2022ApJ...934...65K

We explore the validity of the recently reported fundamental plane (FP) relation of γ-ray pulsars using 190 pulsars included in the latest 4FGL-DR3 catalog. This sample number is more than twice as large as that of the original study. The FP relation incorporates four parameters, i.e., the spin-down power, $\dot{{ \mathcal E }}$, the surface magnetic field, B_⋆, the total γ-ray luminosity, L_γ, and a spectral cutoff energy, ϵ_cut. The derivation of ϵ_cut is the most intriguing one because ϵ_cut depends on the proper interpretation of the available phase-averaged spectra. We construct synthetic phase-averaged spectra, guided by the few existing phase-resolved ones, to find that the best-fit cutoff energy, ϵ_c1, corresponding to a purely exponential cutoff (plus a power-law) spectral form, is the parameter that optimally probes the maximum cutoff energy of the emission that originates from the core of the dissipative region, i.e., the equatorial current sheet. Computing this parameter for the 190 4FGL pulsars, we find that the resulting FP relation, i.e., the γ-ray luminosity in terms of the other observables, reads as ${L}_{\gamma }={10}^{14.3\pm 1.3}{({\epsilon }_{{\rm{c}}1}/\mathrm{MeV})}^{1.39\pm 0.17}$${({B}_{\star }/{\rm{G}})}^{0.12\pm 0.03}{(\dot{{ \mathcal E }}/\mathrm{erg}\,{{\rm{s}}}^{-1})}^{0.39\pm 0.05}\,\mathrm{erg}\,{{\rm{s}}}^{-1};$ this is in good agreement with both the empirical relation reported by Kalapotharakos et al. and the theoretically predicted relation for curvature radiation. Finally, we revisit the radiation reaction limited condition, to find it is a sufficient but not necessary condition for the theoretical derivation of the FP relation. However, the assumption of the radiation reaction limited acceleration reveals the underlying accelerating electric field component and its scaling with $\dot{{ \mathcal E }}$.