Periodic satellite orbits of the Broucke-Hadjidemetriou-Hénon family of three-body system with unequal masses


 Triple systems are common and key objects in astronomy. The three-body problem has received much more attention in recent years [1–3]. All observed periodic triple stars systems [4–6] belong to the Broucke-Hadjidemetriou-Hénon’s (BHH) family [7–9]. The BHH orbits are a family of periodic orbits of the three-body system with the simplest topological free group word [10] a, while Jankovíc and Dmitrasinovíc [1] gained 58 equal-mass BHH satellite orbits which have free group words ak (k > 1), where k is the topological exponent. However, the BHH satellite orbits with equal mass is lack of realistic meaning because they do not exist in practice. Here we present 419743 new BHH orbits and 179253 new BHH satellites (k > 1) for three-body system with unequal mass. Especially, 48761 among the 179253 new BHH satellites are stable and have unequal masses. It suggests that these 48761 stable BHH satellites could be found by the observation. Besides, for the three-body system with equal mass at a fixed energy, it was demonstrated that the relationship between the angular momentum (L) and topological scaled period (T/k) of the BHH satellites is the same as that of the BHH orbits [1]. However, we found that this does not hold for the three-body system with unequal mass. Our findings have broad impact for the astrophysical scenario: they could inspire the theoretical and observational study of the triple system, the formation of triple stars [11], the gravitational waves pattern [12] and the gravitational waves observation [13] of the triple system.

by the observation. Besides, for the three-body system with equal mass at a fixed energy, it was demonstrated that the relationship between the angular momentum (L) and topological scaled period (T /k) of the BHH satellites is the same as that of the BHH orbits 1 . However, we found that this does not hold for the three-body system with unequal mass. Our findings have broad impact for the astrophysical scenario: they could inspire the theoretical and observational study of the triple system, the formation of triple stars 11 , the gravitational waves pattern 12 and the gravitational waves observation 13 of the triple system.
The triple systems are common and key objects in astronomy 3 . It can help us to understand the formation and evolution of multiple star systems 11 . The three-body problem can be traced back to Newton in 1680s, but is still an open question in astrophysics today, mainly because it is not an integrable system 14 and besides has the sensitivity dependance on initial condition (SDIC) 15 , i.e. butterfly-effect that broke a new field of scientific research, i.e. chaos. Even today the three-body problem is still one of central issues for scientists 3 . Especially, periodic orbits of triple system play an important role since they are "the only opening through which we can try to penetrate in a place which, up to now, was supposed to be inaccessible", as pointed out by Poincaré 15 . However, since the famous three-body problem was first put forward, only three families of periodic orbits were found in about three hundred years: (1) the Lagrange-Euler family discovered by Lagrange and Euler in the 18th century; (2) the Broucke-Hadjidemetriou-Hénon (BHH) family 7-9 ; (3) the figure-eight family, discovered numerically by Moore 16 in 1993 and then proofed by Chenciner & Montgomery 17 in 2000, until 2013 whenŠuvakov and Dmitrašinović 18 numerically found 13 distinct periodic orbits of the three-body system with equal mass. In recent years, numerically search-ing for periodic orbits of the three-body system has been received much attention [19][20][21][22][23] . Šuvakov 19 reported the satellites of the figure-eight periodic orbit with equal mass. Especially, more than six hundred new families of periodic orbits of equal-mass three-body system were found by Li and Liao 20 using a new numerical strategy, namely the clean numerical simulation (CNS) 24-26 that can give the convergent/reliable numerical solution of chaotic systems in a long enough duration. Li et al. 21 further used the CNS to obtain 1223 new families of periodic orbits of three-body system with two equal-mass bodies. All of these greatly enrich our knowledge of the famous three-body problem.
With the topological classification method 10 , the so-called BHH orbits have the simplest topology (free group word w = a), while the BHH satellites have more free group words w = a k , where k is the topological exponent. In theory, Janković and Dmitrašinović 1 numerically gained 58 BHH satellites (k > 1) with equal mass, and found that the relationship between the scaleinvariant angular momentum (L) and the topologically rescaled period (T /k) is the same for both of the BHH orbits (k = 1) and satellites (k > 1). On the other side, all practically observed periodic triple star systems belong to the BHH orbits (k = 1). This fact raises a question of whether we can observe any BHH satellites (k > 1) in our universe. Unfortunately, all previously reported BHH satellites have equal mass, which are impossible to be observed in practice. Janković and Dmitrašinović 1 also mentioned the importance of the realistic case of three different masses. In this letter, we will investigate the BHH orbits and satellite orbits with unequal mass.
Let us consider a three-body system in the Newtonian gravitational field. Without loss of generality, let the Newtonian gravitational constant G = 1. As shown in Figure 1, the three bodies have collinear initial configuration for the BHH family of periodic orbits: , and their initial velocities are orthogonal to the line determined by the three bodies: Due to the homogeneity of the potential field of the three-body system, there is a scaling The known periodic orbits of the BHH family and their satellites with equal mass 1, 7, 9 have zero total momentum, i.e., m 1ṙ1 + m 2ṙ2 + m 3ṙ3 = 0. Using the scaling law, we can transform the initial conditions of the known periodic orbits of the BHH family and their satellites to the initial positions  Table 1. and the initial velocitieṡ We use the numerical continuation method 27 and clean numerical simulation [24][25][26] (see Methods) to gain the BHH orbits (k = 1) and their satellites (k > 1) with unequal mass. Note that all of the known BHH orbits (k = 1) and satellites (k > 1) are "relative periodic orbits": after a period, these relative periodic orbits will return to initial conditions in a rotating frame of reference. So, there is an individual rotation angle θ for each relative periodic orbit. Table 1: Initial conditions and periods T of some BHH satellites of three-body system with unequal mass in case of r 1 (0) = (x 1 , 0), r 2 (0) = (1, 0), r 3 (0) = (0, 0),ṙ 1 (0) = (0, v 1 ), Here m i , x i and v i are the mass, initial position and velocity of the ith body, θ is the rotation angle of relative periodic orbits, and k is the topological power of periodic orbits, respectively.  with unequal mass are shown in Figure 2. All of the six BHH relatively periodic satellites are linearly stable. Their initial conditions, periods and topological powers (k) are listed in Table 1. It should be emphasized that we can modify these 200686 stable new BHH orbits and satellites with unequal mass to an arbitrary accuracy by means of the above-mentioned numerical strategy and the CNS. For example, we further obtained much more accurate initial conditions of the six BHH relatively periodic satellites (listed in Table 1) with return distance d < 100 −100 , as shown in the corresponding supplementary file. From the viewpoint of accuracy, all of these numerical solutions have no essential difference from "closed-form" analytic solutions that however unfortunately do not exist for three-body problem in general.
With rescaling to the same energy E = −1/2, Janković and Dmitrašinović 1 found that, in case of equal mass, the relationship between the scale-invariant angular momentum (L) and the topologically rescaled period (T /k) is the same for both of the BHH orbits (k = 1) and satellites (k > 1), as shown in Figure 3(a), where k is the topological exponent of periodic orbits. However, for our newly-found periodic orbits with unequal masses (at the same energy E = −1/2), it is found that the relationship between the scale-invariant angular momentum (L) and period (T /k) of the BHH satellites (k > 1) is different from that of the BHH orbits (k = 1), as illustrated in be emphasized that all practically observed periodic triple star systems belong to the family of BHH orbits (k = 1), but up to now BHH (relative periodic) satellites (k > 1) with unequal mass were never reported before even in theory. Therefore, these 48761 stable BHH (relative periodic) satellites (k > 1) with unequal masses have important meaning in practice: many among them could be found by the observation. Besides, for the three-body system with equal mass at a fixed energy, Janković and Dmitrašinović 1 numerically demonstrated that the relationship between the angular momentum (L) and topological period (T /k) of the BHH satellites (k > 1) is the same as that of the BHH orbits (k = 1). However, we found that this does not hold for the three-body system with unequal mass. Therefore, our study has important meanings not only in practice but also in theory: they could greatly deepen our understandings and enrich our knowledge for the practical observation of the BHH satellites (k > 1), formation of multiple stars, the gravitational waves pattern, the gravitational waves observation of the triple system, and so on.

