Satellites of the Broucke-Hadjidemetriou-Hénon family of periodic unequal-mass three-body orbits
aa r X i v : . [ n li n . C D ] J u l Satellites of the Broucke-Hadjidemetriou-H´enon family of periodicunequal-mass three-body orbits
Xiaoming Li , and Shijun Liao , ∗ MOE Key Laboratory of Disaster Forecast and Control in Engineering, School of Mechanics and ConstructionEngineering, Jinan University, Guangzhou 510632, China Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology,Cambridge, Massachusetts 02139, USA School of Physics and Astronomy, Shanghai Jiaotong University, Shanghai 200240, China Center of Advanced Computing, School of Naval Architecture, Ocean and Civil Engineering, Shanghai JiaotongUniversity, Shanghai 200240, China ∗ Corresponding author: [email protected]
Abstract
The Broucke-Hadjidemetriou-H´enon’s (BHH) orbits are a family of periodic orbits of the three-body system withthe simplest topological free group word a , while the BHH satellites have free group words a k ( k > ), where k isthe topological exponent. Jankovi´c and Dmitraˇsinovi´c [Phy. Rev. Lett. 116, 064301 (2016)] reported 57 new BHHsatellites with equal mass and found that at a fixed energy the relationship between the angular momentum ( L ) andthe topologically rescaled period ( T /k ) is the same for both of the BHH orbits ( k = 1 ) and the BHH satellites ( k > ).In this letter, we report 419,743 new BHH orbits ( k = 1 ) and 179,253 new BHH satellites ( k > ) of the three-bodysystem with unequal mass, which have never been reported, to the best of our knowledge. Among these newly-found598,996 BBH orbits and satellites, about 33.5% (i.e., 200,686) are linearly stable and thus many among them mightbe observed in practice. Besides, we discover that, for the three-body system with unequal mass at a fixed energy,relationship between the angular momentum ( L ) and topologically rescaled period ( T /k ) of the BHH satellites ( k > )is different from that of the BHH orbits ( k = 1 ). I. INTRODUCTION
The three-body problem can be traced back to Newton in 1680s, but is still an open question in astrophysics andmathematics today, mainly because it is not an integrable system [1] and besides has the sensitivity dependanceon initial condition (SDIC) [2] that broke a new field of scientific research, i.e. chaos. Even today the three-bodyproblem is still one of central issues for scientists [3]. Especially, periodic orbits of triple system play an important rolesince they are “the only opening through which we can try to penetrate in a place which, up to now, was supposedto be inaccessible”, as pointed out by Poincar´e [2]. However, since the famous three-body problem was first putforward, only three families of periodic orbits were found in about three hundred years: (1) the Lagrange-Euler familydiscovered by Lagrange and Euler in the 18th century; (2) the Broucke-Hadjidemetriou-H´enon (BHH) family [4–6]; (3)the figure-eight family, discovered numerically by Moore [7] in 1993 and then proofed by Chenciner & Montgomery [8]in 2000, until 2013 when ˇSuvakov and Dmitraˇsinovi´c [9] numerically found 13 distinct periodic orbits of the three-bodysystem with equal mass. In recent years, numerically searching for periodic orbits of the three-body system has beenreceived much attention [10–13]. ˇSuvakov [10] 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 Liand Liao [11] using a new numerical strategy, namely the clean numerical simulation (CNS) [14–16] that can give theconvergent/reliable numerical solution of chaotic systems in a long enough duration. Li et al. [12] further used theCNS to obtain more than one thousand new families of periodic orbits of three-body system with two equal-massbodies. All of these greatly enrich our knowledge of the famous three-body problem.With the topological classification method [17], the 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. Recently, Jankovi´cand Dmitraˇsinovi´c [18] reported 57 BHH satellites ( k >
