On quasi-satellite periodic motion in asteroid and planetary dynamics
OOn quasi-satellite periodic motion in asteroid andplanetary dynamics
G. Voyatzis , K. I. Antoniadou Department of Physics, Aristotle University of Thessaloniki, 54124, Thessaloniki, Greece NaXys, Department of Mathematics, University of Namur, 8 Rempart de la Vierge, 5000 Namur, [email protected], [email protected]
The final publication is available athttps://link.springer.com/article/10.1007/s10569-018-9856-2
Abstract
Applying the method of analytical continuation of periodic orbits, we study quasi-satellite motion in the framework of the three-body problem. In the simplest, yet nottrivial model, namely the planar circular restricted problem, it is known that quasi-satellite motion is associated with a family of periodic solutions, called family f , whichconsists of 1:1 resonant retrograde orbits. In our study, we determine the critical orbits offamily f that are continued both in the elliptic and in the spatial model and compute thecorresponding families that are generated and consist the backbone of the quasi-satelliteregime in the restricted model. Then, we show the continuation of these families inthe general three-body problem, we verify and explain previous computations and showthe existence of a new family of spatial orbits. The linear stability of periodic orbitsis also studied. Stable periodic orbits unravel regimes of regular motion in phase spacewhere 1:1 resonant angles librate. Such regimes, which exist even for high eccentricitiesand inclinations, may consist dynamical regions where long-lived asteroids or co-orbitalexoplanets can be found. keywords The term quasi-satellite (QS) motion refers to retrograde satellite motion, which takes placeoutside of the Hill’s sphere (Mikkola and Innanen, 1997). In the framework of the three-bodyproblem (TBP), QS motion is a special case of 1:1 mean-motion resonance or, with anotherterm, co-orbital motion. In the planar circular restricted TBP such a type of motion has beenidentified by the existence of the family f of periodic orbits (Broucke, 1968; H´enon, 1969; Benest,1974). In the planar general TBP, where co-orbital planetary motion is considered, a familyof periodic QS orbits has been computed by Hadjidemetriou et al. (2009); Hadjidemetriou andVoyatzis (2011), and was called family S .Special interest for QS-orbits, which are called also distant retrograde orbits (DRO), isgrowing for the design of spacecraft missions around moons or asteroids (Perozzi et al., 2017;Minghu et al., 2014). A first application was achieved for the Phobos program (Sagdeev andZakharov, 1989) for which QS orbits were computed by Kogan (1989). In the last twenty1 a r X i v : . [ a s t r o - ph . E P ] S e p ears, much attention has been given to QS asteroid motion. Evidence for the existence ofstable QS-type motion around giant planets of our Solar system has been found after havingconsidered analytical or semi-analytical perturbative methods (Mikkola and Innanen, 1997;Namouni, 1999; Nesvorn´y et al., 2002; Mikkola et al., 2006; Sidorenko et al., 2014; Pousseet al., 2017) or numerical integrations (Wiegert et al., 2000; Christou, 2000a,b). A lot ofstudies have also focused on particular observed asteroids and Centaurs e.g. the 2002 VE68around Venus (Mikkola et al., 2004) and the 2015 BZ509 around Jupiter (Namouni and Morais,2018). Furthermore, greatly interesting is the long-term stability of near-Earth asteroids, whichare located in the QS regime, like e.g. 2004GU9, 2006FV35 and 2013LX28 (Connors et al.,2004; Morais and Morbidelli, 2006; Wajer, 2010; Connors, 2014) or of Earth-trojans (Dvoraket al., 2012).Exosolar planets in co-orbital motion, although having not been discovered yet, they canconstitute exceptional planetary configurations, which include exomoons (Heller, 2018), ex-otrojans or, possibly, quasi-satellite-like orbits, where planets of equivalent masses may evolveinto (Giuppone et al., 2012; Funk et al., 2013; Leleu et al., 2017; Lillo-Box et al., 2018). Stud-ies on the possible existence and stability of such planetary configurations were performed