Onset of Electron Captures and Shallow Heating in Magnetars

Technical Summary: Onset of Electron Captures and Shallow Heating in Magnetars

Problem Statement
Magnetars, a subclass of neutron stars characterized by ultra-strong magnetic fields ($10^{14}–10^{15}$ G, potentially reaching $10^{17}–10^{18}$ G internally), exhibit outbursts and persistent X-ray luminosity exceeding their rotational energy limits. While crustal deformations due to magnetic stress are a known heating mechanism, heat sources in the shallow crust are required to avoid excessive neutrino cooling. An alternative mechanism involves the dissipation of magnetic energy into heat via electron captures by nuclei in the outer crust, triggered by the decay of the magnetic field and the subsequent compression of the crust. Previous studies had estimated the heat release and source location primarily in the "strongly quantizing" regime, where all electrons are confined to the lowest Landau–Rabi level. This paper addresses the need for accurate analytical treatments across the entire range of magnetic field strengths, from the weakly quantizing regime to the most extreme quantizing conditions.

Methodology
The authors develop a comprehensive theoretical framework for the equation of state (EoS) and thermodynamics of the magnetar outer crust (densities between ionization and neutron drip).

• Equation of State: The crust is modeled as fully ionized nuclei embedded in a relativistic electron gas. The electron motion perpendicular to the magnetic field is quantized into Landau–Rabi levels. The authors derive analytical expansions for electron density and pressure that are valid for arbitrary magnetic field strengths ($B_\star = B/B_{rel}$), bridging the gap between the weakly quantizing limit (many levels populated) and the strongly quantizing limit (only the lowest level populated).
• Initial Composition: The initial nuclear composition of the outer crust is determined by minimizing the Gibbs free energy per nucleon. Analytical formulas are derived to calculate the pressure and density at the interfaces between adjacent crustal layers for both zero-field and finite-field scenarios.
• Electron Capture Thresholds: The onset of electron captures is determined by the condition of constant Gibbs free energy per nucleon. The authors derive analytical formulas for the threshold electron Fermi energy, density, and pressure, accounting for the magnetic field's influence on the electron gas and the ion masses.
• Heat Release: The maximum heat released per nucleus is calculated by considering the difference in baryon chemical potentials between the initial nucleus and the final nucleus after two successive electron captures (assuming the intermediate daughter nucleus is unstable). The calculation includes first-order electrostatic corrections and nuclear data.
• Nuclear Data: The calculations utilize experimental atomic masses from the 2016 Atomic Mass Evaluation, supplemented by predictions from the Brussels-Montreal atomic mass model HFB-24 (based on the BSk24 energy-density functional). Excitation energies are taken from IAEA Nuclear Data.

Key Contributions

1. Universal Analytical Formulas: The paper provides highly accurate analytical formulas for the threshold density ($\bar{n}_\beta$) and pressure ($P_\beta$) for the onset of electron captures, as well as for the maximum heat released ($Q$). These formulas are valid for arbitrary magnetic field strengths, eliminating the need for purely numerical solutions in intermediate regimes.
2. Extended Regime Validity: While previous work focused on the strongly quantizing regime ($\nu_{max}=0$), this study demonstrates that the weakly quantizing regime ($\nu_{max} \gg 1$) and the intermediate regime are also amenable to accurate analytical approximations. The error analysis shows these expansions remain precise (errors $< 1\%$ for density and $< 0.02\%$ for pressure in many cases) even as the field approaches the strongly quantizing threshold.
3. Comprehensive Nuclear Parameters: The authors calculate and tabulate the necessary nuclear parameters ($\gamma_e^{1\to2}$, $\gamma_e^\beta$, and $Q^{(0)}$) for a wide range of transitions (ground-state to ground-state, ground-state to excited-state, and light elements like C and O) using the HFB-24 model.
4. Initial Crust Composition: The study extends the determination of the initial equilibrium composition of the outer crust to the full range of magnetic field strengths ($B=0$ to $10^{17}$ G), identifying specific field strengths where new nuclides appear or disappear.

Results

• Accuracy: Numerical comparisons between the exact solutions (using the full summation over Landau levels) and the approximate analytical formulas confirm high precision. For the threshold density, errors are typically a few percent at specific field strengths corresponding to exact Landau level fillings but drop significantly in the strongly quantizing regime. The pressure approximations are found to be even more reliable, with errors generally below 0.02%.
• Heat Release: The heat released per nucleon exhibits small quantum oscillations dependent on the magnetic field strength but is, to a first approximation, effectively independent of the magnetic field (consistent with the field-independent nature of nuclear masses at these field strengths).
• Composition Changes: The presence of strong magnetic fields alters the sequence of equilibrium nuclides in the outer crust. The paper identifies specific dimensionless field strengths ($B_\star$) where transitions occur, such as the appearance of $^{88}$Sr at $B_\star \approx 858$ or the disappearance of $^{66}$Ni at $B_\star \approx 67$.
• Neutron Delayed Emission: The framework allows for the identification of conditions where electron captures are accompanied by neutron emission, occurring when the threshold for neutron emission is lower than that for pure electron capture.

Significance
The paper establishes a robust, consistent set of microscopic inputs for modeling the magneto-thermal evolution of neutron stars. By providing accurate analytical formulas valid across all magnetic field regimes, the work facilitates the calculation of shallow heating rates and the location of heat sources without relying on computationally expensive numerical integrations for every magnetic field configuration. The results, combined with previously published equations of state and superfluid properties using the same HFB-24 model, offer a unified theoretical basis for understanding the thermal behavior of magnetars, particularly regarding the energy release from electron captures triggered by magnetic field decay. The full numerical results are made available to support further modeling efforts.