Absorption of Gamma-Ray Photons in a Vacuum Neutron Star Magnetosphere: I. Electron-Positron Pair Production

The production of electron-positron pairs in a vacuum neutron star magnetosphere is investigated for both low (compared to the Schwinger one) and high magnetic fields. The case of a strong longitudinal electric field where the produced electrons and positrons acquire a stationary Lorentz factor in a short time is considered. The source of electron-positron pairs has been calculated with allowance made for the pair production by curvature and synchrotron photons. Synchrotron photons are shown to make a major contribution to the total pair production rate in a weak magnetic field. At the same time, the contribution from bremsstrahlung photons may be neglected. The existence of a time delay due to the finiteness of the electron and positron acceleration time leads to a great reduction in the electron-positron plasma generation rate compared to the case of a zero time delay. The effective local source of electron-positron pairs has been constructed. It can be used in the hydrodynamic equations that describe the development of a cascade after the absorption of a photon from the cosmic gamma-ray background in a neutron star magnetosphere.

💡 Research Summary

The paper investigates the production of electron‑positron pairs in a vacuum neutron‑star magnetosphere when high‑energy gamma‑ray photons from the cosmic background are absorbed. The authors consider both weak magnetic fields (B ≪ B_cr, where B_cr ≈ 4.4 × 10¹³ G) and fields approaching the Schwinger limit, and they focus on the regime where a strong longitudinal electric field (E_∥) accelerates the newly created particles to a stationary Lorentz factor γ₀ on a very short timescale.

In a weak field the primary source of secondary photons is synchrotron radiation emitted as the freshly created particles quickly lose their transverse momentum and settle into the ground Landau level. The synchrotron photon energy k_syn = (3/2) B θ γ² depends on the instantaneous pitch angle θ and the particle Lorentz factor γ. The number of synchrotron photons emitted by a single particle, N_syn, is derived by integrating the synchrotron emissivity over the particle’s deceleration trajectory; the result depends on the initial photon energy k_i, the angle χ between the photon momentum and the magnetic field, and the magnetic field strength B. The absorption condition for a photon in a magnetic field, k_i χ² ≈ a Λ (with a = 4/3 B and Λ ≈ 10), forces χ to be very small, which simplifies the angular integration.

Curvature photons, by contrast, become important only after particles have been accelerated to the quasi‑steady Lorentz factor γ₀. Their characteristic energy is k_curv = (3/2) γ₀³/ρ, where ρ is the curvature radius of the magnetic field line. The curvature‑photon spectrum follows the standard expression involving the Macdonald function K_{5/3}.

The particle distribution is modeled as a delta‑function at γ₀: F(γ) = (n_+ + n_−) δ(γ − γ₀), reflecting the rapid acceleration in the strong E_∥. The pair‑creation rate q_p(γ) is linked to the photon production rate q_ph(k) = q_curv(k) + q_syn(k) through a self‑consistent integral equation (Eq. 13). Because q_syn itself is proportional to q_p, the equation contains a feedback loop. By normalizing the total particle number density to unity (n_+ + n_− = 1) the authors obtain an explicit expression for the number of pairs produced per unit time per particle, Q(γ_min) = ∫_{γ_min}^∞ q_p(γ) dγ.

A crucial new element is the inclusion of a finite acceleration time τ_st = γ₀/E_∥. This time delay means that secondary photons cannot be absorbed instantaneously after the primary pair is created; consequently the cascade growth is significantly slower than in models that assume zero delay. The authors define an “effective local source” Q_eff that incorporates this delay and show, through numerical estimates, that Q_eff can be reduced by one to two orders of magnitude compared with the instantaneous‑absorption approximation for typical magnetospheric parameters (E_∥ ≈ 10⁻⁴ B, B ≈ 10⁻² B_cr).

Bremsstrahlung radiation produced by the longitudinal acceleration (intensity W_brem ∝ α E_∥²) is shown to be negligible compared with curvature radiation (W_curv ∝ α γ⁴/ρ²) for the Lorentz factors (γ₀ ≈ 10⁷–10⁸) and curvature radii (ρ ≈ 10⁶ cm) relevant to neutron‑star magnetospheres. Therefore bremsstrahlung can be omitted from the cascade model.

The paper concludes that (1) synchrotron photons dominate pair production in weak magnetic fields, (2) a strong longitudinal electric field quickly drives particles to a stationary γ₀, (3) the finite acceleration time introduces a substantial delay that suppresses the overall plasma generation rate, and (4) the derived effective source term Q_eff provides a realistic input for hydrodynamic equations describing the development of a cascade after the absorption of a cosmic‑gamma‑ray photon. By integrating the effects of synchrotron radiation, curvature radiation, and acceleration‑time delay, the work offers a more complete and physically consistent description of pair creation in vacuum neutron‑star magnetospheres, which is essential for understanding the intermittent radio emission observed in certain pulsars and magnetars.