Nucleosynthesis simulations for the production of the p-nuclei $^{\text{92}}$Mo and $^{\text{94}}$Mo in a Supernova type II model

We present a nucleosynthesis sensitivity study for the $\gamma$-process in a Supernova type II model within the NuGrid research platform. The simulations aimed at identifying the relevant local production and destruction rates for the p-nuclei of molybdenum and at determining the sensitivity of the final abundances to these rates. We show that local destruction rates strongly determine the abundance of $^{92}$Mo and $^{94}$Mo, and quantify the impact.


Introduction
The p-nuclei between 74 Se and 196 Hg can be produced under explosive conditions in a sequence of photodissociation reactions and subsequent β-decays starting at s-and r-process seeds [1]. Most of the p-nuclei are about two orders of magnitude less abundant than other stable isotopes belonging to the same element. Relevant exceptions are the isotopes 92,94 Mo and 96,98 Ru [2]. Their production in stars is a puzzle for nuclear astrophysics since present models underproduce these p-nuclei by orders of magnitude. According to recent stellar model calculations, 94 Mo is mainly synthesized via the (γ,n) photodisintegration chain starting from the more neutron-rich and stable molybdenum isotopes [3]. Some of the light p-nuclei, like the neutron magic isotope 92 Mo, may also be synthesized by proton capture reactions [4].
We investigate the nucleosynthesis of 92 Mo and 94 Mo in a Supernova type II model. We present results on the sensitivity of rates of direct production or destruction reactions of the two Mo isotopes.

PPN simulations using a Supernova type II model
For the post-processing nucleosynthesis (PPN) simulations we use a stellar progenitor with an initial mass of 25 M ⊙ and metallicity Z = 0.02 [5]. The p-process nucleosynthesis is calculated by using classic Supernova type II trajectories [6]. In the model, a shock front passes through the Ne/O burning zone, which is subdivided into 14 mass layers. Figure 1 shows the temperature profiles, which a e-mail: goebel@physik.uni-frankfurt.de describe the astrophysical environment of each layer as a function of time. The innermost mass shell is the hottest and densest environment with a peak temperature of 3.45 GK. The maximum value drops to 1.79 GK for the outermost mass layer. The shock front reaches the mass layers successively. The temperature rapidly increases to the maximum value and afterwards drops slowly.  The complete nucleosynthesis is calculated using the post-processing code PPN within the NuGrid research framework [7,8]. NuGrid offers a software framework for nucleosynthesis simulations in relevant astrophysical environments. A large network of over 5,000 isotopes and more than 60,000 reactions is used. Most of the nuclear EPJ Web of Conferences reaction rates adopted are taken from the JINA Reaclib Database V1.1 [9].

Nucleosynthesis fluxes
The relative time-integrated nucleosynthesis fluxes of all mass layers for the isotopes 92 Mo and 94 Mo are shown in Figures 2 and 3. For each isotope, the sum of the production fluxes is normalized to 100%, and the destruction fluxes are scaled with the same factor. Fluxes smaller than 1% are not shown. If the sum of destruction fluxes is less than 100%, there is a net yield in the simulation. Otherwise the isotope gets depleted. The sum of the production fluxes is larger than the sum of the destruction fluxes for both isotopes. The isotope 92 Mo has the largest net yield with 100% production and 93% destruction.
The isotope 92 Mo is mainly produced by photodisintegration of 93 Mo. The reaction 92 Mo(γ,p) is the main destruction path followed by 92 Mo(n,γ) reactions. The reaction 95 Mo(γ,n) strongly contributes to the production fluxes of 94 Mo. The isotope 94 Mo is equally destroyed by (γ,n) and (n,γ) reactions.
The reactions accounting for a significant fraction of the nucleosynthesis fluxes are investigated in the rate variation study in section 4.

Abundances of 92 Mo and 9Mo depending on local production and destruction rates
The PPN simulations using a Supernova type II model aim at identifying the relevant reactions for the production and destruction of the Mo p-nuclei, and at quantifying the effect of rate changes on the final abundances. Previous nuclear sensitivity studies varied the reaction rates for all reactions of a certain type in the network, as well as for all reactions involving the p-nuclei simultaneously [3,6]. In this work, we explore the impact of single rates directly producing or destroying 92 Mo and 94 Mo. The effect of single rate variations on the final abundances of the p-nuclei 92 Mo and 94 Mo was determined by multiple simulations of the same scenario using different reaction rates. The rates were multiplied/divided by factors 2 and 5 for reactions involving a neutron ((γ,n) and (n,γ)) or a proton ((γ,p) and (p,γ)), and by factors 5 and 10 for reactions involving an α-particle ((γ,α) and (α,γ)). The factors were chosen according to the commonly stated uncertainties of the reaction rates ( [3] and references therein). In the PPN simulation, the rate taken from the library is calculated for the current temperature in the astrophysical environment and then multiplied by the factor set by the user.
This work shows the dependencies for the rates where the maximum abundance ratio is larger than 5% within the applied rate variations. Figure 4 shows the final abundance of 92 Mo as a function of the factor applied to one production or destruction rate. The abundance of 92 Mo strongly depends on the 92 Mo(γ,p) reaction rate, which is the main destruction path. The abundance of 92 Mo drops with increasing rate and vice versa (Figure 4, top panel). Neutron dissociation reactions are the main production path of 92 Mo (Figure 4, bottom panel). The 94 Mo(γ,n) reaction shows a strong impact on the abundance of 92 Mo. Variations of the reaction rate of 93 Mo(γ,n) hardly affect the 92 Mo abundance since the unstable isotope 93 Mo has to be produced first.

The case of 92 Mo
The 92 Mo abundance drops if the reaction rate of 94 Mo(γ,α) increases. The ratio of the reaction rates of 94 Mo(γ,α) and 94 Mo(γ,n) is larger at high temperatures, hence, the nucleosynthesis path is altered producing less 92 Mo via (γ,n) reactions.

The case of 94 Mo
Photodisintegration reactions starting from heavier Mo isotopes produce 94 Mo.
The destruction reaction 94 Mo(γ,n) mainly determines the final abundance of 94 Mo. In the simulation, a small fraction of the nucleosynthesis fluxes proceeds via 94 Mo(γ,α). An increase of the

Summary and outlook
The impact of variations of local production and destruction rates on the p-nuclei 92 Mo and 94 Mo was investigated. The abundance of 92 Mo is mainly determined by the 92 Mo(γ,p) reaction rate, the 94 Mo abundance by the 94 Mo(γ,n) and 94 Mo(γ,α) reaction rates. Most rates taken from JINA Reaclib Database V1.1 are calculated within the Hauser-Feshbach model [10]. New evaluated data can be compared to these rates and new final abundances of the Mo p-nuclei can be obtained from the results presented here.
In the γ-process, reactions on heavy nuclei (Z > 42) may determine the reaction flow to the light p-nuclei and, hence, the abundance of 92 Mo and 94 Mo. Future studies foresee a global sensitivity study of reaction rates in different p-nuclei production scenarios including Supernovae type Ia.