forthe co-planar case. For instance, a numerical study for the stability of exotrojans is given inSchwarz et al. (2009). Hadjidemetriou et al. (2009) computed a stable family, S , of orbits whichpass smoothly from planetary to satellite type orbits. In Giuppone et al. (2010), the stabilityregions are examined and, besides family S , new stable families, called anti-Lagrange solutions,are given by using a numerical averaging approach. In Hadjidemetriou and Voyatzis (2011),the existence of anti-Lagrange solutions is confirmed in the general TBP and it is shown thatmigration from planetary to satellite type of motion is possible under the effect of Stoke’s-likedissipative forces. On the other hand, tidal forces may cause instabilities in co-orbital motionand planetary collisions (Rodr´ıguez et al., 2013). Analytical and semi-analytical treatment ofthe phase space structure of co-orbital motion has been given in Robutel and Pousse (2013);Leleu et al. (2017) by constructing appropriate averaged Hamiltonians for the 1:1 resonance.In an inertial frame of reference, QS motion is described by intersecting orbits of the smallbodies and therefore, a resonant mechanism is necessary for avoiding close encounters (Mikkolaand Innanen, 1997; Mikkola et al., 2006). Studying the system in the framework of the TBPmodel in a rotating frame, periodic orbits are of major importance for understanding theunderlying resonant dynamics. They consist exact positions of resonances and should appearas equilibrium points of a model, where the fast component of the motion is averaged. Linearlystable periodic solutions are associated with the existence of a foliation of invariant tori, whichform a regime of long-term stability, where the resonant arguments librate regularly. For the1:1 resonant QS motion, the main resonant argument is the angle θ = λ − λ , where λ i , i = 1 , f of the planar circular restricted TBP, we obtain bifurcations and compute the familiesof periodic orbits for more complicated versions of the TBP up to the general spatial TBP.The results for the planar models verify previous studies. Additionally, the three-dimensionalmotion is also studied, both in the restricted and in the general model. In Sect. 2, the restrictedmodel is addressed, while in Sect. 3, we apply continuation with respect to the mass of thesmallest body and, thus, approach the families of the general TBP. In Sect. 4, we present thespatial families of the general model for various planetary masses and, finally, we conclude inSect. 5. 2 QS periodic motion in the restricted three-body model(RTBP)
In the restricted case, we consider the system Sun - Planet - asteroid, where the Sun and thePlanet (primary bodies) have masses m and m , respectively, and revolve around their centreof mass O in a Keplerian orbit. We consider the classical rotating frame Oxyz , where the
Oxy plane coincides with the inertial one that contains the orbit of the primaries, the x -axisis directed along the direction line Sun - Planet and Oz is perpendicular to the plane Oxy .In this frame, and by considering the mass normalisation m + m = 1, the mass parameter µ = m and the gravitational constant G = 1, the motion of the massless asteroid (body 2) isdescribed by the Lagrangian function L R = T R − U R , (1)where T R = ( ˙ x + ˙ y + ˙ z ) + ( x ˙ y − y ˙ x ) ˙ υ + ( x + y ) ˙ υ , U R = − − µr − µr ,r = (cid:112) ( x + µr ) + y + z , r = (cid:112) ( x − (1 − µ ) r ) + y + z , and υ = υ ( t ) is the true anomaly and r = r ( t ) the mutual distance of the primaries alongtheir relative Keplerian orbit.In the following, we will refer also to the planetocentric, barycentric and heliocentric os-culating orbital elements of the orbits, a i (semi-major axis), e i (eccentricity), (cid:36) i (longitude ofpericenter), Ω i (longitude of ascending node) and λ i (mean longitude), where the index i = 1and 2 refers to the planet and the asteroid, respectively. Considering the primaries moving in a circular orbit with unit mutual distance ( e = 0, a = 1)and the asteroid on the plane Oxy ( z = 0) we obtain the planar circular restricted three-bodyproblem, where