Methods
Briefly speaking, the numerical continuation method can be used to gain solution of a differential where λ a physical parameter, called "natural parameter". Assume that u 0 is a solution at a natural parameter λ = λ 0 . Using the solution u 0 at λ = λ 0 as an initial guess, a new solution u ′ can be obtained at a new natural parameter λ = λ 0 + ∆λ through the Newton-Raphson method 29,30 and the clean numerical simulation (CNS) [24][25][26] , if the increment ∆λ is small enough to make sure iterations convergence. Note that the CNS 24-26 is a numerical strategy to obtain reliable numerical simulation of chaotic systems in a given time of interval. The CNS is based on an arbitrary high order Taylor series method 31, 32 and the multiple precision arithmetic 33 , plus a convergence check using an additional computation with even smaller numerical error. The CNS 24-26 is used here mainly because three-body system is chaotic in general.
First of all, using the known BHH orbits (k = 1) and satellites (k > 1) with equal mass (m 1 = m 2 = m 3 = 1) as initial guesses and m 1 as a natural parameter of the continuation method, we obtain new periodic orbits with various m 1 by continually correcting the initial conditions x 1 , v 1 , v 2 , the period T and the rotation angle θ. Then, using these periodic solutions with m 1 ̸ = 1, m 2 = m 3 = 1 as initial guesses and m 2 as a natural parameter of the continuation method, we similarly gain periodic orbits for different values of m 2 . In this way, we can obtain the corresponding BHH (relative periodic) orbits (k = 1) and satellites (k > 1) with unequal mass m 1 ̸ = m 2 ̸ = m 3 ,