this final stage of stellar evolution are known as presupernova evolution. They follow the late-stage stellar evolution, and proceed until core densities of about  g cm and temperatures between 5 and 10K are reached. Until this point, modeling stellar evolution requires the consideration of extensive nuclear reaction networks, but is simplified by the fact that neutrinos need only be treated as a sink of energy and lepton number (due to their immediate escape). At later time and towards the collapse, this is no longer valid: As the weak interaction rates increase with the increasing density, the neutrino mean free paths shorten, so that the neutrinos eventually proceed from phases of free streaming, towards diffusion, and trapping. An adequate handling of the transitions between these transport regimes necessitates a detailed time- and space-dependent bookkeeping of the neutrino distributions in the core (see Ch. 8). During collapse, electron capture, accompanied by neutrino emission, dominates over electron antineutrino emission because the positron abundance is very low under electron-degenerate conditions. Later in the evolution the electron degeneracy is partially lifted, and in addition to the electron flavor neutrinos, also heavy neutrinos, and and their antiparticles, are usually included in numerical simulations of core collapse and postbounce evolution.

Advantageously, the temperature during the collapse and explosion are high enough that the matter composition is given by nuclear statistical equilibrium (NSE), i.e. without the need of reaction networks for the strong and electromagnetic interactions. The transition from a rather complex global nuclear reaction network, involving many neutron, proton and fusion reactions and their inverses, to a quasi-statistical equilibrium, in which reactions are fast enough to bring constrained regions of the nuclear chart into equilibrium, to final and global nuclear statistical equilibrium is extensively discussed by Hix and Thielemann (1996, 1999b); Hix et al (2007). In the late stages of Si-burning and the early collapse phase, weak interactions are dominated by electron captures on protons and nuclei. These are important equally in controling the neutronization of matter and, in a large portion, also the stellar energy loss. Due to their strong energy dependence , the electron capture rates increase rapidly during the collapse while the density and the temperature increase (the electron Fermi energy scales with , see 2).

The main weak interaction processes during the final evolution of a massive star are electron capture and -decays. Their determination requires the calculation of Fermi and Gamow-Teller (GT) transitions. While the treatment of Fermi transitions (important only for -decays) is straightforward, a correct description of the GT transitions is a difficult problem in nuclear structure physics. In astrophysical environments, nuclei are fully ionized. Therefore, electron capture occurs from the continuum of the degenerate electron plasma, and energies of the electrons are high enough to induce transitions to the Gamow-Teller resonance. Shortly after the discovery of this collective excitation, Bethe et al (1979) recognized its importance for stellar electron capture. -decay converts a neutron inside the nucleus into a proton and emits an electron. In a degenerate electron gas, with fully populated levels up to the Fermi energy , all decays which would produce electrons with smaller energies than are not possible (blocked). Then, the decay rate of a given nuclear state is greatly reduced or even completely blocked at high densities. However, there is another pathway, as high temperatures populate a distribution of nuclear states: If an excited and thermally populated state of the decaying nucleus is connected by large GT transition probabilities to low-lying states in the daughter nucleus, producing electrons above the Fermi energy, such transition path can contribute significantly to the stellar -decay rates. The importance of these states in the parent nucleus for -decay in astrophysical environments was first recognized by Fuller et al (1980, 1982, 1985).

Recent experimental data on GT distributions in iron group nuclei, measured in charge exchange reactions, show that the GT strength is strongly quenched (reduced), compared to the independent-particle-model value, and fragmented over many states in the daughter nucleus. An accurate understanding of these effects is essential for a reliable evaluation of the stellar weak-interaction rates, particularly for the stellar electron-capture rates (Fuller et al, 1980; Langanke and Martínez-Pinedo, 2000). The nuclear shell-model is the only known tool to reliably describe GT distributions in nuclei. When comparing the shell-model based rates (by Langanke and Martinez-Pinedo) with the those from Fuller et al., one finds that the shell-model based rates are almost always smaller at the relevant temperatures and densities, caused by the above mentioned quenching of the Gamow-Teller strength, and by a systematic misplacement of the energy of the Gamow-Teller resonance.

Figure 6: Comparison of the center values of (left), the iron core sizes (middle) and the central entropy (right) for 11–40 M stars between the WW models and the ones using the shell model weak interaction rates (LMP) (Heger et al, 2001a). The lower parts define the changes in the 3 quantities between the LMP and WW models.

The influence of these shell-model rates on the late-stage evolution of massive stars has been investigated by Heger et al (2001a, b) , and compared to earlier calculations (Woosley and Weaver, 1995). Fig. 6 illustrates the consequences of the shell model weak interaction rates for presupernova models in terms of the three decisive quantities: the central electron or proton to nucleon ratio , the entropy, and the iron core mass. The central values of at the onset of core collapse increased by 0.01-0.015 for the new rates. This is a significant effect. For example, a change from = 0.43 in the Woosley & Weaver model for a 20 M star to = 0.445 in the new models increases the respective Chandrasekhar mass by about 0.075 M (see Equ.3). The new models also result in lower core entropies for stars with M, while for M, the new models actually have a slightly larger entropy. The Fe-core masses are generally smaller, where the effect is larger for more massive stars ( M), while for the most common supernovae ( M) the reduction is by about 0.05 M (the Fe-core is here defined as the mass interior to the point where the composition is dominated by more than 50% of Fe-group elements with ). This reduction seems opposite to the expected effect due to slower electron capture rates in the new models. It is, however, related to changes in the entropy profile during shell Si-burning which reduces the growth of the iron core just prior to collapse.

