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ARTICLE TYPE

Escaping orbits in the N-body ring problem

Juan F. Navarro* | M. C. Martínez–Belda

1Department of Applied Mathematics,University of Alicante, Alicante, Spain

Correspondence*Juan F. Navarro, Carretera San Vicente delRaspeig s/n, 03690, Alicante. Email:jf.navarro@ua.es

Present AddressCarretera San Vicente del Raspeig s/n,03690, Alicante

Summary

The aim of this paper is to start a numerical exploration of the escape in the N-

body ring problem. When the energy of the orbits is larger than the escape energy,

test particles may escape through any of the N openings of the potential well. We

have computed the “basins" of escape towards the different directions by means of

Poincaré sections. We have also analyzed the proportion of escaping orbits and the

direction of escape.

KEYWORDS:

N-body ring problem, Escape, Celestial Mechanics

1 INTRODUCTION

The analysis of the behaviour of escaping particles from dynamical systems has attracted the interest of many scientists overthe years, and it is also a present active field of research. Likewise, the N-body problem is one of the most important issuesin Celestial Mechanics. This dynamical system describes the motion of a particle attracted by the gravitational field of N + 1bodies. The N-body ring problem is a simplification of the scenario, consisting of N equidistant primaries of equal masses onthe periphery of a circle and a central primary of different mass. The primaries move around the centre of mass of the systemwith constant angular velocity. The interest in this problem has increased during the last three decades, due to the discovery ofrings around all the gas giant planets in the Solar System. In addition, the existence of natural satellites orbiting some asteroidsand recent missions such as the one of The China National Space Administration, which is planning to launch a spacecraft toexplore one of Earth’s nearest non-moon neighbours, make this model a useful tool, since it can provide additional qualitativeinformation.

The ring configuration in the N-body problem was first introduced by Maxwell in his study of Saturn’s rings. 19 Maxwellconsidered that the rings consist of a great number of small particles orbiting Saturn at a common radius and uniformly distributedabout a circle of this radius, and proved, among other things, that the relative equilibrium formed by an n-gon of small equalmasses in orbit around a single large mass is stable. In 1889, Tisserand 28 reformulated Maxwell’s analysis and presented arelation between the mass of each ring particle and the number of them in order that the system be linearly stable. Willerding 29

continued Maxwell study, analyzing the behaviour of density waves appearing in narrow planetary rings in a general centralpotential, and Salo and Yoder 23 examined the stability of a many-body system in an anular arrangement. Goudas 15 used the sameprinciple to explain the Saturn’s rings by means of its N-dipole system. In 1999, Kalvouridis derived the equations of motionof a particle of negligible mass under the gravitatory influence of a planar configuration consisting of N primaries arranged inequal arcs on an ideal ring and a central body of different mass located at the centre of mass of the system. 16,17 Some yearslater, Barrabés 4 and co-workers analyzed a limit case of this problem, when N is taken very large and the mass of each of theN primaries is very small.

The aim of this work is to start an investigation of the escape of a particle from the N-body ring configuration. The escapesof a particle from a dynamical system is a subject that has attracted much interest in the last decades. 1,2,3,5,6,7,8,10,11,12,20,25,30,31

When the energy of a particle is larger than the energy of escape, the curve of zero velocity opens and many particles escape

This article has been accepted for publication and undergone full peer review but has not been through the copyediting,typesetting, pagination and proofreading process, which may lead to differences between this version and the Version ofRecord. Please cite this article as doi: 10.1002/cmm4.1067

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from the potential well. However, there are regions of orbits that never escape, or escape after a very long time. For each valueof the energy larger than the energy of escape, there is a highly unstable periodic orbit that bridge the minimum openings of thecurve of zero velocity. The sets of escaping orbits are limited by the orbits asymptotic to the Lyapunov orbits.

