References

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.

  Class, number and I.A(0.5) 0.2009656237 0.2431076328 19.0134164290 19.086 16 I.A(0.5) 0.2138410831 0.0542938396 83.8472907647 118.112 104 I.B(0.75) 0.4101378717 0.1341894173 121.0976361440 183.067 102 I.A(2) 0.6649107583 0.8324167864 12.6489061509 42.121 8 II.D(2) 0.3057224330 0.5215124257 8.8237067653 64.567 12 I.A(4) 0.9911981217 0.7119472124 17.6507807837 276.852 24  

Table 1: The initial velocities and periods of some newly-found periodic orbits of the three-body system with unequal mass and zero angular momentum in the case of , and , when and , where is its scale-invariant period, is the length of the free group word (element). Here, the superscript indicates the case of the initial conditions with isosceles collinear configuration, due to the fact that there exist periodic orbits in many other cases.
Figure 1: (color online.) Brief overview of the six newly-found families of periodic three-body orbits.

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).

Figure 2: (color online.) The periodic orbits with the same free group element for = 0.5, 0.75 and 1.

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.

Figure 3: (color online). The scale-invariant average period versus . Symbols: computed results; line: .

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 ”

  Class and number I.A(0.5) 0.2869236336 0.0791847624 4.1761292190 4.538 4 I.A(0.5) 0.3420307307 0.1809369236 13.9153339459 9.063 8 I.A(0.5) 0.3697718457 0.1910065395 25.9441952945 13.095 12 I.A(0.5) 0.2009656237 0.2431076328 19.0134164290 19.086 16 I.A(0.5) 0.2613236072 0.2356235235 28.4358575383 23.513 20 I.A(0.5) 0.1908428490 0.1150772110 15.9682350284 22.361 20 I.A(0.5) 0.1579313682 0.0949852732 14.5766076405 22.363 20 I.A(0.5) 0.0979965852 0.0369408875 15.6059191780 27.112 24 I.A(0.5) 0.3589116510 0.0578397225 35.2777168591 27.120 24 I.A(0.5) 0.2066204352 0.1123859298 22.8770013381 30.956 28 I.A(0.5) 0.3095805649 0.1012188182 37.8353981553 