Figure 7: Evolution of the value in the center of a 15  star (left panel) and a 25  star (right panel) as a function of time until bounce. The dashed line shows the evolution in the Woosley and Weaver models (WW) (Woosley and Weaver, 1995), while the solid line shows the results using the shell-model based weak-interaction rates of Langanke and Martínez-Pinedo (LMP). The two most important nuclei in the determination of the total electron-capture rate, for the calculations adopting the shell model rates, are displayed as a function of stellar evolution time.

The evolution of during the presupernova phase is plotted in Fig. 7. Weak processes become particularly important in reducing below 0.5 after oxygen depletion ( s and s before core collapse for the 15 M and 25 M stars, respectively) and begins a decline, which becomes precipitous during Si-burning. Initially electron captures occur much more rapidly than beta-decays. As the shell model rates are generally smaller, the initial reduction of is smaller in the new models. The temperature in these models is correspondingly larger as less energy is radiated away by neutrino emission. An important feature of the new models is shown in the left panel of Fig. 7. For times between and s before core collapse, increases due to the fact that -decay becomes competitive with electron capture after Si-depletion in the core and during shell Si-burning. The presence of an important -decay contribution has two effects (Aufderheide et al, 1994). Obviously it counteracts the reduction of in the core, but also acts as an additional neutrino source, causing a stronger cooling of the core and a reduction in entropy. This cooling can be quite efficient, as often the average neutrino energy from the -decays involved is larger than for the competing electron captures. As a consequence the new models have significantly lower core temperatures. At later stages of the collapse -decay becomes unimportant again as an increased electron Fermi energy blocks/reduces its role. The shell model weak interaction rates predict the presupernova evolution to proceed along a temperature-density- trajectory where the weak processes involve nuclei rather close to stability which will permit to test these effects in the next-generation radioactive ion-beam facilities.

Fig. 7 identifies the two most important nuclei (the ones with the largest value for the product of abundance times rate) for the electron capture during various stages of the final evolution of 15 M and 25 M stars. An exhaustive list of the most important nuclei for both electron capture and beta-decay during the final stages of stellar evolution for stars of different masses is given in Heger et al (2001b). In total, the weak interaction processes shift the matter composition to smaller values and hence more neutron-rich nuclei, subsequently affecting the nucleosynthesis. Its importance for the elemental abundance distribution, however, strongly depends on the location of the mass cut in the supernova explosion. It is currently assumed that the remnant will have a larger baryonic mass than the Fe-core, but smaller than the mass enclosed by the O-shell (Woosley et al, 2002). As the reduction of occurs mainly during Si-burning, it is essential to determine how much of this material will be ejected.

4 Supernovae from Massive Stars and the Role of Radioactivity

4.1 The Explosion Mechanism

Supernova explosions are an application of numerical astrophysical modelling that has a long tradition. Continued improvements of the models are motivated by the following points: (i) open questions regarding the explosion mechanism; (ii) availability of observations for inidividual supernova explosions; (iii) interesting input physics that tests matter under conditions that are not accessible on earth; (iv) visibility in light and other photon wavelengths, cosmic rays, neutrino emission, decay gamma-rays of radioactive products, perhaps gravitational wave emission; (v) visibility on cosmological distances with improving statistical information on the events and (vi) their impact on the interstellar matter (e.g. abundances of metal-poor stars) and Galactic evolution.

As discussed in the previous sections, the death of massive stars M proceeds through several evolutionary and dynamical phases. At first, the modeling of a star must include the evolution through all nuclear burning stages until the resulting inner iron core grows beyond the maximum mass which can be supported by the dominant pressure of the degenerate electron gas. At this point, the inner stellar core enters a dynamical phase of gravitational collapse, during which it compactifies by orders of magnitude. The nuclear saturation density (i.e. the density of stable nuclei g cm) is exceeded at the center of the collapse and a protoneutron star (PNS) is formed. The dynamical time scale reduces from a few hundreds of milliseconds at the onset of collapse to a few milliseconds after the core has bounced back at nuclear densities (see Fig. 8 from Liebendörfer et al (2003).

Figure 8: A sequence of density profiles of a 13 M star before and after core bounce. For such a relatively low mass supernova with a small Fe-core the bounce occurs at a maximum density of less than twice nuclear matter density. At the bounce one recognizes the size of the homologous core (with roughly constant density). Thereafter the emergence of an outward moving density (shock) wave can be witnessed.

The ensuing accretion phase onto the protoneutron star with fluid instabilities and radiative transfer phenomena, like the transport of neutrinos, is not well understood. It may last seconds and can therefore be interpreted as a second evolutionary stage (much longer than the dynamical or transport time scale). Eventually it will lead to the observed vigorous supernova explosion, a dynamic phase where heavy elements are produced by explosive nucleosynthesis in an outward propagating shock wave. The processed matter is mixed by fluid instabilities and ejected into the interstellar medium, where it contributes to Galactic evolution. The remaining PNS at the center enters another evolutionary phase during which it cools by neutrino emission and contracts or even collapses to a black hole in a last dynamical phase.

