TYPE Ia SUPERNOVAE AS SITES OF THE p-PROCESS: TWO-DIMENSIONAL MODELS COUPLED TO NUCLEOSYNTHESIS

Among the nuclei heavier than ⁵⁶Fe, there is a class of 35 nuclides called p-nuclei, which are typically 10–1000 times less abundant than the s- or r-isotopes in the solar system. They cannot be synthesized by neutron-capture processes since they are located on the neutron-deficient side of the valley of β-stability. The astrophysical origin of p-nuclei has been studied for 50 years, starting from the pioneering works by Cameron (1957) and Burbidge et al. (1957). They suggested that a combination of proton captures and photodissociations of s- and r-seed nuclei could produce p-nuclei at temperatures between 2 and 3 × 10⁹ K. About 20 years later, different attempts were made to explain the synthesis of all p-nuclei in one astrophysical site (Audouze & Truran 1975; Arnould 1976; Woosley & Howard 1978). Those authors suggested that the largest fraction of p-isotopes in the solar system should be created by photodisintegration (the so-called γ-process) reactions operating upon a distribution of s-process seeds synthesized in the earlier evolutionary stages of the progenitor. They suggested Type II supernovae (hereafter SNe II) as the astrophysical site of the p-process and demonstrated that the p-process is extremely sensitive to temperature and timescales. Detailed calculations of p-process nucleosynthesis in SNe II have been performed by many authors, e.g., Woosley & Howard (1990), Rayet et al. (1990, 1995), Rauscher et al. (2002), Hayakawa et al. (2006, 2008), and Farouqi et al. (2009). In these studies, core-collapse supernovae have been considered the best candidates for reproducing the solar abundances of the bulk of the p-isotopes. However, it has been shown that the "gamma-process" scenario suffers from a strong underproduction of the most abundant p-isotopes, ^{92, 94}Mo (see, e.g., Fisker et al. 2009) and ^{96, 98}Ru. For these nuclei, alternative processes and sites have been proposed, either strong neutrino fluxes in the deepest layers of SN II ejecta (Frölich et al. 2006), or rapid proton-captures in proton-rich, hot matter accreted onto the surface of a neutron star (e.g., Schatz et al. 2001), with the νp-process in neutrino-driven winds of SNe II (e.g., Frölich et al. 2006; Pruet et al. 2006; Wanajo 2006; Wanajo et al. 2011a, 2011b). Woosley et al. (1990) additionally introduced a ν-process to reproduce ¹³⁸La and ^180mTa. Recent multidimensional SN II models show that the ejecta can become proton-rich for several seconds (Fisher et al. 2010), and the importance of the nucleosynthesis (including the νp-process) in the reverse shock has been studied in detail (Wanajo et al. 2011a; Roberts et al. 2010; Arcones & Janka 2011). Fujimoto et al. (2007) performed calculations of the composition of magnetically driven jets ejected from a collapsar, based on magnetohydrodynamic simulations of a rapidly rotating massive star during core collapse. They found that not only does the r-process successfully operate in the jets (with a pattern inside the jets similar to that of the r-elements in the solar system), but also p-nuclei are produced without the need of s-seeds. Light p-nuclei, such as ⁷⁴Se, ⁷⁸Kr, ⁸⁴Sr, and ⁹²Mo, are abundantly synthesized in the jets, together with ¹¹³In, ¹¹⁵Sn, and ¹³⁸La. They claim that the amounts of p-nuclei in the ejecta are much larger than those in core-collapse supernovae. More recently, a different possibility for synthesizing light p-nuclei has been presented by Wanajo et al. (2011a) and Arcones & Montes (2011). These authors explored the possibility of synthesis of p-nuclei in proton-rich winds of SNe II. Other possible scenarios for the production of p-nuclei have been proposed by Iwamoto et al. (2005) considering hypernovae, by Fujimoto et al. (2003) suggesting accretion disks around compact objects, and by Nishimura et al. (2006) investigating jet-like explosions.

The p-process has also been suggested to occur in the outermost layers of Type Ia supernovae (SNe Ia), based on a delayed detonation model (Howard & Meyer 1993), He-detonation models for a sub-Chandrasekhar mass white dwarf (WD; Goriely et al. 2005; Arnould & Goriely 2006), and by Kusakabe et al. (2005, 2011) based on the W7 carbon deflagration model (Nomoto et al. 1984). All these authors considered both solar abundances as seeds for the p-process and s-enhanced seeds. Goriely et al. (2002), presenting one-dimensional He-detonating, sub-Chandrasekhar mass carbon–oxygen white dwarf models (hereafter CO-WD), and Goriely et al. (2005), presenting three-dimensional explosion models, concluded that sub-Chandrasekhar He-detonation models are not an efficient site for the synthesis of p-nuclei.

Kusakabe et al. (2005, 2011) analyzed the effects of different s-seed distributions on p-production. They derived the s-seed distribution by assuming an exponential distribution of neutron exposures with two choices of the mean exposure τ₀ = 0.30 mb⁻¹, which best reproduces the main component in the solar system (Arlandini et al. 1999, the classical model), or τ₀ = 0.15 mb⁻¹, which gives rise to an s-process distribution decreasing with increasing atomic mass number A. The accuracy of the treatment of the outer zones of the SN Ia is fundamental for p-nuclei production, and the sparse zoning in the outermost layers of the W7 model has to be taken into account for thorough p-process nucleosynthesis calculations. As we will describe in this paper, our multidimensional SN Ia models can follow quite accurately even the outermost parts of the star.

We have calculated p-process nucleosynthesis with high-resolution two-dimensional hydrodynamic models of SNe Ia considering both a pure deflagration (similar to Röpke et al. 2006) and delayed detonations of different explosion strengths (similar to those presented in Kasen et al. 2009). We also have calculated p-process nucleosynthesis for SNe Ia of metallicity lower than solar. The adopted SN Ia models are detailed in Section 2. The tracer particle method of calculating the nucleosynthesis in multidimensional simulations (see, e.g., Travaglio et al. 2004; Maeda et al. 2010), together with the nuclear reaction network, is described in Section 3. We consider different s-process seed distributions, as detailed in Section 4. In Section 5, we show an analysis of the production mechanisms of all p-nuclei. The results for the p-process production, depending on the different SN Ia models adopted and on the s-process distribution tested, are discussed in Section 6. Finally, conclusions and work in progress are described in Section 7.

SNe Ia are associated with thermonuclear explosions of WD stars that are composed of carbon and oxygen. A favored scenario is that the explosion is triggered once the WD approaches the Chandrasekhar mass. In the single-degenerate scenario to which we refer here, this happens due to accretion of material from a main-sequence or evolved companion star in a binary system. We follow the explosion phase of Chandrasekhar-mass WD in two-dimensional hydrodynamic simulations. The numerical method employed has been described in detail by Reinecke et al. (1999), Röpke & Niemeyer (2007), Röpke & Schmidt (2009), and Röpke (2005).

For the explosion, a number of scenarios have been suggested (see Hillebrandt & Niemeyer 2000 for a review). Here, we focus on pure deflagrations and delayed detonations. In the former, the burning front propagates subsonically throughout the explosion. It is subject to instabilities that drive turbulence due to which the flame is strongly accelerated beyond its laminar burning speed. However, the pure deflagration model is not considered a candidate for most of the normal SNe Ia as it falls short of reproducing the necessary explosion energies and ⁵⁶Ni masses (Röpke et al. 2007), although it may be an explanation for the peculiar sub-class of 2002cx-like SNe Ia (Phillips et al. 2007). A transition of the flame propagation mode from subsonic deflagration to supersonic detonation in a late stage of the burning constitutes the delayed detonation model (Khokhlov 1991). The necessary deflagration-to-detonation transition (DDT) may arise from strong turbulence in the so-called distributed burning regime (e.g., Röpke 2007; Woosley 2007; Woosley et al. 2009), but rigorous evidence for its occurrence in SNe Ia is still lacking and details of its physical mechanism are uncertain. Nonetheless, delayed detonations successfully reproduce main observables of normal SNe Ia (Mazzali et al. 2007; Kasen et al. 2009) and are therefore considered a standard model for these events.

Our hydrodynamic simulations do not follow the evolution of the WD toward ignition, but start at the onset of the explosion. Here, we ignite a cold (T = 5.0 × 10⁵ K) Chandrasekhar-mass WD in hydrostatic equilibrium with a chemical composition that results from three different evolutionary phases of the progenitor, i.e., central He-burning, He-shell burning during the early asymptotic giant branch (AGB) phase, and He-shell burning during the thermally pulsing AGB phase (Domínguez et al. 2001). According to Piro & Bildsten (2008) and Jackson et al. (2010), we assume that as soon as the accreting WD approaches the Chandrasekhar, mass carbon-burning starts to occur in the core. The energy released by this burning drives convection (the so-called simmering phase) that lasts for about 1000 years before the explosion. The extension of this convective zone is not well determined, and the most external zones beyond 1.2 M_☉ may well remain unmixed (Piro & Bildsten 2008). Note that the zone where the bulk of the p-isotopes is produced is the most external one, with an extension of the order of ∼0.1 M_☉.