36.122 32 I.A(0.5) 0.2935606362 0.2168613674 54.5846159117 41.571 36 I.A(0.5) 0.2614113685 0.1097599351 39.2849176561 45.204 40 I.A(0.5) 0.3049866810 0.0979042378 46.1065937257 45.210 40 I.A(0.5) 0.1644199050 0.0637816144 29.0215071279 45.244 40 I.A(0.5) 0.2698142826 0.0360688014 37.8687787781 45.458 40 I.A(0.5) 0.1451647294 0.0318334148 30.5079373557 49.973 44 I.A(0.5) 0.3467747647 0.0474429378 59.7722919460 49.973 44 I.A(0.5) 0.3025694869 0.0951546278 54.5401904272 54.294 48 I.A(0.5) 0.2726720005 0.0478754379 46.2148464304 54.543 48 I.A(0.5) 0.2997637007 0.0934329270 62.7105115603 63.376 56 I.A(0.5) 0.2747511246 0.0544869553 54.5744279508 63.626 56 I.A(0.5) 0.2867479329 0.0521752523 57.0633556930 63.626 56 I.A(0.5) 0.2172290935 0.0383448898 45.1666009592 63.631 56 I.A(0.5) 0.3108794721 0.1023369865 71.4799545005 67.698 60 I.A(0.5) 0.2979925625 0.0918951185 70.9842467059 72.456 64 I.A(0.5) 0.2366779591 0.0914177522 56.6833946453 72.485 64 I.A(0.5) 0.1628551705 0.0589464762 46.2097799724 72.484 64 I.A(0.5) 0.2763361520 0.0588302447 62.9447137981 72.706 64 I.A(0.5) 0.1936757357 0.0730232621 49.7181917085 72.488 64 I.A(0.5) 0.3017504100 0.1030778699 77.7653686390 76.789 68 I.A(0.5) 0.1671144104 0.0438815944 49.1675240282 77.185 68 I.A(0.5) 0.3274705985 0.0612651208 84.0143824473 77.185 68 I.A(0.5) 0.2668455153 0.0138391891 63.2419174415 77.282 68 I.A(0.5) 0.3220251063 0.0754954232 87.6821712800 81.574 72 I.A(0.5) 0.2965579937 0.0906370328 79.2439520137 81.536 72 I.A(0.5) 0.2775882955 0.0619333069 71.3240509215 81.786 72 I.A(0.5) 0.3558062278 0.0405108521 108.4971611336 86.349 76 I.A(0.5) 0.3060017590 0.0986219478 88.0890690057 85.876 76 I.A(0.5) 0.2689229383 0.0312527426 71.5697332821 86.372 76 I.A(0.5) 0.1317126561 0.0254909293 51.5736578935 86.350 76 I.A(0.5) 0.1428972736 0.0445901978 55.4783856796 90.717 80 I.A(0.5) 0.3132151994 0.1046181562 96.6632360131 90.178 80 I.A(0.5) 0.2954964679 0.0895434067 87.5354654697 90.615 80 I.A(0.5) 0.2749526022 0.0648656500 78.6571356819 90.866 80
 

