The annular gap model under a rotating dipole field approximation: simulating gamma-ray light curve

Abstract

A more realistic description of the magnetosphere is crucial for understanding the radiation emitted by pulsars. In this paper, we revisit the annular gap model by employing a rotating dipole field, which is more realistic than the static dipole field, as an approximation of the magnetic structure of the pulsar magnetosphere. Compared with the static dipole field approximation, the open field-line region, including both the core and annular gaps, is significantly enlarged, and the two regions become asymmetric with respect to the fiducial plane. We apply this model to three young gamma-ray pulsars with distinct light-curve morphologies, PSRs J0631$+$1036 (single peak), J1709$-$4429 (double peaks), and J1048$-$5832 (three peaks). Using viewing geometries constrained by radio polarization measurements, the annular gap model within the rotating dipole field successfully reproduces the main morphological features of their gamma-ray light curves above 0.1 GeV. Our model provides a framework for interpreting pulsar high-energy emission, which can be used to analyze the emission properties of high-energy pulsars.

Figures (7)

• Figure 1: An example of a rotating dipole magnetosphere plotted using Crab pulsar parameters and $\alpha=30^\circ$. The red and black curves represent the critical field lines and the last open field lines, respectively. The colored surface and the grey cylindrical surface depict the null charge surface and the light cylinder, respectively.
• Figure 2: The diagram illustrates the footprints of the last open field lines (in black) and the critical field lines (in red), calculated using the Crab pulsar parameters and various $\alpha$, under both the rotating dipole approximation (dark shading) and the static dipole approximation (light shading). $\Theta_{\rm p}$ and $\Phi_{\rm p}$ are the magnetic colatitude and magnetic azimuth angle for footprints, respectively.
• Figure 3: This figure shows the heights, $r_{\rm N}$, from the pulsar center to the intersection of the null charge surface with the last open field line, calculated using the Crab pulsar parameters and various $\alpha$, for both the static (light color) and rotating (dark color) dipole approximations. $\Phi_{\rm p}$ is the magnetic azimuth angle of footprints for these last open field lines. The blue vertical line marks where $\Phi_{\rm p} = 0^{\circ}$.
• Figure 4: The figure shows the values of $E_{||}$ computed along a given field line using parameters of the Crab pulsar and different values of $\alpha$ under the rotating (dark shading) and the static (light shading) dipole field approximation. The magnetic colatitude and magnetic azimuth of the footprint of this field line are $\frac{\Theta_{p,o}(0)+\Theta_{p,c}(0)}{2}$ and $0^\circ$, respectively, where $\Theta_{p,o}(0)$ and $\Theta_{p,c}(0)$, respectively, denote the magnetic colatitude of the footprints of the last open and the critical field line whose footprint magnetic azimuths are $0^\circ$, under the rotating dipole field approximation. Two red vertical lines indicate the height from the pulsar center to the intersection of the null‑charge surface and the last open field line whose footprint has a magnetic azimuth of $0^\circ$ under the rotating (dark shading) and the static (light shading) dipole field approximation, respectively.
• Figure 5: Joint fitting of radio polarization and gamma-ray light curves for PSR J0631+1036. Top-left: The lower panel displays the total intensity (black solid line), total linear polarization (red dashed line), and circular polarization (blue dashed line). The upper panel shows the PPA as black dots with error bars, along with the best-fit RVM result plotted as a red curve. Top-right: The reduced $\chi^2$ of the fit is represented by the blue color scale, with red contours marking the 1$\sigma$, 2$\sigma$, and 3$\sigma$ confidence levels. Bottom-left: The upper panel presents a gamma-ray photon sky map spanning two rotational periods, and the lower panel shows the observed (red dots with error bars) and simulated(black solid line) gamma-ray light curves as well as the time-aligned radio light curves(light-green solid line), spanning two rotational periods. The vertical and horizontal red dashed lines represent the phase of the fiducial plane and the line-of-sight angle, respectively. Bottom-right panel: the gamma ray radiation height corresponding to the simulation is shown.