Gamma-Ray Millisecond Pulsars: Off-pulse Emission Characteristics, Phase-Resolved Pseudo-Luminosity--Cutoff Energy Correlation, and High-energy Pulsed Emission

Abstract

We investigate the $γ$-ray emission from 38 millisecond pulsars using 15 years of Fermi-LAT Pass 8 data in the 0.3--500 GeV range. Off-pulse intervals defined objectively with the Bayesian Blocks algorithm reveal significant off-pulse emission from 15 sources. Ten exhibit clear spectral cutoffs indicative of magnetospheric origin, while the remaining five show no compelling evidence for non-magnetospheric origins, as their off-pulse emission is spatially unresolved and inconsistent with hadronic, inverse Compton, or intrabinary contributions, implying a likely magnetospheric origin. We perform phase-resolved spectral fits for these 15 sources. In 11 of them, the cutoff energy $E_{\rm cut}$ varies markedly with rotation phase and correlates positively with the phase-resolved photon counts. Defining a phase-resolved pseudo-luminosity, these 11 pulsars follow a linear relation between $\log_{10}L$ and $\log_{10}E_{\rm cut}$, with slope $α= 2.31^{+0.22}_{-0.25}$, consistent with curvature-radiation predictions from the equatorial current sheet ($α\approx 2.29$). The same relation appears in the bright pulsar J0614$-$3329, implying the same emission mechanism across all rotational phases. We detect pulsed emission above 10 GeV from 19 sources, and a significant fraction of these also exhibit robust off-pulse emission. The coexistence of robust off-pulse flux and pulsed emission extending to high energies challenges standard outer-gap models. While other frameworks can also produce off-pulse flux, the phase-resolved $L$--$E_{\rm cut}$ correlation could provide a key diagnostic, and our measured slope may provide new evidence supporting the equatorial current sheet scenario as an important $γ$-ray emission mechanism in millisecond pulsars.

Figures (8)

• Figure 1: Unweighted pulse profile of PSR J0614$-$3329 for one rotation periods is shown in the black solid line. The red histogram represent the light curve obtained through the Bayesian Block algorithm, and the grey area shows the off-pulse phase range $0.04-0.22$.
• Figure 2: Spectral energy distributions of MSPs with TS of off-pulse emission exceeding 25. The blue curves and the red dotted lines are the fitting results of ExpCutoff model and PL model, respectively. The statistical errors with 1$\rm \sigma$ are shown. Upper limit is computed when TS is lower than 4.
• Figure 3: Off-pulse TS map of a $4^{\circ}$×$4^{\circ}$ region centered on MSP J0034$-$0534. The green cross symbol indicates the catalog position of the pulsar. The result of the spatial extension test, $\rm TS_{ext}$, is shown in the lower left corner. The color bar denotes the scale of TS values.
• Figure 4: Phase-resolved cutoff energy ($E_{\rm cut}$, black points with $1\sigma$ error bars) together with $\gamma$-ray light curve from $Fermi$-LAT for the eleven MSPs with a significant correlation ($P < 0.05$) between $E_{\rm cut}$ and average photon counts. The phase-resolved analysis is performed with the photon index $\Gamma$ tied across all bins for each pulsar. The black histogram illustrates the pulsar light curve, and the red histogram indicates the phase bins determined by the Bayesian Blocks algorithm. The Spearman rank correlation coefficient ($R$) and $P$-value ($P$) for the correlation between $E_{\rm cut}$ and average photon counts are indicated in each panel.
• Figure 5: Linear correlation between the logarithm of the cutoff energy, $\log_{10}(E_{\rm cut}/\mathrm{GeV})$, and the logarithm of the pseudo-luminosity, $\log_{10}(L)$, using data from all phase bins of some MSPs. Left panel: Fit to the combined data from all 11 MSPs. The inset shows the posterior probability distribution for the fitted slope $\alpha$, with the median (solid line), $1\sigma$ (dashed black lines), and $2\sigma$ (dashed red lines) confidence intervals indicated. Middle panel: Fit for the bright source, J0614$-$3329, alone. Right panel: Fit for the sample excluding J0614$-$3329.