The main objective of this paper is to perform a numerical exploration of the N-body ring problem in order to determinewhich orbits escape and which remain trapped in the potential well, locating the basins of escape leading to different escapechannels, and trying to connect them with the corresponding escape times of the orbits.

The present article is organized as follows. In Section 2 we describe the properties of the potential of the N-body ringproblem. The numerical method used in order to determine the nature (trapped/escaping) of orbits is described in Section 3. Inthe following Section, we conduct a thorough analysis of a set of initial conditions of orbits presenting in detail all the numericalresults of our computations. Our article ends with Section 5, where the conclusions and the discussion of this research arepresented.

2 EQUATIONS OF MOTION

The N-body ring problem consists of a central body, P0, with mass m0 located at the center of mass of the system, and N smallbodies of equal masses, m, arranged at equal distances among them on the periphery of a circle or, equivalently, at the vertices ofa regular polygon. 18 These N bodies are rotating on its own plane around the mass center. We are interested in the motion of aparticle S of negligible mass moving under the resultant gravitational action of this system. We will refer to the configuration’smassive bodies as “primaries". 18

We assume that the peripheral primaries rotate at constant angular velocity, here taken as unity. One of the existing axes ofsymmetry of the primaries’ configuration is taken as the x-axis of a synodic coordinate system Oxyz, that is, a reference framerigidly attached to the primaries, with origin O at the center of mass of the system. After the transformation of the physicalquantities to dimensionless ones, and considering the side of the regular polygon as unity, the equations which describe theplanar motion of the small body S are16,17

x = 2y +)U)x

, y = −2x +)U)y

. (1)

The Jacobian integral of motion is given byC = 2U (x, y) − (x2 + y2) , (2)

where C is the Jacobian constant, and

U (x, y) =12(x2 + y2) +

1Δ

(�r0

+N∑

�=1

1r�

)

.

Here, � = m0∕m is the ratio of the central mass to a peripheral one,

r0 =√x2 + y2

is the distance from the central body to the test particle, and

r� =√

(x − x∗� )2 + (y − y∗� )

2 ,

for � = 1, 2,… , N , are the distances of the particle from the peripheral primaries. The quantities x∗� and y∗� are the coordinatesof the peripheral primaries,

x∗� =1

2 sin �cos(2(� − 1)�) , y∗� =

12 sin �

sin(2(� − 1)�) ,

and Δ is given byΔ =M(Λ + �M2) ,

where

Λ =N∑

�=2

sin2 � cos((N∕2 + 1 − �)�)

sin2((N + 1 − �)�)=

N∑

�=2

sin2 �sin((� − 1)�)

,

andM =

√2(1 − cos ) = 2 sin � .

In these formulas, is the angle between the central and two succesive peripheral primaries, and � = ∕2 = �∕N .

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The curves of zero velocity can be obtained through

C = 2U (x, y) = x2 + y2 +2Δ

(�r0

+N∑

�=1

1r�

)

. (3)

These curves form the boundaries that the test particle can not cross. In Figure 1, we show the curves of zero velocity for N = 5,� = 2 and several values of the Jacobi constant C .

There is a critical value of the Jacobi constant, denoted by Ce, such that, for smaller values of C , the potential well opens up toinfinity and test particles may escape. Due to the symmetries of the potential, the well opens up at N places in the configurationspace (see Figure 2). For each value of C smaller than Ce, there is a highly unstable periodic orbit that bridges the minimumopenings of the curve of zero velocity, bouncing back and forth between the two walls of the pass. Such orbits are called“Lyapunov orbits" and they are always unstable. Their most important property is that any orbit crossing them outwards movesalways outwards and escapes from the system. These orbits intersect perpendicularly the lines defined by

y = tan(2(� − 1)�)x , � =�N,

for � = 1,… , N . We have numbered the openings of the potential well, as well as the associated Lyapunov orbits (�� , � =1,… , 5) as detailed in Figure 2. Inside the region defined by the Lyapunov orbits, there are sets of orbits that escape, and setsof orbits which remain trapped.