the Sun and the Planet are fixed on the x -axis at position − µ and 1 − µ ,respectively (Murray and Dermott, 1999). We have r = 1 and ˙ υ = 1, thus, the system (1) isautonomous of two degrees of freedom and possesses the Jacobi integral C J or the equivalent“energy integral” h given by h = 12 (cid:0) ˙ x + ˙ y (cid:1) − (cid:0) x + y (cid:1) − − µ (cid:112) ( x + µ ) + y − µ (cid:112) ( x − µ ) + y = − C J / . (2)In this model, QS orbits are associated with the existence of a family of symmetric periodicorbits, called family f (Broucke, 1968; H´enon and Guyot, 1970; Pousse et al., 2017). Thisfamily tends to the H´enon’s family E +11 of generating orbits which starts from a third-speciesorbit, as µ → µ , family f starts with orbits that encircle the planet at average distance that approaches zero. As thedistance from the planet increases, the family terminates at a collision orbit with the Sun. For µ = 0 . f are shown in the rotating frame in Fig. 1a. They canbe assigned to initial conditions x , y = ˙ x = 0 and ˙ y , where, from Eq. (2), ˙ y = ˙ y ( x , h ).Assuming the interval − µ < x < − µ , the orbits of family f can be mapped to a uniquevalue x , which can be used as the parameter of the family. The characteristic curve x − h ofthe family is shown in Fig. 1b. 3igure 1: The family f of periodic orbits for µ = 0 . a Orbits in the rotating frame, where S and P indicate the Sun and the Planet, respectively. b The characteristic curve x − h of thefamily f . c The eccentricity and d the semi-major axis of the orbits along family f computedfor the heliocentric, barycentric and planetocentric reference system. The grey zone indicatesthe QS b domain defined in Pousse et al. (2017), which separates the heliocentric quasi-satellite(QS h ) orbits from retrograde satellite orbits (sRS)When the motion of the massless body takes place close to the Planet, where the gravita-tional perturbation of the Sun is assumed relatively very small, we obtain almost Keplerianretrograde satellite orbits. When the orbits of the family f are quite distant from the Planet,the gravitation of the Sun dominates and the orbits are almost Keplerian planetary-type orbits.So, we can describe the orbits by using osculating orbital elements and by computing themin a planetocentric system (for satellite orbits) or in a heliocentric system (for planetary typeorbits). The distinction between the two types of orbits cannot be strictly defined. In panels(c) and (d) of Fig. 1, we present the eccentricity, e , and the semi-major axis, a , for the orbitsof family f considering a heliocentric, a barycentric and a planetocentric reference system. Forall orbits the longitude of perihelion is (cid:36) = 0 ◦ . In the plots, we present also the regions sRS(retrograde satellite orbits), QS h (heliocentric quasi-satellite orbits) and the separation greyregion (called QS b , binary quasi-satellite), which are defined in Pousse et al. (2017). The rightborder of the grey region, measured from the planet position, is the Hill’s radius R H , whichcan alternatively be used for the distinction between QS h and sRS orbits. We can observethat as sRS orbits approach the Planet ( x → − µ ) their planetocentric eccentricity tendsto zero. The heliocentric eccentricity of QS h orbits increases and tends to 1, as we approachthe collision orbit with the Sun. The slope of the increasing eccentricity, as x decreases, isalmost equal to 1, because x is the perihelion distance and a ≈
1, as it is shown in panel(d). Apparently, the periodic orbits of the family f indicate the exact position of the 1:1 meanmotion resonance of QS orbits (Sidorenko et al., 2014).All periodic orbits of family f are linearly (horizontally) stable for µ < . x − ˙ x ( y = 0, ˙ y >
0) for different energy levels. Foreach case the initial position x and the corresponding eccentricity of the QS periodic orbit isindicated. The maximum amplitude of librations of the resonant angle θ , which is computedfor the last invariant curve (approximately) of the island of stability, is indicated. Strong chaosoccurs outside of the islandswe present in Fig. 2 surfaces of section y = 0 ( ˙ y >
