One family of 13315 stable periodic orbits of the non-hierarchical unequal-mass triple system
OOne family of 13315 stable periodic orbits of the non-hierarchical unequal-mass triple system
Xiaoming Li , , Xiaochen Li , & Shijun Liao , MOE Key Laboratory of Disaster Forecast and Control in Engineering. School of Mechanics andConstruction Engineering, Jinan University, Guangzhou 510632, China Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technol-ogy, Cambridge, Massachusetts 02139, USA School of Civil Engineering and Transportation, South China University of Technology,Guangzhou 510641, China Department of Engineering Science, University of Oxford, Oxford OX1 3PJ, UK Center of Advanced Computing, School of Naval Architecture, Ocean and Civil Engineering,Shanghai Jiaotong University, Shanghai 200240, China School of Physics and Astronomy, Shanghai Jiaotong University, Shanghai 200240, China
The three-body problem has been studied for more than three centuries
1, 2 , and has receivedmuch more attention in recent years . It shows complex dynamical phenomena due to themutual gravitational interaction of the three bodies. Triple systems are common in astron-omy, but all observed periodic triple systems are hierarchical up till now . It is traditionallybelieved that bound non-hierarchical triple systems are almost unstable and disintegrate intoa stable binary system and a single star , and thus stable periodic orbits of non-hierarchicaltriple systems are rather scarce. Here we report one family of 13315 stable periodic orbits a r X i v : . [ n li n . C D ] J u l f the non-hierarchical triple system with unequal mass. Compared with the narrow massregion (only − ) of the stable figure-eight solution , our newly-found stable periodic orbitscan have fairly large mass region. It is found that many of these newly-found stable periodicorbits have the mass ratios close to those of the hierarchical triple systems that have beenmeasured by the astronomical observation. It implies that these stable periodic orbits of thenon-hierarchical triple system with distinctly unequal masses can be quite possibly observedin practice. Our investigation also suggests that there should exist an infinite number of stableperiodic orbits of non-hierarchical triple systems with distinctly unequal masses. Obviously,these stable periodic orbits of the non-hierarchical unequal-mass triple system have broadimpact for the astrophysical scenario: they could inspire the theoretical and observationalstudy of the non-hierarchical triple system, the formation of triple stars , the gravitationalwaves pattern and the gravitational waves observation of the non-hierarchical triple sys-tem. Triple systems are common and key objectives in astrophysics . Although the three-bodyproblem has been investigated for more than three hundred years
1, 2 , it is still a challenging andopen question for astrophysicist because of its inherent chaotic characteristics . Currently, basedon the assumption of ergodicity, Stone and Leigh gave a statistical solution to the non-hierarchicalchaotic three-body system. It is traditionally believed that bound non-hierarchical triple systemsare always unstable and disintegrate into a stable binary system and a single star . Therefore,periodic orbits of the three-body problem are extremely precious since they are the only wayto penetrate the fortress which was previously considered to be inaccessible . However, only2hree families of periodic orbits had been found in more than 300 years until ˇSuvakov and Dmi-traˇsinovi´c numerically found 13 distinct periodic orbits of the three-body problem with equalmass in 2013. Li and Liao further found more than six hundred new families of periodic orbitsof the three-body system with equal mass. Li et al. gained more than one thousand new familiesof periodic orbits of the three-body system with two equal-mass bodies. Among the about twothousand new families of periodic orbits of the three-body system, dozens of linear stable periodicorbits were found for the non-hierarchical triple system
15, 16 , however, some of them have threeequal-mass bodies and the others have two equal-mass bodies . The famous figure-eight solu-tion
17, 18 of the equal-mass triple system is non-hierarchical and linear stable . Unfortunately,the stable mass region of the figure-eight solution is very narrow (only − ) . That is to say thefigure-eight solution is stable only when three bodies have almost equal mass, so the probability ofobserving this periodic orbit is extremely low in practice. So far, non-hierarchical periodic triplestars have not been found in the astronomical observation yet.In this letter, we focus on periodic orbits of non-hierarchical triple system with unequal masses. The motion of the Newtonian planar three-body problem is described by the differentialequations ¨ r i = (cid:88) j =1 ,j (cid:54) = i Gm j ( r j − r i ) | r i − r j | , (1)where m i and r i are mass and position of the i th body ( i = 1 , , , G is the Newtonian gravityconstant, respectively. Without loss of generality, we set the gravitational constant G = 1 byproperly choosing a characteristic mass M , a characteristic spatial length R and a characteristictime T ∗ . 