1) with equal mass. Especially, it was found [18] that therelationship 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 > equal mass. The BHH satellites with unequal mass have never been reported yet, to the best of our knowledge.In this letter, we investigate the BHH orbits ( k = 1) and satellites ( k >
1) with unequal mass. In §
2, the basicideas of the numerical continuation method are briefly introduced. In §
3, we numerically search for the BHH orbitsand satellites with unequal mass, and show the corresponding results. Brief conclusions are given in § xy O ✒✞☛✁ ✄(cid:0) (1 ✂ ☎✆✝✟ ✝✠ ✡☞✌✟ ✍☞✎✏ ✑ ✓✔✕✖✕ ✗ ✔✘✖✘✔✙ FIG. 1. (color online.) The initial configuration of the three-body system. Here m , m and m denote the mass of body-1, body-2 and body-3, respectively. The corresponding initial velocities are v , v and v = − ( m v + m v ) /m , and thecorresponding initial positions are ( x , II. THE INITIAL CONFIGURATION AND NUMERICAL METHOD
Let us consider a three-body system in the Newtonian gravitational field. Without loss of generality, let theNewtonian gravitational constant G = 1. As shown in Figure 1, the three bodies have collinear initial configurationfor the BHH family of periodic orbits: r (0) = ( x , r (0) = ( x , r (0) = ( x , r (0) = (0 , v ), ˙ r (0) = (0 , v ), ˙ r (0) = (0 , v ).Due to the homogeneity of the potential field of the three-body system, there is a scaling law : r ′ = α r , v ′ = v / √ α , t ′ = α / t , energy E ′ = E/α and angular momentum L ′ = √ αL . The known periodic orbits of the BHH family andtheir satellites with equal mass [4, 6, 18] have zero total momentum (i.e., m ˙ r + m ˙ r + m ˙ r = 0). Using the scalinglaw, we can transform the initial conditions of known periodic orbits of the BHH family and their satellites to theinitial positions r (0) = ( x , , r (0) = (1 , , r (0) = (0 , , (1)and the initial velocities ˙ r (0) = (0 , v ) , ˙ r (0) = (0 , v ) , ˙ r (0) = (cid:18) , − m v + m v m (cid:19) . (2)We use the numerical continuation method [19] to gain the BHH orbits ( k = 1) and their satellites ( k >
1) with unequal mass. Briefly speaking, the numerical continuation method can be used to gain periodic solutions of thenonlinear differential system ˙ u = F ( u , λ ) , (3)where λ a physical parameter, called “natural parameter”. Assume that u is a solution at a natural parameter λ = λ . Using the solution u at λ = λ as an initial guess, a new periodic orbit u ′ can be obtained at a new naturalparameter λ = λ + ∆ λ through the Newton-Raphson method [20, 21] and the clean numerical simulation (CNS)[14–16] if the increment ∆ λ is small enough to make sure iterations convergence. The CNS is a numerical strategy toobtain reliable numerical simulation of chaotic systems in a given time of interval. The CNS is based on an arbitraryhigh order Taylor series method [22, 23] and the multiple precision arithmetic [24], plus a convergence check using anadditional computation with even smaller numerical error.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 individualrotation angle θ for each relative periodic orbit.First of all, using the known BHH orbits ( k = 1) and satellites ( k >
1) with equal mass ( m = m = m = 1) asinitial guesses and m as a natural parameter of the continuation method, we obtain new periodic orbits with various m by continually correcting the initial conditions x , v , v , T and the rotation angle θ . Then, using these periodicsolutions with m = 1, m = m = 1 as initial guesses and m as a natural parameter of the continuation method, wesimilarly gain periodic orbits for different values of m . In this way, we can obtain the corresponding BHH (relativeperiodic) orbits ( k = 1) and satellites ( k >
1) with unequal mass m = m = m = 1. x y -1.5 -1 -0.5 0 0.5 1 1.5-1.5-1-0.500.511.5 (a) x y -3 -2 -1 0 1 2-2-1012 (b) x y -6 -5 -4 -3 -2 -1 0 1-3-2-10123 (c) x y -6 -5 -4 -3 -2 -1 0 1-3-2-10123 (d) x y -10 -8 -6 -4 -2 0-4-3-2-101234 (e) x y -27 -24 -21 -18 -15 -12 -9 -6 -3 0-6-4-20246 (f) FIG. 2. (color online.) The stable BHH satellites ( k >
