More than six hundred new families of Newtonian periodic planar collisionless three-body orbits

The famous three-body problem can be traced back to Isaac Newton in the 1680 s. In the 300 years since this "three-body problem"was first recognized, only three families of periodic solutions had been found, until 2013 when ˇSuvakov and Dmitraˇsinovi′c [Phys.Rev. Lett. 110, 114301(2013)] made a breakthrough to numerically find 13 new distinct periodic orbits, which belong to 11 new families of Newtonian planar three-body problem with equal mass and zero angular momentum. In this paper, we numerically obtain 695 families of Newtonian periodic planar collisionless orbits of three-body system with equal mass and zero angular momentum in case of initial conditions with isosceles collinear configuration, including the well-known figure-eight family found by Moore in 1993, the 11 families found by ˇSuvakov and Dmitraˇsinovi′c in 2013, and more than 600 new families that have never been reported, to the best of our knowledge. With the definition of the average period T = T=Lf, where Lf is the length of the so-called "free group element", these 695 families suggest that there should exist the quasi Kepler's third law T* ≈ 2:433 ± 0:075 for the considered case, where T*= T|E|^(3/2) is the scale-invariant average period and E is its total kinetic and potential energy,respectively. The movies of these 695 periodic orbits in the real space and the corresponding close curves on the "shape sphere"can be found via the website: http://numericaltank.sjtu.edu.cn/three-body/three-body.htm.

Natural coupled orbit-attitude periodic motions in the perturbed-CRTBP including radiated primary and oblate secondary

This study investigated periodic coupled orbit-attitude motions within the perturbed circular restricted three-body problem(P-CRTBP)concerning the perturbations of a radiated massive primary and an oblate secondary.The radiated massive primary was the Sun,and each planet in the solar system could be considered an oblate secondary.Because the problem has no closed-form solution,numerical methods were employed.Nevertheless,the general response of the problem could be non-periodic or periodic,which is significantly depended on the initial conditions of the orbit-attitude states.Therefore,the simultaneous orbit and attitude initial states correction(SOAISC)algorithm was introduced to achieve precise initial conditions.On the other side,the conventional initial guess vector was essential as the input of the correction algorithm and increased the probability of reaching more precise initial conditions.Thus,a new practical approach was developed in the form of an orbital correction algorithm to obtain the initial conditions for the periodic orbit of the P-CRTBP.This new proposed algorithm may be distinguished from previously presented orbital correction algorithms by its ability to propagate the P-CRTBP family orbits around the Lagrangian points using only one of the periodic orbits of the unperturbed CRTBP(U-CRTBP).In addition,the Poincarémap and Floquet theory search methods were used to recognize the various initial guesses for attitude parameters.Each of these search methods was able to identify different initial guesses for attitude states.Moreover,as a new innovation,these search methods were applied as a powerful tool to select the appropriate inertia ratio for a satellite to deliver periodic responses from the coupled model.Adding the mentioned perturbations to the U-CRTBP could lead to the more accurate modeling of the examination environment and a better understanding of a spacecraft's natural motion.A comparison between the orbit-attitude natural motions in the unperturbed and perturbed models was also conducted to show this claim.