Table S. 1: Initial conditions and periods of the periodic three-body orbits for class I.A in the case of , and , when and and by means of the search grid in the interval , where is its scale-invariant period, is the length of the free group element.

  Class and number I.A(0.5) 0.2153858894 0.0499108529 64.5903822323 90.874 80 I.A(0.5) 0.2770512888 0.1061164594 87.0243497655 94.961 84 I.A(0.5) 0.3213655238 0.0897415420 103.6411578855 94.961 84 I.A(0.5) 0.2706252379 0.0398693716 79.9076627886 95.459 84 I.A(0.5) 0.2786174957 0.2292777524 131.7357122012 103.058 88 I.A(0.5) 0.2469591661 0.0904031712 80.4191164038 99.719 88 I.A(0.5) 0.3228250487 0.0968450484 110.3291503458 99.273 88 I.A(0.5) 0.2954791318 0.1096218279 98.7725671166 99.273 88 I.A(0.5) 0.2945920946 0.0886163420 95.8143486677 99.694 88 I.A(0.5) 0.1945402732 0.0706830008 68.4003313978 99.727 88 I.A(0.5) 0.1889789414 0.0744407920 70.7522226916 104.113 92 I.A(0.5) 0.3011607127 0.0965243843 104.0576058514 104.046 92 I.A(0.5) 0.2938398485 0.0878089007 104.0941488385 108.772 96 I.A(0.5) 0.3323949017 0.0892628147 125.2963737946 108.361 96 I.A(0.5) 0.1303739201 0.0110880478 64.8930053804 109.158 96 I.A(0.5) 0.3388203286 0.0534486578 130.4329576227 113.548 100 I.A(0.5) 0.1528319560 0.0357128745 70.2840899540 113.548 100 I.A(0.5) 0.2931864677 0.0871031920 112.3665360659 117.850 104 I.A(0.5) 0.2712558612 0.0702892333 101.3437101345 118.101 104 I.A(0.5) 0.3227971415 0.0926025930 129.6770893175 117.455 104 I.A(0.5) 0.2862860474 0.1079517198 112.0449969617 117.455 104 I.A(0.5) 0.3078743981 0.0999722100 121.7962271737 117.455 104 I.A(0.5) 0.2138410831 0.0542938396 83.8472907647 118.112 104 I.A(0.5) 0.2672229781 0.0183673134 96.9436256746 118.196 104 I.A(0.5) 0.2762141023 0.1007012568 110.7531694029 122.210 108 I.A(0.5) 0.3151460645 0.0881775189 129.0111010994 122.210 108 I.A(0.5) 0.2816347029 0.1165460881 115.5464074541 121.753 108 I.A(0.5) 0.1897313629 0.0569798644 82.3194421170 122.548 108 I.A(0.5) 0.2365156031 0.0783479856 94.4496491279 122.548 108 I.A(0.5) 0.1892308321 0.0662636060 85.7557681263 126.964 112 I.A(0.5) 0.3167473355 0.0564409230 130.6376396627 127.209 112 I.A(0.5) 0.3063566174 0.0988758502 130.0686378697 126.543 112 I.A(0.5) 0.2686058325 0.0293186861 105.2715049896 127.286 112 I.A(0.5) 0.3115828668 0.1029439863 138.7657196429 130.849 116 I.A(0.5) 0.1926049152 0.0732402826 89.8940701833 131.356 116 I.A(0.5) 0.2678790398 0.0910268984 117.9225138007 136.004 120 I.A(0.5) 0.2769745628 0.1073466261 124.4856938122 135.627 120 I.A(0.5) 0.2817808026 0.0707431739 121.7428951384 136.256 120 I.A(0.5) 0.2887331369 0.1103223170 135.3850884074 139.941 124 I.A(0.5) 0.2820080025 0.0568015273 124.4362856628 140.873 124 I.A(0.5) 0.1635241804 0.0417506200 88.9883269271 140.761 124 I.A(0.5) 0.2167821354 0.0426860123 100.0973395584 140.885 124 I.A(0.5) 0.3111417615 0.0939330679 151.0501281052 144.714 128 I.A(0.5) 0.2821974598 0.0715109410 130.1650998679 145.334 128 I.A(0.5) 0.2277884543 0.0646616649 107.9084014726 145.349 128 I.A(0.5) 0.2708795826 0.0409737605 121.9472637204 145.460 128 I.A(0.5) 0.2744503104 0.0976059036 133.9471506576 149.452 132 I.A(0.5) 0.3114904704 0.0864935535 154.6502149192 149.452 132 I.A(0.5) 0.1794922768 0.0605219047 101.6064275160 154.197 136 I.A(0.5) 0.3128476865 0.1042021457 163.9473666361 153.329 136 I.A(0.5) 0.2718250586 0.0447887187 130.2931785836 154.545 136 I.A(0.5) 0.2773004224 0.0612425070 138.4568988662 159.033 140 I.A(0.5) 0.1882207909 0.0737375490 110.5070750985 162.981 144 I.A(0.5) 0.2829061443 0.0727791246 147.0223286431 163.488 144 I.A(0.5) 0.2176616157 0.0285927952 115.8358005700 163.643 144  