For our DDT-a model, we use a CO-WD structure presented by Domínguez et al. (2001) (Table 1), with Z = 0.02 and a progenitor mass of M = 1.5 M_☉. The sensitivity to the different CO-WD structure (obtained evolving models with different main-sequence mass) and the uncertainties in the extension of the simmering phase suggest that a uniform C/O ratio with 50% mixture for the burning can be a good approximation. Concerning the initial Y_e, two different setups are used; the first one, marked with a, assumes Y_e = 0.499, i.e., close to solar metallicity⁶ Z_☉. Setup b assumes Y_e = 0.4995 (corresponding to ∼1/20 Z_☉). All simulations were performed with 512 × 1024 non-uniform moving computational grid cells adopting the nested-grid technique of Röpke & Hillebrandt (2005).

Table 1. Results of the Hydrodynamic SN Ia Explosion Simulations: Asymptotic Kinetic Energy E^asym_kin and Final Composition (Iron Group Elements: IGEs, Intermediate-mass Elements: IMEs)

Model	Easymkin (1051 erg)	$M(\mathrm{IGE{\rm s}})$	$M(\mathrm{IME{\rm s}})$	M(C)	M(O)
DEF-a	0.3522	0.497	0.081	0.357	0.472
DDT-a	1.3027	0.767	0.509	0.018	0.113
DDT-b	1.2903	0.759	0.522	0.018	0.114
DDTw-a	1.1142	0.531	0.644	0.046	0.187
DDTs-a	1.5017	1.142	0.215	0.007	0.044

Note. All masses are given in solar masses M_☉.

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In all models discussed in this paper, the thermonuclear burning front is ignited in multiple sparks (Röpke et al. 2006) near the center of the WD. Specifically, we chose the setup of model DD2D_iso_05 of Kasen et al. (2009; 90 ignition kernels of 6 km radius randomly placed in the radial direction according to a Gaussian distribution with a width of 150 km) for all but one model, where we employ the ignition kernel distribution of model DD2D_iso_01 (20 ignition kernels of the same radius placed according to a Gaussian radial distribution with a standard deviation of 150 km).

The chosen WD setups and the explosion scenarios considered result in the following five different hydrodynamic explosion models.

• 1.  
  DEF-a. A pure deflagration model assuming the WD setup a.
• 2.  
  DDT-a. A delayed detonation model assuming the WD setup a and the deflagration-to-detonation criterion dc2 of Kasen et al. (2009). Detonations are triggered at any point on the deflagration flame where a Karlovitz number of Ka = 250 is reached while the fuel density is in the range 0.6 ≲ ρ_fuel/(10⁷ g cm⁻³) ≲ 1.2.
• 3.  
  DDT-b. A delayed detonation model assuming the WD setup b and the same deflagration-to-detonation criterion dc2 of Kasen et al. (2009).
• 4.  
  DDTw-a. A delayed detonation model assuming the WD setup a and the deflagration-to-detonation criterion dc4 of Kasen et al. (2009), i.e., detonations are this time triggered at any point on the deflagration flame where a Karlovitz number of Ka = 1500 is reached while the fuel density is in the range 0.6 ≲ ρ_fuel/(10⁷ g cm⁻³) ≲ 1.2.
• 5.  
  DDTs-a. A delayed detonation model assuming the WD setup a, the ignition kernel distribution DD2D_iso_01, and the deflagration-to-detonation criterion dc2 of Kasen et al. (2009).

The results of the hydrodynamic explosion simulations are summarized in Table 1. Those of the delayed detonation models are not identical to (but are reasonably close to) the results of the corresponding models presented in Kasen et al. (2009) where different initial WD models were used. The pure-deflagration model produces little ⁵⁶Ni and the explosion energy is particularly low because—as is characteristic for pure deflagration models—burning ceases quickly after nuclear statistical equilibrium (NSE) is no longer reached in the ashes, and only about a tenth of a solar mass of intermediate-mass elements is produced. More than half of the WD mass remains unburned. Since the iron group nuclei are synthesized at rather high densities, neutronization is efficient and a large fraction of the iron group nuclei are Fe and Ni stable isotopes rather than ⁵⁶Ni. As expected, the DD2D_iso_05 ignition kernel setup in combination with the dc2 DDT criterion leads to an explosion strength (E^asym_kin ∼ 1.3 × 10⁵¹ erg) and a ⁵⁶Ni mass (∼0.5 M_☉) that are typical for "normal" SNe Ia, and the effect of the two WD models (a versus b) is of secondary importance for the global explosion characteristics. The two additional models, DDTw-a and DDTs-a, were computed to explore the range of explosion strengths that is observed for normal SNe Ia. With ⁵⁶Ni masses of 0.301 M_☉ and 0.951 M_☉ they capture the extreme limits of the range (or perhaps even exceed them slightly).

The multidimensional SN Ia simulations described above assume instant burning of the initial C+O material once crossed by a deflagration or detonation front (which is represented as a discontinuity applying the level-set method). The microphysical details of the burning are not resolved. Instead, the fuel material is converted to an ash composition according to the fuel density ahead of the front. The ash material is modeled by a mixture of ⁵⁶Ni and α-particles representing NSE that is dynamically adjusted according to the thermodynamic conditions during the simulation and by a hypothetical (but representative) intermediate-mass nucleus (A = 30, E_bind = 6.8266 × 10¹⁸ erg g⁻¹). At the lowest fuel densities above the burning threshold, burning from carbon to oxygen (which also mimics burning to neon) is included. The neutronization of the material is treated independently of this composition by advecting Y_e as a passive scalar and taking into account its evolution due to electron capture reactions in the NSE material. This coarse treatment of nuclear reactions is sufficient to account with sufficient precision for the energy release that drives the explosion dynamics. A more detailed treatment, however, is necessary to analyze the specific nucleosynthetic processes that are the focus of our present work. To this end, a number of Lagrangian tracer particles that record thermodynamic trajectories are passively advected with the hydrodynamic flow in the explosion simulation. On that basis, a post-processing step is performed using an extended network up to ²⁰⁹Bi as described in the following section. The masses of the species in the ejecta quoted above (and shown in Table 1) are therefore only an estimate. More reliable values are obtained in the nucleosynthetic postprocessing step.

The multidimensional hydrodynamic scheme we use follows the explosion by means of an Eulerian grid. In order to follow over time the temperature and density evolution of the fluid, we introduce a Lagrangian component in the form of tracer particles. During the hydrodynamic simulation, they are advected by the flow, recording the T and ρ history along their paths. The nuclear post-processing calculations are then performed separately for each particle. The tracer particle method for nucleosynthesis calculations for core-collapse SNe was first introduced by Nagataki et al. (1997), and for SNe Ia by Travaglio et al. (2004, 2005).

For the present work we use 51,200 tracer particles, uniformly distributed in mass coordinate. Each tracer represents the same mass of ≃2.73 × 10⁻⁵ M_☉ (= M_WD/51,200). The distribution of the tracer particles for our standard DDT-a model is shown in Figure 1 in several snapshots illustrating the evolution. We compare the density distribution obtained in the hydrodynamic simulation with the distribution of the tracers. Different colors are used for different ranges of peak temperature of the tracers (i.e., T_peak, the maximum T reached by a tracer throughout the entire explosion). The tracers marked in black are responsible for the Fe-group production. With their maximum temperature above T₉  ⩾ 7 (where T₉ is the temperature in units of 10⁹ K) they reach NSE and most of the nucleosynthesis goes to ⁵⁶Ni or Fe-group nuclei. The gray tracers are instead the main producers of the lighter α-isotopes. In red, green, and blue, we plot the tracers with T₉  ⩽ 3.7, the main contributors to the p-process nuclei. Close to the surface of the WD, the peak temperature reached during the explosive phase is not high enough for the nuclei to attain the nuclear statistic equilibrium condition, although significant transmutation of heavy elements into p-process nuclei occurs. The three ranges within this group, 3.0<T₉  ⩽ 3.7 (red), 2.4 < T₉  ⩽ 3.0 (green), and 1.5 ⩽ T₉  ⩽ 2.4 (blue), are connected to three different behaviors we identify for the production of p-nuclei (see Section 5). This figure shows that there are tracers with low T_peak (red) in the inner part of the star, resulting from low-density burning in the deflagration regime. However, the bulk of the red tracers is located in the outermost part of the star, together with green and blue tracers, whereas in our models most of the mass has been accreted from a companion and is burned in the delayed detonation models at low densities in the detonation phase.

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Recently, Seitenzahl et al. (2010) demonstrated that in two-dimensional SN Ia simulations with 80² tracers, all isotopes up to Mo with abundances higher than 10⁻⁵ are reproduced with an accuracy of 5% (with the exception of ²⁰Ne). We also performed a resolution study, which is discussed in Section 6.5. The nucleosynthesis of the main species obtained with the tracer particle method (i.e., ⁵⁶Fe, ¹²C, ¹⁶O as well as the mass involved for the p-process nucleosynthesis) is summarized in Table 2. In the last column we report for comparison the abundances for the W7 SN Ia model (Iwamoto et al. 1999).

