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M. Guzzo, E. Lega, On the identification of multiple close encounters in the planar circular restricted three-body problem, Monthly Notices of the Royal Astronomical Society, Volume 428, Issue 3, 21 January 2013, Pages 2688–2694, https://doi.org/10.1093/mnras/sts225
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Abstract
We describe a technique which allows us to numerically detect orbits of the planar circular restricted three-body problem with multiple close encounters with the secondary mass for values of the Jacobi constant |${\cal C}< 3+\mu ^2-4\mu$|. We find that these orbits are organized in structures which, on Poincaré surfaces of sections, appear as a hierarchy of lobes. The detection of multiple close encounters has implications in cometary dynamics as well as in the study of asteroids with potential impact risk with the Earth.
1 INTRODUCTION
The study of multiple close encounters of a small body with a planet or a natural satellite appears in several studies of cometary dynamics, of asteroids which in the future could impact the Earth and of the space flight dynamics (see Greenberg, Carusi & Valsecchi 1988; Chodas 1999; Valsecchi 1999; Valsecchi et al. 2003; Valsecchi 2005; Koon et al. 2007 and references therein). Close encounters are studied in the literature with different techniques: mathematical studies tackle the problem with techniques combining regularizations, hyperbolic dynamics and variational methods (see, for example, Henrard 1980; Bolotin & MacKay 2000; Font, Nunes & Simó 2002); studies with astronomical motivations mainly use numerical integrations, often in combination with some approximations of the close encounters of the three-body problem such as the Öpik theory (Öpik 1976; Valsecchi 2002). In practice, astronomical studies require techniques which find orbits with multiple close encounters, working in a broad class of models and situations. In this paper we consider the planar circular restricted three-body problem, without assuming further approximations to treat the close encounters, for small values of the mass ratio μ = 10−6 (suitable for the comparison with the Kepler problem) and μ = 0.000 9537 (representative of the Jupiter–Sun mass ratio), at values of the Jacobi constant |${\cal C} <3+\mu ^2-4\mu$| which correspond to hyperbolic encounters with the secondary mass (see Section 3).
We explore surfaces of sections, which are of frequent use in astronomical studies,1 with a numerical technique which combines the fast Lyapunov indicator (FLI hereafter; Froeschlé, Lega & Gonczi 1997; Froeschlé, Guzzo & Lega 2000; Guzzo, Lega & Froeschlé 2002) and the regularization of the three-body problem (as in Celletti et al. 2011; Lega, Guzzo & Froeschlé 2011). We remark that FLI techniques developed with model problems can be suitably adapted to the more complicate models of astronomical interest (see Guzzo 2005, 2006; Villac 2008). The typical use of the FLI to investigate the dynamics of a system consists in its computation on a grid of initial conditions regularly spaced on a well-chosen surface of section. The representation of the values of the FLI on the surface of section provides the so-called FLI map of the section. The FLI method we use in this paper is a modification of the FLIs which have been used so far to study the three-body problem (Celletti et al. 2011; Lega et al. 2011). In the following we will call this modified FLI method: regularized fast Lyapunov indicator (RFLI hereafter).
In this paper, using the RFLI, we find orbits with multiple encounters with the secondary mass in neighbourhoods of resonant periodic orbits. The RFLI analysis reveals that orbits with multiple close encounters, among all chaotic resonant motions, appear organized in curves with a hierarchical lobe structure, each lobe of the hierarchy corresponding to an additional close encounter. The lobe hierarchy appears from a sequence of snapshots of the resonance obtained by computing the RFLI on the surface of section in intervals of time containing different multiples of the period T of the reference resonant periodic orbit. The numerical detection of the lobe hierarchy does not require strong restrictions on the mass ratio, and it could be exported also to more complicate models, accounting, for example, the spatial problem and other perturbing bodies. We remark that some kinds of hierarchies in the studies of multiple close encounters have been detected also in previous numerical studies (Chodas 1999; Valsecchi et al. 2003) as well as in analytic studies of Poincare’s second species solutions of the three-body problem, such as Font et al. (2002).