The critical value Ce of the Jacobi constant can be calculated by following the procedure described by Caranicolas. 9 To thatend, the equation of the curves of zero velocity is rearranged as

F (x, y) = x2 + y2 +2Δ

(�r0

+N∑

�=1

1r�

)

− C = 0 . (4)

The points of the x − y plane where the curves of zero velocity open are the saddle points of (4). First, we solve the system

)F)x

= 2x −2Δ

(

�x

r30+

N∑

�=1

x − x∗�r3�

)

= 0 ,

)F)y

= 2y −2Δ

(

�y

r30+

N∑

�=1

y − y∗�r3�

)

= 0 , (5)

which provides the critical points of (4). These points are also the equilibria of the system.In the present work, we have used the parameters � = 2 andN = 5. The equilibria of the system, which are found numerically,

are arranged in equal arcs on concentric circumferences centered at the origin. In Figure 3 we have depicted the curves that result,in our particular case, for the partial derivatives, Fx and Fy, as well as the circumferences containing the equilibria. Accordingto Kalvoridius 16, we have only three different zones if N ≤ 6, which we denote as A1, C1 and C2, with N equilibria in eachone. All points that belong to a particular zone are characterized by the same Jacobi constant. We shall refer to as CA1

, CC1and

CC2. Zone A1 consists of the equilibria which lie on the lines connecting the central body with a peripheral primary and they are

located between them. Zone C2 consists of the equilibria which lie on the same lines as those of the zone A1 but they are outsidethe peripheral primaries. Finally, zone C2 consists of the equilibria which lie in the regions between the central body and twosuccessive peripheral primaries, but outside the peripheral primaries. The values of the Jacobi constants are CA1

= 4.176823126,CC2

= 3.642984268 and CC1= 3.971595480, and the radii of the circumferences are rA1

= 0.4880617835, rC2= 1.062230640

and rC1= 1.312776448, respectively.

The saddle points of (4) are those of the critical points satisfying the condition

S =

()2F)x2

)()2F)y2

)

−

()2F)x)y

)2

< 0 ,

and correspond to the critical energy values CA1and CC1

. The other 5 critical points are relative extrema and lead to the valueCC2

. For every value of the energy C smaller than CC1the potential well opens up to infinity and, therefore, we can conclude that

Ce = CC1. In our computations, we have taken C = 3.96, a value of the Jacobi constant slightly smaller than the critical one.

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FIGURE 1 Curves of zero velocity for N = 5, � = 2 and C = 4.2 (upper-left panel), C = 4 (upper-right panel), C = 3.96(lower-left panel) and C = 3.9 (lower-right panel). We show in gray the regions where the particle motion is impossible tohappen.

3 INTEGRATION OF THE EQUATIONS OF MOTION

The study of the escape of a particle from the potential well of the N-body ring problem requires the integration of the equationsof motion of the problem over long spans of time. Thus, an accurate integration of these equations is needed in order to explorethe long-term behavior of the solutions of the problem. Navarro 21 adapted the ideas of Deprit and Price 14 and Steffensen27 tothe equations of motion of the N-body ring problem. The fourth-order system of differential equations was replaced there by asystem of 6 + 2N first-order differential equations. Navarro proved that the system is integrable by means of recurrent power

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FIGURE 2 Curves of zero velocity for N = 5, � = 2 and C = 3.96 (left). There is a highly unstable periodic orbit (right)bridging the minimum openings of the curve of zero velocity.

series (RPS), and also that these series are convergent for any set of initial conditions, excluding those corresponding to binarycollisions. He also checked the benefits of the integration by RPS over short spans of time, against more classical methods, likethe Runge–Kutta of order 4 (RK4), the Runge–Kutta–Fehlberg method (RKF) (with constant and variable step size) and theDormand–Prince (DOPRI5) (with constant and variable step size). Navarro and Vargas 22 analyzed the integration by RPS overlong spans of time for large values of N and compared the results against some other methods. They showed that if the initialconditions of the problem lead to a regular orbit, the long term numerical solution is reliable.

