Orbit classification in the Hill problem: I. The classical case
NNonlinear Dynamics manuscript No. (will be inserted by the editor)
Orbit classification in the Hill problem - I. The classical case
Euaggelos E. Zotos
Received: 8 November 2016 / Accepted: 17 March 2017 / Published online: 27 March 2017
Abstract
The case of the classical Hill problem is numeri-cally investigated by performing a thorough and systematicclassification of the initial conditions of the orbits. More pre-cisely, the initial conditions of the orbits are classified intofour categories: (i) non-escaping regular orbits; (ii) trappedchaotic orbits; (iii) escaping orbits; and (iv) collision or-bits. In order to obtain a more general and complete viewof the orbital structure of the dynamical system our explo-ration takes place in both planar (2D) and the spatial (3D)version of the Hill problem. For the 2D system we numeri-cally integrate large sets of initial conditions in several typesof planes, while for the system with three degrees of free-dom, three-dimensional distributions of initial conditions oforbits are examined. For distinguishing between ordered andchaotic bounded motion the Smaller ALingment Index (SALI)method is used. We managed to locate the several boundedbasins, as well as the basins of escape and collision and alsoto relate them with the corresponding escape and collisiontime of the orbits. Our numerical calculations indicate thatthe overall orbital dynamics of the Hamiltonian system isa complicated but highly interested problem. We hope ourcontribution to be useful for a further understanding of theorbital properties of the classical Hill problem.
Keywords
Hill problem · Escape dynamics · Fractal basinboundaries
The process where test particles escape from Hamiltoniansystems is surely one of the most intriguing problems in
Department of Physics, School of Science,Aristotle University of Thessaloniki,GR-541 24, Thessaloniki, GreeceCorresponding author’s email: [email protected] nonlinear dynamics (e.g., [8–10, 42, 43]). In the case wherethe energy of escape has a finite value escape is possible. Inparticular, when the particles have values of energy higherthan the critical energy of escape they are able to find theopenings, or exit channels, in the equipotential surface andtherefore escape to infinity. Systems with exit channels arealso known as open or leaking Hamiltonian systems. Theliterature is replete with studies on such open Hamiltoniansystems (e.g., [4, 12, 17, 30, 31, 35, 50–52, 60, 61, 63, 70]).The topic of leaking Hamiltonian systems however is byfar less explored than the closely related issue of chaoticscattering. In many previous works the chaos theory hasbeen successfully used in order to investigate and explainthe phenomenon of chaotic scattering (e.g., [5–7, 14–16, 18,21–28, 44–48]). Needles to say that all the above-mentionedcitations on both issues of chaotic scattering and open Hamil-tonian systems are exemplary rather than exhaustive.The restricted three-body problem (RTBP for short) isan excellent example of an open Hamiltonian system withescapes (e.g., [57, 58]). Over the last decade or so a largenumber of studies have been devoted on orbit classificationin the RTBP. It all started with the pioneer works of Nagler[33, 34] where initial conditions of orbits were classified asbounded, escaping or collision. Moreover, bounded orbitswere further classified into orbital families by taking intoaccount the type of the motion of the test particle around theprimary bodies. In the same vein, orbit classification in theRTBP with perturbations has also been performed in [64,65] where we investigated the influence of the oblateness,while in [66] we explored how the radiation pressure a ff ectsthe orbital content of the dynamical system.Similarly, orbit classification has also been conductedin celestial mechanics and dynamical astronomy regardingplanetary systems. In particular, the orbital dynamics of theEarth-Moon system with two degrees of freedom using ascattering region around the Moon has been studied in [13]. a r X i v : . [ n li n . C D ] J u l Euaggelos E. Zotos
In [69] the numerical investigation has been expanded intothree dimensions thus exploring the orbital structure of thethree degrees of freedom Earth-Moon system. Furthermore,very recently the escape and collision dynamics in the Saturn-Titan and in the Pluto-Charon binary planetary systems havebeen revealed in [67] and [68], respectively, by classifyinginitial conditions of orbits in the configuration space.The Hill limiting case is in fact a simplified modificationof the RTBP which focus on the vicinity of the secondary(e.g., [20, 38, 39]). This allow us to study the motion of thetest particles in the neighborhood of the equilibrium points L and L . At this point it should be emphasized that theHill approximation is valid only when the mass of the sec-ondary is much smaller compared with the mass of the pri-mary body. One can directly obtain the Hill model from theclassical RTBP by translating the origin to the center of thesecondary and also by rescaling the coordinates by a factor µ / . In [49] the authors performed an extensive investiga-tion of the planar Hill problem by computing several homo-clinic and heteroclinic connections. In this work, which isthe first part of a series of papers, we shall perform a thor-ough and systematic orbit classification in the classical Hillproblem. To our knowledge, there are no previous detailedand systematic numerical studies regarding orbit classifica-tion in the classical Hill problem. On this basis, the new in-formation presented in this paper is exactly the contributionas well as the novelty of this research work. In the followingpapers of the series we will explore how several perturba-tions (i.e., the oblateness and the radiation pressure) influ-ence the orbital structure of the dynamical system.The present paper is organized as follows: In Section 2we present in detail the