3ontgomery proofed that all three-body orbits of zero angular momentum have syzygies(i.e., collinear instant of three bodies) except for the Lagrange’s solution. Thus, it is reasonableto consider initial conditions with the collinear configuration . In this letter, we investigate theunequal-mass triple system with the initial positions r (0) = ( x , , r (0) = ( x , , r (0) =( x , and the initial velocities ˙ r (0) = (0 , v ) , ˙ r (0) = (0 , v ) , ˙ r (0) = (0 , v ) , which areperpendicular to the straight line formed by three bodies.The first step to achieve our goal is to find periodic orbits of the equal-mass triple systemwith the collinear initial condition configuration mentioned above. We numerically search forperiodic orbits of the three-body problem with equal mass and zero angular momentum by meansof the grid search method, the Newton-Raphson method
24, 25 and the numerical strategy, namely theclean numerical simulation (CNS) (see Methods). We find that one equal-mass periodic orbithas good stability. The initial condition of this periodic orbit is r (0) = ( x , , r (0) = (1 , , r (0) = (0 , , ˙ r (0) = (0 , v ) , ˙ r (0) = (0 , v ) , ˙ r (0) = (0 , − ( m v + m v ) /m ) , where x = − . , v = 1 . , v = 0 . and the period T = 7 . and m = m = m = 1 . Note that it holds m v x + m v x + m v x =0 . Using the homotopy classification method
13, 30 , the free group element of this periodic orbit is bABabaBAba . This periodic orbit has the same free group element with the moth-I orbit , buttheir periodic orbits are different. Note that, for the astrophysical three-body system, the massesof the bodies are rarely equal. Thus, using this as a starting point, we investigate periodic orbits ofthe unequal-mass triple system by means of the numerical continuation method (see Methods).4 y -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.5-0.4-0.3-0.2-0.100.10.20.30.40.5 a x y -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.5-0.4-0.3-0.2-0.100.10.20.30.40.5 b x y -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.5-0.4-0.3-0.2-0.100.10.20.30.40.5 c Figure 1: Three newly-found stable periodic orbits of the non-hierarchical triple systems withdifferent masses and period: (a) m = 0 . , m = 0 . , m = 1 and T = 5 . ; (b) m = 0 . , m = 0 . , m = 1 and T = 6 . ; (c) m = 0 . , m = 0 . , m = 1 and T = 6 . . Body-1: blue line; Body-2: red line; Body-3: black line.Starting from the periodic orbit of the equal-mass triple system mentioned above, we obtain periodic orbits in the region of m ∈ [0 . , . and m ∈ [0 . , . with a fixed mass m = 1 by means of the continuation method (see Methods). The periodic orbits are outputtedwith the mass interval δm = 0 . . The detailed initial conditions and periods are listed in thesupplementary data. Three examples of these periodic orbits are shown in Figure 1. Their initialconditions and periods of the three periodic orbits are listed in Table 1.Due to the homogeneity of the potential field for the three-body problem, there is a scalinglaw: r (cid:48) = α r , v (cid:48) = v / √ α , t (cid:48) = α / t and energy E (cid:48) = E/α and angular momentum L (cid:48) = √ αL .The scale-invariant average period ¯ T ∗ = ( T /k ) | E | / is approximately equal to a constant forperiodic orbits of the three-body problem with equal mass
14, 32 , where k is the number of freegroup words of periodic orbits. For the family of periodic orbits bABabaBAba , we always have5he number of the free group words k = 10 . For the newly-found periodic unequal-mass orbits,Figure 2a shows that the scale-invariant average period ¯ T ∗ = ( T /k ) | E | / depends on the massof bodies. The multiple linear regression for these periodic orbits is ( T /k ) | E | / = 2 . m +1 . m − . . The standard error of this multiple linear regression is . . It indicates thatthe scale-invariant average period ¯ T ∗ = ( T /k ) | E | / is approximately linear to m and m forthis family of periodic orbits. Jankovi´c and Dmitraˇsinovi´c found that the scale-invariant angularmomentum is a function of topologically rescaled period for the Broucke-Hadjidemetriou-H´enonfamily of periodic triple orbits with equal mass. For our newly-found family of periodic orbits, itis demonstrated that the scale-invariant angular momentum L | E | / varies among different masses m and m as shown in Figure 2b. It implies that the scale-invariant angular momentum alsodepends on the mass of bodies for this family of periodic orbits of the unequal-mass triple system.Note that some regions of the Figure 2 is blank. It suggests that no periodic orbits can be foundthere because the orbits of the three-body system might have collision in that mass region.Stability is an important property for periodic orbits because only stable triple system canprobably be observed. The stability of periodic orbits of the three-body system can be investigatedaccording to the characteristic multipliers of the monondromy matrix . Due to the fixed centerof mass, the dimension of the planar three-body problem can be reduced to eight. We employ atheorem proofed by Kepela and Sim´o to determine the linear stability of periodic orbits of three-body problem through the monondromy matrix. With