We have performed the integration of the equations of motion of the N-body problem by means of the RPS method, as thismethod is much more accurate than RK4, RKF and DOPRI5. 21 Moreover, RPS achieves the smallest computational time, com-pared with the other three methods. The Jacobi constant associated with the RPS solution of the equations of motion remainedthe same to 17 significant figures. Only RKF and DOPRI5 with variable step sizes achieve a similar accuracy in the Jacobiconstant, but RPS performs the numerical integration 100 and 1000 times faster than DOPRI5 and RKF, respectively.

The equations of motion described by equation (1) are given by

x = q ,

y = p ,

x = q = 2p +)U)x

,

y = p = −2q +)U)y

, (6)

that is,

x = q ,

y = p ,

q = 2p + x −1Δ

(

�x

r30+

N∑

�=1

x − x∗�r3�

)

,

p = −2q + y −1Δ

(

�y

r30+

N∑

�=1

y − y∗�r3�

)

. (7)

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-1.8

-1.2

-0 .6

0

0.6

1.2

1.8

-1.8 -1.2 -0 .6 0 0 .6 1.2 1.8

y

x

FIGURE 3 Partial derivatives, Fx and Fy, for � = 2 and N = 5. The critical points are arranged in equal arcs on concentriccircumferences centered at the origin.

With the introduction of the quantities

R0 = �1

r30, R� =

1r3�, (8)

for � = 1,… , N , the derivatives of R0 and R� are obtained through

r0R0 = −3R0r0 , r�R� = −3R�r� , (9)

with � = 1,… , N .Hence, the equations (7) may be replaced by a system of 6 + 2N differential equations,

x = q ,

y = p ,

r0r0 = xx + yy ,

r� r� = (x − x∗� )x + (y − y∗� )y ,

r0R0 = −3R0r0 ,

r�R� = −3R�r� ,

q = 2p + x −1Δ

(

xR0 +N∑

�=1

(x − x∗� )R�

)

,

p = −2q + y −1Δ

(

yR0 +N∑

�=1

(y − y∗� )R�

)

, (10)

where � = 1,… , N . Equations (10) lend themselves to an integration by recurrent power series (RPS).The RPS method allows us to determine an optimum value of the integration step to secure the desired accuracy of the

numerical solution. 21 The precision of the method has been fixed to � = 10−23, and the number of term series to 26.

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x

x.

FIGURE 4 Region of allowed initial conditions in the phase (x, x), for N = 5 and C = 3.96.

4 ANALYSIS OF THE NUMERICAL RESULTS

In order to explore the escape dynamics of a test particle in the N-body ring problem, we need to define samples of initialconditions of orbits. To this end, we have defined a grid of initial conditions regularly distributed in the area allowed by thevalue of the energy. Our investigation is focused on the (x, x) space, taking as surface of section the hyperplane y = 0. The stepsize of the initial conditions along the x and x axes has been fixed to 0.0025, that is, we have created a grid dense enough ofequally spaced initial conditions of orbits to analyze the system. Therefore, we consider orbits with initial conditions (x0, x0)with y0 = 0, while the initial value of y0 is obtained through the equation

y0 = +√

2U (x0, 0) − C − x20 .

Thus, the initial conditions must be taken in the domain D defined by the equation

D = {(x0, x0) ∈ ℝ2 ∶ U (x0, 0) − x20 ≥ C} .

In Figure 4, we represent (in grey) this domain.We have performed the numerical integration over T = 102, 103 and 104 time units as a maximum time of numerical inte-

gration, to analyze the distribution of times of escape of the test particle. The escape time tesc is defined as the time a particleneeds to cross one of the Lyapunov orbits with velocity pointing outwards.