derivation and the basic dynamicalproperties of the mathematical model. All the computationaltechniques we used in order to classify the initial conditionsof the orbits are described in Section 3. In the followingSection, a thorough and systematic numerical investigationtakes place which allow us to reveal the overall orbital struc-ture of the Hill system with two and three degrees of free-dom. Our paper ends with Section 6 where the discussionand the main conclusions of our research are given. The classical Hill problem is derived from the circular re-stricted three-body problem if we make some scale changesand if we also take the limiting case where the mass ratio µ = m / ( m + m ) tends to zero ( µ → ff ective potential function of thecircular restricted three-body problem is Ω ( X , Y , Z ) = (1 − µ ) r + µ r + (cid:16) X + Y (cid:17) , (1) where ( X , Y , Z ) are the coordinates of the test particle, while r = (cid:113) ( X − µ ) + Y + Z , r = (cid:113) ( X − µ + + Y + Z , (2)are the distances of the test particle from the two primaries.According to [54], in a rotating system of reference theequations of motion are Ω X = ∂Ω∂ X = ¨ X − Y ,Ω Y = ∂Ω∂ Y = ¨ Y + X ,Ω Z = ∂Ω∂ Z = ¨ Z . (3)We now place the origin of the coordinates at the centerof the secondary and we change the scale of lengths by afactor of µ / X = µ − + µ / x , Y = µ / y , Z = µ / z . (4)Here we would like to note that the notations ( X , Y , Z ) and( x , y , z ), regarding the coordinates for the RTBP and the Hillproblem, respectively are according to [37].Applying the above-mentioned transformations to (1) weobtain1 µ / (cid:32) Ω − (cid:33) = x − z + r + O ( µ / ) , (5)with r = (cid:112) x + y + z .Taking the limit of the right-hand side of (5) for µ → W ( x , y , z ) = x − z + r . (6)More details about the derivation of the potential func-tion of the Hill problem are given in the Appendix.The corresponding equations of motion are W x = ∂ W ∂ x = (cid:32) − r (cid:33) x = ¨ x − y , W y = ∂ W ∂ y = − yr = ¨ y + x , W z = ∂ W ∂ z = − (cid:32) + r (cid:33) z = ¨ z . (7)The Hill problem admits a Jacobi integral of motion J = W ( x , y , z ) − (cid:16) ˙ x + ˙ y + ˙ z (cid:17) , (8)where J is the new Jacobi constant which is related to the Ja-cobi constant C of the restricted three-body problem throughthe relation C = + µ / J . (9) rbit classification in the Hill problem - I. The classical case 3 Fig. 1 (a-left): The isoline contours of the e ff ective potential function W in the configuration ( x , y ) plane ( z = J L are shown in red. (b-right): Thethree-dimensional equipotential surface of the e ff ective potential function 2 W ( x , y , z ) = J , when J = .
25. The orbits can leak out through the exitchannels 1 and 2 of the equipotential surface passing either through L or L , respectively. (For the interpretation of references to colour in thisfigure caption and the corresponding text, the reader is referred to the electronic version of the article.) In order to locate the positions of the equilibrium pointsof the system we have to solve the system of di ff erentialequations W x = W y = W z = . (10)The Hill problem has only two equilibrium points which arelocated on the x axis at x L = ± / / (see Fig. 1a). The valueof the Jacobi integral at the equilibrium points is J L = / and this is a critical value. This is because for J < J L thezero velocity surfaces are open and two symmetrical chan-nels (exits) are present near the equilibrium points L and L through which the test particles can escape from the interiorregion ( − x L ≤ x ≤ x L ) of the system. Therefore we may ar-gue that J L plays, in a way, the role of the energy of escapeand this is exactly why it is a critical value of the Jacobiintegral. In Fig. 1b we present a plot of the three dimen-sional equipotential surface 2 W ( x , y , z ) = J , when J = . x direction, while channel 2 in-dicates escape towards the positive x direction. In order to reveal the orbital structure of the Hill problem weneed to define sets of initial conditions whose properties willbe identified. For obtaining a more complete and sphericalview regarding the nature of orbits we are going to exploreboth the two-dimensional (2D) and the three-dimensional(3D) Hill systems. In this work we decided to classify or-bits with initial conditions in the configuration space only. Inparticular, for several values of the Jacobi integral J we de-fine dense uniform grids of initial conditions inside the cor-responding zero velocity surface. For the 2D system ( z = × x , y ) are numeri-cally integrated. For all 2D orbits ˙ x =
0, while the initialvalue of ˙ y is always obtained from the Jacobi integral ofmotion (8), as ˙ y = ˙ y ( x , y , ˙ x , J ) >
0. In the same vein,for the 3D system, grids of 100 × ×
100 initial condi-tions ( x , y , z ) are numerically integrated. For all 3D orbits˙ x = ˙ z =
0, while the initial value of ˙ y is always obtainedfrom the energy integral (8), as ˙ y = ˙ y ( x , y , z , ˙ x , ˙ z , J ) >
0. In both cases, the initial conditions of the orbits lie in-side the interior region (which is the scattering region) with R = (cid:113) x + y + z < x L .The classification of the initial conditions of the orbits inthe Hill problem is a rather demanding task if we take intoaccount that the configuration space extends to infinity. In Euaggelos E. Zotos this study we shall classify initial conditions of orbits intofour main categories:1. Orbits that move in bounded, or non-escaping regularorbits inside the interior region.2. Orbits that move in trapped chaotic orbits inside the in-terior region.3. Orbits that escape from the interior region passing eitherthrough L or L .4. Orbits that collide with the secondary located at the ori-gin of the coordinates.Our next task is to define appropriate numerical criteria inorder to distinguish between the above-mentioned four typesof motion. An orbit is considered to escape from the systemwhen x < − x L − δ , or when x > x L + δ , where δ = .
