Mass parameter and the bounds on redshifts and blueshifts of photons emitted from geodesic particle orbiting in the vicinity of regular black holes

We obtain the mass parameter for a class of static and spherically symmetric regular black holes (BHs) (namely Bardeen, Hayward and Ay{o}n-Beato-Garc{i}a BHs) which are solutions of Einsteins field equations coupled to nonlinear electrodynamics (NED) in terms of redshifts and blueshifts of photons emitted by geodesic particles (for instance, stars) orbiting around these BHs. The motion of photons is not governed by null geodesics for these type of spacetime geometries which reflects the direct effects of the electrodynamic nonlinearities in the photon motion; hence, an effective geometry needs to be constructed to study null trajectories [Phys. Rev. D61, 045001 (2000)]. To achieve the above, we first study the constants of motion from the analysis of the motion of both geodesic particles moving in stable circular orbits and photons ejected from them and reaching a distant observer (or detector) in the equatorial plane for the above mentioned regular BHs. The relationship between red/blueshifts of photons and the regular BH observables is presented. We also numerically find the bounds on the photon shifts for these regular BH cases.