3.1. p-process Nucleosynthesis

The p-process nucleosynthesis is calculated using a nuclear network with 1024 species from neutron and proton up to ²⁰⁹Bi combined with neutron, proton, and α-induced reactions and their inverse. The code used for this work was originally developed and presented by Thielemann et al. (1996). We use the nuclear reaction rates based on the experimental values and the Hauser–Feshbach statistical model NON-SMOKER (Rauscher & Thielemann 2000). Theoretical and experimental electron capture and β-decay rates are from Langanke & Martínez-Pinedo (2000).

The currently favored production mechanism for those isotopes is photodisintegration of intermediate and heavy nuclides. The γ-process becomes effective during the explosive O-burning phase, at T₉ ⩾ 2.5, and starts with the photodisintegration of stable seed nuclei. During the photodisintegration period, proton, neutron, and α-emission channels (γ, n), (γ, p), and (γ, α) compete with each other and with β-decays of nuclides far from stability.

In Figure 2, we plot the abundance variations for proton, neutron, and ⁴He as a function of time for selected tracer particles, at different peak temperatures, in the range between T₉ = 1.5 and T₉ = 3.7. Comparing with the analogous plot presented in Figure 3 of Kusakabe et al. (2011), we find similar abundances for p, n, and ⁴He, but the timescales in our explosion model are longer. A difference to note, however, is the double peak for some of the selected tracers. This can be attributed to the fact that our model involves both a deflagration phase and a detonation phase, whereas Kusakabe et al. (2011) employ a pure deflagration model.

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In Table 3, we list the 35 p-isotopes with their relative abundance (in %) to the respective elements in the solar system. In Table 4, we list additional isotopes, for which we obtain an important p-contribution—among them the s-only isotopes ⁸⁰Kr and ⁸⁶Sr and the neutron magic ⁹⁰Zr. We also include the neutron-rich isotope ⁹⁶Zr, which is substantially produced by neutron capture via the ²²Ne(α, n)²⁵Mg chain during the carbon-burning phase (see Sections 5 and 6.2 for a more detailed discussion). In the same table, we additionally list the neutron magic isotopes ⁸⁶Kr, ⁸⁷Rb, ⁸⁸Sr, and ⁸⁹Y, which are very abundant in the s-seed and are not affected by photodisintegrations (see Section 6.2). Consequently, these isotopes have to be considered as relics of the s-process seeds.

Table 3. p-Nuclides

Isotope	% Isotopic Abundance	Note	(Xi/Xi, ☉)/(144Sm/144Sm☉) a
74Se	0.87		0.846
78Kr	0.354		0.716
84Sr	0.56		3.463
92Mo	15.84		0.880
94Mo	9.04		0.149
96Ru	5.51		1.244
98Ru	1.87		1.643
102Pd	0.96		1.092
106Cd	1.215		1.129
108Cd	0.875		0.410
113In	4.28	b	0.032
112Sn	0.96		0.751
114Sn	0.66		0.372
115Sn	0.35	b	0.001
120Te	0.089		0.506
124Xe	0.126		0.997
126Xe	0.115		1.289
130Ba	0.101		1.118
132Ba	0.0097		0.838
138La	0.091		0.041
136Ce	0.193		0.425
138Ce	0.25		0.551
144Sm	3.09		1.000
152Gd	0.20	c	0.017
156Dy	0.0524		0.332
158Dy	0.0902		0.178
162Er	0.136		0.433
164Er	1.56	d	0.116
168Yb	0.135		1.353
174Hf	0.18		1.230
180mTa	0.0123	e	0.156
180W	0.135		2.633
184Os	0.018		1.156
190Pt	0.0127		0.385
196Hg	0.146		1.581

Notes. ^aSynthesized mass in the DDT-a model normalized to ¹⁴⁴Sm. For the results of this column, see Section 6.2. ^b113In and ¹¹⁵Sn, see the text for discussion. Indication for the r-process contribution (Dillmann et al. 2008b) is discussed in the text. ^c152Gd, s-only isotope (Arlandini et al. 1999), due to the radiogenic s-branching from ¹⁵¹Sm. ^d164Er, s-only isotope (Arlandini et al. 1999) due to the branching at ¹⁶³Dy that is stable at terrestrial conditions, but becomes unstable at stellar temperatures (Takahashi & Yokoi 1987). ^e180mTa, s-only isotope (Arlandini et al. 1999) due to the branching at ¹⁷⁹Hf. It is stable at terrestrial conditions and becomes unstable at stellar temperatures (Takahashi & Yokoi 1987). The s-process feeds ∼50% of the solar ^180mTa. Another substantial contribution to this isotope comes from the νp-process in SNe II (Woosley et al. 1990).

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Table 4. s-Nuclides with p-Contribution

Isotope	% Isotopic Abundancea	Note	(Xi/Xi, ☉)/(144Sm/144Sm☉)b
80Kr	11.7	c	0.803
86Kr	27.0	d	0.142
86Sr	47.0	c	0.786
87Rb	35.3	d	0.181
88Sr	92.3	d	0.418
89Y	92.0	d	0.412
90Zr	72.2	e	0.905
96Zr	55.0	f	2.142

Notes. ^aArlandini et al. (1999). ^bSynthesized mass in the DDT-a model normalized to ¹⁴⁴Sm. For the results of this column, see Section 6.2. ^c80Kr and ⁸⁶Sr, s-only isotopes. In this work, we find an important contribution from the p-process. ^d86Kr, ⁸⁷Rb, ⁸⁸Sr, and ⁸⁹Y are relics of the s-process seeds. ^e90Zr is a neutron magic nucleus at N = 50. In this work, we find an important contribution from the p-process. ^f96Zr in our SN Ia models gets an important contribution by neutron capture during ²²Ne-burning.

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The p-process nucleosynthesis occurs in SNe Ia only if there is an s-process enrichment; therefore, it is essential to determine the s-process enrichment in the exploding WD. In the single-degenerate progenitor model assumed here, there are two sources of s-enrichment: (1) during the AGB phase leading to the formation of the WD, thermal pulses in which s-isotopes are produced occur (TP-AGB phase; see, e.g., Domínguez et al. 2001; Straniero et al. 2006) and (2) thermal pulses during the accretion phase enrich the matter accumulated on the WD (Iben 1981; Iben & Tutukov 1991; Howard & Meyer 1993; Kusakabe et al. 2011). The s-enrichment of the WD in a layer of ∼0.1 M_☉ deriving from the past AGB history, prior to the accretion phase, is convectively mixed into the WD (see the discussions of the simmering phase in Piro & Bildsten 2008; Piro & Chang 2008; Chamulak et al. 2008). This dilutes the s-seeds so that their abundances are too low to produce significant yields of p-isotopes. In addition, s-process nucleosynthesis can occur in the H-rich matter accreted by the CO-WD, due to recurrent He-flashes (Iben 1981), with neutrons mainly produced by the ¹³C(α, n)¹⁶O reaction.

We investigate the influence of different s-process abundance distributions for Z = 0.02 and Z = 0.001 (see Figures 3 and 4) as seeds of p-process nucleosynthesis. These distributions are obtained from s-process nucleosynthesis calculations with a post-process method (Gallino et al. 1998; Bisterzo et al. 2010). In the AGB scenario, there is a general consensus (Gallino et al. 1998 and references therein) that the main neutron source is the reaction ¹³C(α, n)¹⁶O. In order to activate it, partial mixing of protons from the envelope down into the C-rich layers is required (physical causes of this mixing are discussed by many authors, e.g., Hollowell & Iben 1988; Herwig et al. 1997; Goriely & Mowlawi 2000; Langer et al. 1999; Herwig et al. 2003; Denissenkov & Tout 2003; Cristallo et al. 2009). However, the mass involved and the profile of the ¹³C-pocket still have to be considered as free parameters, given the difficulty of a realistic treatment of the hydrodynamic behavior at the H/He discontinuity. A series of constraints has been obtained by comparing spectroscopic abundances in s-enriched stars at different metallicities with AGB model predictions (see, e.g., Busso et al. 2001; Bisterzo et al. 2010). The spread in the s-process yields at each metallicity has been modeled parametrically by varying the ¹³C concentration in the pocket from 0 up to a factor of two times the standard value of ∼4 × 10⁻⁶ M_☉ of ¹³C (Gallino et al. 1998, ST case), and is indicated in Figures 3 and 4 as ST, ST×2 (standard ¹³C-pocket multiplied by a factor of two), ST/2 (standard ¹³C-pocket divided by a factor of two), etc. For this work we varied the ¹³C-pocket concentration in order to obtain a flat s-seed distribution, or a non-flat s-seed distribution peaked to the lighter or to the heavier s-nuclei (see Figures 3 and 4), for both Z = 0.02 and Z = 0.001.