Table S. 2: Initial conditions and periods of the periodic three-body orbits for class I.A in the case of , and , when and and by means of the search grid in the interval , where is its scale-invariant period, is the length of the free group element.

  Class and number I.A(0.5) 0.3242506160 0.0631949983 180.1794835948 167.981 148 I.A(0.5) 0.2684432435 0.0282692084 138.9734917593 168.200 148 I.A(0.5) 0.2778616536 0.0625774643 146.8402992438 168.113 148 I.A(0.5) 0.3051413550 0.0931938899 174.3071401892 171.964 152 I.A(0.5) 0.1629885654 0.0522640671 109.4669718191 172.377 152 I.A(0.5) 0.1462535243 0.0217299404 105.3228173310 172.730 152 I.A(0.5) 0.2906493900 0.0841512530 161.9536048399 172.315 152 I.A(0.5) 0.3307845107 0.0282510145 186.8989238269 172.762 152 I.A(0.5) 0.2734339548 0.0504407955 147.0001395753 172.712 152 I.A(0.5) 0.1864663107 0.0314460085 116.6973116737 177.222 156 I.A(0.5) 0.3054857593 0.0982549763 180.3047709490 176.296 156 I.A(0.5) 0.1556984762 0.0375406067 110.2475386504 177.124 156 I.A(0.5) 0.2693793982 0.0338185849 147.3066947737 177.288 156 I.A(0.5) 0.1759573432 0.0582869204 118.5242718315 181.429 160 I.A(0.5) 0.3063956054 0.0665241524 179.3796751444 181.667 160 I.A(0.5) 0.1844792914 0.0664186417 124.2585706817 185.835 164 I.A(0.5) 0.2788305752 0.0647737865 163.6184241261 186.271 164 I.A(0.5) 0.3181202953 0.0262108895 189.0552591547 186.385 164 I.A(0.5) 0.2760366688 0.1001106086 172.0404786725 190.126 168 I.A(0.5) 0.2901045319 0.0834748902 178.4639514916 190.469 168 I.A(0.5) 0.2710039691 0.0415005602 163.9870417061 195.461 172 I.A(0.5) 0.2792522152 0.0656898524 172.0127626343 195.350 172 I.A(0.5) 0.2797022972 0.0856462692 179.7442254623 199.546 176 I.A(0.5) 0.1897491024 0.0464972761 133.3930483814 199.814 176 I.A(0.5) 0.2955617289 0.0815450298 190.9002964594 199.546 176 I.A(0.5) 0.2991146914 0.0928880688 196.3997389850 199.208 176 I.A(0.5) 0.2753246510 0.0561181124 172.0902983279 199.958 176 I.A(0.5) 0.1608928391 0.0423566492 128.6102391540 204.338 180 I.A(0.5) 0.2858473085 0.1123019709 194.7380952367 203.092 180 I.A(0.5) 0.1632254337 0.0372831480 128.8601185998 204.338 180 I.A(0.5) 0.2203920025 0.0720677540 149.5147366218 204.249 180 I.A(0.5) 0.1747313962 0.0572750530 135.8934495248 208.659 184 I.A(0.5) 0.2841883501 0.0749607778 189.2225836059 208.874 184 I.A(0.5) 0.2799959773 0.0672515431 188.8112079689 213.506 188 I.A(0.5) 0.1477960023 0.0331559448 130.9407612438 213.493 188 I.A(0.5) 0.2843861676 0.0752851658 197.6692863336 217.951 192 I.A(0.5) 0.2166248717 0.0438265532 155.0134625948 218.138 192 I.A(0.5) 0.2141903816 0.0534864829 158.0856536626 222.606 196 I.A(0.5) 0.2550856649 0.0780342905 185.5037127808 227.045 200 I.A(0.5) 0.2027586331 0.0547976409 156.8933680291 227.054 200 I.A(0.5) 0.2735238924 0.0507318572 197.3931765300 231.797 204 I.A(0.5) 0.2174237597 0.0354281968 170.8625404977 240.899 212 I.A(0.5) 0.1702863136 0.0542463019 163.7282370536 254.054 224 I.A(0.5) 0.2114296338 0.0633519827 183.7049132401 258.716 228 I.A(0.5) 0.2040531625 0.0308354022 183.6432714881 268.146 236 I.A(0.5) 0.2128981089 0.0560440442 196.4519555242 277.080 244 I.A(0.5) 0.1914804176 0.0167542245 198.8845878317 300.002 264  

Table S. 3: Initial conditions and periods of the periodic three-body orbits for class I.A in the case of , and , when and and by means of the search grid in the interval , where is its scale-invariant period, is the length of the free group element.