1. At thispoint, it should be clarified that the tolerance δ was includedso as to avoid the unstable Lyapunov orbits [32], locatednear the equilibrium points, to be incorrectly classified asescaping orbits. A collision with the secondary occurs when R < R col , where R col = − .Our preliminary calculations suggest that a considerableportion of the initial conditions correspond to bounded or-bits, which can be either regular or chaotic. Therefore wedecided to distinguish between initial conditions of regu-lar non-escaping and trapped chaotic motion. Over the yearsmany dynamical indicators have been introduced for distin-guishing between regular and chaotic orbits. We chose to usethe Smaller ALingment Index (SALI) method [53], whichhas been proved a very fast and accurate tool. The mathe-matical definition of SALI is the followingSALI(t) ≡ min(d − , d + ) , (11)where d − ≡ (cid:107) w ( t ) − w ( t ) (cid:107) and d + ≡ (cid:107) w ( t ) + w ( t ) (cid:107) are thealignments indices, while w ( t ) and w ( t ), are two deviationvectors which initially point in two random directions. Fordistinguishing between ordered and chaotic motion, all wehave to do is to compute the SALI along a time interval t max of numerical integration. In particular, we track simultane-ously the time-evolution of the main orbit itself as well asthe two deviation vectors w ( t ) and w ( t ) in order to com-pute the SALI.The determination of the nature of an orbit is obtainedfrom the final value of the SALI at the end of the numeri-cal integration. In particular, if SALI > − the motion isregular, while if SALI < − the motion is chaotic. If thefinal value of SALI lies in the interval [10 − , − ] then wehave the case of a “sticky” orbit and further numerical inte-gration is required for obtaining the true nature of the orbit.In our study the maximum time of the numerical integration Sticky orbits need an extremely long time interval in order to moveaway from the invariant sticky tori [36]. Therefore, sticky orbits be-have, for long time interval, as regular ones before revealing their truechaotic nature. ( t max ) was set to be equal to 10 time units. Orbits that re-main inside the interior region after integrating them for t max are considered as non-escaping regular or trapped chaotic,according to their final value of SALI.The set of the equations of motion (7) as well as thecorresponding variational equations, needed for the compu-tation of the SALI, have been forwarded integrated usinga double precision Bulirsch-Stoer FORTRAN 77 algorithm(e.g., [40]) with a variable time step. Throughout our nu-merical calculations the error regarding the conservation ofthe Jacobi integral of motion (8) was generally smaller than10 − , while in some cases it was observed to be smallerthan 10 − . For those initial conditions of orbits which moveinside a region of radius R < − , thus leading to colli-sion with the secondary, the Lemaitre’s global regularizationmethod (e.g., [54]) has been applied. All graphical illustra-tions presented in this paper have been created using the lat-est version 11 of the software Mathematica (cid:114) (e.g., [59]). In this section we shall classify orbits with initial conditionsin the configuration space into the four categories analyzedin the previous section. Parallel to the classification we shallalso record the time scale (or time period) of the collisionand the time scale of the escape. Our aim is to numericallyexplore the orbital properties of the Hill problem for severalvalues of the Jacobi integral of motion.In the following we shall present color-coded diagrams(CCDs) in which each pixel is assigned a specific color ac-cording to the particular type of the nature of the orbit. TheseCCDs are modern types of color-coded maps where the phasespace is a complex mixture of basins of escape, collisionbasins and regions of bounded motion. By the term “basin”we refer to a set of initial conditions which lead to a certainfinal state (collision, escape or bounded motion). Our pre-liminary numerical calculations indicate that bounded mo-tion is almost always present. Generally speaking, the vastmajority of the bounded basins correspond to initial condi-tions of regular orbits, where an adelphic integral of motionis present. This additional integral poses new restrictions tothe available phase space and therefore it prevents them fromescaping to infinity.4.1 Results for the 2D systemWe begin our investigation with the two-dimensional (2D)system when z = ˙ z =
0. In Fig. 2 we present, using a richcollection of CCDs, the evolution of the orbital structure ofthe configuration ( x , y ) plane as the value of the Jacobi con-stant decreases. The outermost black solid line denotes thezero velocity curve which is defined as 2 W ( x , y ) = J . We rbit classification in the Hill problem - I. The classical case 5 Fig. 2
Evolution of the orbital structure of the configuration ( x , y ) space with decreasing value of the Jacobi integral of motion. The color code isas follows: non-escaping regular orbits (blue), trapped chaotic orbits (yellow), escaping orbits through L (red), escaping orbits through L (green),and collision orbits (cyan). (For the interpretation of references to colour in this figure caption and the corresponding text, the reader is referred tothe electronic version of the article.) observe that when J = . J L , the vast majority of the config-uration plane is covered either by stability islands or colli-sion basins. The remaining area is a highly fractal mixtureof escaping orbits. Inside this fractal area it is impossible When we state that an area is fractal we simply suggest that it hasa fractal-like geometry without conducting any specific calculationsregarding the fractal dimensions as in [3]. to predict through which channel a test particle will escape.This is true because even a slight change on the initial con-ditions of an orbit leads the test particle to escape from theopposite channel, which is of course a classical indicationof chaotic motion. As the value of the Jacobi integral of mo-tion reduces even further (remember that at the same timethe total orbital energy increases) the following importantchanges take place on the configuration ( x , y ) plane: Euaggelos E. Zotos
Fig. 3
Distribution of the corresponding escape and the collision time of the orbits with initial conditions on the configuration ( x , y ) space for thevalues of Jacobi integral of motion presented in Fig. 2. The bluer the color, the larger the escape or the collision time. Initial conditions of bothnon-escaping regular orbits and trapped chaotic orbits are shown in white. (For the interpretation of references to colour in this figure caption andthe corresponding text, the reader is referred to the electronic version of the article.) – The two symmetrical escape channels near the equilib-rium points become more and more wide, as the ener-getically forbidden regions are reduced. – Several basins of escape start to emerge inside the es-cape regions, while the fractality of the same regions isreduced. In particular, for J < . – The area of the basins of escape grows rapidly and for J < . x , y ) plane. Itis observed that for relatively low values of the Jacobiintegral of motion ( J <
