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Frozen orbits are always important foci of orbit design because of theirvaluable characteristics that their eccentricity and argument of pericentreremain constant on average. This study investigates quasi-circular frozenorbits and examines their basic nature analytically using two differentmethods. First, an analytical method based on Lagrangian formulations isapplied to obtain constraint conditions for Martian frozen orbits. Second, Lietransforms are employed to locate these orbits accurately, and draw thecontours of the Hamiltonian to show evolutions of the equilibria. Both methodsare verified by numerical integrations in an 80\times80 Mars gravity field. Thesimulations demonstrate that these two analytical methods can provide accurateenough results. By comparison, the two methods are found well consistent witheach other, and both discover four families of Martian frozen orbits: threefamilies with small eccentricities and one family near the criticalinclination. The results also show some valuable conclusions: for the majorityof Martian frozen orbits, argument of pericentre are kept at 270 degreesbecause J3 has the same sign with J2; while for a minority of ones with lowaltitude and low inclination, argument of pericentre are possible to be kept at90 degrees because of the effect of the higher degree odd zonals; for thecritical inclinations cases, argument of pericentre can also be kept at 90degrees. It is worthwhile to note that there exist some special frozen orbitswith extremely small eccentricity, which could provide much convenience forreconnaissance. Finally, the stability of Martian frozen orbits is estimatedbased on the trace of the monodromy matrix. The analytical investigations canprovide good initial conditions for numerical correction methods in the morecomplex models.

Analytical investigations of quasi-circular frozen orbits in the Martian gravity field