While initially such calculations were performed in spherical symmetry and therefore lacked the consistent treatment of turbulent motion, presently performed research is done with multidimensional supernova models (Hix et al, 2003; Liebendörfer et al, 2005; Marek et al, 2005; Burrows et al, 2006a; Sumiyoshi et al, 2007; Langanke et al, 2008; Marek and Janka, 2009). The main ingredients are radiation (neutrino) transport, (relativistic) hydrodynamics, and the nuclear equation of state at such high densities. Recent progress has been made in the exploration of multidimensional hydrodynamics with idealized input physics. A refreshing view on the supernova mechanism has recently been suggested by pointing out that in certain axisymmetric simulations vibrational (so called PNS g-)modes are excited so that sound waves are emitted into the heating region. These sound waves are postulated to revive the stalled shock by dissipation of sound energy (Burrows et al, 2006a). Other efforts explore the role of magnetic fields and rotation in two-dimensional simulations with simplified input physics. One kind of proof-of-principle models is carried out in spherically symmetric approaches. The assumption of spherical symmetry is for many supernovae not compatible with observational constraints. However, one important advantage of spherically symmetric models is that sophisticated treatments of the neutrino-matter interactions can be included and that the neutrino spectra and transport are correctly treated in general relativistic space-time. Models of this kind try to address the question of how many neutrinos are emerging from the compactification of an inner stellar core, how is their emission distributed as a function of time and how do these neutrino fluxes generically affect the cooling, heating, or nucleosynthesis in the outer layers of the star without the complication of 3D dynamical fluid instabilities (Liebendörfer et al, 2003, 2004; Fischer et al, 2009b, a). The attempt to combine all these aspects with forefront methods is ongoing in order to achieve the final goal of understanding the multi-D explosion mechanism with up to date microphysics from the equation of state to all neutrino and nuclear interactions131313For a review of the corresponding tools see Ch. 8.).

The phase of stellar core collapse has intensively been studied in spherically symmetric simulations with neutrino transport. The crucial weak processes during the collapse and postbounce evolution are , , , , , , , , , , , . Here, a nucleus is symbolized by its mass number and charge , denotes either a neutron or a proton and represents any neutrino or antineutrino. We note that, according to the generally accepted collapse picture (Bethe 1990; Bethe et al (1979)), elastic scattering of neutrinos on nuclei is mainly responsible for the trapping, as it determines the diffusion time scale of the outwards streaming neutrinos. Shortly after trapping, the neutrinos are thermalized by energy downscattering, experienced mainly in inelastic scattering off electrons. The relevant cross sections for these processes are discussed in Martínez-Pinedo et al (2006). The basic neutrino opacity in core collapse is provided by neutrino scattering off nucleons. Depending on the distribution of the nucleons in space and the wavelength of the neutrinos, various important coherence effects can occur: Most important during collapse is the binding of nucleons into nuclei with a density contrast of several orders of magnitude to the surrounding nucleon gas. Coherent scattering off nuclei dominates the scattering opacity of neutrinos (and scales with ). Moreover, these neutrino opacities should be corrected by an ion-ion correlation function, this occurs if the neutrino wavelength is comparable to the distances of scattering nuclei and quantum mechanical intererence effects appear (Sawyer, 2005; Burrows et al, 2006b). Even if current core collapse models include a full ensemble of nuclei in place of the traditional apprach with one representative heavy nucleus, it remains non-trivial to adequately determine correlation effects in the ion mixture. Depending on the Q-value of an electron-capturing nucleus, neutrinos are emitted with a high energy of the order of the electron chemical potential/Fermi energy. As the neutrino opacities scale with the squared neutrino energy, the initially trapped neutrinos will downscatter to lower energies until the diffusion time scale becomes comparable to the thermalization time scale. The thermalization in current collapse models occurs through neutrino-electron scattering because the energy transfer per collision with the light electron is more efficient than with the heavier nucleons. The contribution of inelastic scattering of neutrinos off heavy nuclei depends on the individual nuclei and affects only the high-energy tail of the neutrino spectrum.

Goldreich and Weber (1980) have shown that only the inner (see the definition in Eq. 3) undergo a homologous collapse (), while at the edge of this core the velocity becomes supersonic and a fraction of the free-fall velocity. The inner core, falling at subsonic velocities where matter can communicate with sound speed, cannot communicate with the free-falling envelope. After the neutrinos are trapped, electron captures and neutrino captures are in equilibrium () and the total lepton fraction stays constant. stops to decrease and stops shrinking. Typical values (with the most recent electron capture rates (Langanke et al, 2003) of are found in numerical collapse calculations (Hix et al, 2003; Marek et al, 2005) which correspond to M. As soon as nuclear densities are reached at the center of the collapsing core, repulsive nuclear forces dominate the pressure in the equation of state. The collapse comes to a halt and matter bounces back to launch an outgoing pressure wave through the core. It travels through the subsonic inner core and steepens to a shock wave as soon as it faces supersonic infall velocities. Hence the matter in the PNS remains at low entropy k per baryon while the supersonically accreting layers become shock-heated and dissociated at entropies larger than k per baryon. Numerical simulations based on standard input physics and accurate neutrino transport exclude the possibility that the kinetic energy of the hydrodynamical bounce at nuclear densities drives a prompt supernova explosion because of dissociation and neutrino losses.

This can be seen in Fig. 8 presenting spherically symmetric calculations of a 13 M star. The inner core contains about 0.6 M of the initial Fe-core. The transition to free nucleons occurred only in this inner, homologous core and the outward moving shock runs through material consiting of Fe-group nuclei. The dissociation takes 8.7 MeV/nucleon or erg g. Based on initial shock energies of erg, this is sufficient for passing through 0.25-0.5 M and leads in essentially all cases to a stalling of the prompt shock. Only recently a possible exception was found (Sagert et al, 2009). If a hadron-quark phase transition occurs in the collapsed core at the appropriate time, releasing additional gravitational binding energy in the form of neutrinos from this second collapse, prompt explosions can be attained.

While core collapse determines the state of the cold nuclear matter inside the PNS, the mass of the hot mantle around the PNS grows by continued accretion. The infalling matter is heated and dissociated by the impact at the accretion front and continues to drift inward. At first, it can still increase its entropy by the absorption of a small fraction of outstreaming neutrinos (heating region). Further in, where the matter settles on the surface of the PNS, neutrino emission dominates absorption and the electron fraction and entropy decrease significantly (cooling region). The tight non-local feedback between the accretion rate and the luminosity is well captured in computer simulations in spherical symmetry that accurately solve the Boltzmann neutrino transport equation for the three neutrino flavors. All progenitor stars between main sequence masses of and M showed no explosions in simulations of the postbounce evolution phase (Liebendörfer et al, 2003). This indicates that the neutrino flux from the PNS does not have the fundamental strength to blow off the surrounding layers for a vigorous explosion.