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In Figure 3, we show for two metallicities a typical flat s-process distribution (i.e., the isotopes produced mainly by s-process nucleosynthesis all have the same overabundance with respect to the solar abundances). The upper panel is a flat s-process distribution similar to the one presented by Arlandini et al. (1999, "stellar model") as a best fit to the solar main component. In Arlandini et al. (1999), a full stellar evolutionary model is followed along the TP-AGB phase, using an updated network of neutron captures and beta decay rates (Takahashi & Yokoi 1987; Bao et al. 2000) updated with more recent experimental determinations in the KADONIS database (Dillmann et al. 2006).

The larger number of thermal pulses that in principle can be realized during mass accretion more easily allows the s-nuclei to achieve an asymptotic distribution. The difference with respect to the Kusakabe et al. (2011) case B, for a flat distribution in the heavy s-only isotope overabundances (about 7000 as compared to our 2000), may be ascribed to different reasons. First, their choice of the neutron capture Maxwellian averaged cross section (MACS) is based on the use of the current 30 keV data. At 30 keV, the MACS of ⁵⁶Fe, the major seed for the s-process, is 11.7 ± 0.4 mb (Bao et al. 2000). However its temperature dependence strongly departs from the usual 1/v rule. Actually, the MACS is almost flat between 100 keV and 7 keV (Beer et al. 1992, see their Figure 3). In AGB stellar modes, the major neutron source is the ¹³C(α,n)¹⁶O reaction. Neutrons are released at about 8 keV. Using a value of 30 keV instead is equivalent to doubling the effective MACS of ⁵⁶Fe. Second, Kusakabe et al. derived the s-process distribution using a simplified exponential distribution of neutron exposures. This is in agreement with the classical analysis of the s-process, using for the mean neutron exposure parameter the value τ₀ = 0.30 mb⁻¹. The result is a flat distribution for the s-only isotopes beyond A  ∼ 90 (Ulrich 1973).

The classical analysis operates at a fixed temperature and neutron density assumed parametrically and using an unbranched s-process flow, not accounting for the much more complex astrophysical situation occurring during recurrent thermal pulses in AGB stars. There, two neutron sources operate at different thermal conditions. The major neutron exposure is activated at kT ∼ 8 keV by the ¹³C(α,n)¹⁶O reaction in radiative conditions between two subsequent convective thermal instabilities in the top layers of the He intershell (the so-called ¹³C-pocket). A second neutron exposure results from the marginal activation of the ²²Ne(α,n)²⁵Mg reaction in the convective instability at kT ∼ 23 keV. Moreover, the classical analysis in principle works only for an asymptotic distribution of neutron exposures over a series of identical thermal instabilities and with a constant overlap factor between adjacent pulses. Besides the different thermal and physical conditions, the neutron density is also far from being constant. For an exhaustive comparison see Gallino et al. (1998). Additionally, Kusakabe et al. (2011, their case A) did not consider the neutron poison effect of all light isotopes below ³²S, which in particular ignores ²⁵Mg, the most important competitor of ⁵⁶Fe in neutron capture.

In Figure 4, we present different s-process distributions, used as s-seeds for this work. In the upper panel, we show an s-distribution peaked at the light nuclei (with A between ∼75 and ∼90). This resembles the pattern obtained from a classical mean neutron exposure with τ₀ = 0.15 mb⁻¹. Similar conditions have been analyzed by Kusakabe et al. (2011). In the middle panel an analogous distribution is shown for Z = 0.001. The lower panel of Figure 4 shows a distribution peaked at heavy isotopes (A  ⩾ 150), with an important contribution at ²⁰⁸Pb. In Section 6, we discuss in detail the consequences for p-process nucleosynthesis using these different distributions of s-seeds.

The production mechanism of the various p-nuclei can be understood through an analysis of the corresponding nuclear flows. In Figures 5–8, we plot the behavior of the different nuclei versus peak temperature for a solar metallicity case (i.e., DDT-a) using the flat s-process distribution shown in Figure 3 (upper panel). In Figure 5, we show ¹²C, ¹⁶O, ²⁰Ne, and ²²Ne abundances as a function of T_peak for the tracers in the temperature range to produce p-nuclei. Starting from the cold outer layers of the star, at T₉ ≃ 1.4 ²²Ne burns through (α, n) reaction, becoming the most important source of neutrons. This happens at about 0.6 s after ignition of the SN Ia. At T₉  ≃ 2, ¹²C burns mainly via (¹²C,¹²C) channels making ²³Na, ²⁰Ne, α and p-nuclei. At T₉ ≃ 2.6, photodisintegration of ²⁰Ne via ²⁰Ne(γ,α)¹⁶O becomes efficient, thus increasing the amount of available ¹⁶O. At T₉ ≃ 3.2, photodisintegration of ¹⁶O becomes efficient. The range of T₉ chosen for Figure 5 is related to the T₉ range where the p-process nucleosynthesis occurs.

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In Figures 6 and 7, we show the final abundances of all 35 p-isotopes listed in Table 3 as a function of peak T. In Figure 8, we plot the abundance behavior of the isotopes listed in Table 4, i.e., the s-only ⁸⁰Kr and ⁸⁶Sr (upper panel), ⁸⁶Kr, ⁸⁷Rb, ⁸⁸Sr, ⁸⁹Y (middle panel) to be considered as relics of s-process seeds with a small contribution from ²²Ne burning, and the two special ⁱZr isotopes ⁹⁰Zr and ⁹⁶Zr (lower panel).

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When T₉ ≃ 2.2, protons are released by burning of ¹²C, we find that the first p-isotope produced at the lowest T₉ is ^180mTa. A behavior similar to ^180mTa is observed for ¹⁸⁴Os, but at somewhat higher temperatures (2.4 ≲ T₉ ≲ 2.5).

A bit further inside the star, where peak temperatures reach T₉  ≃ 2.3–2.4, the p-isotopes ¹⁵⁸Dy, ¹⁶⁴Er, ¹⁸⁰W, ¹⁷⁴Hf, ¹⁶⁸Yb, ¹⁹⁰Pt, and ¹⁹⁶Hg are produced. This group of isotopes shows a second peak of production at higher T, T₉  ≃ 2.7. Also ¹⁵²Gd belongs to this group of p-isotopes, but here the second abundance peak at T₉  ≃ 2.7 is much lower with respect to the first abundance peak at T₉  ≃ 2.4. Note that ¹⁵²Gd is mainly produced by the s-process, as recalled in Section 6.2.

When T₉ exceeds ∼ 3, (γ, p), (γ, α), and (γ, n) become dominant and the lightest p-isotopes are produced. ⁷⁴Se, ⁷⁸Kr, and ⁸⁴Sr are synthesized in a wide range of peak temperatures (3.1 ≲ T₉ ≲ 3.6). Other light p-isotopes such as ^{92, 94}Mo, ^{96, 98}Ru, ¹⁰²Pd, ¹⁰⁶Cd, and ^{112, 114}Sn are also produced mostly at T₉ ⩾ 3.

In Figure 8, we show the abundance behavior as a function of peak temperature for all isotopes reported in Table 4, i.e., those where we find (as will be discussed in detail below) an important contribution from p-process nucleosynthesis. From this figure it is evident that ⁸⁰Kr and ⁸⁶Sr (upper panel), and ⁹⁰Zr (lower panel) are first destroyed by ²²Ne burning and subsequently produced as p-nuclei in the ¹²C-burning phase, at T₉ higher than 2. In contrast, the isotopes in the middle panel, i.e., ⁸⁶Kr, ⁸⁷Rb, ⁸⁸Sr, and ⁸⁹Y, retain almost their initial values (with small increases in ⁸⁶Kr and ⁸⁷Rb) during the ²²Ne- and ¹²C-burning phases, while they are destroyed by photodisintegration at T₉ ⩾ 3. Finally, ⁹⁶Zr (lower panel) is produced at T₉  ≃ 1.6 due to ²²Ne burning; in these conditions the neutron density is high and the neutron capture channel on ⁹⁵Zr is open. Note that the production yield of ⁹⁶Zr directly depends on the uncertain theoretical Maxwellian neutron capture cross section (MACS) of the unstable ⁹⁵Zr. In our network we have adopted a factor of two lower than the one reported in the compilation of Bao et al. (2000), taken as the average between the Bao et al. (2000) prescription and the much lower recent theoretical evaluation by the TALYS network of the Brussels database (http//www.astro.ulb.ac.de). Several sources may contribute to the cosmic origin of the most neutron-rich Zr isotope ⁹⁶Zr (2.8% of solar Zr), i.e., the weak s-process in massive stars, the main s-process in AGB stars of low and intermediate mass where the ²²Ne neutron source is partially activated, and the weak r-process during SN II explosion. A further source that can no longer be discarded is now associated with the p-process in SN Ia explosions. Clearly, an experimental estimate of the MACS of the unstable ⁹⁵Zr is of paramount importance.

In this section, we present p-process nucleosynthesis results obtained with different SN Ia models, exploring different s-process distributions and investigating the metallicity effect on the p-nuclei production.

