Kepler’s third law of n-body periodic orbits in a Newtonian gravitation field

One of the central and most vivid problems of celestial mechanics in the 18th and 19th centuries was the motion description of the Sun-Earth-Moon system under the Newtonian gravitation field (Figure 1(a)). Notable work was done by Euler (1760), Lagrange (1776), Laplace (1799), Hamilton (1834), Liouville (1836), Jacobi (1843), and Poincare (1889) ´ [1] and Xia (1992) [2]. The study of the motion between the two bodies was solved by Kepler (1609) and Newton (1687) early in the 17th century. For the elliptic periodic orbit of 2- body system, Kepler’s third law of the two-body system [3] is given by T|E| 3/2 = π√ 2 Gm1m2 √ m1m2 m1+m2 , where the gravitation constant, G = 6.673 × 10−11m3 kg−1 s −2 , the orbit period, T, the total energy of the 2-body system, |E|, and point masses m1 and m2 (Figure 1(b)). However, the 3-body system (Figure 1(c)) cannot be solved analytically because unlike the 2-body problem, the 18 variables that describe the system cannot be reduced to a single variable. Simplification of the two-body problem was allowed by invariance and conserved quantities as “first integrals”. It was proven impossible to reduce the 18 variables of the 3-body problem in order to produce an analytic solution. Notwithstanding that the analytic solution cannot be found, it is possible to find a numerical solution for the 3- body problem, in which the study of the periodic 3-body orbit has received particular attention in recent years [4-11]