0) escape basins dominate occu-pying almost the entire configuration space. – The extent of the main stability island located at the leftside of the ( x , y ) plane ( x <
0) is gradually reduced andwhen J < rbit classification in the Hill problem - I. The classical case 7 ular motion whatsoever. Additional smaller stability is-lands appear and disappear with decreasing value of J . – The collision basins are mainly located around the sta-bility island. For J < . – With increasing orbital energy it is seen that the vast ma-jority of the escaping orbits leak out through exit channel2. In fact at the highest energy level studied ( J = − . L is almost four times larger than the areaof initial conditions of orbits that escape through L .Therefore we may say that in the 2D system exit channel2 seems to be much more preferable. – Our computations indicate that in the 2D Hill systemthere is no numerical evidence of trapped chaotic mo-tion.In the following Fig. 3 we demonstrate how the escapeand the collision time of the orbits are distributed on the con-figuration ( x , y ) space for the same values of the Jacobi inte-gral shown in Fig. 2. Light reddish colors correspond to fastescape or collision orbits, dark blue / purple colors indicatelarge escape and collision time, while white color denote re-gions of non-escaping regular motion and trapped chaoticmotion. We note that the scale on the color bar is logarith-mic varying always from -2 to 4. Inspecting the distribu-tion of the escape time of orbits it is rather easy to associatethe stable manifold of the non-attracting invariant set withmedium escape time. Similarly, the largest observed escaperates are directly linked with the presence of sticky motion inthe vicinity of boundaries of the several stability islands (seealso [11]). It is interesting to emphasize that when J = . J < . J ( J > .
2) about halfof the configuration ( x , y ) plane is covered by initial condi-tions of non-escaping regular orbits. However as the valueof J decreases the percentage of non-escaping orbits is alsoreduced and for J < ff erence is that for J < Fig. 4
Evolution of the percentages of all types of orbits with initialconditions in the configuration ( x , y ) plane, as a function of the Jacobiconstant J . (For the interpretation of references to colour in this figurecaption and the corresponding text, the reader is referred to the elec-tronic version of the article.) which leak out through L initially increases, while for J < L increases, almost linearly, andat the highest energy level studied ( J = − .
5) they dominatecovering about 80% of the ( x , y ) plane. Taking into consid-eration the above-mentioned analysis, we may reasonablyconclude that in the 2D Hill problem non-escaping regularorbits is the most populated type of orbits at low energy lev-els, while at high enough values of the energy escaping or-bits dominate. In particular we observed that the probabilityof an orbit to leak out through L is much more higher thanthrough L . Therefore we may claim that exit channel 2 ismore preferable in the 2D system.The CCDs presented in Fig. 2 can provide useful infor-mation regarding the orbital structure of the 2D system how-ever, only for some specific values of the Jacobi integral ofmotion. In order to overcome this limitation we shall adoptthe H´enon’s method [19] thus using the section y = ˙ x = y >
0. This mean that all the initial conditions of the 2D or-bits will be launched from the x axis, with x = x , parallelto the y axis. Therefore, only orbits with pericenters on the x axis will be included, while the value of the Jacobi constant J can now be used as an ordinate. Using this method we areable to monitor the evolution of the orbital structure of the2D Hill system using a continuous spectrum of energy val-ues, rather than a few discrete ones. The orbital structure ofthe ( x , J ) plane when J ∈ [ − . , J L ) is presented in Fig. 5a, Euaggelos E. Zotos
Fig. 5 (a-left): Orbital structure of the ( x , J ) plane when J ∈ [ − . , J L ). The color code is the same as in Fig. 2. (b-right): Distribution of thecorresponding escape and collision time of the orbits with initial conditions on the ( x , J ) plane. (For the interpretation of references to colour inthis figure caption and the corresponding text, the reader is referred to the electronic version of the article.) Fig. 6
Evolution of the percentages of all types of orbits with initialconditions on the ( x , J ) plane, as a function of the Jacobi constant J .(For the interpretation of references to colour in this figure caption andthe corresponding text, the reader is referred to the electronic versionof the article.) while in Fig. 5b the distribution of the corresponding escapeand collision time of the orbits is given.With a closer look at Fig. 5a we can distinguish a whitevertical line located at x =
0. We decided to exclude theinitial condition x = R = (cid:113) x + y is zero (remember that in the( x , J ) plane all orbits have y =
0) and therefore is smallerthan the defined collision radius R col = − . This mean thatall orbits with x = x <
0, a main stability islands is present. As we proceedto lower values of J however its area is constantly reducedand when J < J L . Basins ofescape composed of initial conditions of orbits that escapethrough L are much more prominent with respect to thosecorresponding to escape through L . It is interesting to notethat Fig. 5a indicates how the fractality of the several basinboundaries varies not only as a function of the Jacobi con-stant but also of the spatial variable. In particular, we canobserve that the fractality of the basin boundaries, which isof course related to the unpredictability, migrates from theupper right side of the ( x , J ) plane for relatively high valuesof J (or in other words low values of the total orbital en-ergy) to the lower left side of the same plane for low valuesof the Jacobi constant. According to Fig. 5b near the vicin-ity of those fractal basin boundaries the highest values of theescape time of the orbits have been reported.In Fig. 6 we illustrate how the percentages of all typesof orbits with initial conditions on the ( x , J ) plane of Fig.5 evolve as a function of the value of the Jacobi integral ofmotion. We see that the evolution of the rates of all typesof orbits are more or less similar to that observed earlier in rbit classification in the Hill problem - I. The classical case 9 Fig. 4. The main di ff erences between the two plots, whichcorrespond to initial conditions of orbits in di ff erent types ofplanes, that occur for relatively high energy levels ( J < L in the ( x , J ) plane seems to saturate at a lower value (at about5%) than in the ( x , y ) plane which was equal to 20%, (ii) thepercentage of escaping orbits through L is more dominantin the ( x , J ) plane as it occupies more than 95% of the sameplane, while in the configuration ( x , y ) plane it was found tobe lower than 80%.4.2 Results for the 3D systemOur numerical investigation continues with the full 3D sys-tem. Fig. 7 presents a collection of CCDs using three dimen-sional distributions of orbits with initial condition inside the( x , y , z ) space for the same energy levels of Fig. 2. We ob-serve that in this case the grids composed of the initial condi-tions of the orbits are in fact three dimensional solids whichmeans that only their outer shell is visible. Three dimen-sional distributions of initial conditions of orbits have alsobeen recently used in order to reveal the orbital structure ofthe 3D Earth-Moon system [69]. The corresponding distri-butions of the escape and collision time of the 3D orbits aregiven in Fig. 8, where for the initial conditions of the non-escaping regular orbits as well as the trapped chaotic oneswe have used transparent white color in order to be able toinspect, in a way, the interior of the solids.If we want to examine in more detail the interior regionof the 3D grids we have to use a tomographic-style approachas in [60]. According to this approach we take the cuts onthe ( x , y ) plane for several levels of the z coordinate. Thismethod however is very costly as for every 3D grid we needseveral 2D cuts on the ( x , y ) plane. For saving space we de-cided to present for every energy level a cut on the ( x , z )plane. Therefore all 3D orbits with initial conditions on the( x , z ) plane have y = ˙ x = ˙ z =
0, while the initial valueof ˙ y is always obtained from the Jacobi integral of motion(8). There is also one more reason justifying our choice re-garding the type of the 3D cuts. The z coordinate is directlyrelated with the 3D motion of orbits, therefore we decided touse a continuous spectrum of z values rather than a few cutsfor specific values of z as in [60]. Furthermore, the CCDsof the 2D system presented earlier in Fig. 2 are in fact the z = x , z ) space,as the value of the Jacobi integral decreases, is revealed throughthe rich collection of the CCDs presented in Fig. 9, whilethe corresponding distribution of the escape and collisiontime of the orbits is given in Fig. 10. The outermost black solid line denotes the zero velocity curve which is definedas 2 W ( x , y = , z ) = J . It is seen that the orbital structure isvery di ff erent with respect to that observed in the 2D system.For high values of the Jacobi constant non-escaping regularmotion dominates, while initial conditions of escaping or-bits are mainly confined to the right side of the ( x , z ) planes.A very interesting phenomenon is the presence of trappedchaos. Trapped chaotic orbits have also been observed inother types of 3D open Hamiltonian systems. For instance,in [62] we detected a considerable amount of trapped chaoticorbits when we investigated the orbital properties of an opentidally limited star cluster, using a three dimensional dynam-ical model. On the other hand, in the 3D Earth-Moon systemwhich was explored in [69] we did not find any numericalevidence of trapped chaos. The most important changes thatoccur on the ( x , z ) plane as the value of the Jacobi constantdecreases are the following: – The area composed of initial conditions correspondingto non-escaping regular orbits is constantly reduced. It isseen that for J < . x , z ) plane, while for negative valuesof the Jacobi integral of motion they completely disap-pear. – Collision basins exist mainly inside the vast regular re-gions thus creating a complicated mixture of initial con-ditions of both collision and non-escaping regular orbits.We found that for J < . – Initial conditions corresponding to trapped chaotic or-bits are visible only for relatively high values of the Ja-cobi constant, very close to the critical energy value J L .Our computations suggest that for J < . x , z ) plane and therefore they arenot visible. – As we move away from the critical value of the energyto higher levels of energy the basins of escape start to ex-pand. Initially they are mainly located to the right side ofthe ( x , z ) plane however as the value of J decreases theymigrate also to the left side of the same plane. Finally atextremely low levels of J and particularly for negativevalues basins of escape cover the entire ( x , z ) plane. – Our numerical calculations strongly indicate that the twoexit channels, even though they are symmetrical, theyare not equiprobable. Indeed it was found that the testparticles have the tendency to leak out mainly from theexit channel near L . Similar behavior was also observedin the 2D system. It is interesting to note that for ex-tremely low values of the Jacobi integral of motion ( J <
0) almost the entire ( x , z ) plane is dominated by initialconditions of orbits which escape through channel 2. Fig. 7
Evolution of the orbital structure of the configuration ( x , y , z ) space with decreasing value of the Jacobi integral of motion. The color codeis the same as in Fig. 2. (For the interpretation of references to colour in this figure caption and the corresponding text, the reader is referred to theelectronic version of the article.) However, a small basin of escape corresponding to chan-nel 1 is still present near the origin.Exploiting the numerical integration of the three dimen-sional distributions of initial conditions of orbits shown inFig. 7 we managed to demonstrate in the diagram of Fig. 11the evolution of the percentages of all types of orbits as afunction of the value of the Jacobi integral of motion. It isseen that at high enough values of J , just above the energyof escape, about 80% of the configuration ( x , y , z ) space is covered by initial conditions of non-escaping regular orbits.As the value of J decreases however the rate of regular or-bits also decreases until it completely disappears for J < J > . rbit classification in the Hill problem - I. The classical case 11 Fig. 8
Distribution of the corresponding escape and the collision time of the orbits with initial conditions in the configuration ( x , y , z ) space forthe values of Jacobi integral of motion presented in Fig. 7. The bluer the color, the larger the escape or the collision time. Initial conditions of bothnon-escaping regular orbits and trapped chaotic orbits are shown in transparent white. (For the interpretation of references to colour in this figurecaption and the corresponding text, the reader is referred to the electronic version of the article.) low values of the Jacobi constant, even though the corre-sponding percentage is lower than 0.01%. The percentage ofescaping orbits through L initially increases until J = . J the tendency is reversed. On the otherhand, the rate of orbits that leak out through L is constantlyincreases. At the highest energy level studied, that is for J = − .