0) for some energy values in the QS h regime. Outside the islands of stability strongly chaotic motion occurs. For the regular orbitsthe resonant angle θ = λ − λ librates. The maximum amplitude of libration, θ max , whichcorresponds to the orbit of the last invariant tori of the island region, increases as the orbitsbecome more distant from the Planet. This has been shown also by Pousse et al. (2017) withthe use of an average model. However, due to the absence of chaos in the averaged model, themaximum amplitude of libration is overestimated. Assuming the primaries moving in an eccentric orbit ( e (cid:54) = 0, a = 1), the system (1) becomesnon-autonomous and periodic in time with period T (cid:48) equal to the period of the revolutions ofthe primaries, namely in our units T (cid:48) = 2 π . Concerning the continuation of periodic orbits fromthe circular to the elliptic model, this is possible for the periodic orbits of the circular model,which have period T c = pq T (cid:48) , with p and q being prime integers. Starting from such a periodicorbit, which corresponds to e = 0, we can obtain by analytic continuation monoparametricfamilies of periodic orbits for e (cid:54) = 0. Along these families, the period of orbits is constant, T = q T c = 2 pπ , and q defines the multiplicity of the orbits (Broucke, 1969). Particularly, twodistinct families are generated according to the initial location of the primary (perihelion oraphelion).In Fig. 3a, we present the variation of the period along the family f of the circular model.We have added an axis showing the eccentricity e , which can be used also as the parameterof the family f in the QS h regime. Considering the case of simple periodic orbits ( q = 1), weobtain the periodic orbit B ce of period T = 2 π , eccentricity e = 0 . a = 1 . f . This orbit can be assumed as a bifurcation (or generating)orbit for a family of periodic QS orbits in the PE-RTBP. Certainly, many other cases of highermultiplicity can be obtained in QS h regime. Lidov and Vashkov’yak (1994) and Voyatzis et al.(2012) used the PE-RTBP and the elliptic Hill model, respectively, and studied multiple QSperiodic orbits ( q ≥ E p (for (cid:36) = 0, υ (0) = 0) and E a (for (cid:36) = π , υ (0) = π ),which bifurcate from B ce . All orbits of family E p are horizontally and vertically unstable. Thecontinuation of the family becomes computationally very slow for e > .
44, where e → a The period T of orbits along the family f . b The vertical stability index along thefamily f . The vertical line indicates the location of Hill’s radius, R H , which is the right borderof the QS b region (gray zone).Figure 4: The families E p and E a of the PE-RTBP of simple multiplicity ( T = 2 π ). Char-acteristic curves with parameter the eccentricity of the primaries ( e ) in the plane a e − a and b e − e . Blue solid segments indicate horizontal and vertical stability, dotted parts arehorizontally stable but vertically unstable and red parts are both horizontally and verticallyunstable 6igure 5: Projections of the spatial family F on the planes a x − T , b x − z and c e − i .The family starts from the v.c.o. B cs . The orbit B sce is the orbit of F with period T = 2 π Family E a continues up to e = 1 (rectilinear model, see Voyatzis et al. (2018b)) and alongit the asteroid’s eccentricity, e , initially decreases. At e = 0 .
825 it takes its minimum value( ∼ . e →
1, they seem to become unstable. Since B ce is vertically unstable (see section 2.3), family E a also starts with vertically unstable periodic orbits, but at e = 0 . B es ) the orbits turn into vertically stable. So, B es is a vertical critical orbit and is a potentialgenerating orbit for a family of the spatial model, as we will see in Sect. 2.4.We mention that the families E a and E p have been also computed by Pousse et al. (2017),as sets of equilibrium solutions (called G e (cid:48) QS, and G e (cid:48) QS, , respectively) by using a numericallyaveraged model. Their results are in a very good agreement with those presented in Fig. 4.A worthy noted difference is that G e (cid:48) QS, is stable up to e ≈ .