Improved electron capture rates on heavy nuclei overcame the idealized blocking of Gamow-Teller transitions in the traditionally applied single-particle model. In the single-particle picture of nuclei the so-called pf-shell is filled for or for protons or neutrons respectively. Neutron numbers beyond require a filling of the gd-orbits. If during core collapse nuclei () become so neutron-rich that nuclei with and dominate the NSE composition, electron capture would require the conversion of an proton to a neutron as all neutron orbits are filled. This Pauli-blocked transition would lead to the dominance of electron capture on free protons rather than nuclei and under such conditions. The recent finding, that configuration mixing and finite temperature effects result in unfilled neutron orbits, removes this Pauli-blocking and results in the fact that under these condition electron capture rates on nuclei dominate those on free protons (Langanke et al, 2003). Thus, there are two effects due to the new set of electron capture rates: 1. at low densities for less neutron-rich nuclei the total amount of electron capture is reduced with an improved description of Gamow-Teller transitions (see the discussion of the early collapse phase in Sect. 3), 2. at high densities in the late collapse phase the total amount of electron capture is enhanced, leading to smaller and values than before. Such changes caused a reduction of homologous core sizes down to  M (see discussion above and Hix et al (2003)). This faster deleptonization in the collapse phase in comparison to captures on free protons alone thus resulted in a smaller inner core at bounce.

Taking all this improved physics into acount leads in the entire simulations (i.e. all mass zones invoved) to conditions in densities , electron abundance and entropy per baryon, where properties like the equation of state or other microscopic physics is needed in current supernova simulations. Fig. 9 provides this information for a simulation of a 20 M star (Liebendörfer et al, 2009).

Figure 9: Overview of the conditions attained in a simulation of the collapse, bounce, and explosion (artificially induced) of a 20 M star. Shown are two histograms of the occurrence of conditions as a function of density , specific entropy and electron fraction . The shading of a given bin corresponds to in arbitrary units, where the integral over mass is performed over the mass of matter whose thermodynamic state at a given time falls into the bin. The integral over time extends over the duration of a simulation. Hence, regions of dark shading correspond to states that are experienced by considerable mass for an extended time, while light or absent shading corresponds to conditions that are rarely assumed in the supernova simulation. The vertical black line indicates the nuclear density. The horizontal black line indicates an entropy of 3 k /baryon beyond which ions are dissociated. It clearly separates the conditions of cold infalling matter on the lower branch from the conditions of hot shocked matter on the upper branch.

Moreover, a comparison of the effects of the only two publicly available equations of state by Lattimer and Douglas Swesty (1991) and Shen et al (1998b, a) is required. In simulations of massive progenitors that do not explode and exceed the maximum stable mass of the accreting neutron star in the postbounce phase, it was demonstrated that the neutrino signal changes dramatically when the PNS collapses to a black hole (Fischer et al, 2009b). Depending on the stiffness of the equation of state or the accretion rate from the external layers of the progenitor star, this can happen at very different time after bounce. Hence, the neutrino signal carries a clear imprint of the stiffness of the equation of state and the accretion rate to the observer of neutrinos.

The detailed treatment of the neutrino transport and interactions is of great importance for the nucleosynthesis. This has been shown in several recent studies (Fröhlich et al, 2006a, b; Pruet et al, 2005, 2006; Wanajo, 2006). This also opens an opportunity to investigate neutrino flavor oscillations among electron, muon and tau neutrinos. On the one hand side the long term explosion runs achieve (low) density structures that allow for MSW (Mikheyev-Smirnov-Wolfenstein effect) neutrino flavor oscillations in the outer layers (Wolfenstein, 1978; Mikheyev and Smirnov, 1985). These may give additional hints on the expansion velocity and density distribution in case that the neutrinos can be observed from a near-by supernova. On the other hand, collective flavor transitions have recently been postulated in regions where the neutrino density exceeds the electron density (Duan et al, 2006, 2007; Fogli et al, 2007). This condition will be achieved in the evacuated zone that surrounds the PNS after the onset of an explosion. The impact of these collective neutrino flavor oscillations on the neutrino heating during the shock expansion, the neutrino wind, and the nucleosynthesis are important open points that require a detailed investigation under consideration of accurate neutrino transport and spectra.

Figure 10: Illustration of the early accretion phase in a three-dimensional simulation with a resolution of zones and the isototropic diffusion source approximation for 3D neutrino transport (Liebendörfer et al, 2009). Shown are density contours as black lines for a 15 M star from Woosley and Weaver (1995). Left: The color indicates the specific entropy and the cones the direction of the velocity. Right: The color refers to the magnetic field strength and the cones to its direction. The cool high-density interior of the PNS and the hot low-density accreted matter behind the standing accretion front are clearly distinguishable.

The difficulty to reproduce explosions in spherically symmetric models of core-collapse and postbounce evolution stimulated the consideration of numerous modifications and alternatives to this basic scenario, mostly relying on multi-dimensional effects that could not be treated in spherical symmetry. It was discussed whether convection inside the PNS could accelerate the deleptonization and increase the neutrino luminosity (Wilson and Mayle, 1993). The convective overturn between the PNS and shock front was shown to increase the efficiency of neutrino energy deposition (Herant et al, 1994). Asymmetric instabilities of the standing accretion shock (Blondin et al, 2003; Foglizzo, 2009) may help to push the shock to larger radii and g-mode oscillations of the PNS may contribute to neutrino heating by the dissipation of sound waves between the PNS and the shock (Burrows et al, 2006a). Moreover, it has been suggested that magnetic fields have an impact on the explosion mechanism (Kotake et al, 2006). Most of the above-mentioned modifications of the explosion mechanism are essentially of a three-dimensional nature. In order to illustrate the complexity of the crucial accretion phase we show in Fig. 10 a slice through a three-dimensional simulation of core-collapse and postbounce evolution of a recent run (Liebendörfer et al, 2008). Its input physics uses the Lattimer-Swesty equation of state (Lattimer and Douglas Swesty, 1991) and a parameterization of the neutrino physics for the collapse phase (Liebendörfer et al, 2005). The treatment of neutrino cooling and heating in the postbounce phase is based on multi-group diffusion (the isotropic diffion source approximation of Liebendörfer et al (2009)).

