The 1223 new periodic orbits of planar three-body problem

with unequal mass and zero angular momentum

Xiaoming Li and Yipeng Jing and Shijun Liao

School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiaotong University, China

Dept. of Astronomy, School of Physics and Astronomy, Shanghai Jiaotong University, China

Ministry-of-Education Key Laboratory in Scientific and Engineering Computing, Shanghai 200240, China

* The corresponding author: sjliao@sjtu.edu.cn

Abstract We present 1349 families of Newtonian periodic planar three-body orbits with unequal mass and zero angular momentum and the initial conditions in case of isosceles collinear configurations. These 1349 families of the periodic collisionless orbits can be divided into seven classes according to their geometric and algebraic symmetries. Among these 1349 families, 1223 families are entirely new, to the best of our knowledge. Furthermore, some periodic orbits have the same free group element for different mass. We find that the scale-invariant average period linearly increases with the mass of one body when the masses of the other two are constant. It is suggested that the masses of bodies may play an important role in periodic three-body systems. The movies of these periodic orbits are given on the website http://numericaltank.sjtu.edu.cn/three-body/three-body-unequal-mass.htm .

The study of periodic three-body problem has received lots of attention in recent years. In 2013, Šuvakov and Dmitrašinović Šuvakov and Dmitrašinović (2013) found 13 new distinct collisionless periodic orbits of Newtonian planar three-body problem with equal mass and zero angular momentum. In 2014, Dmitrašinović et al. Dmitrašinović et al. (2014) investigated gravitational waves from these periodic three-body systems. In the same year, Iasko and OrlovIasko and Orlov (2014) found nine new close to periodic orbits of three-body problem with equal mass and Šuvakov (Dmitrašinović and Šuvakov, 2014) found eleven solutions in the vicinity of the figure-eight orbits with equal mass. In 2015, Hudomal Hudomal (2015) reported 25 families of periodic orbits Newtonian planar three-body problem with equal mass, including the 11 families found in Šuvakov and Dmitrašinović (2013). In 2016, Rose Rose (2016) gained 90 periodic planar collisionless orbits with equal mass in case of isosceles collinear configurations. Recently, Li and Liao Li and Liao (cept) found more than six hundred new families of planar three-body problem with equal mass. These researchers focused on periodic three-body problem with equal mass. However, the three-body problem with unequal mass is more general. In 2002, Galán et al. Galán et al. (2002) studied stability properties of figure-eight Moore (1993)Chenciner and Montgomery (2000) as the masses are unequal. Doedel et al. Doedel et al. (2003) gained periodic orbits from the figure-eight as the mass of one body is varied. In 2015, Yan et al. Yan et al. (2015) investigated the spatial isosceles three-body problem with unequal masses in case of one body moves up and down on a vertical line. However, little attention has been paid to search for the collisionless periodic orbits of the planar three-body problem with unequal mass and zero angular momentum. In this paper, we present results of collisionless periodic orbits in Newtonian planar three-body problem with unequal mass and zero angular momentum.

The motions of Newtonian planar three-body system can be described by Newtonian second law and gravitational law: where and denote vector position and mass of the th body , denotes the Newtonian gravity coefficient, and the dot denotes the derivative with respect to the time , respectively. We consider a planar three-body system with zero angular momentum and unequal mass () in the case of and the initial conditions in case of the isosceles collinear configurations:

(1) |

With these configurations, if with denotes a periodic orbit with the period of a three-body system, then

(2) |

has a same periodic orbit with the period for arbitrary . Therefore, without loss of generality, we consider and is varied. Note that 11 families were found by Šuvakov and Dmitrašinović Šuvakov and Dmitrašinović (2013) and more than six hundred new families of periodic orbits were found by Li and Liao (Li and Liao, cept) in case of . In this paper, we search for periodic orbits in case of .

For any given , the orbits are determined by the four parameters . Write . A periodic solution with the period is the root of the equation , where is unknown.

Firstly, we use the grid search method to find approximated initial conditions to satisfy the equation . We set the initial positions and and we search for initial conditions in a square plane: and . We set 4000 points in each dimension and thus have 16 million grid points in the square search plane. With these different 16 million initial conditions, the motion equations are integrated up to the time by means of the ODE solver dop853 developed by Hairer et al. (Hairer et al., 1993), which is based on an explicit Runge-Kutta method of order 8(5,3) in double precision with adaptive step size control. The corresponding initial conditions and the period are chosen as the candidates when the return proximity function

(3) |

is less than .

Secondly, we improve these candidates by means of the Newton-Raphson method (Farantos, 1995; Lara and Pelaez, 2002; Abad et al., 2011). At this stage, the motion equations are solved numerically by means of the same ODE solver dop853 Hairer et al. (1993). The candidates are corrected until the level of the return proximity function (3) is less than . As mentioned by in Li and Liao Li and Liao (cept), many periodic orbits might be lost by means of traditional algorithms in double precision. Thus, we further integrate the motion equations by means of “Clean Numerical Simulation” (CNS) (Liao, 2009, 2014; Liao and Wang, 2014; Liao and Li, 2015) with negligible numerical noises in a long enough interval of time, which is based on the arbitrary order of Taylor expansion method (Barton et al., 1971; Corliss and Chang, 1982; Chang and Corhss, 1994; Barrio et al., 2005) in arbitrary precision (Oyanarte, 1990; Viswanath, 2004), plus a convergence check by means of an additional computation with even smaller numerical noises. A periodic orbit is found when the level of the return proximity function (3) is less than . Note that the initial positions will depart from a little.

