Constraining millisecond pulsar geometry using time-aligned radio and gamma-ray pulse profile

Since the launch of the Fermi Gamma-Ray Space Telescope, several hundred gamma-ray pulsars have been discovered, some being radio-loud and some radio-quiet with time-aligned radio and gamma-ray light curves. In the second Fermi Pulsar Catalogue, 117 new gamma-ray pulsars have been reported based on three years of data collected by the Large Area Telescope on the Fermi satellite, providing a wealth of information such as the peak separation~$\Delta$ of the gamma-ray pulsations and the radio lag~$\delta$ between the gamma-ray and radio pulses. We selected several radio-loud millisecond gamma-ray pulsars with period~$P$ in the range 2-6~ms and showing a double peak in their gamma-ray profiles. We attempted to constrain the geometry of their magnetosphere, namely the magnetic axis and line-of-sight inclination angles for each of these systems. We applied a force-free dipole magnetosphere from the stellar surface up to the striped wind region -- well outside the light cylinder -- to fit the observed pulse profiles in gamma-rays, consistently with their phase alignment with the radio profile. In deciding whether a fitted curve is reasonable or not, we employed a least-square method to compare the observed gamma-ray intensity with that found from our model, emphasising the amplitude of the gamma-ray peaks, their separation, and the phase lag between radio and gamma-ray peaks. We obtained the best fits and reasonable parameters in agreement with observations for ten millisecond pulsars. Eventually, we constrained the geometry of each pulsar described by the magnetic inclination~$\alpha$ and the light-of-sight inclination~$\zeta$. We found that both angles are larger than approximately~$45^{\rm o}$.