the monodromy matrix, we can gain theequation as follows: T − ( α − T + β − α + 8 = 0 , (2)6 able 1: Initial conditions and periods T of three stable periodic orbits for the non-hierarchical three-body system in the case of r (0) = ( x , , r (0) = (1 , , r (0) = (0 , , ˙ r (0) = (0 , v ) , ˙ r (0) = (0 , v ) , ˙ r (0) = (0 , − ( m v + m v ) /m ) when G = 1 . m m m x v v T m m (T/k)|E| a m m L|E| b Figure 2: The contour map of the scale-invariant average period and scale-invariant angular mo-mentum of newly-found periodic orbits: (a) The contour map of the average scale-invariant period ¯ T ∗ = ( T /k ) | E | / in the m - m plane, where E , T , k is total energy, period and the numberof free group words of periodic orbits, respectively; (b) The contour map of the scale-invariantangular momentum L | E | / in the m - m plane, where L is angular momentum.7 m m = m Figure 3: The stability region of periodic orbits in the m - m plane. Shadowing domain: stableperiodic orbits.where α = trace ( A ) = (cid:80) i =1 a ii , β = (cid:80) ≤ i
17, 18 is very narrow (only − ) . So, the mass region of the8ewly-found stable non-hierarchical periodic orbits is fairly large and their masses have apparentdifferences. For instance, for the stable non-hierarchical periodic orbit m = 0 . , m = 0 . and m = 1 , we have its mass ratio m /m ≈ . and m /m = 0 . . A recent observed hierarchical triple system has masses . , . and . M (cid:12) , corresponding to mass ratios . and . . Itshould be emphasized that the mass ratios of our newly-found stable non-hierarchical periodicorbits are close to the mass ratios of the hierarchical triple system which has been measured bythe astronomical observation. This implies that our newly-found stable non-hierarchical periodicorbits are likely to be observed in astronomy.Since the dimensionless quantities are used in above numerical results, the variables can berescaled to applications of stellar dynamics through GM T ∗ /R = 1 , where M , T ∗ and R is thecharacteristic mass, time and length and G is the Newtonian gravitational constant. If we choose M = M (cid:12) and R = 10 AU, then T ∗ = (cid:113) R GM ≈ years. For instance, with these units of quantities,the stable non-hierarchical periodic orbit with m = 0 . M (cid:12) , m = 0 . M (cid:12) and m = M (cid:12) hasperiod for about years. Note that the hierarchical triple system HD 188753 has period for 25years and semi-major axis for 11.8 AU . Thus, our newly-found stable non-hierarchical triplesystems have similar size and period with the observed hierarchical triple system.There may be two reasons why the non-hierarchical periodic triple stars has not been foundin the astronomical observation yet. On one hand, the accurate positions and motions of the non-hierarchical systems were not easy to determine because they are complicated and far away fromthe earth. On the other hand, there were few periodic non-hierarchical unequal-mass triple systems9ound in theoretical and numerical study before. Fortunately, Gaia mission has produced high-precision measurements of positions and motions of nearly 1.7 billion stars which provide resourceto study non-hierarchical periodic triple systems. This implies that our newly-found stable non-hierarchical periodic orbits are likely to be observed in near future.In this letter, we present one family of periodic orbits for non-hierarchical triplesystem with unequal masses. Surprisely, among these periodic orbits of this family, periodic orbits are linear stable in a large mass region. Most of them have fairly different masses,which implies that our newly-found stable periodic orbits are likely to be observed in practice.Note that we only consider here one family of the periodic orbits with the free group element bABabaBAba , but found stable ones among the periodic orbits. Note also thathundreds of families of periodic equal-mass three-body orbits were found : similarly, each ofthem as a starting point might lead to thousands of stable periodic orbits of the non-hierarchicaltriple system with unequal-mass. Therefore, in theory, there should exist an infinite number ofstable periodic orbits of non-hierarchical triple systems with distinctly unequal mass. Our newly-found stable periodic orbits of the non-hierarchical unequal-mass triple system have broad impactfor the astrophysical scenario: they could inspire the theoretical and observational study of thenon-hierarchical triple system, the formation of triple stars , the gravitational waves pattern andthe gravitational waves observation of the non-hierarchical triple system.10 ethodsClean Numerical Simulation. The clean numerical simulation (CNS) is a numerical strategyto gain reliable numerical simulation of chaotic dynamical systems, such as the three-body system.The CNS is based on an arbitrary Taylor series method and multiple-precision arithmetic ,plus a convergence verification by means of an additional computation with smaller numericalnoise. Li and Liao
14, 41 found that many periodic orbits of three-body problem might be lost byusing conventional numerical algorithms in double precision. Thus, here we apply the CNS tointegrate the differential equations of the three-body system.
