Frozen apsidal line orbits around tiaxial Moon with coupling quadrupole nonlinearity

In this research paper, new families of frozen orbits of a satellite orbiting the oblate as well as triaxial Moon are investigated. The Hamiltonian of the problem is constructed including the zonal harmonic coefficients of the Moon's gravity field up to J4 and its triaxiality term J2,2. Using two successive canonical Lie transforms, the short and long periodic terms are eliminated from the Hamiltonian. The secular terms are retained up to first order plus the coupling quadrupole nonlinearity. New families of the critical roots of inclination are revealed, one of them is very close the polar orbits and the other is near to the usual critical inclination. The variations in the critical inclination due to the change in the eccentricity, in the semi-major axis and in the argument of periapsis are studied. A family of frozen apsidal line orbits is obtained and then is represented graphically. To guarantee these orbits, the solution for the periapsis argument is solved. This actually set out some restrictions on choosing the inclination satisfying the frozen argument of periapsis orbits. The perturbations in the critical inclination become significant for the high lunar orbits.