6.1. p-process in WD with AGB Progenitor

We present the results of the nucleosynthesis calculation for DDT-a, i.e., a delayed-detonation model of a solar metallicity Chandrasekhar-mass WD. We first consider the s-process enrichment of the accreting WD as a result of its past AGB history. In this way the mass of ∼0.1 M_☉ enriched in s-process elements produced during the AGB phase is spread out over the WD core (following Piro & Bildsten 2008). The resulting nucleosynthesis is presented in Figure 9. Nucleosynthesis calculations with the full network give a ⁵⁶Fe yield of 0.584 M_☉. This is consistent with what is typically expected for a standard SN Ia (Contardo et al. 2000; Stritzinger et al. 2006). The detonation burns the WD almost completely, and only little ¹²C and ¹⁶O remain in the ejecta. The average p-process enrichment we obtain is by a factor of about 50 to 100 below the ⁵⁶Fe production. From the rough estimate that SNe Ia contribute 2/3 of the total Galactic ⁵⁶Fe, we conclude that this model cannot account for a significant fraction of the solar p-nuclei.

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6.2. p-process in Delayed Detonation Models with s-process in the Accreted Material

The mass accreted in a close binary system from a hydrogen-rich envelope of a companion onto the CO-WD can be enriched in s-process material. As discussed by many authors in the literature, e.g., Sugimoto & Fujimoto (1978), Iben (1981), Nomoto (1982a, 1982b), Iben & Tutukov (1991), and more recently Kusakabe et al. (2011), the accretion rate in the Chandrasekhar-mass progenitor scenario should be sufficiently high to avoid a detonation of He. Recurrent thermal-pulses during accretion, however, are likely to occur (Iben 1981). This s-process rich material will act as seed for p-process nucleosynthesis during the explosion. In Figure 10, we present our results obtained from the DDT-a model (upper panel) and the DDT-b model (middle and lower panels). Both models, presented in Section 2, involve delayed detonations. The s-seeds used are the ones presented in the three panels of Figure 3. The nucleosynthesis yields for A ⩾ 40 from model DDT-a (using the s-distribution plotted in Figure 3 and marked there as ST×2) are given in Table 5. Since the SN Ia model used is the same as the one presented in the previous section, the resulting nucleosynthesis for nuclei below the Fe group does not differ significantly. The production ratio of the synthesized isotopes normalized to the production ratio of ¹⁴⁴Sm (the isotope that has the most similar ratio over solar with respect to ⁵⁶Fe over solar) is listed in the last column of Table 3 for the "traditional" 35 p-nuclei, and in the last column of Table 4 for the other heavy nuclides with an important contribution from the p-process in SNe Ia. The problem of the production factors of all isotopes listed in Table 4 will be treated separately, considering the various contributions at the solar system formation through a Galactic Chemical Evolution treatment as in Travaglio et al. (2004).

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Table 5. Synthesized Mass (M_☉) in the SN Ia DDT-a Model (Figure 10, Upper Panel)

Species	Abundance	Species	Abundance	Species	Abundance	Species	Abundance	Species	Abundance	Species	Abundance
12C	1.5671D-02	50Cr	5.4047D-04	84Sr	4.5067D-07	115Sn	1.0004D-09	145Nd	1.8189D-10	176Hf	8.6456D-10
13C	9.2987D-08	52Cr	1.1575D-02	86Sr	1.8400D-06	116Sn	6.0745D-09	146Nd	7.1862D-10	177Hf	7.1442D-11
14N	2.1701D-05	53Cr	1.5783D-03	87Sr	1.4327D-07	117Sn	2.9810D-10	148Nd	4.0872D-10	178Hf	4.3106D-10
15N	2.1673D-08	54Cr	6.6240D-06	88Sr	8.3871D-06	118Sn	6.9873D-09	150Nd	4.9146D-09	179Hf	1.3962D-10
16O	1.4423D-01	55Mn	1.4643D-02	89Y	1.9993D-06	119Sn	6.6589D-10	144Sm	1.3524D-08	180Hf	1.5056D-09
17O	3.0220D-06	54Fe	1.1991D-01	90Zr	5.6021D-06	120Sn	7.5572D-09	147Sm	1.4712D-10	180mTa	8.1889D-13
18O	2.5498D-08	56Fe	5.8375D-01	91Zr	7.3390D-08	122Sn	6.7889D-09	148Sm	6.1696D-11	181Ta	2.8885D-10
19F	7.1068D-10	57Fe	1.5874D-02	92Zr	2.8426D-07	124Sn	2.6818D-08	149Sm	1.5191D-10	180W	9.6280D-10
20Ne	5.6727D-03	58Fe	4.0790D-05	94Zr	2.3997D-07	121Sb	1.2773D-09	150Sm	4.9988D-11	182W	8.4175D-10
21Ne	1.8643D-06	59Co	6.6528D-04	96Zr	7.7244D-07	123Sb	1.9351D-09	152Sm	1.3103D-10	183W	1.3810D-10
22Ne	3.7666D-04	58Ni	6.4124D-02	93Nb	4.9745D-08	120Te	3.0647D-09	154Sm	1.2263D-09	184W	1.8243D-10
23Na	1.1737D-04	60Ni	6.7264D-03	92Mo	2.5027D-07	122Te	7.4450D-09	151Eu	1.4435D-11	186W	1.3116D-09
24Mg	2.0306D-02	61Ni	7.7360D-05	94Mo	1.6391D-07	123Te	4.2524D-10	153Eu	1.3177D-10	185Re	1.1175D-10
25Mg	1.9054D-04	62Ni	5.2097D-04	95Mo	4.9595D-08	124Te	1.4025D-09	152Gd	1.9977D-11	187Re	2.2337D-10
26Mg	1.9362D-04	64Ni	4.6122D-06	96Mo	3.8125D-08	125Te	2.7803D-10	154Gd	5.7821D-11	184Os	3.0456D-10
27Al	8.2725D-04	63Cu	1.5072D-05	97Mo	9.1247D-09	126Te	1.9648D-09	155Gd	1.3780D-11	186Os	2.5331D-10
28Si	3.3130D-01	65Cu	6.5753D-06	98Mo	3.2686D-08	128Te	2.1474D-09	156Gd	9.6601D-11	187Os	4.1744D-11
29Si	1.1974D-03	64Zn	6.4154D-06	100Mo	1.4678D-08	130Te	1.1202D-08	157Gd	2.8455D-11	188Os	1.3941D-10
30Si	2.5425D-03	66Zn	3.1710D-05	96Ru	1.4367D-07	127I	1.6683D-09	158Gd	1.3021D-10	189Os	3.3415D-11
31P	5.8732D-04	67Zn	1.2094D-06	98Ru	6.5865D-08	124Xe	8.2858D-09	160Gd	7.4789D-10	190Os	1.7001D-10
32S	1.3914D-01	68Zn	2.9505D-06	99Ru	1.3524D-08	126Xe	9.7036D-09	159Tb	1.8889D-10	192Os	1.4298D-09
33S	3.3104D-04	70Zn	2.5358D-07	100Ru	1.6813D-08	128Xe	1.4251D-08	156Dy	1.3456D-10	191Ir	1.0247D-10
34S	3.5648D-03	69Ga	1.1747D-06	101Ru	1.0647D-09	129Xe	6.7265D-10	158Dy	1.2516D-10	193Ir	3.6770D-10
36S	5.6228D-07	71Ga	3.6326D-07	102Ru	1.4012D-09	130Xe	1.1922D-09	160Dy	6.0440D-10	190Pt	1.4588D-10
35Cl	1.5530D-04	70Ge	2.3483D-06	104Ru	7.8273D-09	131Xe	5.0030D-10	161Dy	9.2518D-11	192Pt	3.6798D-10
37Cl	3.2379D-05	72Ge	2.4812D-06	103Rh	1.3001D-09	132Xe	2.3573D-09	162Dy	1.1898D-10	194Pt	6.3110D-10
36Ar	2.3840D-02	73Ge	3.6625D-08	102Pd	1.8434D-08	134Xe	2.7812D-09	163Dy	9.6694D-11	195Pt	3.1544D-10
38Ar	1.4908D-03	74Ge	3.5906D-07	104Pd	4.8247D-09	136Xe	1.0643D-08	164Dy	9.6427D-10	196Pt	2.3987D-10
40Ar	2.2358D-07	76Ge	1.0567D-06	105Pd	6.7008D-10	133Cs	1.9744D-09	165Ho	1.9165D-10	198Pt	1.4554D-09
39K	9.8071D-05	75As	1.9919D-07	106Pd	1.4563D-09	130Ba	8.1235D-09	162Er	2.8935D-10	197Au	6.8591D-10
40K	8.9145D-08	74Se	4.0425D-07	108Pd	1.4541D-09	132Ba	5.8825D-09	164Er	9.0568D-10	196Hg	1.8911D-09
41K	5.9740D-06	76Se	8.9696D-07	110Pd	1.0164D-08	134Ba	4.9567D-09	166Er	5.6222D-10	198Hg	2.4552D-09
40Ca	2.0560D-02	77Se	7.6909D-09	107Ag	5.1444D-10	135Ba	1.1355D-09	167Er	8.5695D-11	199Hg	3.5610D-10
42Ca	3.3420D-05	78Se	3.4351D-07	109Ag	1.4341D-09	136Ba	5.1733D-09	168Er	1.2140D-10	200Hg	2.4156D-09
43Ca	2.1826D-07	80Se	2.1413D-07	106Cd	2.7891D-08	137Ba	1.7809D-09	170Er	7.1327D-10	201Hg	5.2630D-10
44Ca	1.5950D-05	82Se	8.2393D-07	108Cd	7.2676D-09	138Ba	7.1886D-08	169Tm	2.5781D-10	202Hg	2.2432D-09
46Ca	8.8764D-08	79Br	4.8509D-08	110Cd	2.3744D-08	138La	2.7045D-11	168Yb	8.5902D-10	204Hg	3.5097D-09
48Ca	1.5001D-08	81Br	2.1398D-07	111Cd	8.9753D-10	139La	6.6975D-09	170Yb	1.1057D-09	203Tl	8.4790D-10
45Sc	4.6983D-07	78Kr	1.0037D-07	112Cd	2.1349D-09	136Ce	1.4674D-09	171Yb	1.0724D-10	205Tl	2.9824D-09
46Ti	1.9035D-05	80Kr	7.5378D-07	113Cd	1.4926D-10	138Ce	2.5269D-09	172Yb	5.9715D-10	204Pb	1.9311D-09
47Ti	1.2317D-06	82Kr	3.7126D-07	114Cd	2.3586D-09	140Ce	1.6945D-08	173Yb	1.2626D-10	206Pb	4.8271D-09
48Ti	4.4708D-04	83Kr	2.1573D-08	116Cd	1.3732D-08	142Ce	7.8332D-09	174Yb	3.5181D-10	207Pb	6.0427D-09
49Ti	3.6257D-05	84Kr	2.9946D-07	113In	1.6635D-09	141Pr	2.7194D-09	176Yb	1.3185D-09	208Pb	3.3585D-08
50Ti	2.3077D-07	86Kr	1.1256D-06	115In	1.2077D-09	142Nd	1.2709D-08	175Lu	1.8984D-10	209Bi	1.8919D-10
50V	7.8373D-08	85Rb	2.4553D-07	112Sn	3.6123D-08	143Nd	7.0663D-10	176Lu	1.1738D-10
51V	1.2992D-04	87Rb	3.8945D-07	114Sn	1.1931D-08	144Nd	1.0278D-09	174Hf	6.2584D-10