  Class and number I.B(0.5) 0.2374365149 0.2536896353 8.5581422789 7.262 6 I.B(0.5) 0.2707702758 0.2974619413 19.9858290667 11.480 10 I.B(0.5) 0.1804341862 0.0774390466 10.5764781985 15.808 14 I.B(0.5) 0.0548520001 0.3291535443 20.9927052014 19.231 14 I.B(0.5) 0.2817159946 0.3093138094 31.1291374576 15.374 14 I.B(0.5) 0.2679384847 0.0246961144 16.8511048757 20.457 18 I.B(0.5) 0.2674226718 0.2139289499 23.9372167355 20.903 18 I.B(0.5) 0.2878430093 0.3151477978 42.2778567687 19.118 18 I.B(0.5) 0.3030963188 0.0966599779 25.1036320618 24.876 22 I.B(0.5) 0.1099852485 0.0308448543 14.5241562996 24.961 22 I.B(0.5) 0.2291294485 0.2119828182 24.7437423714 25.221 22 I.B(0.5) 0.3692649167 0.0417694147 34.2771859316 24.965 22 I.B(0.5) 0.2737871583 0.0515706441 25.1965811355 29.542 26 I.B(0.5) 0.2988140236 0.0926281921 33.4227154398 33.958 30 I.B(0.5) 0.2788426894 0.0601828236 33.7944313342 38.623 34 I.B(0.5) 0.2790832581 0.2276230751 50.6192127242 39.773 34 I.B(0.5) 0.2163072511 0.0457824324 27.4550707427 38.627 34 I.B(0.5) 0.2631918196 0.0971022505 36.9742814503 43.036 38 I.B(0.5) 0.1940565085 0.0716923953 29.5220753291 43.054 38 I.B(0.5) 0.2790460219 0.0652447476 41.9537927832 47.703 42 I.B(0.5) 0.1166234512 0.0062473070 27.8908904718 47.784 42 I.B(0.5) 0.2941922385 0.0882015308 49.9756827423 52.116 46 I.B(0.5) 0.1482620649 0.0469161409 32.2082149709 52.162 46 I.B(0.5) 0.3383350929 0.0633820096 60.3349978637 52.165 46 I.B(0.5) 0.3086986007 0.1005760508 58.8260422524 56.455 50 I.B(0.5) 0.2804820502 0.0682367913 50.3545241044 56.781 50 I.B(0.5) 0.2928957198 0.0867806393 58.2512450352 61.194 54 I.B(0.5) 0.3121801357 0.0954940881 68.9135784414 65.543 58 I.B(0.5) 0.2815530004 0.0703169410 58.7667019161 65.859 58 I.B(0.5) 0.2702462061 0.0720377706 56.4003264775 65.859 58 I.B(0.5) 0.2131987197 0.0555522173 46.7195262030 65.865 58 I.B(0.5) 0.1832565230 0.0627595131 46.7521066510 70.290 62 I.B(0.5) 0.2919082649 0.0856563914 66.5195343482 70.272 62 I.B(0.5) 0.2703631586 0.0386904791 58.8880393273 70.458 62 I.B(0.5) 0.1859380756 0.0731374234 50.3601763743 74.679 66 I.B(0.5) 0.2489094150 0.0992895950 61.2653969718 74.663 66 I.B(0.5) 0.2823888686 0.0718580778 67.1889348357 74.936 66 I.B(0.5) 0.1615485916 0.0277748182 46.9823741559 74.980 66 I.B(0.5) 0.2911309221 0.0847384936 74.7822861358 79.349 70 I.B(0.5) 0.1771098445 0.0784887228 52.5734020100 79.040 70 I.B(0.5) 0.1838565013 0.0752899792 53.1629113586 79.040 70 I.B(0.5) 0.2143486812 0.0918266864 58.0738450139 79.038 70 I.B(0.5) 0.3247862862 0.0695897171 85.9588123017 79.387 70 I.B(0.5) 0.1610459823 0.0516375322 50.2179896145 79.382 70 I.B(0.5) 0.2722601340 0.0464077813 67.2341769532 79.543 70 I.B(0.5) 0.3192498609 0.0879575434 90.1594809343 83.709 74 I.B(0.5) 0.2830620767 0.0730518241 75.6195474605 84.014 74 I.B(0.5) 0.2905035826 0.0839717300 83.0409561905 88.427 78 I.B(0.5) 0.3002585565 0.1059795568 88.9553085721 88.032 78 I.B(0.5) 0.2174816082 0.0342966208 62.8453538957 88.634 78  

Table S. 4: Initial conditions and periods of the periodic three-body orbits for class I.B in the case of , and , when and and by means of the search grid in the interval , where is its scale-invariant period, is the length of the free group element.