8, while the equivalent family E a is stable up to e →
1. Also, the perturbative approach used by Mikkola et al. (2006)showed the stability of QS planar orbits under the perturbation caused by the elliptic orbitof the primaries and by assuming small inclinations. However, it could not provide the aboveperiodic solutions, due to the high eccentricities.
Vertical stability and three-dimensional families emanating from the short and long periodplanar families of co-orbital trojan-like orbits have been studied extensively (see e.g. Perdioset al. (1991); Hou and Liu (2008)). Here, we consider QS co-orbital motion and examine theplanar orbits of family f with respect to their vertical stability. For each orbit of period T wecompute the index b v = | trace∆( T ) | , (3)where ∆( T ) is the monodromy matrix of the vertical variations (H´enon, 1973). The orbit isvertically stable iff | b v | <
2. Orbits with | b v | = 2 are called vertical critical orbits (v.c.o.) andthey can be analytically continued to the spatial problem, for z (0) (cid:54) = 0 or ˙ z (0) (cid:54) = 0.In Fig. 3b, we present the index b v along the family f . The index b v exhibits a minimumat the orbit located at the Hill’s radius. Then, in the sRS regime, the index increases, but b v < h regime, b v is close to the critical value 2 for a long interval,but actually exceeds the critical value at the orbit B cs , where x ≈ . e ≈ . B cs , we can apply numerical continuation by using differential correctionsand obtain a family of spatial periodic orbits in the spatial circular model ( e = 0, z (cid:54) = 0).We call this family F with its orbits being identified by the non-zero initial conditions ( x , ˙ y , z ) beside y = ˙ x = ˙ z =0. In Fig. 5a, we present the evolution of the period T along the family(using x as parameter). T increases monotonically and takes the value 2 π at x ≈ .
16. This7igure 6: The families H p and H a of the SE-RTBP on the projection space e − e − i . Alongthe families a ≈ . a = 1). Family H p is linearly unstable, while H a is stableorbit, denoted by B sce , is potential for continuation in the spatial elliptic model. In panel (b),we obtain that z takes a maximum value for x = 0 .
183 and then decreases towards zero as x → ∼ ◦ . The orbit B sce has inclination 22 . ◦ . All orbits ofthe family F are linearly stable with (cid:36) = 0. We compute the index given by Eq. (3) for the elliptic planar model too, and particularly, forthe orbits of the families E p and E a . As we can see from Fig. 3b, the orbit B ce , where thesefamilies originate, is vertically unstable. Thus, both families start with vertically unstableorbits. As we mentioned in Sect. 2.2, E p is whole vertically unstable, while E a becomesvertically stable after the v.c.o. B es , which can be analytically continued to the spatial problemproviding a family of spatial periodic orbits (Ichtiaroglou and Michalodimitrakis, 1980). Also,as we mentioned in Sect. 2.3, the spatial orbit B sce is also a potential orbit for a continuationwith e (cid:54) = 0 and with the Planet located initially at its perihelion or aphelion.By considering the planet at the perihelion, the continuation of the orbit B sce for e (cid:54) = 0provides the family H p . This family is linearly unstable and extends up to very high eccentric-ities, e ≈ .
