This contribution is a continuation of the work introduced during the 3AF-OPTRO2022 Conference [1]. We present here an updated starting procedure for the more general case where the target, D, is at long orbital distance to the tracker, S. Here, S is also assumed capable to reach D with very narrow laser beams. Due to the large orbital distance from S to D, only a small number of shooting directions can coincide with the LOS directions to reach D. Therefore, this work can help to implement

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... omous and secure FSO communications between some particular LEO emitters and GEO receivers as well as to achieve other similar goals. Since the orbital distance between S and D may be very large, the only realistic starting strategy for the emitter to be autonomous is to determine the instants, if any, at which the observed angular motion of the receiver from the emitter becomes zero, see [1]. Then we show that even assuming that S is capable to determine these particular configurations with the highest accuracy available at present, the success of the subsequent shooting actions can only be guaranteed by using both the p-N tracking formulae in [2] and the relative p-N equations of D with respect to S introduced here. This is because unlike current procedures, the implementation of these formulae and equations, accounts for the gravitational gradient along the trajectories of the narrow laser beams from S to D, see [1]. Additionally, our updated method accounts for the different behavior of the clocks on board of S and D, as this is the case in the usual procedures implemented for accurate space navigation and geolocation. Finally, we note that the new equations approach equations in [1] in case that the trajectories of S and D become close, so that the difference in the respective corrections to the current predicted locations are of higher order. Hence, both the starting strategy and the equations introduced here can become a part of the general procedure presented in [3,4] to deal with arbitrary shooting actions. Refere [...]