  Class and number I.B(0.5) 0.2913964797 0.0479668990 80.7592905815 88.627 78 I.B(0.5) 0.2960647431 0.0951352334 90.5465691204 92.793 82 I.B(0.5) 0.2836170353 0.0740064147 84.0569632087 93.091 82 I.B(0.5) 0.2679928311 0.0763276805 79.4113786013 93.090 82 I.B(0.5) 0.2899831050 0.0833218568 91.2952846897 97.504 86 I.B(0.5) 0.3080355719 0.1000897653 100.8065558913 97.122 86 I.B(0.5) 0.1751508803 0.0576966004 63.5831421300 97.522 86 I.B(0.5) 0.2341908442 0.0837260648 75.0584041949 97.525 86 I.B(0.5) 0.2750441739 0.0553294312 83.9531808683 97.709 86 I.B(0.5) 0.3442855217 0.0202360484 113.5049294912 97.768 86 I.B(0.5) 0.3138785714 0.1054199562 109.2112794427 101.417 90 I.B(0.5) 0.2840833689 0.0747873466 92.4999175257 102.168 90 I.B(0.5) 0.1918308588 0.0726163450 69.5933264452 101.922 90 I.B(0.5) 0.2967400478 0.0807114885 102.3821088719 106.581 94 I.B(0.5) 0.1671740666 0.0535233977 68.2728344880 106.607 94 I.B(0.5) 0.3062135700 0.0987733794 109.0789742536 106.210 94 I.B(0.5) 0.2852343987 0.1063718482 100.6294915837 106.209 94 I.B(0.5) 0.3169586307 0.0723491117 111.3358326383 106.613 94 I.B(0.5) 0.2760984711 0.0582105683 92.3235074759 106.790 94 I.B(0.5) 0.2977432358 0.0916649424 108.5470086473 110.955 98 I.B(0.5) 0.2844794178 0.0754369919 100.9465701326 111.245 98 I.B(0.5) 0.3249845265 0.0202864165 116.6138500488 111.383 98 I.B(0.5) 0.2694680508 0.0342913622 92.5876783108 111.373 98 I.B(0.5) 0.2891747738 0.0822826126 107.7974854583 115.658 102 I.B(0.5) 0.2885302862 0.0933767620 113.1508640370 120.034 106 I.B(0.5) 0.2666426613 0.0786440441 102.3898928785 120.321 106 I.B(0.5) 0.2848187359 0.0759836961 109.3953593575 120.322 106 I.B(0.5) 0.1977132984 0.0525053406 82.0586018240 120.336 106 I.B(0.5) 0.2707787504 0.0405404036 100.9274275269 120.459 106 I.B(0.5) 0.2449409545 0.0850729560 99.3632704793 124.756 110 I.B(0.5) 0.1773510110 0.0248360734 80.6083002912 124.987 110 I.B(0.5) 0.2888572586 0.0818643534 116.0480068229 124.734 110 I.B(0.5) 0.2181512383 0.0833244460 94.9920652210 129.161 114 I.B(0.5) 0.2904092593 0.1083815356 125.0092177942 128.699 114 I.B(0.5) 0.2851109699 0.0764474074 117.8446993799 129.398 114 I.B(0.5) 0.2703553256 0.0851792459 116.2605246289 133.811 118 I.B(0.5) 0.2985222840 0.0787345637 129.2429556733 133.811 118 I.B(0.5) 0.3138600006 0.0908763661 140.5350980635 133.464 118 I.B(0.5) 0.3135219744 0.1049833885 142.8584537333 132.992 118 I.B(0.5) 0.3098615763 0.0844309289 141.5123187727 138.190 122 I.B(0.5) 0.2853633746 0.0768427234 126.2931104761 138.475 122 I.B(0.5) 0.2176335170 0.0299228620 98.1757759947 138.640 122 I.B(0.5) 0.2883498729 0.0811835015 132.5546327228 142.888 126 I.B(0.5) 0.1749022486 0.0766378437 94.0590542640 142.271 126 I.B(0.5) 0.1636510704 0.0474589703 90.6558775234 142.963 126 I.B(0.5) 0.2150995281 0.0509041474 101.7118032345 143.121 126 I.B(0.5) 0.2822851529 0.0922191464 135.1262821798 147.271 130 I.B(0.5) 0.2855816329 0.0771807489 134.7392792763 147.552 130 I.B(0.5) 0.2947320378 0.0887615187 141.6521915390 147.271 130 I.B(0.5) 0.3344339309 0.0263022352 162.8324323639 147.763 130  

Table S. 5: Initial conditions and periods of the periodic three-body orbits for class I.B in the case of , and , when and and by means of the search grid in the interval , where is its scale-invariant period, is the length of the free group element.