91 and e → i ≈ ◦ . The initial segment of H p ispresented in Fig. 6.By continuing the periodic orbits B es for i (cid:54) = 0 and B sce for e (cid:54) = 0 (with the Planet ataphelion at t = 0), we obtain a unique stable family, denoted by H a (Fig. 6). Thus, family H a forms a bridge linking the families E a and F . The inclination along the family monotonicallyincreases from B es ( i = 0 ◦ ) to B sce ( i = 22 . ◦ ).8 From the restricted to the general three-body problem(GTBP)
We consider the general three-body problem in a configuration “Sun - Planet - third body”,where the third body is a second planet or a satellite. We will use the indices 0, 1 and 2 forreferring to the three bodies, respectively. In the GTBP, it is m (cid:54) = 0 and, based on an inertialframe OXY Z , with O being the centre of mass, we can assume a rotating frame Gxyz wherei) the origin G is the centre of mass of m and m ii) Gz -axis is parallel to OZ and iii) thebodies m and m move always on the plane Gxz (Michalodimitrakis, 1979). We denote by υ the angle between the axes Gx and OX . In this rotating frame, the Lagrangian is written L G = 12 M (cid:0) ˙ x + ˙ z + x ˙ υ (cid:1) + M T R − U G , (4)where in T R the coordinates x , y are denoted now by x , y and M = ( m + m ) m m , M = ( m + m ) m m + m + m , U G = − (cid:88) i,j =0 m i m j r ij ( i (cid:54) = j ) ,r = (1 + a ) ( x + z ) , a = m m ,r = ( ax + x ) + y + ( az + z ) , r = ( x − x ) + y + ( z − z ) , We use the mass normalisation m + m + m = 1 and note that for m → L = (0 , , p υ ), where p υ = ∂ L G /∂ ˙ υ , is also conserved andprovides z , ˙ z and ˙ υ as functions of the variables ( x , x , y and z ) and their time derivatives(Michalodimitrakis, 1979; Katopodis, 1986; Antoniadou and Voyatzis, 2013). In the following,we will present the characteristic curves with respect to their osculating orbital elements thatcorrespond to the initial conditions. For symmetric periodic orbits the initial angles ∆ (cid:36) = (cid:36) − (cid:36) , ∆Ω = Ω − Ω and θ = λ − λ are either equal to 0 or π . Also, we will refer to themutual inclination of the small bodies, ∆ i . According to Hadjidemetriou (1975), all periodic orbits of the PC-RTBP (where m = 0) canbe continued for 0 < m (cid:28) T , provided that their period is not aninteger multiple of the period of the primaries, i.e. T (cid:54) = 2 kπ , k ∈ N , in our normalisation. Adirect deduction is that all QS periodic orbits of family f are continued to the P-GTBP exceptthe orbit B ce , which is the generating orbit of the families E a and E p in the PE-RTBP. Also,all periodic orbits of the PE-RTBP ( e (cid:54) = 0) are continued to the P-GTBP, but with differentperiods (Ichtiaroglou et al., 1978; Antoniadou et al., 2011).In order to obtain the families formed in the P-GTBP, we firstly perform the continuationof an orbit of the family E a (or E p ) with respect to m and we get a periodic orbit for aparticular value m (cid:54) = 0. Then, by keeping fixed all the masses we perform continuation inthe P-GTBP by using as parameter the variable x . Following this procedure, we obtain thefamilies g ( f , E a ) and g ( f , E p ). The characteristic curves of the families in the eccentricityplane are shown in Fig. 7 for m = 10 − and m = 10 − . In panel (a), a part of them nearthe singular point B ce is presented beside the families of the planar RTBP. The family f ofthe PC-RTBP is presented by the line e = 0 and is separated into two segments, f and f , by the orbit B ce . The transition of the characteristic curves from the restricted to thegeneral problem has been discussed first by Bozis and Hadjidemetriou (1976) and has been9igure 7: Characteristic curves of the families g ( f , E a ) and g ( f , E p ) for m = 10 − and m = 10 − on the eccentricities plane. a The characteristic curves close to the singular point B ce . The families f = f ∪ f , E a and E p of the restricted problem are also presented. b Thetotal segments of the families (notice the logarithmic horizontal axis) including the segment g s ( f , E a ) of satellite orbits. Blue (red) colour indicates horizontal stability (instability)found in other resonances, 1:2 (Voyatzis et al., 2009) and 1:3 and 3:2 (Antoniadou et al., 2011).Particularly, as m (cid:54) = 0, the family E a and the family segment f join smoothly forming thefamily g ( f , E a ) and the family E p joins the family segment f forming the family g ( f , E p ).At the neighbourhood of the singular point B ce , we obtain a gap between the two generatedfamilies. The formation of the two distinct families at this eccentricity domain causes a changein the topological structures in phase space, which may be related to that obtained in Leleuet al. (2017). The v.c.o. B cs and B es of the restricted problem, are continued for m (cid:54) = 0 asthe orbits gB cs and gB es , respectively.In Fig. 7b, we present the complete characteristic curves of the families in the P-GTBP.We use the scale log ( e ), in order to emphasize the structure for e ≈
