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With the advent of multi-wavelength electromagnetic observations of neutronstars, spanning many decades in photon energies, from radio wavelengths up toX-rays and $\gamma$-rays, it becomes possible to significantly constrain thegeometry and the location of the associated emission regions. In this work, weuse results from the modelling of thermal X-ray observations of PSR~J0030+0451from the NICER mission and phase-aligned radio and $\gamma$-ray pulse profilesto constrain the geometry of an off-centred dipole able to reproduce thelight-curves in these respective bands simultaneously. To this aim, we deduce aconfiguration with a simple dipole off-centred from the location of the centreof the thermal X-ray hot spots and show that the geometry is compatible withindependent constraints from radio and $\gamma$-ray pulsations only, leading toa fixed magnetic obliquity of $\alpha \approx 75\deg$ and a line of sightinclination angle of $\zeta \approx 54\deg$. We demonstrate that an off-centreddipole cannot be rejected by accounting for the thermal X-ray pulse profiles.Moreover, the crescent shape of one spot is interpreted as the consequence of asmall scale surface dipole on top of the large scale off-centred dipole.

Constraining the magnetic field geometry of the millisecond pulsar PSR~J0030+0451 from joint radio, thermal X-ray and $\gamma$-ray emission