5, escaping orbits through channel 2 cover about 86% of the configuration space, while at the same time or-bits that escape through exit channel 1 occupy only 13% ofthe same space. Thus is becomes more that evident that theescape process in the 3D Hill system is a very complicatedphenomenon.The CCDs presented in both Figs. 7 and 9 reveal the or-bital properties of the 3D system however for a fixed valueof the Jacobi integral of motion. In order to obtain a more
Fig. 9
Evolution of the orbital structure of the ( x , z ) space with decreasing value of the Jacobi integral of motion. The color code is the same asin Fig. 2. (For the interpretation of references to colour in this figure caption and the corresponding text, the reader is referred to the electronicversion of the article.) complete view of the orbital structure of the 3D Hill sys-tem, we shall try to gather information of a continuous spec-trum of energy values. To achieve this we will adopt the ap-proach used previously for the 2D system. In particular, forspecific values of z we define dense uniform grids of ini-tial conditions on the ( x , J ) plane with y = ˙ x = ˙ z = y is obtained fromthe energy integral (8). Fig. 12a shows the case when z = . z is relatively low and thereforethe motion of the orbits takes place very close to the two-dimensional ( x , y ) plane. When z = . x , J ) plane seems to weaken, since initial conditionsof non-escaping regular orbits emerge inside it. Furthermoreit is seen that basins composed of bounded regular orbits ex-pand also at the right side of the plane, while the extent of rbit classification in the Hill problem - I. The classical case 13 Fig. 10
Distribution of the corresponding escape and the collision time of the orbits with initial conditions on the ( x , z ) space for the values ofJacobi integral of motion presented in Fig. 9. (For the interpretation of references to colour in this figure caption and the corresponding text, thereader is referred to the electronic version of the article.) the basins of initial conditions of orbits that escape through L is reduced. The outermost black solid line is the limitingcurve which is defined as f ( x , J ; z ) = W ( x , y = , z = z ) = J . (12)In panel (c) where z = . z = . x , J ) plane. It should beclarified that in this case (the 3D system) the vertical whiteline corresponding to initial conditions of orbits inside thecollision radius of the secondary is not present. This is truebecause all four values of z are such ( z > .
01) so thedistance R = (cid:113) x + y + z is always larger than the col-lision radius, even when x = y =
0. Looking the four
Fig. 11
Evolution of the percentages of all types of orbits with initialconditions in the configuration ( x , y , z ) space, as a function of the Ja-cobi constant J . (For the interpretation of references to colour in thisfigure caption and the corresponding text, the reader is referred to theelectronic version of the article.) panels of Fig. 12(a-d) it becomes evident that the energeti-cally permissible area on the ( x , J ) plane is reduced as theinitial value of the z coordinate increases. The distributionof the corresponding escape and collision time of the or-bits with initial conditions on the ( x , J ) plane is presentedin Fig. 13(a-d). We see in panel (c) that the escape time ofthe orbits with initial conditions in the vicinity of the fractalbasin boundaries are extremely high corresponding to morethan 1000 time units. On the other hand, all orbits with ini-tial conditions very close to the equilibrium point L (outerright side of the plots) escape almost immediately within thefirst couple of time steps of the numerical integration.We close our numerical investigation by presenting inFig. 14(a-d) the evolution of the percentages of all types oforbits with initial conditions on the ( x , J ) plane for the fourcases discussed in Fig. 12(a-d). One may observe that inall cases escaping orbits completely dominate for extremelylow values of the Jacobi integral of motion, or in other wordsfor high values of the total orbital energy. For high valuesof J on the other hand, non-escaping regular orbits is themost populated type of orbits in the first three cases, ex-cept the case where z = .