Section 2 of this paper is fully dedicated to the description of the RFLI detection of multiple close encounters and their organization in hierarchical lobe structures. In Section 3 we provide all the details about the FLI used for the study of multiple close encounters. We also provide a motivation for the use of the RFLI variant for |${\cal C} <3+\mu ^2-4\mu$|. Conclusions are reported in Section 4.
2 DETECTION OF ORBITS WITH MULTIPLE CLOSE ENCOUNTERS
The RFLI is computed for all the points of the grid W up to the same integration time T and is then represented with a colour scale on the window of the |$(x,\dot{x})$| variables, so that yellow corresponds to the highest values for the |${\rm RFLI}((x,\dot{x}), w,T)$|.2 We will refer to the colour representation of the RFLI value on the selected window of Σμ, C, as an RFLI map of the section.
In Lega et al. (2011) we extensively discussed why the FLIs are suitable to detect close encounters and collisions, as well as other resonant structures of the three-body problem. In Section 3 we provide a detailed motivation why the variant RFLI is suitable to detect the close encounters for C < 3 + μ2 − 4μ. Precisely, among all the initial conditions |$(x,\dot{x})$| on the selected window W ⊆ Σμ, C, those which have a close encounter in the time interval [−T, T] are characterized by the highest values of |${\rm RFLI}((x,\dot{x}),w,T)$|, that is, are identified by the yellow points of the RFLI map. In the case μ = 0, the yellow points of the RFLI map provide the initial conditions of orbits which pass very close or through (x, y) = (1, 0) in the time interval [−T, T]. We refer to Section 3 for all the technical details about the indicators. In the following we discuss the results about the detection of the multiple close encounters.
Let us denote by T* the period of the reference periodic orbit with initial condition |$(x_*,\dot{x}_*)\in \Sigma _{\mu .C}$|. In the Kepler approximation, the orbit with initial condition |$(x_*,\dot{x}_*)$| passes through (1, 0) for t = t0 + kT*, |$k\in {\mathbb {Z}}$|, where t0 is the time of the first passage. In Fig. 1 we report RFLI maps of the section Σμ, C for μ = 10−6, C = 2.7, in a window including a periodic orbit of the Kepler problem of period T* = 5(2π), obtained by integrating numerically the three-body problem in different time intervals which include different multiples of T*. For reference, we also report (top-left panel) the RFLI map obtained by integrating numerically the same initial conditions in the Kepler approximation μ = 0.
Top-left panel; μ = 0, T = 30: the RFLI is computed by numerically integrating the initial conditions on a grid of |$(x,\dot{x})\in \Sigma _{\mu ,C}$| in the Kepler approximation μ = 0, on the time intervals [0, T], [−T, 0]. Therefore, the yellow curves on the RFLI map, determined by the highest values of the RFLI, correspond to the initial conditions on the section which pass through (1, 0) for some time t ∈ [−T, T]. Points on the section which correspond to periodic orbits with suitably short period may be found at the crossing of two yellow curves, corresponding to the highest values of the RFLI obtained with the forward and the backward integration, respectively. This is also the case of the reference periodic orbit, which appears at the crossing of two arc-shaped yellow curves.
Top-right panel; μ = 10−6, T = 30: we still appreciate two arc-shaped yellow curves which cross close to |$(x_*,\dot{x}_*)$|. These curves, determined by the highest values of the RFLI, correspond to the initial conditions on the section which pass very close to the singularity at (1 − μ, 0) for some time t ∈ [−T, T]. These arc-shaped curves are close to those detected in the μ = 0 case and will be denoted by L(1) curves. The box represents the zoomed area which will be considered in the bottom panels.
Bottom-left panel; μ = 10−6, T = 60: by extending the integration time from T = 30 to 60, we see the appearance of a lobe structure of orbits which have a close encounter in the extended time interval. The lobe structure contains the initial conditions on the section that perform a revolution around the primary before having an approximate collision in the time interval [30, 60] (or in the time interval [−60, −30]) and will be denoted by L(2).