1) of the three-body system with unequal mass in a rotating system.Blue line: body-1; red line: body-2; black line: body-3. The corresponding physical parameters are given in Table I.TABLE I. Initial conditions and periods T of some BHH satellites of three-body system with unequal mass in 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 ). Here m i , x i and v i arethe mass, initial position and velocity of the i th body, θ is the rotation angle of relative periodic orbits, and k is the topologicalpower of periodic orbits, respectively.No. m m m x v v T θ k (a) 0.44 0.87 1 -1.21992948117021 -0.992252134619392 -0.513024298255905 9.1758282973000 0.500325594634030 3(b) 0.1 0.2 1 -2.59038883768724 -0.619538016547757 -0.865730420457027 23.1822105206534 0.110299604356735 5(c) 0.64 0.36 1 -5.34303779563735 -0.320961649402539 -0.737471717803478 36.3965789125352 0.085296971469750 7(d) 0.4 0.7 1 -6.19095126372906 -0.330208056009860 -0.703472163111217 40.8464178849168 0.107927979415387 9(e) 0.6 0.8 1 -9.25316717310693 -0.235487010686519 -0.683551074764106 60.3350581589708 0.086983638647642 13(f) 0.82 0.9 1 -25.5854144048497 -0.093120912929298 -0.673916187504190 204.731304275836 0.056232585939249 48
III. RESULTS
Starting from the 16 known Broucke’s periodic orbits ( k = 1 with equal mass) [4], the 45 known H´enon’s periodicorbits ( k = 1 with equal mass) [6] and the 58 known BHH satellites ( k > new periodic orbits of the three-body system with unequal mass for m ∈ [0 . , m ∈ [0 . , m = 1, respectively. Totally, we gain 419,743 new BHH orbits ( k = 1 with unequal mass) and 179,253 newBHH satellites ( k > unequal mass). Note that all of them are retrograde, say, the binary system and the thirdbody move in opposite direction. The corresponding initial conditions, the periods and the rotation angles of thesenew BBH orbits and satellites with unequal mass are given in the supplementary data. The return distance of theseperiodic orbits and satellites satisfies d = vuut X i =1 (( r i ( T ) − r i (0)) + ( ˙ r i ( T ) − ˙ r i (0)) ) < − in a rotating frame of reference, where T is the period. Note that Broucke and Boggs [25] gave dozens of the BHHorbits with unequal mass (their ratios of mass are different from ours), but neither have any BHH satellites with T/k L Broucke’s orbits (k=1)Henon’s orbits (k=1)Satellites (k>1) (a)
T/k L Broucke’s orbits (k=1)Henon’s orbits (k=1)Satellites (k>1) (b)
T/k L Broucke’s orbits (k=1)Henon’s orbits (k=1)Satellites (k>1) (c)
T/k L Broucke’s orbits (k=1)Henon’s orbits (k=1)Satellites (k>1) (d)
FIG. 3. (color online.) The angular momentum ( L ) versus the topological rescaled period ( T /k ) for BHH periodic orbits andtheir satellites at fixed energy E = − / m = m = m = 1; (b) m = 0 . m = 0 . m = 1; (c) m = 0 . m = 0 . m = 1; (d) m = 0 . m = 0 . m = 1. unequal mass been reported, to the best of our knowledge. It should be emphasized that, among our newly-found598,996 BHH orbits and satellites with unequal mass, there are 151,925 stable BHH orbits ( k = 1) and 48,761 stableBHH satellites ( k > unequal mass (i.e.,200,686) are stable, and thus many among them might be observed in practice. The stability of these periodic orbitsand satellites is marked by “S” in the supplementary data. Six new BHH satellites with unequal mass are shown inFigure 2. All of the six orbits are linearly stable. Their initial conditions, periods and topological powers are listedin Table I. Note that these orbits are relatively periodic, say, their orbits are closed curves in a rotating frame ofreference.With rescaling to the same energy E = − /
