Amplitude of the Lidov-Kozai i and e oscillations in asteroid families

Asteroid families were used to study secular perturbations induced by the Lidov-Kozai mechanism (LKM). The LKM represents coupled long-period oscillations of the inclination i and the eccentricity e. These oscillations depend on the argument of the perihelion omega and become substantial for high inclinations and large eccentricities. After excluding classical secular perturbations, the LKM oscillations of the elements became visible very clearly in the distributions of orbital elements (sin i,omega) and (e,omega). These oscillations can be approximated by the functions A(sin) i(sin) (2 omega + 90 omega) and Aesin (2 omega -90 degrees), respectively, and the amplitudes of the oscillations A(sin) i and A(e) can be easily obtained by the least-squares method. By excluding the LKM oscillations, we can calculate the proper elements i(p) and e(p). Asteroid families that have different proper inclinations and eccentricities were used to study the amplitudes of the LKM i and e oscillations. As a result, it was found that the net amplitude A = root A(sin) (2)(i) + A(e)(2) increases with increasing ip and ep and can be approximated by a power law of the product e(p)sin i(p). If the amplitude A is known, the amplitudes of the e and i oscillations can be calculated as A(e) = Acos a and Asin i = Asin a, where tan alpha = -e(p)(1 - sin(2) i(p))/sin i(p)(1 - e(p)(2)). It follows that the relationship between the amplitudes is approximately described as A(sini)/A(e) approximate to e(p)/sin i(p).