Bottom-right panel; μ = 10−6, T = 90: by further extending the integration time from T = 60 to 90, we see the appearance of other lobe structures, which we denote by L(3), which contain the initial conditions on the section which have a close encounter in the time interval [60, 90]. The approximate close encounter occurs after the orbit performs two revolutions around the primary.
We select three initial conditions, which we denote by l(1), l(2), l(3), on the different close encounter structures L(1), L(2), L(3) detected by the RFLI, and we study in more detail their dynamics.
Fig. 1 shows that the set of initial conditions on the section which have a close encounter in some time t appears as a hierarchy of lobes L(n) which can be labelled by the number n − 1 of revolutions around the primary characterizing the close encounter. In Fig. 2, we increase the integration time to T = 150 to show how the hierarchy of lobes can be increased by increasing further the number of revolutions around the primary, thus providing the concrete possibility of selecting initial conditions which perform a close encounter after a desired number of revolutions.
2.1 Orbits with multiple close encounters
As a matter of fact, because of the peculiar location of the lobes L(n) on the section, it is possible to choose initial conditions on a lobe L(n) which are very close to other lobes L(n′), L(n′′), …. These initial conditions correspond to multiple close encounters occurring after n − 1, n′ − 1, n′′ − 1 revolutions around the primary, the depth of each close encounter decreasing by increasing the distance from the corresponding lobe.
For example, since the structures L(1), L(2), L(3) detected in Fig. 1 are quite close on the FLI map, it is possible to choose initial conditions on L(1), L(2), L(3) related to orbits with one, two or three close encounters. These close encounters are characterized by different depths, that is, by different distances r2 between the two bodies, and occur in zero, one or two revolutions around the primary. At this regard, let us consider the initial condition l(3) ∈ L(3) (see bottom-right panel): since l(3) ∈ L(3), the orbit with initial condition l(3) has a close encounter after two revolutions around the primary, at a certain distance which we denote by r(3)2. But l(3) is also very close to L(2), so the orbit has a close encounter also after one revolution around the primary at a distance r(2)2. Since l(3) is very close to L(2), but it does not belong to L(2), we expect r(3)2 < r2(2). Since l(3) is also close to L(1), the orbit has a close encounter before completing the first revolution at a distance r(1)2, again with r(3)2 < r2(1) and possibly r(3)2 < r2(2) < r(1)2. These multiple close encounters are shown in Fig. 3. Precisely,
Top panel: we plot the projection on the x, y plane of three orbits with initial conditions l(1) ∈ L(1), l(2) ∈ L(2) and l(3) ∈ L(3) marked by bullets in the bottom-right panel of Fig. 1. The initial conditions on each lobe have been chosen according to the highest value of the RFLI. We denote by l(1)a, by l(2)a, l(2)b and by l(3)a, l(3)b, l(3)c the sequence of the closer encounters of the three orbits occurring at every revolution around the primary mass. We remark that, only in correspondence with the last close encounter, the orbits show sharp deviations (well represented in the zoom box within Fig. 3) typical, for example, of the so-called Poincaré second species solutions. For l(2) and l(3), such a deep close encounter occurs after one and two revolutions, corresponding to one and two additional close encounters, respectively.
Bottom panel: we plot the time evolution of the |${\rm RFLI}((x,\dot{x}),w,t)$| for the three different orbits and for t in the interval [0, 90]. We mark with l(1)a, with l(2)a, l(2)b and with l(3)a, l(3)b, l(3)c the time corresponding to the close encounters marked in the top panel. It appears that the RFLI increases sharply with time when a close encounter occurs.
2.2 The Jupiter–Sun mass ratio
We here consider a value of the mass ratio μ = 0.000 9537, corresponding to the Jupiter–Sun mass ratio. In Figs 4 and 5 we report the detection of the hierarchical structure of lobes L(n). In the bottom-right panel of Fig. 4 we mark with bullets the initial conditions l(1), l(2), l(3), l(4) on L(1), L(2), L(3), L(4) of orbits having one, two, three and four close encounters, respectively, occurring after zero, one, two and three revolutions around the Sun.
