Frozen orbits around a variable mass

Our work is concerning to find the frozeen orbit conditions for the motion about an oblate varying mass centeral body. The Hamiltonians due to the vriation of the centeral body mass and J₂ zonal harmonics, were constructed in terms of the Delaunay canonical variables. Since the time is appearing exeplicitly in the Hamiltonian function, the phase space was extended by introducing a new pair of canonical variables, \(\left({l}_{4}, {L}_{4}\right)\) $\left(l_4,,,L_4\right)$ . The first one is, \({l}_{4}\) $l_4$ , assigned as the variable mass while the second, \({L}_{4}\) $L_4$ , is its conjugate momentum. The Hamiltonian of the problem was developed in the expandable form, in terms of J₂ as small parameter, to be suitable for the used transformation method. Short and long periodic terms were eliminated, in successive, to obtain the secular perturbation effects. The conditions to freeze the orbital elements were derived and graphically illustrated.