# ZERGOF: Zonal Earth Repeat Ground-track Orbits Finder

For mission designing tasks the whole procedure can be stated as follows. First,
a simple analytical approximation provides initial conditions of the approximate
nominal RGT orbit. These initial conditions are amenable to improvement with differential
corrections until finding a periodic orbit of a low-order, say gravity
model. The fast computation of the corresponding family of periodic orbits of
this model reveals the details of the real phase space and permits a better choice
of initial conditions. Then, new differential corrections converge to a true repeat orbit
of a higher-order gravity model. Finally, if required, the solution can be re-adjusted
by propagating the family of periodic orbits of the higher-order gravity problem.

Mission design typically starts from the experiment requirements, which constrain the orbital parameters to a subset of limited values. Then, a first order repeat ground track design is achieved including J2 to approximate the nominal solution. Further refinements of the orbital elements—usually in the presence of a medium degree zonal model, but sometimes including drag and lunar and solar perturbations—will provide the nominal orbit. The refinement procedure engineers normally use is based on interactive trial and error corrections that converge to a good nominal set of orbital elements. Here “good” indicates that the satellite does not drift substantially from the repeat ground trace over a certain interval of time. This refinement is generally performed via small adjustments to the semimajor axis and the eccentricity in a manual iterative sequence.

However, it is known that periodic solutions of the artificial satellite problem exist for either zonal or full gravity models when the problem is formulated in a rotating frame attached to the central body. These periodic orbits repeat exactly their ground track on the surface and, hence, are ideal candidates as nominal orbits for RGT missions. In addition, the technique provides the stability character and average orbital elements of the RGT orbit, which are the average values of the instantaneous orbital elements over one orbit period computed by Simpson’s rule. Most importantly perhaps, the technique is simple to automate and mission designers can select from full families of RGT orbits.

Mission design typically starts from the experiment requirements, which constrain the orbital parameters to a subset of limited values. Then, a first order repeat ground track design is achieved including J2 to approximate the nominal solution. Further refinements of the orbital elements—usually in the presence of a medium degree zonal model, but sometimes including drag and lunar and solar perturbations—will provide the nominal orbit. The refinement procedure engineers normally use is based on interactive trial and error corrections that converge to a good nominal set of orbital elements. Here “good” indicates that the satellite does not drift substantially from the repeat ground trace over a certain interval of time. This refinement is generally performed via small adjustments to the semimajor axis and the eccentricity in a manual iterative sequence.

However, it is known that periodic solutions of the artificial satellite problem exist for either zonal or full gravity models when the problem is formulated in a rotating frame attached to the central body. These periodic orbits repeat exactly their ground track on the surface and, hence, are ideal candidates as nominal orbits for RGT missions. In addition, the technique provides the stability character and average orbital elements of the RGT orbit, which are the average values of the instantaneous orbital elements over one orbit period computed by Simpson’s rule. Most importantly perhaps, the technique is simple to automate and mission designers can select from full families of RGT orbits.

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