Initially, spherically symmetric supernova models were the most realistic among all feasible computer representations of the event. With increasing observational evidence for the complexity of the explosions (Hamuy, 2003) their primary purpose shifted from a realistic representation to the identification and understanding of the basic principles of the explosion mechanism. After the emergence of axisymmetric simulations with sophisticated and computationally intensive spectral neutrino transport (Buras et al, 2003; Walder et al, 2005) spherically symmetric models still have several assets. In the following subsection we will first describe purely phenomenological calculations based on artificially induced explosions via a “piston” or energy deposition in terms of a “thermal bomb”, purely in order to discuss nucleosynthesis effects. We will then also discuss still artificial explosions in spherical symmetry, however resulting from a “self-consistent” treatment including neutrino transport, which permits to analyse the effect of neutrinos on the nucleosynthesis of the innermost ejecta.

4.2 Nucleosynthesis in Explosions

Major Explosive Burning Processes

Despite considerable improvements of stellar models and numerical simulations in recent years, some fundamental problems remain in nucleosynthesis predictions. It has become evident that certain evolution aspects can only be followed in models going beyond one-dimensional simulations, such as convection, rotation, and the explosion mechanism. However, it is still not feasible to directly couple full reaction networks, containing several thousand nuclei, to multi-dimensional hydrodynamic calculations due to the lack of required computing power, even in modern computers. Thus, postprocessing after explosion models with parameterized networks still remains an important approach. One-dimensional models can directly accommodate increasingly larger networks but they cannot capture all of the necessary physics. As outlined in the previous subsection, it has become apparent that a self-consistent treatment of core collapse supernovae in 1D does not lead to successful explosions when using presently known input physics while 2D models show some promise. There are strong indications that the delayed neutrino mechanism works combined with a multi-D convection treatment for unstable layers (possibly with the aid of rotation, magnetic fields and/or still existent uncertainties in neutrino opacities). Therefore, hybrid approaches using certain parameterizations or approximations have been and are still necessary when predicting the nucleosynthetic yields required for the application described above.

Supernova nucleosynthesis predictions have a long tradition. All of these predictions relied on an artificially introduced explosion, either via a piston or a thermal bomb introduced into the progenitor star model. The mass cut between the ejecta and the remnant does not emerge from this kind of simulations but has to be determined from additional conditions. While the usage of artificially introduced explosions is justifiable for the outer stellar layers, provided we know the correct explosion energy to be dumped into the shock front (on the order of 10 erg seen in observations), it clearly is incorrect for the innermost ejected layers which should be directly related to the physical processes causing the explosion. This affects the Fe-group composition, which has been recognized as a clear problem by many groups (Woosley and Weaver, 1995; Thielemann et al, 1990, 1996; Nakamura et al, 1999, 2001; Nomoto et al, 2006). The problem is also linked to the so-called neutrino wind, emitted seconds after the supernova explosion, and considered as a possible source of the  process to produce the heaviest elements via neutron captures (Qian and Woosley, 1996), as will be discussed below.

Figure 11: Mass fractions of a few major nuclei after passage of the supernova shockfront through a star with an intial mass of 20 M. Matter outside 2M is essentially unaltered. Mass zones further in experience explosive Si, O, Ne, and C-burning. For ejecting 0.07M of Ni the mass cut between neutron star and ejecta is required to be located at 1.6M.

Given the above detailed discussion of the physics, problems and options regarding core collapse supernovae, we will adopt the following approach in order to predict the most reliable nucleosynthesis predictions for the ejecta in a 1D spherically symmetric treatment. The multiplication of neutrino capture cross sections on nucleons with a free parameter in 1D spherically symmetric calculations can mimic the enhanced energy deposition which multi-D models show. The free parameter is tuned to give correct explosion energies and Ni yields for a number of well known supernovae. This approach provides clear predictions for the mass cut between the remaining neutron star and the ejecta. It also includes the effect neutrinos can have on the correct in the ejecta and the related nucleosynthesis. In the outer explosively burning layers, essentially only the energy in the shock front matters. The behavior of these zones can be easily understood from the maximum temperatures attained in the radiation bubble and for a first discussion we will just focus on these features, which can also be obtained with an artifically induced thermal bomb treatment.

For a given/known and density , the most significant parameter in explosive nucleosynthesis is the temperature, and a good prediction for the composition can already be made by only knowing , without having to perform complex nucleosynthesis calculations. Weaver and Woosley (1980) already recognized, that matter behind the shock front is strongly radiation dominated. Assuming an almost homogeneous density and temperature distribution behind the shock (which is approximately correct, one can equate the supernove energy with the radiation energy inside the radius of the shock front

(4)
Figure 12: Mass fractions of the dominant nuclei in zones which experience -rich freeze-out. Notice the relatively large amounts of Zn and Cu nuclei, which originate from -captures on Ni and Co. One can recognize their strong decrease beyond 1.66M, which goes parallel with the decrease of the He-abundance and other -nuclei such as Ca, Ti, Cr, and Fe. Nuclei which would dominate in a nuclear statistical equilibrium like Ni stay constant or increase even slightly. The increase of all nuclei with at 1.63M and the decrease of nuclei with NZ is due to the change in Y in the original stellar model before collapse (see also Fig.11)