6. In this case escaping orbitsdominate but this is actually a numerical artifact created bythe strange geometry of the limiting curve (see Fig. 12d).Here we would like to clarify that the evolution of the per-centages of trapped chaotic orbits was not included in allfour cases because the corresponding percentage was mea-sured to be always extremely low (lower than 0.1%). Taking into account all the results given in Fig. 14(a-d) we may rea-sonably conclude that the behavior according to which forhigh enough values of the energy the probability an orbit toescape through exit channel 2 is considerably much higherthan trough L , that initially observed in the 2D system, isstill valid also in the 3D system. For 3-dof Hamiltonian systems the general situation for mod-erate perturbation is the following: There are many surviv-ing KAM tori which fill a part of the energy shell whosemeasure is larger than zero and even can be large. These in-variant surfaces are 3 dimensional and therefore in the 5 di-mensional energy shell they do not divide anything. I.e. thecomplement of all these tori is connected. On the other hand,there is an infinity of resonance zones which contain unsta-ble behaviour. In principle, all these resonance zones (finechaos strips) are connected and form a single web of reso-nances (Arnold web). For perturbation di ff erent from zero,i.e. for systems which are not integrable, the measure of thisweb of resonance zones also is larger than zero. It is a frac-tal web of an infinity of small layers. Accordingly, at verylarge times an orbit starting in any initial point of this webcan reach any other point of the web, even when it usuallymay take an extremely long time. The motion in the weblooks similar to di ff usive motion and for moderate pertur-bation the di ff usion constant is very small. The surviving 3dimensional tori form a kind of fractal of obstacles for thegeneral motion in the space between the tori.Now if we imagine the situation where there is also asmall escape channel for escape. This escape channel is incontact with the web of chaotic resonance zones. Therefore,any orbit starting in any of the resonance zones will finallyescape after a possibly extremely large time. However, whenwe observe for a finite time only (this may be a large time,even large compared to the life time of the universe) thenwe only observe the di ff usive like chaotic motion inside ofthe web and for most orbits we do not see the final escape.For all practical purposes we have the impression havingobserved trapped chaos.In the CCDs presented in Fig. 9 we observed that for en-ergy levels just above the critical energy of escape J L thereis an amount of trapped chaotic orbits. The phenomenonof trapped chaos has been also observed in other types ofHamiltonian systems (e.g., [29, 62]). According to classicalchaos theory however, chaotic orbits do not admit a third in-tegral of motion. Therefore for time t → ∞ they will fill allthe available phase space inside the zero velocity surface,unless of course they escape. In other words, all trappedchaotic orbits will eventually leak out passing through oneof the two escape channels, given enough time of numeri-cal integration. In Fig. 15 we present a characteristic exam- rbit classification in the Hill problem - I. The classical case 15 Fig. 12
The orbital structure of the ( x , J ) plane when (a-upper left): z = . z = .
2, (c-lower left): z = .
4, and (d-lowerright): z = .
6. The color code is the same as in Fig. 2. (For the interpretation of references to colour in this figure caption and the correspondingtext, the reader is referred to the electronic version of the article.) ple of a trapped chaotic orbit with initial conditions: x = . y = z = . x = ˙ z =
0, whilethe initial value of ˙ y > J = . time units. It is seen that the orbit does escape through exit channel 1, after about208934 time units.To check whether all trapped chaotic orbits, observed inthe corresponding CCD (when J = . time units. Our re- Fig. 13
Distribution of the corresponding escape and the collision time of the orbits with initial conditions on the ( x , J ) plane for the casespresented in Fig. 12. (For the interpretation of references to colour in this figure caption and the corresponding text, the reader is referred to theelectronic version of the article.) sults are illustrated in Fig. 16a where one can observe thatall these initial conditions, that were initially classified astrapped chaotic ones for 10 time units, they have escapedcreating a highly fractal escape mixture. The distribution ofthe corresponding escape time of the 3D orbits is given inpanel (b) of Fig. 16, where this time the values on the colorbar vary from 4 to 6 (remember that the scale of the accom-panying color bar is in logarithmic scale). The same proce-dure was carried out for all the other sets of trapped chaoticorbits reported in the CCDs for lower values of the Jacobiconstant. We found that all of these orbits eventually escapethrough one of the two channels however after extremelylong time of numerical integration ( t esc ∈ (10 , )). There-fore, we may reasonably conclude that in the 3D Hill sys- tem long-lived (or temporarily trapped) chaotic orbits existfor values of energy above but very close to the energy ofescape.In 2-dof systems we have 3 dimensional energy shellsand the 2 dimensional KAM tori separate these energy shellsin disjoint inner and an outer parts. In general, there arechaotic layers also inside of the KAM tori. But these lay-ers are disconnected from the outside. Therefore, chaos inthese inside layers can never reach the exit and consequentlyit can never escape. Thus it is truly trapped chaos for infi-nite time in the strict mathematical sense. Here it should beemphasized that none of the examined initial conditions ofthe 3D trapped chaotic orbits in the CCDs has z =
0. Thisfact supports our findings of the 2D system, where there was rbit classification in the Hill problem - I. The classical case 17
Fig. 14
Evolution of the percentages of all types of orbits with initial conditions on the ( x , J ) plane when (a-upper left): z = . z = .
2, (c-lower left): z = .
4, and (d-lower right): z = .
6. (For the interpretation of references to colour in this figure caption and thecorresponding text, the reader is referred to the electronic version of the article.) no numerical indication of 2D (temporarily) trapped chaoticmotion whatsoever (e.g., [1]).
