Influence of magnification threshold on pixel lensing optical depth, event rate and time scale distributions towards M 31

Pixel lensing is the gravitational microlensing of light from unresolved stars contributing to the luminosity flux collected by a single pixel. A star must be sufficiently magnified, that is, the lens impact parameter must be less than a threshold value uT if the excess photon flux in a pixel is to be detected over the background. Assuming the parameters of the Isaac Newton Telescope and typical observing conditions, we present maps in the sky plane towards M 31 of threshold impact parameter, optical depth, event number and event time scale, analyzing in particular how these quantities depend on uT in pixel lensing searches. We use an analytical approach consisting of averaging on uT and the star column density the optical depth, microlensing rate and event duration time scale. An overall decrease in the expected optical depth and event number with respect to the classical microlensing results is found, particularly towards the high luminosity M 31 inner regions. As expected, pixel lensing events towards the inner region of M 31 are mostly due to self-lensing, while in the outer region dark events dominate even for a 20% MACHO halo fraction. We also find a far-disk/near-disk asymmetry in the expected event number, smaller than that found by [CITE]. Both for self and dark lensing events, the pixel lensing time scale we obtain is $\simeq$$1{-}7$ days, dark events lasting roughly twice as long as self-lensing events. The shortest events are found to occur towards the M 31 South Semisphere. We also note that the pixel lensing results depend on $\langle u_{\rm T} \rangle$ and $\langle u_{\rm T}^{~2} \rangle$ values and ultimately on the observing conditions and telescope capabilities.