It is well-known that, if denotes a periodic orbit with the period of a three-body system, then

(4) |

is also a periodic orbit with the period for arbitrary . Thus, through coordinate transformation and then the scaling of the spatial and temporal coordinates, we can always enforce (-1,0), (1,0) and (0,0) as the initial positions of the body-1, 2 and 3, respectively. In this way, the periodic orbits are only dependent upon two physical parameters (,), the initial velocity of Body-1.

We identify these periodic orbits by means of Montgomery’s topological identification and classification method (Montgomery, 1998). The positions and of the three-body corresponds to a unit vector in the so-called “shape sphere” with the Cartesian components

where , and the hyper-radius . Then a periodic orbit is associated with a closed curve on the shape sphere, which can be characterized by its topology with three punctures (two-body collision points). With one of the punctures as the “north pole”, the sphere can be mapped onto a plane by a stereographic projection. And a closed curve can be mapped onto a plane with two punctures and its topology can be described by the so-called “free group element” (word) with letters (a clockwise around right-hand side puncture), (a counter-clockwise around left-hand side puncture) and their inverses and . For details, please refer to Dmitrašinović and Šuvakov (2014).

The periodic orbits can be divided into different classes on basis of their geometric and algebraic symmetries Šuvakov and Dmitrašinović (2013). There are two types of geometric symmetries in the shape space: (I) the reflection symmetries of two orthogonal axes — the equator and the zeroth meridian passing through the “far” collision point; and (II) a central reflection symmetry about one point — the intersection of the equator and the aforementioned zeroth meridian. Besides, Šuvakov and Dmitrašinović Šuvakov and Dmitrašinović (2013) mentioned three types of algebraic exchange symmetries for the free group elements: (A) the free group elements are symmetric with and , (B) free group elements are symmetric with and , and (C) free group elements are not symmetric under either (A) or (B). However, in this paper, we find a new algebraic symmetry class (D) with free group words symmetric under and . And then (C) can be regarded as free group words are not symmetric under one of (A), (B) and (D). We observe that the algebraic symmetry class (D) always corresponds to the geometric class (II) for present orbits.

Within the period , we find 565, 401, 237, 85, 35, 17 and 9 families of periodic orbits in case of = 0.5, 0.75, 2, 4, 5, 8 and 10, respectively. Among these 1349 families of periodic orbits, 1223 families of periodic orbits are entirely new, to the best of our knowledge. These 1349 families of the periodic collisionless orbits can be divided into seven classes: I.A, I.B, I.C, II.A, II.B, II.C amd II.D, as listed in Table S I-LXXXI in Supplementary material see the Supplementary material (). The initial velocities of the periodic orbits are listed in Tables S I-XXIX in see the Supplementary material (). The free group elements of the 1349 families are listed in Table S XXIX-LXXXI in see the Supplementary material (). Due to the limited length, only six new families are listed in Table 1, and their orbits in real space are shown in FIG. 1. Note that Doedel et al. Doedel et al. (2003) found periodic orbits which have the same topology with the figure-eight as the mass of one body is varied. Here we also find that some periodic orbits have the same free group element for different . For instance, as shown in FIG. 2, the periodic orbits have the same free group element () for = 0.5, 0.75 and 1. The movies of these periodic orbits are given on the website: http://numericaltank.sjtu.edu.cn/three-body/three-body-unequal-mass.htm .

As mentioned by Li and Liao Li and Liao (cept), the scale-invariant average period is approximately equal to a universal constant in case of , i.e. , with the definition of the average period , where is the length of free group element of periodic orbit of a three-body system. Here the scale-invariant average period is equal to 1.14, 1.77, 2.43, 5.39, 11.53, 14.64, 23.99 and 30.25 for = 0.5, 0.75, 2, 4, 5, 8 and 10, respectively. It is found that the scale-invariant average period agree well with the formula as shown in FIG. 3. The standard deviation is = 0.135. It is suggested that the scale-invariant average period linearly increases with .

In this paper, we totally find 1349 families of Newtonian periodic planar three-body orbits with unequal mass and zero angular momentum and the initial conditions in case of isosceles collinear configurations. These 1349 families of the periodic collisionless orbits can be divided into seven classes according to their geometric and algebraic symmetries. Among these 1349 families, 1223 families are entirely new, to the best of our knowledge. Furthermore, some periodic orbits have the same free group element for different . It is found that the scale-invariant average period linearly increases with . It is suggested that the masses of bodies may play an important role in periodic three-body systems.

This work was carried out on TH-2 at National Supercomputer Centre in Guangzhou, China. It is partly supported by National Natural Science Foundation of China (Approval No. 11432009).

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Supplementary information for

“The 1223 new periodic orbits of planar three-body problem

with unequal mass and zero angular momentum ”