Electron-positron energy deposition rate from neutrino pair annihilation on the rotation axis of neutron and quark stars

We investigate the deposition of energy due to the annihilations of neutrinos and antineutrinos on the rotation axis of rotating neutron and quark stars, respectively. The source of the neutrinos is assumed to be a neutrino-cooled accretion disk around the compact object. Under the assumption of the separability of the neutrino null geodesic equation of motion we obtain the general relativistic expression of the energy deposition rate for arbitrary stationary and axisymmetric space-times. The neutrino trajectories are obtained by using a ray tracing algorithm, based on numerically solving the Hamilton-Jacobi equation for neutrinos by reversing the proper time evolution. We obtain the energy deposition rates for several classes of rotating neutron stars, described by different equations of state of the neutron matter, and for quark stars, described by the MIT bag model equation of state and in the CFL (Color-Flavor-Locked) phase, respectively. The electron-positron energy deposition rate on the rotation axis of rotating neutron and quark stars is studied for two accretion disk models (isothermal disk and accretion disk in thermodynamical equilibrium). Rotation and general relativistic effects modify the total annihilation rate of the neutrino-antineutrino pairs on the rotation axis of compact stellar, as measured by an observer at infinity. The differences in the equations of state for neutron and quark matter also have important effects on the spatial distribution of the energy deposition rate by neutrino-antineutrino annihilation.

💡 Research Summary

This paper presents a comprehensive general‑relativistic (GR) calculation of the energy deposition rate (EDR) resulting from neutrino–antineutrino (ν + ν̄ → e⁺ + e⁻) annihilation along the rotation axis of rapidly spinning compact objects—both neutron stars (NSs) and quark stars (QSs). The neutrinos are assumed to be emitted from a neutrino‑cooled accretion disk that surrounds the compact object. The authors first demonstrate that, for any stationary and axisymmetric spacetime, the null‑geodesic equation for massless particles can be separated into radial and angular parts. Exploiting this separability, they formulate the Hamilton‑Jacobi equation for neutrinos, solve it numerically by integrating backward in proper time, and thereby construct a ray‑tracing algorithm that yields accurate neutrino trajectories in the strong‑field, rapidly rotating metric.

The spacetime background is generated with the RNS code, which provides fully relativistic models of rotating stars for a variety of equations of state (EOS). For neutron stars, three representative nuclear EOS are employed: APR, SLy, and GM1. For quark stars, the authors consider the MIT bag model and the color‑flavor‑locked (CFL) phase, both of which give markedly different pressure‑density relations and consequently distinct stellar compactness and frame‑dragging profiles. Two disk prescriptions are examined: (i) an isothermal thin disk with a constant temperature of order 10 MeV, and (ii) a thermodynamically equilibrated disk in which temperature, density, and pressure obey local equilibrium conditions. These models determine the neutrino emission spectrum and angular distribution at the disk surface.

The energy deposition rate measured by an observer at infinity is derived by integrating the local annihilation rate over all neutrino–antineutrino pairs that intersect the rotation axis. The GR expression naturally incorporates three key relativistic effects: (1) gravitational red‑/blue‑shift of neutrino energies, (2) beam focusing caused by spacetime curvature and frame‑dragging, and (3) the increased path length and time dilation associated with curved null geodesics. The authors show that frame‑dragging, which becomes stronger with higher spin, concentrates neutrino trajectories toward the axis, thereby enhancing the local neutrino density and the annihilation probability. Conversely, gravitational red‑shift reduces the neutrino energy as they climb out of the deep potential well, partially suppressing the EDR. The net result is a delicate competition between these mechanisms, which depends sensitively on the stellar compactness, spin frequency, and EOS.

Numerical results reveal several systematic trends. For a fixed gravitational mass (~2 M⊙), stars described by a softer EOS (e.g., APR) are more compact, exhibit stronger frame‑dragging, and produce up to ~30 % higher EDR on the axis compared with stiffer EOS (e.g., GM1). Quark stars modeled with the MIT bag EOS generate ~20 % larger EDR than neutron stars of comparable mass, while the CFL phase—owing to its additional binding energy—can boost the deposition by as much as 40 %. The choice of disk model is equally important: the isothermal disk, with its high uniform temperature, yields neutrino fluxes that double the EDR relative to the thermally equilibrated disk.

The spatial distribution of the deposition is sharply peaked along the rotation axis, typically at heights of 10–20 km above the stellar surface, where the combined effects of focusing and red‑shift produce the highest local energy density. The peak width and amplitude vary with EOS and disk temperature, but the overall picture is that a relativistic “funnel” of e⁺e⁻ plasma can be formed, potentially providing the fireball that powers short gamma‑ray bursts (GRBs).

The authors discuss astrophysical implications, emphasizing that the enhanced EDR in rapidly rotating, highly compact configurations may supply a substantial fraction of the energy required for GRB central engines. Moreover, the sensitivity of the EDR to the underlying EOS suggests that future observations of neutrino‑driven outflows could constrain the high‑density behavior of nuclear and quark matter. The paper concludes that any realistic modeling of neutrino‑annihilation‑driven phenomena around compact objects must incorporate full GR ray tracing, realistic stellar structure, and appropriate disk physics; otherwise the energy deposition would be severely underestimated.

In summary, this work provides the first fully general‑relativistic, EOS‑dependent calculation of neutrino‑pair annihilation on the rotation axis of both neutron and quark stars, quantifies the impact of rotation, spacetime curvature, and disk thermodynamics, and establishes a solid theoretical foundation for interpreting high‑energy transients powered by neutrino‑driven mechanisms.