0. All orbits of g ( f , E a )are horizontally stable. They are also vertically stable except those in the segment between thev.c.o. gB cs and gB es , which are vertically unstable. The continuation to the general problemdoes not affect the horizontal and vertical stability for sufficiently small values of m . Theorbits of the family g ( f , E p ) which are continued from f are horizontally stable, while itssegment that originates from the E p family consists of unstable orbits. All orbits of g ( f , E p )are vertically unstable. We mention also that for g ( f , E a ) it is θ = 0 and ∆ (cid:36) = π , while for g ( f , E p ) it is θ = ∆ (cid:36) = 0.In Fig. 7b, we also present the family segment g s ( f , E a ) of satellite periodic orbits whichcontinues the family g ( f , E a ). In the eccentricity plane we obtain a cusp, where these familiesmeet, which can be assumed as a border between planetary type orbits (like in QS h domain)and satellite orbits (like in sRS domain) (Hadjidemetriou et al., 2009; Hadjidemetriou andVoyatzis, 2011). However, in the variables of the rotating frame the two families join smoothly.Along the family g s ( f , E a ), the eccentricity e seems to increase rapidly and takes values > e and e tend to zero. All orbits of g s ( f , E a ) are both horizontally and vertically stable, in consistencywith the stability of the family segment of f , where they originate from.10 .2 The spatial general problem (S-GTBP) Similarly to the planar problem, all orbits of the SC-RTBP are continued to the S-GTBPif their period is not an integer multiple of the period of the primaries (Katopodis, 1979).Also, the periodic orbits of the SE-RTBP ( e (cid:54) = 0) are generically continued to the S-GTBP(Ichtiaroglou et al., 1978). Subsequently, all orbits of family F are continued for m (cid:54) = 0 exceptthe orbit B sce , which has a period equal to the period of primaries ( T = 2 π ). This critical orbitseparates the spatial family F into two segments, F and F (Fig. 8) and generates the families H p and H a of the SE-RTBP (see also Fig. 6), which are also continued in the S-GTBP.Figure 8: The families g ( F , H a ) and g ( F , H p ) of the S-GTBP in the projection space e − e − ∆ i . Blue (red) color indicates linear stability (instability). The families of the restrictedproblems and the planar family g ( f , E a ) are also shown (family f extends along the axis e = 0,∆ i = 0 ◦ )Our computations for m = 10 − and m = 10 − show that a similar structure with that ofthe planar case is formed (Fig. 8). Particularly, the continuation of the stable segment F andthe stable family H a constructs the stable family g ( F , H a ), which forms a bridge between thetwo orbits, gB cs and gB es of the planar family g ( f , E a ). Along the formed spatial family theorbits are symmetric with respect to the Oxz -plane and the initial conditions correspond to θ = 0 , ∆ (cid:36) = π, ∆Ω = π. The peak of the bridge corresponds to e = 0 . e = 0 .
803 and the maximum mutualinclination ∆ i = 20 . ◦ .The continuation of the stable segment F and the unstable family H p constructs for m (cid:54) = 0the family g ( F , H p ). The stability changes close to the critical orbit B sce , where there exists11 gap between the two families. Along g ( F , H p ) the initial conditions correspond to θ = 0 , ∆ (cid:36) = 0 , ∆Ω = 0 . Computations of the bridge family g ( F , H a ) are also found in Antoniadou et al. (2014), butwithout explaining its origin. Also, in that paper, computations were performed by startingfrom the v.c.o. of the planar families given in Hadjidemetriou and Voyatzis (2011). However,by using such an approach, we were not able to detect the existence of the family g ( F , H p ). In Sect. 2, we presented the families of periodic orbits of the RTBP for µ = 0 . B cs , B ce , B es and B sce (see Fig.6). By performing numerical computations in the range m ⊕ ≤ µ ≤ m J we found that thereare no structural changes, namely all critical orbits and the corresponding families still existand their stability, either horizontal or vertical, is unaltered. Therefore, the picture depictedin Fig. 6 holds for all values of µ in the above interval at least. The location of the criticalorbits for some values of µ is given in Table 1.Table 1: Location of the critical orbits of the restricted model for some values of the massparameter µB cs B ce B es B sce µ e e e e e i · − .
