2025MNRAS.541.2053Y

Fermi-Large Area Telescope observations revealed that each GeV phase-folded light curve (aka phaseogram) of the Crab, Geminga, Dragonfly, and Vela pulsars consists of two pulses (P1 and P2) and a 'Bridge' between them. There is clearly a 'bump' at the Bridge phase of Vela's pulse profiles, that could also be regarded as the third pulse (P3). Differently, the Crab's, Geminga's, and Dragonfly's Bridges relatively resemble a 'valley floor'. Despite such an apparent difference, it is interesting to investigate whether their Bridge emissions are still within the same general picture as Vela's. Assuming the north-south symmetry, we would expect the fourth component (Bridge2/P4) to exist as well. However, such a hypothetical Bridge2/P4 is not intuitively identified on $\gamma$-ray phaseograms of the Crab, Geminga, Dragonfly, and Vela pulsars. It is also intriguing to hint at the rationale for the non-discovery of Bridge2/P4. Our prototypical toy model is free of assumptions on emission regions or radiation mechanisms. Instead, it assumes a north-south symmetric geometry and one circularly symmetric beam per hemisphere, while taking into account Doppler shifts (the most innovative element), time delays and energy-dependent beam shapes. Tentative compatibility of our fitting results with wind models is reported. Notably, for the Crab pulsar, we found a preliminary qualitative correlation between our model predictions and the Imaging X-ray Polarimetry Explorer (IXPE) results on X-ray polarization. The softer $\gamma$-ray pulsation of the Geminga pulsar is found to span over its full phase. Prompted by systematic evaluations, we outline some potential improvements for our toy model.