Résumé
Extended gamma-ray emission, interpreted as halos formed by the inverse-Compton scattering of ambient photons by electron-positron pairs, is observed toward a number of middle-aged pulsars. The physical origin and actual commonness of the phenomenon in the Galaxy remain unclear. The conditions of pair confinement seem extreme compared to what can be achieved in recent theoretical models. We searched for scenarios minimizing as much as possible the extent and magnitude of diffusion suppression in the halos in J0633+1746 and B0656+14, and explored the implications on the local positron flux if they are applied to all nearby middle-aged pulsars. We used a phenomenological static two-zone diffusion framework and compared its predictions with Fermi-LAT and HAWC observations of the two halos, and with the local positron flux measured with AMS-02. While strong diffusion suppression by 2-3 orders of magnitude at ~100TeV is required by the data, it is possible to find solutions with diffusion suppression extents as small as 30pc for both objects. If all nearby middle-aged pulsars develop such halos, their combined positron flux including the contribution from Geminga would saturate the >100GeV AMS-02 measurement for injection efficiencies that are much smaller than those inferred for the canonical halos in J0633+1746 and B0656+14, and more generally with the values typical of younger pulsar wind nebulae. Conversely, if positrons from other nearby pulsars are released in the interstellar medium without any confinement around the source, their total positron flux fits into the observed spectrum for the same injection efficiencies of a few tens of percent for all pulsars, from kyr-old objects powering pulsar wind nebulae to 100kyr-old objects like J0633+1746 and B0656+14. It seems a simpler scenario to assume that most middle-aged pulsars do not develop halos (abridged).