  Class and number I.B(0.5) 0.3258205442 0.0948987232 170.1648469173 151.183 134 I.B(0.5) 0.2887071810 0.1116133434 146.5524883273 151.183 134 I.B(0.5) 0.3136039168 0.0546949113 153.7578002255 152.216 134 I.B(0.5) 0.2881475675 0.0809077918 140.8129334279 151.965 134 I.B(0.5) 0.2690556508 0.0320231034 126.2887308493 152.287 134 I.B(0.5) 0.2857703636 0.0774702399 143.1821020744 156.629 138 I.B(0.5) 0.2845169706 0.0979582675 149.9521586362 160.710 142 I.B(0.5) 0.3360387142 0.0408766303 180.7765714891 161.357 142 I.B(0.5) 0.2667815506 0.0845599684 138.0065476191 161.041 142 I.B(0.5) 0.2700536880 0.0372395731 134.6253858410 161.374 142 I.B(0.5) 0.1620074707 0.0567511584 105.1701997563 165.454 146 I.B(0.5) 0.2846001734 0.0542198926 147.7254345023 165.875 146 I.B(0.5) 0.2937225183 0.0876854814 158.2087324044 165.427 146 I.B(0.5) 0.2752695212 0.0559644171 142.7112029555 165.875 146 I.B(0.5) 0.2170568347 0.0402768809 117.8057908660 165.889 146 I.B(0.5) 0.3256765948 0.0919189282 189.6279855460 169.365 150 I.B(0.5) 0.2878232371 0.0804604016 157.3437190277 170.118 150 I.B(0.5) 0.2709508831 0.0412767654 142.9671399385 170.460 150 I.B(0.5) 0.2325404803 0.0634100827 128.0692487359 170.359 150 I.B(0.5) 0.2777928735 0.0911581231 157.0896063985 174.505 154 I.B(0.5) 0.1884011033 0.0613519263 120.4032477307 179.229 158 I.B(0.5) 0.2876940968 0.0802805105 165.6171419591 179.195 158 I.B(0.5) 0.2981478673 0.0920364541 175.3905472876 178.871 158 I.B(0.5) 0.2717607209 0.0445435233 151.3128521285 179.545 158 I.B(0.5) 0.2694901377 0.0990271302 157.6461325706 178.868 158 I.B(0.5) 0.2780311897 0.0798240174 163.4425288092 183.859 162 I.B(0.5) 0.1535774570 0.0490036967 114.5530130654 183.703 162 I.B(0.5) 0.3084553159 0.0875379516 191.9788446279 187.951 166 I.B(0.5) 0.2819693119 0.0959873338 173.1019417858 187.951 166 I.B(0.5) 0.2724981752 0.0472634419 159.6637122720 188.629 166 I.B(0.5) 0.2812272902 0.0696965706 167.8845519750 188.499 166 I.B(0.5) 0.2851403139 0.1114556037 183.1661242112 191.850 170 I.B(0.5) 0.2925384369 0.0863782875 183.0243366435 192.661 170 I.B(0.5) 0.1940041780 0.0442273745 132.9965797155 197.594 174 I.B(0.5) 0.2874879398 0.0799909840 182.1812064952 197.349 174 I.B(0.5) 0.2743698810 0.0902336260 179.0226701835 201.738 178 I.B(0.5) 0.2691006601 0.0969556708 180.8849024635 206.107 182 I.B(0.5) 0.1872185341 0.0408400425 136.8385909602 206.678 182 I.B(0.5) 0.2708877517 0.0813214333 182.9605968884 211.089 186 I.B(0.5) 0.2632104741 0.0837121394 182.1621606443 215.501 190 I.B(0.5) 0.2821312739 0.0713901022 193.1416258963 215.731 190 I.B(0.5) 0.2394605930 0.0614826122 168.8258165112 220.374 194 I.B(0.5) 0.1924591719 0.0733547400 150.2890500922 219.658 194 I.B(0.5) 0.2124129905 0.0567064578 159.2830391611 224.832 198 I.B(0.5) 0.2716732997 0.0442108364 193.3516822884 229.546 202 I.B(0.5) 0.1784558095 0.0599818427 156.5079579799 238.104 210 I.B(0.5) 0.2168487625 0.0421574292 172.7342330299 243.143 214 I.B(0.5) 0.2116222333 0.0574328794 178.3098811885 252.068 222 I.B(0.5) 0.1734501113 0.0584514203 172.5037487239 265.336 234 I.B(0.5) 0.1773937613 0.0586759020 173.8818026019 265.336 234 I.B(0.5) 0.1589938384 0.0147834329 168.2376674277 270.519 238 I.B(0.5) 0.1732220167 0.0304907626 193.6476848963 302.170 266  

Table S. 6: Initial conditions and periods of the periodic three-body orbits for class I.B in the case of , and ,