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 ( \ll 1 ) , 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 } .