2, Jankovi´c and Dmitraˇsinovi´c [18] found that, in case of equal mass,the relationship between the scale-invariant angular momentum ( L ) sand the topologically rescaled period ( T /k ) isthe same for both of the BHH orbits ( k = 1) and satellites ( k > k is the topologicalexponent of periodic orbits. However, for our newly-found periodic orbits with unequal masses (at the same energy E = − / L ) and period ( T /k )of the BHH satellites ( k >
1) is different from that of the BHH orbits ( k = 1), as illustrated in Figure 3 (b)-(d). Itsuggests that the relationship between the scale-invariant angular momentum ( L ) and topologically rescaled period( T /k ) of the BHH orbits ( k = 1) and satellites ( k >
1) in general cases of unequal masses m = m = m should bemore complicated than that in the case of the equal mass m = m = m . IV. CONCLUSION
The BHH orbits are a family of periodic orbits of the three-body system with the simplest topological free groupword a , while the BHH satellites have free group words a k ( k > k is the topological exponent. In this paper,starting from the 16 known Broucke’s periodic orbits ( k = 1 with equal mass) [4], the 45 known H´enon’s periodicorbits ( k = 1 with equal mass) [6] and the 58 known BHH satellites ( k > equal mass) [18], we found 419,743new BHH orbits ( k = 1) and 179,253 new BHH satellites ( k >
1) for three-body system with unequal mass, whichhave never been reported, to the best of our knowledge. Among these newly-found 598,996 BBH orbits and satellitesof three-body system with unequal mass, about 33.5% (i.e., 200,686) are linearly stable and thus many among themmight be observed in practice.For the three-body system with equal mass at a fixed energy E = − /
2, it was reported [18] that the relationshipbetween the angular momentum ( L ) and topological period ( T /k ) of the BHH satellites ( k >
1) is the same as thatof the BHH orbits ( k = 1). However, this does not hold for the three-body system with unequal mass, as reported inthis letter.This work was carried out on TH-1A at National Supercomputer Center in Tianjin and TH-2 at National Super-computer Center in Guangzhou, China. It is partly supported by National Natural Science Foundation of China(Approval No. 91752104) and the International Program of Guangdong Provincial Outstanding Young Researcher. [1] Z. E. Musielak and B. Quarles, Rep. Prog. Phys. , 065901 (30pp) (2014).[2] J. H. Poincar´e, Acta Math. , 1 (1890).[3] N. C. Stone and N. W. Leigh, Nature , 406 (2019).[4] R. Broucke, Celestial Mechanics , 439 (1975).[5] J. D. Hadjidemetriou, Celestial Mechanics , 255 (1975).[6] M. H´enon, Celestial mechanics , 267 (1976).[7] C. Moore, Phys. Rev. Lett. , 3675 (1993).[8] A. Chenciner and R. Montgomery, Annals of Mathematics , 881 (2000).[9] M. ˇSuvakov and V. Dmitraˇsinovi´c, Phys. Rev. Lett. , 114301 (2013).[10] V. Dmitraˇsinovi´c and M. ˇSuvakov, American Journal of Physics , 609 (2014).[11] X. Li and S. Liao, SCIENCE CHINA Physics, Mechanics & Astronomy , 129511 (2017).[12] X. Li, Y. Jing, and S. Liao, Publications of the Astronomical Society of Japan , 64 (2018).[13] V. Dmitraˇsinovi´c, A. Hudomal, M. Shibayama, and A. Sugita, Journal of Physics A: Mathematical and Theoretical ,315101 (2018).[14] S. Liao, Tellus A , 550 (2009).[15] S. Liao and P. Wang, Sci. China - Phys. Mech. Astron. , 330 (2014).[16] T. Hu and S. Liao, Journal of Computational Physics , 109629 (2020).[17] R. Montgomery, Nonlinearity , 363 (1998).[18] M. R. Jankovi´c and V. Dmitraˇsinovi´c, Phys. Rev. Lett. , 064301 (2016).[19] E. L. Allgower and K. Georg, Introduction to numerical continuation methods , Vol. 45 (SIAM, 2003).[20] S. C. Farantos, Journal of Molecular Structure: THEOCHEM , 91 (1995).[21] M. Lara and J. Pelaez, Astronomy and Astrophysics , 692 (2002).[22] Y. F. Chang and G. F. Corhss, Computers Math. Applic. , 209 (1994).[23] R. Barrio, F. Blesa, and M. Lara, Computers & Mathematics with Applications , 93 (2005).[24] L. Fousse, G. Hanrot, V. Lef`evre, P. P´elissier, and P. Zimmermann, ACM Transactions on Mathematical Software (TOMS) , 13 (2007).[25] R. Broucke and D. Boggs, Celestial mechanics11