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In Figure 10, we plot the production factor of each isotope i normalized to the ratio of ⁵⁶Fe produced by the model over (⁵⁶Fe)_☉. We note that for many of the p-isotopes the overproduction is at the level of ⁵⁶Fe. Starting from the lightest p-isotopes, ⁷⁴Se, ⁷⁸Kr, ^{92, 94}Mo, ^{96, 98}Ru, ¹⁰²Pd, and ^{106, 108}Cd are produced at the level of ⁵⁶Fe (within a factor of two). In the case of ⁸⁴Sr, mainly ⁸⁵Sr(γ,n)⁸⁴Sr is active to produce ⁸⁴Sr at T₉ ⩾ 2.6. We test the sensitivity of the production of ⁸⁴Sr to the uncertainty of the ⁸⁵Sr(γ,n)⁸⁴Sr rate. Rauscher & Thielemann (2000) estimate 30% uncertainty of the MACS of this rate at kT = 100 keV (the typical temperature for explosive conditions). We find that a small change in the cross section of this reaction changes the final ⁸⁴Sr abundance by a large factor. This has to be carefully taken into account in p-process nucleosynthesis calculations. We also get a high production of ⁸⁶Sr from p-process nucleosynthesis (almost at the level of ⁵⁶Fe). ⁸⁶Sr is an s-only isotope, contributed by both massive stars (weak component) and by low-mass AGB stars (main component). However, we find a substantial p-process contribution to ⁸⁶Sr.

Despite the historical problem of reproducing the solar abundances of ^{92, 94}Mo, ^{96, 98}Ru (these isotopes were found neither in comparable abundance with the other p-isotopes in SN II nor in SN Ia models or in any other stellar site), we find a good agreement of the abundances of these isotopes with all the other p-only isotopes, suggesting SNe Ia as important stellar sites for the synthesis of heavy and light p-nuclei. For ⁹²Mo the most important production channel is ⁹³Mo(γ,n)⁹²Mo. The second most important chain is ⁹⁶Ru(γ,α)⁹²Mo. A small contribution to ⁹²Mo comes from the (p, γ) channel.

⁹⁴Mo is mainly synthesized via the (γ, n) photodisintegration chain starting from ⁹⁸Mo. For T₉  < 3 this is almost the only channel to produce ⁹⁴Mo, while for T₉  ⩾ 3 a contribution of ∼30% also comes from ⁹⁵Tc(γ,p)⁹⁴Mo.

Concerning the two Ru p-only isotopes, we find that almost 90% of ⁹⁶Ru is produced in the chain ⁹⁷Ru(γ,n)⁹⁶Ru, with a small contribution from ¹⁰⁰Pd(γ,α)⁹⁶Ru. In contrast, about 50% of the ⁹⁸Ru is made by ⁹⁹Ru(γ,n)⁹⁸Ru, and ∼50% by ⁹⁹Rh(γ,p)⁹⁸Ru.

The p-contribution to ⁹⁰Zr mainly derives from ⁹¹Nb(γ,p)⁹⁰Zr and ⁹¹Zr(γ,n)⁹⁰Zr.

The other important Zr-isotope in this study, ⁹⁶Zr, the most sensitive isotope to the neutron density, comes mainly from the neutron capture channel ⁹⁵Zr(n,γ)⁹⁶Zr, where the necessary neutrons are supplied by the ²²Ne(α,n)²⁵Mg reaction. ²²Ne, as shown in Figure 5, burns at very low T₉ (≃1.7) in the outermost layers of the star. Therefore, as we will discuss in more detail below, the abundance obtained can be very sensitive to the modeling of the explosion.

Kusakabe et al. (2011) followed the p-process nucleosynthesis adopting the W7 C-deflagration model by Nomoto et al. (1984). When we compare these trends with selected trajectories at T_9peak  ≃ 3 in Figures 1 and 3 of Kusakabe et al. (2011), our results are quite similar in all respects (see below for more details). We also agree with most nucleosynthetic details they discuss. The differences in the total and relative p-process yields may be ascribed to the different numerical accuracy and fine resolution treatment of the explosive nucleosynthesis in the outermost layers, where carbon-burning occurs and where most of the p-nuclei are synthesized. We fully agree with Kusakabe et al. (2011) when they write that "the yields are very sensitive to the temperature and density trajectories." The explosive models we follow, especially for the low-temperature trajectories, and the technique we employ for the distribution and the total number of tracers in the outermost regions are probably a major cause of the relative flat distribution among the light, intermediate, and heavy p-process yields, including the reproduction of the most abundant p-nuclei in the Mo and Ru region. Comparison with previous p-calculation presented by Howard et al. (1991) and with more technical details in Howard & Meyer (1993) is somewhat hampered by their parameter study and subdivision in only 15 typical trajectories of the outer region, where they based the post-process calculations on an SN Ia delayed detonation model by Khokhlov (1991).

In order to explain our result, we select two tracers representative of the highest production of ^{92, 94}Mo. They have been selected with peak temperature T_9peak = 3.075 and 3.180, respectively, corresponding (as one can see in Figure 6) to the maximum production of ⁹²Mo and of ⁹⁴Mo. The two tracers are located in two different zones of the star, one in the outermost zone, with initial ρ_peak  ≃ 4.0 × 10⁷ g cm⁻³ (Figure 11, upper panel, dotted line), and the second one a bit further inside the star, with higher initial ρ_peak  ≃ 5.5 × 10⁸ g cm⁻³ (Figure 11, upper panel, solid line). The inner tracer is first reached by the deflagration wave, and has an initially rather high density. When it is finally reached by the detonation, it has much lower density due to the pre-expansion. The time on the plot starts at 0.8 s (the time at 0 s corresponds to the start of the deflagration wave). In the other panels of this plot are shown the mass fractions of p, n, ⁴He, ¹²C, ¹⁶O, ²⁰Ne, ²³Na, and ²²Ne, and in the lower panels the mass fractions of ⁹²Mo and ⁹⁴Mo. Looking at ⁹²Mo and ⁹⁴Mo (lower panels), one can see that, despite the two different histories of the tracers, these isotopes are produced with very similar abundances in these two tracers. In fact, the initial density and temperature are quite different at t ≃ 0.0 s, but, due to the expansion of the star, they are quite similar at the time of the production of ⁹²Mo and of ⁹⁴Mo. The initial ¹²C set with the same value for all tracers falls abruptly when the detonation wave passes through. Consequently, ²³Na is produced, and ²⁰Ne at the beginning of C-burning is mostly dissociated, producing α and ¹²C. As to ²²Ne, at the time of ⁹²Mo and ⁹⁴Mo formation, it is strongly reduced with respect to the initial value, with a consequently low production of neutrons in the region where the light-p are produced. After a detailed analysis of the nucleosynthesis that deals with the production of the p-nuclei, we can state that an accurate treatment of the outermost region of the SN Ia model is at the base of these results.