We should clarify at this point that orbits that do not escape after a numerical integration of 104 time units are considered asnon-escaping or trapped orbits. Our numerical calculations indicate that the percentage of escaping orbits do not vary signifi-cantly by considering a maximum time of integration T of 103 or 104. Whereas for a maximum integration time of 102 unitsonly 21.3% of orbits escape from the potential well, for T = 103 and 104, the percentages of escaping orbits are 52.03% and52.75%, respectively. Thus, we conclude that considering T = 103 is enough to study the escape in this configuration of theN-body ring problem. Besides, focusing on this value of T , we observe that more than 21% of orbits escape from the well in amaximum time of 100 units, and 50% of the orbits escape in a time lower than 660 units. Indeed, nearly 41% of the total numberof escaping orbits need 100 units of time or less to find one of the N openings in the curves of zero velocity, and consideringa maximum time of 660 units, the vast majority of them (95%) escape. Figure 5 shows a pair of escaping orbits with differenttimes of escape (tesc = 8 (orbit in left panel) and tesc = 94 (orbit in right panel)).

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FIGURE 5 Characteristic examples of escaping orbits in the N-body ring problem for N = 5, C = 3.96 presenting differenttimes of escape: tesc = 8 (left) and tesc = 94 (right).

FIGURE 6 Evolution of 100 p(tn−1, tn), percentage of escaping orbits between tn−1 = 10(n − 1) and tn = 10n (left panel), and100P (tn), percentage of escaping orbits before tn = 10n (right panel), for n = 1, 2,… , 10.

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FIGURE 7 Evolution of 100 p(tn−1, tn), percentage of escaping orbits between tn−1 = 10(n − 1) and tn = 10n (left panel), and100P (tn), percentage of escaping orbits before tn = 10n (right panel), for n = 1, 2,… , 100.

FIGURE 8 Evolution of 100 p(tn−1, tn), percentage of escaping orbits between tn−1 = 10(n − 1) and tn = 10n (left panel), and100P (tn), percentage of escaping orbits before tn = 10n (right panel), for n = 1, 2,… , 1000.

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FIGURE 9 S t ruc t ure of t h e p h as e (x, x) f or N = 5 , C = 3.96 and a m ax i m um t i m e of i nt e g rat i on of T = 102.

26. Sit arski G . R e c urre nt p ow e r s e ri e s i nt e g rat i on of t h e e q uat i ons of c om e t ’ s m ot i on. Acta Astronomica. 1 9 7 9 ; 2 9 : 4 0 1 – 4 1 1 .

27. S t e ff e ns e n JF. O n t h e re s t ri c t e d p rob l e m of t h re e b od i e s . Kgl Danske Videnskab Selskab, Mat Fys Medd. 1 9 5 6 ; 3 0 ( 1 8 ) : 1 7 .

28. Ti s s e rand F. Traité de Mechanique Céleste, Tome II. P ari s : G aut h i e r– V i l l ars 1 8 8 9 .

29. W i l l e rd i ng E. Th e ory of d e ns i t y w ave s i n narrow p l ane t ary ri ng s . Astron Astrophys. 1 9 8 6 ; 1 6 1 : 4 0 3 – 4 0 7 .

30. Z ot os EE. Trap p e d and e s c ap i ng orb i t s i n an ax i al l y s y m m e t ri c g al ac t i c – t y p e p ot e nt i al . PASA. 2 0 1 2 ; 2 9 : 1 6 1 – 1 7 3 .

31. Z ot os EE. Es c ap e d y nam i c s i n a H am i l t oni an s y s t e m w i t h f our e x i t c h anne l s . Nonlinear Studies. 2 0 1 5 ; 2 2 ( 3 ) : 1 – 2 0 .