This equation can be solved for . With K, the lower bound for explosive Si-burning with complete Si-exhaustion, and an induced thermal bomb energy of erg, the result is km. For the evolutionary model by Nomoto and Hashimoto (1988) of a 20M star this radius corresponds to 1.7M, in excellent agreement with the exact hydrodynamic calculation. Temperatures which characterize the edge of the other explosive burning zones correspond to the following radii: incomplete Si-burning (=4, =4980km), explosive O-burning (3.3, 6430), and explosive Ne/C-burning (2.1, 11750). This relates to masses of 1.75, 1.81, and 2.05M in case of the 20M star. The radii mentioned are model independent and vary only with the supernova energy. In the following we present a number of plots which show the different mass fractions as a function of radial mass M, passing outwards through a 20M star through all explosive burning regions.

Figure 13: Mass fractions of nuclei in the zones of incomplete Si-burning M1.74M and explosive O-burning M1.8M. The Si-burning zones are are characterized by important quantities of Fe-group nuclei besides Si, S, Ar, and Ca. Explosive O-burning produces mostly the latter, together with more neutron-rich nuclei like Si, S, Ar etc.
Figure 14: Composition in mass zones of explosive Ne and C-burning. The dominant products are O, Mg, and Si. Besides the major abundances, mentioned above, explosive Ne-burning supplies also substantial amounts of Al, Si, S, Si, and P. Explosive C-burning contributes in addition the nuclei Ne, Na, Mg, Mg, and Mg.

Matter between the mass cut = and the mass enclosed in the radius corresponding to explosive Si-burning with complete Si-exhaustion is indicated with (ex Si-c). Then follows the zone of incomplete Si-burning until (ex Si-i), explosive O-burning until (ex O), explosive Ne/C-burning until (ex Ne), and unprocessed matter from the C/Ne-core is ejected until (C-core). The zones beyond explosive Ne/C-burning (K) are essentially unaltered and the composition is almost identical to the pre-explosive one. When performing such calculations for a variety of progenitors over a range of initial stellar masses, one can analyze the dependence of the mass involved in these different burning regimes as a function initial stellar mass (see Sect. 5).

Results for a 20M star (Nomoto and Hashimoto, 1988) are given as examples for the abundance behavior in a series of Figs 11, 12, 13, 14. It should be mentioned here that this still uses a simplified thermal bomb treatment for the pre-collapse model rather than the results from a 1D spherically symmetric simulation with enhanced neutrino capture rates, which insures an explosion also in 1D. The explosion energy used corresponds to a supernova energy of erg. As mentioned before, this treatment cannot predict a self-consistent explosion and the position of the mass cut between neutron star and ejecta. Only the observation of M of Ni in SN1987A (a 20M star) gives an important constraint, because Ni is produced in the innermost ejected zones. The explosive nucleosynthesis due to burning in the shock front is shown in Fig. 11 for a few major nuclei. Inside 1.7M all Fe-group nuclei are produced in explosive Si-burning during the SN II event. At 1.63M Y changes from 0.494 to 0.499 and leads to a smaller Ni abundance further inside, where more neutron-rich Ni-isotopes share the abundance with Ni. This is an artifact of the gradient in the pre-collapse model which can be changed in a consistent explosion treatment via neutrino interactions with this matter.

In explosive Si-burning only -rich freeze-out and incomplete Si-burning are encountered. Contrary to SNe Ia, densities in excess of gcm, which would result in a normal freeze-out, are not attained in the ejecta of this 20M star (see also Fig.2). The most abundant nucleus in the -rich freeze-out is Ni. For the less abundant nuclei the final -capture plays a dominant role transforming nuclei like Ni, Ni, and Ni into Zn, Zn, and Zn (see Fig.12).

The region which experiences imcomplete Si-burning starts at 1.69M and extends out to 1.74M. In the innermost zones with temperatures close to K there exists still a contamination by the Fe-group nuclei Fe, Ni, Fe, Ni, Co, and Ni. Explosive O-burning occurs in the mass zones up to 1.8M (see Fig.13). The main burning products are Si, S, Ar, Ca, Ar, and S. With mass fractions less than also S, K, Cl, Ca, and Ar are produced. Explosive Ne-burning leads to an O-enhancement over its hydrostatic value in the mass zones up to 2M (see Fig.14).

Explosive Burning off Stability

The p-Process

Up to now we discussed the production of heavy nuclei beyond the Fe-group only via slow neutron captures (the  process) in hydrostatic stellar evolution. A number of proton-rich (p-)isotopes of naturally occurring stable heavy nuclei cannot be produced by neutron captures along the line of stability. The currently most favored production mechanism for those 35 p-isotopes between Se and Hg is photodisintegration ( process) of intermediate and heavy elements at high temperatures in late (explosive) evolution stages of massive stars (Woosley and Howard, 1978; Rayet et al, 1990). However, not all p-nuclides can be produced satisfactorily, yet. A well-known deficiency in the model is the underproduction of the Mo-Ru region, but the region 151A167 is also underproduced, even in recent calculations (Rauscher et al, 2002; Arnould and Goriely, 2003; Rapp et al, 2006; Dillmann et al, 2008). There exist deficiencies in astrophysical modeling and the employed nuclear physics. Recent investigations have shown that there are still considerable uncertainties in the description of nuclear properties governing the relevant photodisintegration rates. This has triggered a number of experimental efforts to directly or indirectly determine reaction rates and nuclear properties for the p/ process (Rauscher, 2006). Here it is important to investigate the sensitivity of the location of the -process path with respect to reaction rate uncertainties.