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Going further to higher mass number, we note an underproduction in the Cd–In–Sn region. The origin of the rare odd isotopes ¹¹³In and ¹¹⁵Sn (and, related to these, ¹¹²Sn and ¹¹⁴Sn) is debated. These rare nuclei are shielded from the two dominant nucleosynthesis processes for heavy elements: the s-process flow proceeds via ¹¹²Cd $\longrightarrow {}^{113}$Cd $\longrightarrow {}^{114}$Cd $\longrightarrow {}^{115}$In $\longrightarrow {}^{116}$Sn, thus bypassing the rare nuclei, except for small branchings of the reaction path at ¹¹³Cd, ¹¹⁵Cd, and ¹¹⁵In, which may contribute to the abundances of ¹¹³In, ¹¹⁴Sn, and ¹¹⁵Sn. The β-decay chains from the r-process region are shielded by the neutron-rich Cd and In, but may also feed the rare nuclei via minor decay branchings at ¹¹³Cd and ¹¹⁵Cd (Rayet et al. 1990; Howard et al. 1991; Theis et al. 1998). In a previous investigation of the s- and r-process components of the Cd-, In-, and Sn-isotopes, Nemeth et al. (1994) suggested that the rare isotopes in the A = 112–115 region may be produced by a complex interplay of s- and r-processes. According to Dillmann et al. (2008b), the s-process contribution to ¹¹³In and ¹¹⁵Sn is excluded as well as the thermally enhanced β-decay by an accelerated decay of the quasi-stable ¹¹³Cd and ¹¹⁵In during the s-process (the mechanism proposed by Nemeth et al. 1994). The most promising scenario suggested by Dillmann et al. (2008b) is related to β-delayed r-process decay chains. Nevertheless, uncertainties in nuclear physics have to be taken into account.

Moving to the intermediate p-isotopes, we note ¹³⁸La and ^180mTa are far below the average p-nuclei production (by a factor of ∼50 and ∼8, respectively). The astrophysical origin of the two heaviest odd-odd nuclei ¹³⁸La and ^180mTa has been discussed over the last 30 years (Woosley & Howard 1978; Beer & Ward 1981; Yokoi & Takahashi 1983; Woosley et al. 1990; Rauscher et al. 2002), more recently by Cheoun et al. (2010). We derive a rather small contribution from SNe Ia. ^180mTa receives an important contribution from the s-process due to the branching at ¹⁷⁹Hf, a stable isotope that becomes unstable at stellar temperatures (Takahashi & Yokoi 1987). There is also a second branching at ^180mTa due to the fact that ¹⁷⁹Hf has a small n-capture branching to ^180mHf, which at T₉  ≃ 0.3 is not thermalized and quickly decays to ^180mTa (Beer & Ward 1981; see also Mohr et al. 2007; Käppeler et al. 2004). In a core-collapse supernova, ^180mTa is synthesized by the neutrino process (Woosley et al. 1990). Independent of the production mechanism, according to Mohr et al. (2007), freeze-out from thermal equilibrium occurs at kT  ≃ 40 keV, and only ∼35% of the synthesized ^180mTa survives in the isomeric state. Consequently, in all supernova results, the yield obtained so far without accounting of the freeze-out effect of excited levels discussed by Mohr et al. (2007) and Hayakawa et al. (2010b) should be decreased by about 2/3. Hayakawa et al. (2010b) positively cited the Mohr et al. (2007) evaluation, writing that the isomeric residual population Ratio R by Mohr et al., i.e., R = P_m/(P_g + P_m) = 0.35 ± 0.04. It was based "from an estimate of the freeze-out temperature without following the time-dependent evolution in detail." The new result by Hayakawa et al. (2010b), i.e., R = 0.38, essentially coincides with R = 0.39 ± 0.01 given by Hayakawa et al. (2010a). The new problem resides in the conclusion given by Hayakawa et al. (2010b), "the main conclusion of the previous study by Hayakawa et al. (2010a) is thus strengthened: the solar abundances of ¹³⁸La and ^180mTa relative to ¹⁶O can be systematically reproduced by neutrino nucleosynthesis and an electron neutrino temperature of kT close to 4 MeV." Taken at face value, this sentence contrasts first with the fact that the s-process already produces about 50% of solar ^180mTa (Mohr et al. 2007), whereas no s-process ¹³⁸La is predicted, and second with our present p-process SN Ia result that further provides extra p-contribution to solar ^180mTa (and to ¹³⁸La). Note that in Figure 10 there are two p-process nuclides at A = 180; the higher one is ¹⁸⁰W (which in turn is only produced by the s-process at the level of 5% (see the review by Käppeler et al. 2011). The conclusion by Hayakawa et al. (2010b) should, however, account for the uncertainties in the stellar neutrino cross sections involved in the production of ^180mTa and ¹³⁸La.

Concerning ¹⁵²Gd and ¹⁶⁴Er, we have already outlined that both isotopes are mainly reproduced by the s-process (Arlandini et al. 1999), through the branching at ¹⁵¹Sm (see updates in Marrone et al. 2006) and through the branching at ¹⁶³Dy, another stable isotope that becomes unstable at stellar temperatures (Takahashi & Yokoi 1987). No production of ¹⁵²Gd is expected by the weak s-process in massive stars since it is strongly destroyed during carbon-shell burning (see The et al. 2007 and references therein). We find that ¹⁵²Gd is produced by a factor of 100 less than the average p-distribution, which confirms the non-p astrophysical origin of this isotope. Also ¹⁶⁴Er is underproduced by one order of magnitude with respect to the heavy p-nuclei. Both ¹⁵²Gd and ¹⁶⁴Er are to be classified as s-only isotopes, not p-only.

6.3. Different s-process Distributions

Using the s-process seeds plotted in Figure 4 (upper and middle panels), corresponding to a non-flat s-distribution, peaked to the lighter s-isotopes with A  ∼ 80–90), the nucleosynthesis results are shown in the two panels of Figure 12. Concerning the p-nuclei, we note that with an s-seed distribution that is flat (see Figure 3, upper and lower panels) or peaked to the lighter s-isotopes (see Figure 4, upper and middle panels), almost all p-nuclei scale linearly with metallicity (within a factor of ∼2), with the exception of the three lightest p nuclei ⁷⁴Se, ⁷⁸Kr and ⁸⁴Sr, and ^180mTa, which show a different behavior. While a decrease of ⁷⁴Se, ⁷⁸Kr, and ⁸⁴Sr by a factor of ∼20 would be expected when changing the metallicity from solar to Z = 0.001, we observe a variation by a factor of only ∼2. For ^180mTa, by decreasing metallicity by a factor of 20 (from solar down to Z = 0.001) we obtain an increase of the p-process contribution of a factor of ∼10, still far below the average p-nuclei abundance obtained for these models.

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Particular attention has to be devoted to the s-distribution plotted in Figure 4 (lower panel) peaked to the heavier s-isotopes. In this case, the p-nuclei on average are produced at the level of ⁵⁶Fe, when normalized to solar abundances (see Figure 10, lower panel).

6.4. ²⁰⁸Pb as Seed of p-nuclei

6.5. Resolution Study

6.6. Pure Deflagration and Different Strengths of Delayed Detonations: Consequences for the p-process

In addition to our "standard case" DDT-a, we perform p-process nucleosynthesis calculations for the other SN Ia models presented in Section 2, i.e., pure deflagration and delayed detonations of different strengths. In Figure 13, we show the mass distribution (in the form of tracer particles) at different T_peak (in the temperature range of p-process nucleosynthesis). In the pure deflagration case (DEF-a) it is clear from the histogram that much mass remains unburned (below T₉  ≃ 1) (as was also discussed in greater detail in Travaglio et al. 2004, where only deflagration models were presented). Consequently, in this case we have a factor of ∼10 more ²²Ne, which results in a much higher neutron abundance and, thus, the nuclei most sensitive to the neutron density (e.g., ⁵⁴Fe, ⁵⁸Ni, ⁹⁶Zr) show major overproduction. It is also important to note that in the DEF-a model, ⁹²Mo, ⁹⁶Ru are lower than the DDT-a model (using the same initial s-process distribution) by a factor of ∼3, and ⁷⁴Se, ⁷⁸Kr, and ⁸⁴Sr by a factor of ∼2. This effect can be attributed to the different distributions of mass in the two models. In fact, the amount of mass (corresponding to the number of tracers) at T₉  ⩾ 3 in the DDT-a model is a factor of ≃3 higher than that in the DEF-a model. For ¹⁶⁸Yb, ¹⁷⁴Hf, ¹⁸⁰W, ¹⁸⁴Os, and ¹⁹⁶Hg, we observe the opposite effect. We see that in the DEF-a model these heavy p-isotopes are more abundant by a factor between two and three with respect to the DDT-a case. From Figure 13 we note that the mass in the range T₉  ∼ 2.5–2.7 (as shown in Figures 6 and 7, this is the T zone with the highest production of these isotopes) is a factor of ∼2 lower for the DDT model with respect to the DEF-a model. From this we conclude that the mass distributions and therefore the underlying SN Ia model can introduce rather important changes in the final p-process nucleosynthesis.