A U T H O R B I O G R A P H Y

Juan F . N a v a r r o com ple ted his PhD deg re e f rom t h e U ni ve rs i t y of Al i c ant e , Al i c ant e , S p ai n, i n 2 0 0 2 . H e i sa re c i p i e nt of t h e Ex t raord i nary Aw ard of t h e U ni ve rs i t y of Al i c ant e f or h i s m as t e r t h e s i s on t h e rot at i on oft h e ri g i d Eart h . Th i s w ork t ook p art i n t h e p roj e c t Pinpoint positioning in a w obbly w orld aw ard e d w i t h t h eDescar tes P ri z e i n 2 0 0 3 , an annual aw ard i n s c i e nc e g i ve n b y t h e Europ e an U ni on t o out s t and i ng s c i e nt i fi cac h i e ve m e nt s re s ul t i ng f rom Europ e an c ol l ab orat i ve re s e arc h . H e i s c urre nt l y P rof e s s or at t h e D e p art m e nt ofApplied Mat hematics, U ni ve rs i t y of Al i c ant e , m e m b e r of t h e S c i e nt i fi c G roup on S p ac e G e od e s y and S p ac eDynamics of t h e U ni ve rs i t y of Al i c ant e , and m e m b e r of t h e I nt e rnat i onal As t ronom i c al U ni on. H i s s c i e nt i fi c

FIGURE 9 Structure of the phase (x, x) for N = 5, C = 3.96 and a maximum time of integration of T = 102.

The probability of escape between times t1 and t2, can be defined by

p(t1, t2) =Nesc

NT,

where Nesc is the number of particles escaping between t1 and t2, out of a total number of particles NT . As commented at thebeginning of this section, the initial conditions of the NT particles are regularly distributed in the region of the (x, x) spaceallowed by the value of the energy. We can also define the quantity P (t) = p(0, t) as the probability of escape before timet. In Figures 6, 7 and 8, we show the quantities 100 p(tn−1, tn) and 100P (tn) for a set of times given by tn = 10 n, for anyn = 1, 2,… , T ∕10, and for a maximum time of integration of T = 102, 103 and 104.

The initial conditions on the surface of section (x, x) that lead to escapes through the various Lyapunov orbits form thecorresponding “basins of escape". In Figures 9 and 10, we show the basins of escape considering a maximum time of numericalintegration of T = 102, 103 and 104 respectively, for the value of the Jacobi constant C = 3.96. We have used different colorsin order to distinguish between the five channels of escape, and initial conditions leading to collisions (black). Initial conditionscolored in green correspond to orbits leaving the potential well through the opening 1. Initial conditions in light blue, dark blue,dark red and yellow correspond to orbits escaping through channels 2, 3, 4 and 5, respectively. We observe large connecteddomains, that lead to the same kind of escape, but also some regions where the different basins are intricately mixed. Theseregions have fractal structure. There are also three main domains in white corresponding to initial conditions which remaintrapped in the potential well. In Figure 11, we show two examples of orbits with initial conditions in the main domains of trappedorbits.

In the connected domains located around values of x in the interval (0, 0.3), we can distinguish two main structures: a primarystructure colored in dark blue, and a secondary structure colored in light blue and infinitely spiraling around the first one. Thesetwo structures correspond to initial conditions of orbits that escape through the exit windows guarded by the Lyapunov periodicorbits �3 and �2, respectively. We also distinguish in this region some other tertiary and quaternary connected structures colored

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FIGURE 10 S t ruc t ure of t h e p h as e (x, x) f or N = 5 , C = 3.96 and a m ax i m um t i m e of i nt e g rat i on of T = 103 (lef t ) andT = 104 ( ri g h t ) .

inter e s t s i nc l ud e d i ff e re nt p rob l e m s i n C e l e s t i al M e c h ani c s , s uc h as t h e s t ud y of t h e rot at i onal m ot i on of t h e Eart h , t h e e s c ap eof p art i c l e s f rom g al ac t i c p ot e nt i al s , and t h e num e ri c al e x p l orat i on of t h e N -body ri ng p rob l e m .