The aim of this paper was to reveal the overall orbital struc-ture of the classical Hill problem. By performing a thor- ough and extensive numerical investigation using large setsof initial conditions, in both the two-dimensional and thethree-dimensional configuration space, we managed to clas-sify orbits into four categories: (i) non-escaping regular, (ii)trapped chaotic, (iii) escaping, and (iv) collision. We alsorelated the several basins of escape and collision with thecorresponding escape and collision time of the orbits. Asfar as we know, this is the first detailed and systematic nu-
Fig. 15
Time-evolution of the x coordinate and the distance R = (cid:112) x + y + z from the center of a trapped chaotic orbit, when J = . merical analysis regarding orbit classification in the classi-cal Hill problem (especially in the 3D configuration space)and therefore this is exactly the contribution as well as thenovelty of our study.A Quad-Core i7 2.4 GHz PC was used for the numericalintegration of the sets of the initial conditions of the orbits.For the completion of every set of initial conditions of orbitswe needed between about 2 minutes and 4 days of CPU time,depending of course on the collision and escape time of theorbits in each case. Here it should be explained that, for sav-ing time, when an orbit escaped or collided the numericalintegration was stopped and proceeded to next available ini-tial condition.In this paper we provide quantitative information regard-ing the orbital structure of the classical planar (2D) and spa-tial (3D Hill problem. The main novel results of our orbitclassification can be summarized as follows:1. At high enough values of the Jacobi integral of motion,or in other words at low values of the total orbital en-ergy non-escaping regular orbits as well as collision or-bits dominate in both the 2D and the 3D system, while initial conditions of escaping orbits were found mainlynear the equilibrium points.2. Escaping orbits are the most populated type of orbits atrelatively high values of the total orbital energy, wherebasins of escape were found to occupy almost the entireconfiguration space.3. As the value of the Jacobi constant decreases the per-centages of both non-escaping regular orbits and colli-sion orbits are reduced. Our computations suggest thatfor negative values of J there is no indication of boundedmotion whatsoever, while on the other hand collisionmotion is still possible.4. In the 3D system, and especially for values of energyabove yet very close to the critical energy of escape,we identified a portion of trapped chaotic orbits. Addi-tional numerical calculations revealed that all these or-bits eventually do escape, while having huge escape pe-riods (more than 10 time units). It should be empha-sized that the phenomenon of (temporarily) trapped chaoswas not observed in the 2D system.5. In both the 2D and the 3D systems we observed thatthe two symmetrical escape channels located near theequilibrium points are not equiprobable . In fact, for lowvalues of the Jacobi constant the vast majority of orbitschoose to escape through exit channel 2, while the per-centage of exit channel 1 remains very low.For very low values of the mass ratio µ the restrictedthree-body problem is well approximated by the the Hillproblem. It should be emphasized that there are numerousprevious works on the field of capture and escape on theRTBP (e.g., [2, 41]) as well as on the Hill problem (e.g., [55,56]). Furthermore, similar systematic classification of initialconditions of orbits, with respect to the present one, has beenperformed in the RTBP in general (e.g., [33, 34, 64–66])and also in specific planetary systems (e.g., [13, 69]) (Earth-Moon system), (e.g., [67]) (Saturn-Titan system), and (e.g.,[68]) (Pluto-Charon system). Therefore it would be very in-teresting if we could determine the similarities and the dif-ferences between the RTBP and the Hill problems. Howeverthere are two main reasons which make this task impossible:(i) the scattering region is not the same in all these works.In some cases the scattering region is extended around bothprimaries, while in some other cases it is confined very close The two saddle points L and L lie along the x axis and the zerovelocity surfaces are symmetric with respect to the three axes. It iseasy to prove that if ( x ( t ) , y ( t ) , z ( t )) is a solution of the Hill problem,( − x ( t ) , − y ( t ) , − z ( t )) is also a solution, of course with appropriately cho-sen momenta (because of the presence of the Coriolis force). Therefore,for every orbit escaping through L there is a symmetric-related orbitescaping through L , because the dynamics of the system must be sym-metric. On this basis, the observed preference for escape through exitchannel 2 is due to the particular choice of the initial conditions. Simi-lar choices of initial conditions of orbits can be found in several earlierrelated papers (e.g., [13, 33, 34, 67–69])rbit classification in the Hill problem - I. The classical case 19 Fig. 16 (a-left): Orbital structure of the ( x , z ) plane for J = . time units. The colorcode is the same as in Fig. 2. (b-right): The distribution of the corresponding escape time of the orbits. (For the interpretation of references tocolour in this figure caption and the corresponding text, the reader is referred to the electronic version of the article.) to the secondary. (ii) the criteria for distinguishing betweenthe several types of the orbits are di ff erent. For instance, inthe present Hill problem there is only one type of escapingorbits, while on the other hand in the previous cases therewere two types of escape (escape toward the primary realmand escape toward the exterior region). Therefore, we be-lieve that if we try to compare all the di ff erent results wemight end to erroneous and / or misleading conclusions dueto the two above-mentioned reasons.Taking into account the novel outcomes of our detailednumerical investigation we could argue that out task hasbeen successfully completed. We hope that the present re-sults to be useful in the field of the orbital dynamics of theHill problem. In the next two papers of the series we willexplore the orbital structure of the Hill problem where theperturbations of the oblateness (Part II) and the radiationpressure (Part III) are present. Acknowledgments
The author would like to express his warmest thanks to thethree anonymous referees for the careful reading of the orig-inal manuscript and for all the apt suggestions and com-ments which allowed us to improve both the quality as wellas the clarity of the paper.
Compliance with Ethical Standards–
Funding: The author states that he has not received anyresearch grants. – Conflict of interest: The author declares that he has noconflict of interest.
Appendix: Derivation of the potential function of the clas-sical Hill problem
The first step is to perform the coordinate transformationgiven in Eq. (4) along with µ = M . Then Eq. (1) becomes apolynomial function Ω = Ω ( x , y , z , M ).It is very easy to prove that Ω lim = lim M → Ω = / Ω into a power series around M = W ( x , y , z , M ) = + M (cid:32) x − z + r (cid:33) , (13)where r = (cid:112) x + y + z .Now for obtaining the potential function W ( x , y , z ) of theclassical Hill problem all we have to do is to eliminate theparameter M for Eq. (13). This can be achieved as W ( x , y , z ) = W − Ω lim M , (14)thus deriving the final form of Eq. (6) W ( x , y , z ) = x − z + r . (15) References
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