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The combined X-ray and $γ$-ray modeling of millisecond pulsars PSR J0030+0451 in the dissipative magnetospheres

Modeling of the NICER X-ray light curves of millisecond pulsars PSR J0030+0451 provides a strong evidence for the existence of non-dipole magnetic fields. We study the X-ray and $γ$-ray emission of PSR J0030+0451 in the dissipative dipole plus off-centred quadrupole magnetospheres. The dissipative FF+AE dipole magnetospheres by combining force-free (FF) and Aristotelian electrodynamics (AE) are solved by a 3D pseudo-spectral method in the rotating coordinate system. We use the FF+AE dipole plus off-centred quadrupole fields with minimum free parameters to reproduce two hotspot configurations found by the NICER observations. The X-ray and $γ$-ray emission from PSR J0030+0451 are simultaneously computed by using a ray-tracing method and a particle trajectory method. The modelled X-ray and $γ$-ray emission is then directly compared with those of PSR J0030+0451 from the NICER and Fermi observations. Our results can well reproduce the observed trends of the NICER X-ray and Fermi $γ$-ray emission for PSR J0030+0451.