$νp$-process in Core-Collapse Supernovae: Imprints of General Relativistic Effects

The origin of a number of proton-rich isotopes in the solar system has been a long-standing puzzle. A promising explanation is the $νp$-process, which is posited to operate in the neutrino-driven outflows that form inside core-collapse supernovae after shock revival. While recent studies have analyzed several relevant physical effects that influence the efficiency of this process, the impact of General Relativity (GR) on it remains unexplored. We perform a comparative analysis of the time-integrated $νp$-process yields in Newtonian and fully GR calculations, using detailed models of time-evolving outflow profiles. The GR effects are seen to suppress the production of seed nuclei, significantly boosting the resulting $p$-nuclide abundances. Our reference GR model, with an 18~$M_\odot$ progenitor, reproduces both the relative and absolute solar system abundances of the entire set of the $p$ nuclides in the mass range $74\leq A\leq102$. The yields are suboptimal in our 12.75~$M_\odot$ GR model, where the outflow transitions to the supersonic regime several seconds into the explosion, suppressing further $p$-nuclide production. In both models, most of the production of the crucial $^{92,94}{\rm Mo}$ and $^{96,98}{\rm Ru}$ $p$ isotopes occurs relatively early, 1–3 seconds after shock revival. In contrast, a large fraction of the shielded isotope $^{92}{\rm Nb}$ is produced in the subsequent ejecta. The impact of GR on this isotope is especially large, with its final abundance boosted by a factor of 25 compared to a Newtonian calculation. In summary, with the GR effects taken into account, the $νp$-process in a sufficiently massive progenitor can provide a unifying explanation for the origin of all $p$ nuclei in the solar system up to $^{102}$Pd.

💡 Research Summary

The paper investigates the impact of General Relativistic (GR) effects on the νp‑process occurring in neutrino‑driven outflows (NDOs) of core‑collapse supernovae (CCSNe), with the aim of assessing whether GR can resolve longstanding discrepancies in the production of proton‑rich isotopes (p‑nuclides) observed in the solar system. The authors develop a fully relativistic, spherically symmetric, steady‑state hydrodynamic framework for the NDO, explicitly deriving the continuity, momentum, and entropy equations in natural units and identifying the individual GR corrections: enhanced gravitational potential, enthalpic mass increase, and neutrino energy blueshift. By reducing these equations to their Newtonian limit, they isolate the physical contribution of each term, enabling a systematic comparison with earlier non‑relativistic studies.

Two progenitor models are examined: an 18 M⊙ star (the benchmark) and a lighter 12.75 M⊙ star. For each, the authors compute outflow solutions both with full GR and in the Newtonian approximation, allowing the outflow to self‑consistently transition between subsonic and supersonic regimes based on boundary conditions derived from the progenitor’s density profile and the evolving proto‑neutron star (PNS) radius, luminosity, and neutrino spectra. The GR calculations reveal that the deeper gravitational well near the PNS leads to a substantial blueshift of emitted neutrinos, increasing the capture cross‑sections of νₑ and (\barνₑ) on free nucleons. Consequently, the rate of νₑ + n → p + e⁻ and (\barνₑ) + p → n + e⁺ reactions rises, supplying a larger neutron flux that fuels (n,p) exchange reactions on seed nuclei. Simultaneously, the GR‑induced increase in entropy (by ~30–40 k_B per baryon) suppresses the triple‑α bottleneck, reducing the formation of iron‑group seed nuclei and preserving a proton‑rich composition favorable for the νp‑process.

Using the SkyNet reaction network, the authors post‑process tracer particle trajectories extracted from a series of time‑dependent snapshots of the outflow. Each trajectory provides temperature, density, and electron fraction (Yₑ) histories, which are fed into the network to compute instantaneous nucleosynthesis rates and integrated yields. The analysis shows that, in the 18 M⊙ GR model, the bulk of the crucial p‑nuclides ⁹²,⁹⁴Mo and ⁹⁶,⁹⁸Ru are synthesized early (1–3 s after shock revival) while the outflow remains subsonic. Later, during the cooling phase (3–10 s), the shielded isotope ⁹²Nb is produced in significant amounts; its final abundance is boosted by a factor of ~25 relative to the Newtonian case. The overall integrated yields for all p‑nuclides in the mass range 74 ≤ A ≤ 102 match both the relative solar isotopic pattern and the absolute abundances, something not achievable without GR. In contrast, the 12.75 M⊙ model transitions to a supersonic flow a few seconds post‑bounce, forming a termination shock that curtails further nucleosynthesis. Consequently, while early‑time production of light p‑nuclides occurs, the later‑time contributions (especially for heavier p‑nuclides and ⁹²Nb) are severely diminished, leading to under‑production relative to solar values.

The paper also explores the sensitivity of the results to the equation of state (EOS) and to individual GR corrections. It confirms that the neutrino blueshift and entropy enhancement are the dominant drivers of the increased p‑nuclide yields, whereas the direct gravitational strengthening plays a secondary role. Comparisons with earlier relativistic outflow studies reveal that inconsistencies in reported entropy increases stem from differing formulations of the relativistic hydrodynamic equations; the authors reconcile these by presenting a unified derivation.

In summary, the study demonstrates that incorporating full General Relativistic physics into the modeling of neutrino‑driven outflows dramatically alters the conditions under which the νp‑process operates. By boosting the neutron supply and raising the entropy, GR suppresses seed‑nucleus formation and enables efficient (n,p) cycling, leading to a robust production of all solar p‑nuclides up to ¹⁰²Pd in a sufficiently massive progenitor. This work establishes GR as an essential ingredient for realistic νp‑process calculations and provides a unified explanation for the origin of the solar system’s proton‑rich isotopes.