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For the pure-deflagration SN Ia model (DEF-a), the results of the nucleosynthesis calculations are shown in Figure 14 (lower panel) for all isotopes with A  ⩾ 40. The s-process initial seeds used are the ones plotted in Figure 3 (upper panel). Figure 14 also gives the results for two other DDT models. As was pointed out by Röpke & Niemeyer (2007), Mazzali et al. (2007), and Kasen et al. (2009), it is possible to cover a wide range of ⁵⁶Ni masses with this class of models. The variation is introduced by different ignition geometries that set the strength of the initial deflagration phase and thus pre-expand the WD prior to detonation triggering to variable degrees. Apart from the DDT-a model discussed in previous sections, we calculate the nucleosynthesis for two additional delayed detonation models, one with a stronger (DDTs-a) and one with a weaker (DDTw-a) detonation phase (details of these models are given in Section 2). For the DDTw-a (Figure 14, upper panel), we note a much higher production of the light and intermediate p-nuclei (up to a factor of ∼5 for ⁸⁴Sr). This is mainly connected to the distribution of tracers, for T₉  ⩾ 3. In fact, as shown in Figure 13, with respect to DDT-a the DDTw-a model has two to four times more tracer particles in the high-T p-region, where most of the light p-nuclei are produced. A smaller effect is seen for the heavier p-nuclei. They are produced at lower T₉ (≃2), where the difference between the two tracer distributions is not as high as in the highest T₉ zones. Nevertheless, using for DDT-a and DDTw-a the same s-process seed abundances, we obtain an almost flat distribution for the resulting p-nuclei, including the lightest ones. It is important to note that in the case of DDTw-a we significantly increase ¹¹³In and ¹¹⁵Sn, producing them at the same level as ⁵⁶Fe (within a factor of ∼2).

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For the DDTs-a model, we find generally lower abundances of the p-nuclei. This is due to the fact that much less mass is in the low-T region where the p-process can occur, and most of the material reaches NSE condition, mainly producing ⁵⁶Fe (as indicated in Table 2). Nevertheless, on average the p-nuclei show an almost flat distribution (deriving from the same distribution of the s-process seeds).

We have presented results of detailed nucleosynthesis calculations for two-dimensional delayed detonation and deflagration SN Ia models, focusing in particular on p-process nucleosynthesis. We used initial abundances of s-nuclei synthesized during the past AGB history of the WD and during the mass accretion phase. During the late AGB phase of the companion, about 0.1 M_☉ of the CO-core become enriched in s-process elements by the effect of recurrent He-shell thermal instabilities. About 1000 years before the explosion, the simmering phase induced by central C-burning dilutes the s-rich material by a factor of ∼10 over the whole WD. The corresponding p-process yield is negligible when integrated over the whole ejecta. In contrast, the s-seeds synthesized during the WD mass accretion phase may give rise to an average p-process overabundance in the ejecta comparable with that of ⁵⁶Fe. Neutron fluxes available for the s-process in the accreted material rely on the activation of the ¹³C(α,n)¹⁶O reaction during the convective He flashes. This applies under the assumption that a small amount of protons are ingested in the top layers of the He intershell, as was suggested by Iben (1981). Protons are captured by the abundant ¹²C and converted into ¹³C via ¹²C(p,γ)¹³N(β⁺ν)¹³C at T ∼ 1 × 10⁸ K. Neutrons are then released in the bottom region of the convective He intershell by ¹³C(α,n)¹⁶O. In contrast to the previous AGB phase, where both the formation of the ¹³C-pocket and the subsequent release of neutrons occur radiatively in the interpulse phase, here the s-process is made directly in the convective shell during thermal instability, similar to the plume mixing by Ulrich (1973). The results of our post-process calculations should be considered as preliminary, given the general difficulty of following with full evolutionary codes the peculiar conditions of thermal pulses during the mass accretion phase.

In the cases of s-seeds from the mass accretion phase, we analyzed different SN Ia models, i.e., delayed detonations of different strengths, and a pure-deflagration model. For all these models we explored different s-process distributions and their consequences for the p-process. We also investigated a metallicity effect of the p-process nucleosynthesis in this scenario, considering models with solar metallicity and 1/20 solar. Despite the fact that studies of s-process nucleosynthesis during the accreting WD phase are led by the need to clarify the effective s-process distribution in these particular conditions, the results presented in this paper for p-process nucleosynthesis are quite significant. Note that a flat s-seed distribution directly translates into a flat p-process distribution whose average production factor scales linearly with the adopted level of the s-seeds. In contrast to previous work on p-process nucleosynthesis in SNe Ia (e.g., Kusakabe et al. 2005, 2011; Goriely et al. 2002, 2005), we demonstrated that we can produce almost all the p-nuclei with similar enhancement factors relative to ⁵⁶Fe, including the puzzling light p-nuclei ^{92, 94}Mo, ^{96, 98}Ru. We found that only the isotopes ¹¹³In, ¹¹⁵Sn, ¹³⁸La, ¹⁵²Gd, and ^180mTa diverge from the average p-process production. Among them, ¹⁵²Gd and ^180mTa have an important contribution from the s-process in AGB stars (Arlandini et al. 1999) or the neutrino process in SNe II (Woosley et al. 1990; Wanajo et al. 2011a). Both ¹¹³In and ¹¹⁵Sn are not fed by the p-process nor by the s-process. For these, we refer to the discussion by Dillmann et al. (2008a).

As far as the Galactic chemical evolution of p-nuclei is concerned, our results lead to the following very preliminar conclusions. Given the assumption of different s-seed distributions (see Figures 3 and 4), for both solar metallicity and 1/20 solar, we could show that the p-nuclei on average are produced at the level of ⁵⁶Fe when normalized to solar abundances (as shown in Figure 10). Even taking a fixed choice of the ¹³C-pocket at different metallicities, since the s-seed distributions peaked at heavier mass numbers are on average dominant, we may infer that the (p/⁵⁶Fe)/(p/⁵⁶Fe))_☉ ratio is always constant. This suggests the primary nature of the p-process. This aspect will be examined thoroughly in a forthcoming paper. From the hypothesis that SNe Ia are responsible for 2/3 of the solar ⁵⁶Fe, and by assuming that our DDT-a model represents the typical SN Ia with a frequency of ∼70% of all SNe Ia (Li et al. 2011), we conclude that they may be responsible for about 50% of the solar abundances of all p-nuclei. Instead, if we consider an average between DDTw-a and DDTs-a models, considering that they represent ∼10% of all SNe Ia (Li et al. 2011), still with a flat s-seed distribution (see Figure 14), we obtain that they may account for 75% of the solar abundances of all p-nuclei. Of course, these are the first rough estimates of how SNe Ia, in principle, can contribute to the solar system composition of p-nuclei. These results must also take into account that SNe II can be potentially important contributors to the galactic p-nuclei. Rayet et al. (1995) showed that about 1/4 of the solar system p-nuclei can be attributed to SN II explosions. Later, Woosley & Heger (2007) showed that p-nuclei can have an appreciable contribution at the solar composition from explosive neon and oxygen burning for a mass number greater than 130, but are underproduced for lighter masses. As recalled in Section 1, recent works on the νp-process in SNe II show that a non-negligible contribution may come from this process to the light p-isotopes.

A more thorough analysis of the role of SNe Ia in the solar composition of p-nuclei is planned.

Finally, we note that recent works (e.g., Sim et al. 2010; Fink et al. 2010; and references therein) discussed the fact that the explosion of sub-Chandrasekhar mass WDs via the double detonation scenario is a potential explanation for SNe Ia (but see Woosley & Kasen 2011). Again, the possibility of p-process production will be explored in a forthcoming paper.

All the tables with the yields obtained for different models presented in this work are available upon request.

We thank S. Cristallo and O. Straniero for many discussions of the feasibility of the s-process during mass accretion. We thank U. Battino for testing the production channels of the various p-isotopes on the occasion of his Master's degree. We thank P. Mohr for an extremely detailed update of the nucleosynthesis of ^180mTa and ¹³⁸La in explosive SN conditions. Thanks are due to the anonymous referee for the comments that helped improve the paper. This work has been supported by B2FH Association. The numerical calculations have also been supported by Regione Lombardia and CILEA Consortium through a LISA Initiative (Laboratory for Interdisciplinary Advanced Simulation) 2010 grant. This work was partially supported by the Deutsche Forschungsgemeinschaft via the Transregional Collaborative Research Center "The Dark Universe" (TRR 33), the Emmy Noether Program (RO 3676/1-1), and the Excellence Cluster "Origin and Structure of the Universe" (EXC 153).