Concerning the astrophysical modeling, only a range of temperatures has to be considered which are related to the explosive Ne/O-burning zones of a supernova explosion (see Figs.13 and 14), where partial (but not complete) photodisintegration of pre-existing nuclei occurs (from prior hydrostatic evolution or inherited metallicity), i.e. at 10K. The  process starts with the photodisintegration of stable seed nuclei that are present in the stellar plasma. During the photodisintegration period, neutron, proton, and -emission channels compete with eachother and with beta-decays further away from stability. In general, the process, acting like “spallation” of pre-existing nucei, commences with a sequence of -reactions, moves the abundances to the proton-rich side. At some point in a chain of isotopes, and/or -reactions become faster than neutron emissions, and the flow branches and feeds other isotopic chains. At late times photodisintegrations become less effective, when decreasing temperatures shift the branching points and make beta-decays more important. Finally the remaining unstable nuclei decay back to stability. The branchings established by the dominance of proton and/or -emission over neutron emission are crucial in determining the radioactive progenitors of the stable p-nuclei and depend on the ratios of the involved reaction rates. Numerous experimental and theoretical efforts have been undertaken to improve the reaction input, especially with respect to open questions in optical potentials for  particles and protons (Gyürky et al, 2006; Kiss et al, 2007, 2008; Yalçın et al, 2009).

Figure 15: Normalized overproduction factors of -process nuclei derived with the Rapp et al (2006) (open squares) and Dillmann et al (2008) (full squares) reaction library. In addition, the results from a range of stellar models (10-25M) from Rayet et al (1995) are given for comparison. A value equal to unity corresponds to relative solar abundances.

Applications of -process network calculations to the temperature profiles of initiated explosions have been performed by Rayet et al (1995); Rapp et al (2006); Dillmann et al (2008). Here, in Fig. 15 we present the results of a 25M mass model (Dillmann et al, 2008) with two reaction rate libraries without and with inclusion of all experimental improvements, existing at that point. It is noticed that the nuclear uncertainties cannot change the underproduction of especially the light p-nuclei. Another process seems to be required to supply these missing abundances.

The p-Process

Neutron-deficient nuclei can also be produced by two other astrophysical nucleosynthesis processes: the  process in X-ray bursts (which, however, does not eject matter into the interstellar medium (Wallace and Woosley, 1981; Schatz et al, 1998; Fisker et al, 2008) and the recently discovered  process in core collapse supernovae (Fröhlich et al, 2006a, b; Pruet et al, 2006; Wanajo, 2006). The  process occurs in explosive environments when proton-rich matter is ejected under the influence of strong neutrino fluxes. This includes the innermost ejecta of core-collapse supernova (Buras et al, 2006; Thompson et al, 2005; Liebendörfer et al, 2008) and possible ejecta from black hole accretion disks in the collapsar model of gamma-ray bursts (Surman et al, 2006). The discussion of these innermost ejected mass zones has been skipped above, when discussing the results for explosive nucleosynthesis in a 20 M star, utilizing a thermal bomb but the pre-collapse stellar conditions with the corresponding . Here, as discussed in the beginning of this subsection, we have boosted the energy deposition efficiencies by enhancing the neutrino and anti-neutrino captures on neutrons and protons in a 1D simulation. While this is not a fully self-consistant treatment, no external (artificial) energy is required to produce a successful explosion with a consistently emerging mass cut between neutron star and ejecta. Moreover, this treatment guarantees provides a that is consistently determined by all weak interactions processes. The result is that explosions are obtained and the neutrino interaction with matter leads to a enhanced beyond 0.5 (see Fig. 16) which overcomes nucleosynthesis problems for the Fe-group encountered previously.

Figure 16: of the innermost ejecta due to neutrino interactions with matter. At high temperatures electrons are not degenerate, thus the reduction of due to electron captures is ineffective. For similar neutrino and antineutrino spectra the neutron-proton mass difference favors over .

The matter in these ejecta is heated to temperatures well above 10 and becomes fully dissociated into protons and neutrons. The ratio of protons to neutrons is mainly determined by neutrino and antineutrino absorptions on neutrons and protons, respectively. Similar neutrino and antineutrino energy spectra and fluxes produce proton-dominated matter in the reactions and , due to the n-p mass difference. When the matter expands and cools, the free neutrons and protons combine into -particles. Later, at temperatures around 510K, -particles assemble into heavier nuclei via unstable intermediate nuclei, e.g. the triple- reaction via unstable Be, but - depending on the entropy and the expansion of matter - only a fraction of those form iron-group nuclei ( -rich freeze-out). In case of a proton-rich environment, there are also still free protons available at the time of the alpha freeze-out. Once the temperature drops to about 210K, the composition of the ejecta consists mostly of He, protons, and iron group nuclei with NZ (mainly Ni) in order of decreasing abundance. Without neutrinos, synthesis of nuclei beyond the iron peak becomes very inefficient due to bottleneck (mainly even-even ) nuclei with long beta-decay half-lives and small proton-capture cross sections. Such a nucleus is Ge. Thus, with the determined by neutrino interactions with free neutrons and protons in the early very hot phase of dissociated nuclei, the nucleosynthesis leads to an - and proton-rich freeze-out which does not stop at Ni but continues up to Ge (which later decays to Zn. This part of the story enables core collapse yeads which produce Fe-group nuclei up to essentially Zn. The effect is seen in the upper portion Fig. 17.

Figure 17: Final abundances in mass zones in the innermost ejecta which experienced neutrino irradiation, leading to proton-rich conditions (). The upper part of the figure shows the neucleosynthesis results in the innermost ejecta of explosive, after -rich and proton-rich freeze-out from Si-burning, normalized to solar after decay. The bottom part of the figure also includes the interaction of anti-electron neutrinos with protons () which produces neutons, permitting the late change of Ge via Ge(Ga. This feature permits further proton captures to produce havier nuclei (the so-called  process. Here matter up to is produced.