M . C . M a r t í n e z – B e l d a com ple ted his PhD deg re e f rom t h e U ni ve rs i t y of Al i c ant e , Al i c ant e , S p ai n, i n 2 0 1 2 .She i s c urre nt l y P rof e s s or at t h e D e p art m e nt of Ap p l i e d M at h e m at i c s , U ni ve rs i t y of Al i c ant e , m e m b e r oft h e S c i e nt i fi c G roup on S p ac e G e od e s y and S p ac e D y nam i c s of t h e U ni ve rs i t y of Al i c ant e . H e r s c i e nt i fi cinter e s t s i nc l ud e d i ff e re nt p rob l e m s i n C e l e s t i al M e c h ani c s , s uc h as t h e s t ud y of t h e rot at i onal m ot i on of t h eEart h or t h e num e ri c al e x p l orat i on of t h e N -body ri ng p rob l e m , and t h e i m p rove m e nt of t h e s t ab i l i t y of t i m eand f re q ue nc y t rans f e r w i t h G NS S .

Juan F. Navarro ET AL 13

FIGURE 10 S t ruc t ure of t h e p h as e (x, x) f or N = 5 , C = 3.96 and a m ax i m um t i m e of i nt e g rat i on of T = 103 (lef t ) andT = 104 ( ri g h t ) .

inter e s t s i nc l ud e d i ff e re nt p rob l e m s i n C e l e s t i al M e c h ani c s , s uc h as t h e s t ud y of t h e rot at i onal m ot i on of t h e Eart h , t h e e s c ap eof p art i c l e s f rom g al ac t i c p ot e nt i al s , and t h e num e ri c al e x p l orat i on of t h e N -body ri ng p rob l e m .

M . C . M a r t í n e z – B e l d a com ple ted his PhD deg re e f rom t h e U ni ve rs i t y of Al i c ant e , Al i c ant e , S p ai n, i n 2 0 1 2 .She i s c urre nt l y P rof e s s or at t h e D e p art m e nt of Ap p l i e d M at h e m at i c s , U ni ve rs i t y of Al i c ant e , m e m b e r oft h e S c i e nt i fi c G roup on S p ac e G e od e s y and S p ac e D y nam i c s of t h e U ni ve rs i t y of Al i c ant e . H e r s c i e nt i fi cinter e s t s i nc l ud e d i ff e re nt p rob l e m s i n C e l e s t i al M e c h ani c s , s uc h as t h e s t ud y of t h e rot at i onal m ot i on of t h eEart h or t h e num e ri c al e x p l orat i on of t h e N -body ri ng p rob l e m , and t h e i m p rove m e nt of t h e s t ab i l i t y of t i m eand f re q ue nc y t rans f e r w i t h G NS S .

FIGURE 10 Structure of the phase (x, x) for N = 5, C = 3.96 and a maximum time of integration of T = 103 (left) andT = 104 (right).

-1.8

-1.2

-0 .6

0

0.6

1.2

1.8

-1.8 -1.2 -0 .6 0 0 .6 1.2 1.8

y

x -1.8

-1.2

-0 .6

0

0.6

1.2

1.8

-1.8 -1.2 -0 .6 0 0 .6 1.2 1.8

y

x

FIGURE 11 Two examples of orbits with initial conditions in the main domains of trapped orbits.

in dark red, yellow and green, but with smaller area. These smaller domains correspond to initial conditions of orbits that escapethrough the exit windows guarded by �4, �5 and �1, respectively.

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FIGURE 12 Characteristic examples of orbits in theN-body right problem, forN = 5,C = 3.96. First row (left): orbit escapingthrough opening 1, first row (right): orbit escaping through opening 2, second row (left): orbit escaping through opening 3,second row (right): orbit escaping through opening 4, third row (left): orbit escaping through opening 5, third row (right): orbitremaining inside the potential well.

