Self-organized criticality in a spherically closed cellular automaton: Modeling soft gamma repeater bursts driven by magnetic reconnection

A new cellular automaton (CA) model is presented for the self-organized criticality (SOC) in recurrent bursts of soft gamma repeaters (SGRs), which are interpreted as avalanches of reconnection in the magnetosphere of neutron stars. The nodes of a regular dodecahedron and a truncated icosahedron are adopted as spherically closed grids enclosing a neutron star. It is found that the system enters the SOC state if there are sites where the expectation value of the added perturbation is nonzero. The energy distributions of SOC avalanches in CA simulations are described by a power law with a cutoff, which is consistent with the observations of SGR $1806–20$ and SGR $1900+14$. The power-law index is not universal and depends on the amplitude of the perturbation. This result shows that the SOC of SGRs can be illustrated not only by the crust quake model but also by the magnetic reconnection model.

Figure 1

Schematic diagrams of poloidal and toroidal magnetic field lines in the magnetosphere of neutron stars and their mapping on a regular dodecahedron ($n=20$) grid. Positive ($+\mathrm{\Delta }B$) and negative ($-\mathrm{\Delta }B$) perturbations are added at the start points and end points of the edges shown as thick lines, respectively.

Figure 2

Cumulative burst energy distributions, $N(>E)$, for the case of $⟨\mathrm{\Delta }B⟩=0.2$. Thick and thin lines represent the models with poloidal and toroidal perturbations, respectively. Dashed and solid lines represent the models with the regular dodecahedron ($n=20$) grid and truncated icosahedron ($n=60$) grid, respectively.

Figure 3

Same as Fig. 2, but for the models with the truncated icosahedron ($n=60$) grid. Solid, dashed, dot-dashed, and dotted lines correspond to the cases with $f_{\mathrm{pl}}=1$ (purely poloidal), 0.5, 0.1, and 0 (purely toroidal), respectively.

Figure 4

Same as Fig. 2, but for the models with the truncated icosahedron ($n=60$) grid and poloidal perturbations. Thick dot-dashed, thick solid, and thick dashed lines correspond to the cases with $⟨\mathrm{\Delta }B⟩=0.1$, 0.2, and 0.4, respectively. Thin lines are power laws $N(>E)\propto E^{1-\gamma }$ with $\gamma =1.44$ (dot-dashed), 1.57 (solid), and 1.74 (dashed).

Figure 5

Power-law index $\gamma$ as a function of the amplitude of perturbation $⟨\mathrm{\Delta }B⟩$ for the models with the truncated icosahedron ($n=60$) grid and poloidal perturbations.