However, the matter is subject to a large neutrino/antineutrino flux from the proto-neutron star. While neutrons are bound in neutron-deficient nuclei and neutrino captures on these nuclei are negligible due to energetics, antineutrinos are readily captured both on free protons and on heavy nuclei on a timescale of a few seconds. As protons are more abundant than heavy nuclei, antineutrino captures occur predominantly on protons, leading to residual neutron densities of cm for several seconds. These neutrons are easily captured by heavy neutron-deficient nuclei, for example Ge, inducing reactions with time scales much shorter than the beta-decay half-life. This permits further proton captures and allows the nucleosynthesis flow to continue to heavier nuclei (see lower part of Fig. 17). The  process (Fröhlich et al, 2006b) is this sequence of -reactions, followed by -reactions or beta-decays, where the neutrons are supplied by antineutrino captures on free protons.

Figure 18: Final abundances in mass zones experiencing the  process, i.e. the innermost ejecta of explosive, -rich freeze-out Si-burning, normalized to solar after decay for two sets of thermonuclear reaction rates/masses. Matter up to can be produced easily.

In Fig.18 we also show  process nucleosynthesis results from the innermost early neutrino wind ejecta produced in the explosion of a 15M star (Janka et al, 2003), also utilized in (Pruet et al, 2006; Fisker et al, 2009), which synthesizes efficiently nuclei even for . Two sets of astrophysical reaction rates were used in the reaction network, both based on theoretical rates from the NON-SMOKER code (Rauscher and Thielemann, 2000, 2004), but once with the latest excited state information and masses from the AME2003 compilation (Audi et al, 2003) and another set also with the latest mass measurements (Weber et al, 2008). Fig.18 shows the final abundances normalized to solar abundances after decay to stability for these two sets of thermonuclear reaction rates. Only nuclei produced in the p-rich ejecta are shown. As is clearly seen, there is no difference in the yields for the two different sets of rates except for a few nuclei in the mass range , namely Sr, Y, and Zr. This can be directly traced back to the large change in the mass of Tc (= -1031 keV). This change in mass leads to an increase in the reaction rate for TcMo at the relevant temperatures and therefore a relative suppression of the opposite capture rate. These results show that the  process can easily produce the light p-nuclei of Mo and Ru, which are deficient in -process calculations. Further processing depends on the expansion (speed) of matter and the overlying mass of ejecta.

The r-Process

A rapid neutron-capture process ( process) in an explosive environment is traditionally believed to be responsible for the nucleosynthesis of about half of the heavy elements above Fe. While in recent years the high entropy (neutrino) wind (HEW) of core-collapse supernovae has been considered to be one of the most promising sites, hydrodynamical simulations still encounter difficulties to reproduce the astrophysical conditions under which this process occurs. The classical waiting-point approximation, with the basic assumptions of an Fe-group seed, an -equilibrium for constant neutron densities at a chosen temperature , over a process duration , and an instantaneous freezeout, has helped to gain improved insight into the systematics of an  process in terms of its dependence on nuclear-physics input and astrophysical conditions (Cowan et al, 1991; Kratz et al, 1993, 2007). This corresponds to a set of quasi-equilibria with each QSE group being represented by an isotopic chain. Taking a specific seed nucleus, the solar -process pattern peaks can be reproduced by a variation/superposition of neutron number densities and durations . Whether the solar -process abundances are fully reproduced in each astrophysical event, i.e., whether each such event encounters the full superposition of conditions required, is a matter of debate (Wasserburg et al, 1996; Pfeiffer et al, 2001; Sneden et al, 2003; Honda et al, 2006; Qian and Wasserburg, 2007; Farouqi et al, 2009, 2010). In realistic astrophysical environments with time variations in and , it has to be investigated whether at all and for which time duration the supposed -equilibrium of the classical approach will hold and how freeze-out effects change this behavior. In general, late neutron captures may alter the final abundance distribution. In this case neutron capture reactions will be important. Also -delayed neutrons can play a role in forming and displacing the peaks after freeze-out.

For many years since Woosley et al (1994); Takahashi et al (1994); Qian and Woosley (1996) the high entropy wind has been considered as the most promising (realistic?) environment, expelled from newly formed (hot) neutron stars in core-collapse supernovae, which continue to release neutrinos after the supernova shock wave is launched. These neutrinos interact with matter of the outermost proto-neutron star layers which are heated and ejected in a continuous wind. The late neutrino flux also leads to moderately neutron-rich matter (Qian and Woosley, 1996) via interactions with neutrons and protons and causes matter ejection with high entropies. (However, there are recent studies (Fischer et al, 2009a) from collapse calculations which predict a proton-rich wind composition for more than the first 10s after collapse.) Problems were encountered to attain entropies sufficiently high in order to obtain high neutron/seed ratios which can produce the heaviest  process nuclei (Thompson et al, 2001; Wanajo et al, 2001; Terasawa et al, 2002). Recent hydrodynamic simulations for core-collapse supernovae support the idea that these entropy constraints can be fulfilled in the late phase (after the initial explosion) when a reverse shock is forming (Fryer et al, 2006; Arcones et al, 2007; Burrows et al, 2007; Janka et al, 2007; Panov and Janka, 2009).

The question is whether such high entropies occur at times with sufficiently high temperatures when an  process is still underway (Kuroda et al, 2008). Exploratory calculations to obtain the necessary conditions for an  process in expanding high-entropy matter have been undertaken by a number of groups (Hoffman et al, 1997; Meyer and Brown, 1997; Otsuki et al, 2000; Wanajo et al, 2001; Terasawa et al, 2002; Wanajo et al, 2004; Yoshida e