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We can also observe three connected domains located at values of x in the interval (0.9, 1.3): green, dark red and yellow,which are also interrelated, and correspond to initial conditions of orbits leaving the potential well through the exit windowsguarded by the Lyapunov orbits �1, �4 and �5, respectively. In this region, we can observe that orbits with initial conditionsdefined by x > 0 (and y > 0) escape mainly through the exit window guarded by the Lyapunov orbit �1. In a lower proportion,there are connected domains of orbits that escape through windows guarded by �5 and �4. Orbits with initial conditions definedby x < 0 (and y > 0) escape mainly through the exit window guarded by the Lyapunov orbit �5, but also by �4 and through theexit channel that corresponds to the Lyapunov orbit �3. This means that orbits with initial conditions in the region defined forvalues of x in the interval (0.9, 1.3) and y = 0 escape mainly through the windows guarded by the Lyapunov orbits �1, �4 and�5. It is evident that orbits with initial conditions inside the exit basins escape from the system very quickly, or in other words,they possess extremely small escape periods. On the contrary, orbits with initial conditions located in the fractal parts of thephase plane need considerable amount of time in order to escape.

In Figure 12, we observe five orbits escaping through openings 1, 2, 3, 4 and 5, and one trapped orbit. The initial conditionsof the escaping orbits have been taken from the corresponding basins of escape. We observe that they present different times ofescape, probably due to the way we have defined the surface of section, which does not fit the circular symmetry of the problem.

5 CONCLUSIONS

In this paper, we have started a numerical exploration of the escape of a particle from the N-body ring configuration, focusingon the case with N = 5 small bodies rotating around a central mass. When the energy of the system is smaller than a criticalvalue, the curve of zero velocity is open and exhibits N channels of escape. We have studied the escape of particles startinginside the potential well for three maximum integration times (T = 102, 103 and 104), in order to locate the basins of escape andto connect them with the escape times of the orbits. To this end, we have defined a grid dense enough in the (x, x) phase space.Our results show that a maximum time of 103 is sufficient to determine the percentage of orbits that escape from the potentialwell, as well as to analyze the structure of the basins of escape. Indeed, we have seen that 95% of the escaping orbits need lessthan 660 units of time to escape. In addition, we have shown that there are several large domains in the phase space with thesame kind of escape, as well as regions where the different types of escape are intricately mixed. These regions seem to exhibitfractal structure, so a more detailed analysis of the phase space must be carried out, testing other surface sections with a betterfitting of the symmetry of the problem.

Conflict of interest

The authors declare no potential conflict of interests.

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AUTHOR BIOGRAPHY

Juan F. Navarro completed his PhD degree from the University of Alicante, Alicante, Spain, in 2002. He isa recipient of the Extraordinary Award of the University of Alicante for his master thesis on the rotation ofthe rigid Earth. This work took part in the project Pinpoint positioning in a wobbly world awarded with theDescartes Prize in 2003, an annual award in science given by the European Union to outstanding scientificachievements resulting from European collaborative research. He is currently Professor at the Department ofApplied Mathematics, University of Alicante, member of the Scientific Group on Space Geodesy and SpaceDynamics of the University of Alicante, and member of the International Astronomical Union. His scientific

interests include different problems in Celestial Mechanics, such as the study of the rotational motion of the Earth, the escapeof particles from galactic potentials, and the numerical exploration of the N-body ring problem.

M. C. Martínez–Belda completed his PhD degree from the University of Alicante, Alicante, Spain, in 2012.She is currently Professor at the Department of Applied Mathematics, University of Alicante, member ofthe Scientific Group on Space Geodesy and Space Dynamics of the University of Alicante. Her scientificinterests include different problems in Celestial Mechanics, such as the study of the rotational motion of theEarth or the numerical exploration of the N-body ring problem, and the improvement of the stability of timeand frequency transfer with GNSS.

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