Numerical searching method.
At the beginning, we numerically search for periodic orbits ofthe three-body problem with equal masses m = m = m = 1 and zero angular momentum.Due to the homogeneity of the potential field for the three-body problem, the initial condition x can be fixed to unit. Then we choose the velocity v = − x v due to zero angular momentum.Without loss of generality, we assume total momentum m ˙ r + m ˙ r + m ˙ r = 0 . Therefore, theinitial positions can be specified as r (0) = ( x , , r (0) = (1 , , r (0) = (0 , and the initialvelocities can be specified as ˙ r (0) = (0 , v ) , ˙ r (0) = (0 , − x v ) , ˙ r (0) = (0 , − v + x v ) .With the initial configuration, the orbits of the three-body problem are determined by twoparameters x and v . According to the numerical searching method of the three-body problem
13, 14 ,the first step is to gain approximated initial values of periodic orbits in a two dimensional space(i.e., the x - v plane). We investigate a region of this plane: x ∈ ( − , and v ∈ (0 , .We employ × uniform grid points as initial conditions in this region. With these initial11onditions, the differential equations (1) are numerically solved by an eight-oder Runge Kutta ODEsolver dop853 developed by Hairer et al. . For each initial condition, the return proximity function d ( y (0) , T ) = min t ≤ T || y ( t ) − y (0) || is calculated up to integration time T = 200 . We choose theinitial conditions and periods T as possible candidates of periodic orbits when the return proximityfunction d ( y (0) , T ) < . .The next step is to improve the precision of the approximate initial conditions of the periodicorbits using the Newton-Raphson method
24, 25 and the clean numerical simulation (CNS) by meansof correcting the parameters x , v and period T . The precision of the initial conditions of theperiodic orbits is improved continually until the level of the return proximity function is less than − . Continuation method.
The numerical continuation method is used to gain periodic solutions ofa nonlinear dynamical system with a natural parameter ˙ u = G ( u , λ ) . (3)Using a known periodic orbit u at λ as initial guess, we can obtain a new periodic orbit u (cid:48) at λ +∆ λ by means of the Newton-Raphson method
24, 25 and the clean numerical simulation (CNS) when ∆ λ is sufficient small to guarantee the convergence of iteration.Because of homogeneity of the potential field of the three-body problem, we can fix theinitial distance of two bodies to unit. Without loss of generality, we consider the case of zeromomentum (i.e., m ˙ r + m ˙ r + m ˙ r = 0 ). The periodic orbits are determined by x , v , v and12 with masses m , m and m . Therefore, the initial positions of three bodies can be described by r (0) = ( x , , r (0) = (1 , , r (0) = (0 , , (4)and the initial velocities can be described by ˙ r (0) = (0 , v ) , ˙ r (0) = (0 , v ) , ˙ r (0) = (0 , − m v + m v m ) . (5)With the fixed masses m = m = 1 , periodic orbits can be obtained by means of thenumerical continuation method for different mass m . Using a periodic orbit with equal mass asa starting point, we apply the Newton-Raphson method and the clean numerical simulation (CNS)to gain a new periodic orbit at m + ∆ m by continually modifying the parameters x , v , v and T , where ∆ m is small enough to guarantee the convergence of iteration. In this way, we can gainperiodic orbits with different mass m (cid:54) = 1 and m = m = 1 .Similarly, using the above periodic orbits with m (cid:54) = 1 and m = m = 1 as starting points,we further employ the Newton-Raphson method and the clean numerical simulation (CNS) to gainperiodic orbits at m + ∆ m by continuously correcting the parameters x , v , v and T , where ∆ m is small enough to guarantee the convergence of iteration. Consequently, we gain periodic orbitsof the triple system with unequal masses m (cid:54) = m (cid:54) = m .Note that the periodic orbits might have nonzero angular momentum since we don’t restrictthe angular momentum.1. Newton, I. Philosophiæ naturalis principia mathematica (Mathematical principles of naturalphilosophy) (London: Royal Society Press, 1687).13. Musielak, Z. E. & Quarles, B. The three-body problem.
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Acknowledgements
This work was carried out on TH-1A at National Supercomputer Center in Tianjinand TH-2 at National Supercomputer Center in Guangzhou, China. It is partly supported by National NaturalScience Foundation of China (Approval No. 11702099 and 91752104) and the International Program ofGuangdong Provincial Outstanding Young Researcher. uthor contributions X.M.L. calculated the periodic orbits and generated the first draft. X.C.L. analysedthe data, discussed the results and modified the letter. S.J.L. discussed the results and revised the letter. Allauthors contributed to the discussion and revision of the final manuscript.
Competing Interests
The authors declare that they have no competing financial interests.
Correspondence