Constant orbit elements under the third body effect

An analysis to determine solutions with constant orbit elements has been carried out through a vectorial formulation of the perturbation equations, under the long-term influence due to the attraction of a disturbing body moving over an inclined elliptical orbit. After having gained a frozen orbital plane by assuming an orbital pole parallel or perpendicular to the perturbing body pole, the feasibility to get a frozen condition also on eccentricity or argument of pericentre has been demonstrated and several solutions have been proposed. Moreover, when the orbital pole is perpendicular to the perturbing body pole, a prime integral of motion, linking orbit eccentricity and argument of pericentre, has been retrieved. This prime integral has permitted the identification of solutions characterised by slow variations of eccentricity. A study to obtain orbits at constant eccentricity or argument of pericentre has also been carried out, regardless of the orbital plane evolution. This has highlighted how, while the solutions with a frozen apsidal line have to be determined by means of numerical methods, not pursued in this paper, the ones characterised by a null variation of eccentricity can be retrieved analytically. Examples, for a probe orbiting Mercury, have also been presented.