Internal Heating in Magnetars: Role of Electron Captures

Technical Summary: Internal Heating in Magnetars: Role of Electron Captures

Problem Statement
Magnetars, characterized by extreme magnetic fields ($B \gtrsim 10^{14}$ G), exhibit persistent X-ray luminosities ($\sim 10^{33} - 10^{35}$ erg/s) and occasional giant flares that exceed their rotational energy output. While crustal deformations induced by magnetic stresses are a widely accepted heating mechanism, they are most effective in the inner crust where melting temperatures are high, yet heat sources are not expected to exist in such deep regions. An alternative mechanism posits that magnetic energy is converted into heat via electron captures by nuclei in the outer crust, driven by the slow decay of the magnetic field which compresses the matter. Previous estimates of this heating relied on the HFB-24 atomic mass model; this study investigates the role of electron captures using updated nuclear data and the HFB-27 atomic mass table to refine predictions of heat release and source locations.

Methodology
The authors model the outer crust of a magnetar as a charge-neutral plasma of nuclei $(A, Z)$ and free electrons. The study assumes the crust is initially in full thermodynamic equilibrium under a strong magnetic field ($B_\star \equiv B/B_{rel} = 2000$, corresponding to $B \approx 8.8 \times 10^{16}$ G). Key methodological steps include:

• Quantization Effects: Landau-Rabi quantization is applied to electrons, assuming they occupy the lowest Landau level due to the high magnetic field and relatively low temperatures ($T \sim 10^8 - 10^9$ K).
• Equilibrium Composition: The initial composition is determined by minimizing the Gibbs free energy per nucleon using an iterative approach.
• Reaction Thresholds: The onset of electron captures is calculated analytically. The threshold pressure $P_\beta$ for a nucleus $(A, Z)$ to capture an electron is derived, accounting for the magnetic field strength and the spatial arrangement of nuclei (Wigner-Seitz approximation).
• Heat Calculation: The heat released per nucleus is estimated by considering the energy difference between the parent and daughter nuclei, including excitation energies of the daughter states. The study calculates heat release for both ground-state-to-ground-state and ground-state-to-excited-state transitions.
• Nuclear Models: Calculations utilize experimental nuclear masses from the 2016 Atomic Mass Evaluation and theoretical predictions from the HFB-27 model (based on self-consistent deformed Hartree-Fock-Bogoliubov calculations). These are compared against previous results obtained with the HFB-24 model.
• Pycnonuclear Fusion: The potential for light element fusion (e.g., Carbon, Oxygen) is considered as an upper bound on heat release in the densest regions of the outer crust.

Key Results

• Dominant Heating Reactions: The study identifies specific electron capture reactions that release the most significant amounts of heat. Ground-state transitions include $^{12}\text{C} \to {}^{12}\text{Be}$ ($\sim 0.14$ MeV/nucleon), $^{16}\text{O} \to {}^{16}\text{C}$ ($\sim 0.15$ MeV/nucleon), and $^{62}\text{Cr} \to {}^{62}\text{Ti}$ ($\sim 0.09$ MeV/nucleon). Ground-state-to-excited-state transitions, such as $^{82}\text{Ge} \to {}^{82}\text{Zn}$ and various Strontium/Krypton transitions, contribute additional heat ($\sim 0.07 - 0.1$ MeV/nucleon).
• Model Comparison: While the HFB-27 and HFB-24 models predict similar primary heat sources, significant discrepancies arise in minor reactions. Specifically, the heat predicted for $^{56}\text{Ti} \to {}^{56}\text{Ca}$ and $^{80}\text{Zn} \to {}^{80}\text{Ni}$ transitions is approximately 2 and 3 times higher, respectively, in the HFB-27 model due to differences in the calculated $Q_{EC}$ values.
• Heat Magnitude and Location: The maximum heat released per nucleon is estimated at $\sim 0.1$ MeV from electron captures and $\sim 1 - 2$ MeV from pycnonuclear fusions. These reactions occur at densities $\rho_\beta \sim 10^{10} - 10^{11}$ g cm$^{-3}$ and pressures $P_\beta \sim 10^{29} - 10^{30}$ dyn cm$^{-2}$.
• Timescales and Power: The timescale for electron captures ($\tau$) across different crustal layers is of the order of the kinematic age of magnetars (a few thousand years). The resulting total heating power is estimated at $W_\infty \sim 10^{35} - 10^{36}$ erg/s, consistent with observed magnetar luminosities and previous cooling simulations.

Significance and Claims
The paper concludes that electron captures and pycnonuclear fusion in the outer crust, driven by magnetic field decay, constitute a robust source of internal heating for magnetars. A key finding is that the maximum heat release is essentially determined by nuclear data (masses and excitation energies) rather than the specific magnetic field strength or the phase state (solid vs. liquid) of the electron-ion plasma. Consequently, the heating estimates remain valid approximations even for neutron stars with lower magnetic fields, though the physical location of the heat sources will shift. The authors note that this mechanism remains effective even if parts of the crust are melted, offering a distinct advantage over models relying on elastic energy dissipation from crustal failure. The results are consistent with neutron-star cooling data and the empirically determined density ranges for heat deposition.