Orbital Dynamics with the Gravitational Perturbation due to a Disk

The secular behavior of an orbit under the gravitational perturbation due to a two-dimensional uniform disk is studied in this paper, through analytical and numerical approaches. We develop the secular approximation of this problem and obtain the averaged Hamiltonian for this system first. We find that, when the ratio of the semimajor axes of the inner orbit and the disk radius takes very small values ($\ll1$), and if the inclination between the inner orbit and the disk is greater than the critical value of $30^\circ$, the inner orbit will undergo the (classical) Lidov-Kozai resonance in which variations of eccentricity and inclination are usually very large and the system has two equilibrium points at $\omega=\pi/2,3\pi/2$ ($\omega$ is the argument of perihelion). The critical value will slightly drop to about $27^\circ$ as the ratio increases to 0.4. However, the secular resonances will not occur for the outer orbit and the variations of the eccentricity and inclination are small. When the ratio of the orbit and the disk radius is nearly $1$, there are many more complicated Lidov-Kozai resonance types which lead to the orbital behaviors that are different from the classical Lidov-Kozai case. In these resonances, the system has more equilibrium points which could appear at $\omega=0,\pi/2,\pi,3\pi/2$, and even other values of $\omega$. The variations of eccentricity and inclination become relatively moderate, moreover, in some cases the orbit can be maintained at a highly inclined state. In addition, a analysis shows that a Kuzmin disk can also lead to the (classical) Lidov-Kozai resonance and the critical inclination is also $30^\circ$.