In the top panel of Fig. 6 we plot the projection on the x, y plane of the four orbits with initial conditions l(1), l(2), l(3), l(4). We denote by l(1)a, by l(2)a, l(2)b, by l(3)a, l(3)b, l(3)c and by l(4)a, l(4)b, l(4)c, l(4)d the sequence of the close encounters of the four orbits occurring at every revolution around the Sun. All the four orbits have sooner or later a deep close encounter with Jupiter, which is well represented in the zoom box within Fig. 6.
In the bottom panel of Fig. 6 we plot the time evolution of the |${\rm RFLI}((x,\dot{x}),w,t)$| for the four different orbits and for t in the interval [0, 25]. We mark with l(1)a, with l(2)a, l(2)b, with l(3)a, l(3)b, l(3)c and with l(4)a, l(4)b, l(4)c, l(4)d the time corresponding to the close encounters marked in the top panel. It appears that the RFLI increases sharply with time for most of the close encounters. For the orbits l(3), l(4), the increase of RFLI with time has a less steep gradient when close encounters l(3)a, l(4)b occur; however, the gradient is still sufficiently high enough to allow the detection of the close encounter.
3 USE OF THE FLI METHOD TO COMPUTE THE COLLISIONS WITH THE SECONDARY MASS
The FLI method (Froeschlé et al. 1997; Guzzo et al. 2002) has been successfully used in several problems to detect chaos related to resonances (Froeschlé et al. 2000; Guzzo et al. 2002), as well as the stable and unstable manifolds (Guzzo 2010; Villac 2008; Guzzo, Lega & Froeschlé 2009; Lega, Guzzo & Froeschlé 2010), and also the tube manifolds of the restricted three-body problem (Lega et al. 2011). In this paper we compute the FLI in the space of the Levi-Civita variables, following the manner of Celletti et al. 2011, but with reference to a fixed physical time (and not to a fixed regularized time). Our choice is useful to compare the effect of different close encounters which occur at about the multiples of the period of the reference periodic orbit.
3.1 Collisions with |$\boldsymbol {C< 3+\mu ^2-4\mu }$| are hyperbolic in the space of the regularized variables
We here explain why the RFLI is suitable to detect the close encounters for the values of the Jacobi constant satisfying C < 3 + μ2 − 4μ.
The limit (13) extends the known result (see for example Henrard 1980) that for μ = 0 and C < 3 + μ2 − 4μ the point (u, u′) = (0, 0) is an hyperbolic equilibrium point of the system (8). For μ > 0, we are not able to identify the collision with a hyperbolic equilibrium point, but nevertheless the limit (13) means that, at collisions, the growth of tangent vectors is exponential in the regularized fictitious time s, and correspondingly it has a sharp increment in the physical time t. The computation of the RFLI over grids of initial conditions up to some time t provides a sharp detection of the set of points of the grid which have one or more close encounters in the time t ≤ T. The approximate collision set appears as the set of highest values of the RFLI.
4 CONCLUSION
Using a numerical technique which combines the FLI and the regularization of the three-body problem, we find orbits with multiple close encounters with the secondary mass for values of the Jacobi constant |${\cal C}<3+\mu ^2-4\mu$|. We show that the set of initial conditions which have a close encounter appears as a hierarchy of lobes on the Poincaré surface of sections. This study is relevant for many astronomical studies, like the dynamics of comets and of asteroids which could impact the Earth; it may be also relevant for applications in space flight dynamics.
MG has been supported by the project CPDA092941/09 of the University of Padova and by the Fondazione CaRiPaRo Project ‘Nonlinear Partial Differential Equations: models, analysis, and control-theoretic problems’. Part of the computations have been done on the ‘Mesocentre SIGAMM’ machine, hosted by the Observatoire de la Côte d’Azur. We thank Claude Froeschlé for many useful discussions.
By denoting with x, y the rotating frame with primary and secondary bodies on the x-axis, for a fixed value C of the Jacobi constant, the section is parametrized by the values of |$x,\dot{x}$| (the other initial conditions are y = 0, |$\dot{y}$| is obtained from the Jacobi constant).
The colour version of all figures can be found in the electronic version of the paper so light grey in the black–white printed version corresponds to yellow in the coloured electronic version.