697 0 .
835 0 .
087 0 .
798 0 .
816 22 . ◦ · − .
698 0 .
836 0 .
087 0 .
799 0 .
816 22 . ◦ .
001 0 .
700 0 .
836 0 .
086 0 .
799 0 .
816 22 . ◦ .
004 0 .
710 0 .
837 0 .
083 0 .
801 0 .
818 22 . ◦ .
010 0 .
723 0 .
838 0 .
077 0 .
806 0 .
821 21 . ◦ In the GTBP, the two planetary masses are involved as parameters. For masses of theorder of Jupiter and less, the location and the stability of families of the planar model do notseem to depend on the individual masses, but only on their ratio ρ = m /m (Hadjidemetriouet al., 2009; Hadjidemetriou and Voyatzis, 2011). Considering m = 0 . ρ = 0 . ρ → ρ , we obtain that the critical orbits gB cs and gB es approacheach other and coincide at a critical value ρ ∗ = 0 . g ( F , H a ) shrinks as ρ increases and disappears at ρ = ρ ∗ (see Fig. 9). The linear stability ofthe family is unaltered.The family g ( F , H p ) is also continued as ρ increases and its continuation is not restrictedby the critical mass-ratio ρ ∗ . It consists of two main segments, a stable and an unstable one,but for ρ (cid:38) .
005 a small unstable segment appears inside the stable one. We note thatin this case the numerical computation of the linear stability is quite ambiguous, since thelinear stability appears very close to the critical case. We used long-term computations of thedeviation vectors, as in Voyatzis et al. (2018a), in order to conclude accordingly.By performing computations for m ⊕ ≤ µ ≤ m J we obtain a similar structure for thefamilies. The values of the critical mass ratio, ρ ∗ , is shown in Table 2. Also in Table 3, wepresent the orbital elements for some representative orbits of the “bridge” family g ( F , H a ) forsome values of ρ .The maximum mutual inclination observed along the families g ( F , H a ) and g ( F , H p ) ispresented in the left panel of Fig. 10 as a function of ρ . Along the “bridge”, the maximum12igure 9: The families g ( F , H a ) and g ( F , H p ) of the S-GTBP in the space e − e − ∆ i forvarious mass-ratios ρ = m m mentioned in the labels and for fixed m = 0 . g ( f , E a ) of the planar general problem are also shown ingrey colourmutual inclination, ∆ i ≈ . ◦ , appears as ρ →
0. For the family g ( F , H p ) the maximum ∆ i increases as ρ increases but the particular orbits seem to become unstable for ρ (cid:38) . Our study concerns the quasi-satellite (QS) motion of the 1:1 mean-motion resonance which canfind various applications in celestial mechanics. We consider both the problems of QS motionof an asteroid in the framework of the RTBP and the QS planetary motion in the framework ofthe GTBP. We focus on the computation of families of periodic orbits by assuming the methodof analytical continuation and applying differential corrections. It is well known that periodicorbits play an important role in the dynamics and their linear stability or instability indicatesin general the existence of regions in phase space with stable or chaotic motion, respectively.For the case of asteroids (massless bodies), we started our study from the planar circularRTBP, where the backbone of QS motion is the horizontally stable family f . The horizontaland vertical stability of this family indicates also the existence of long-term stability, whenconsidering small perturbations by adding a small eccentricity in the motion of the primariesor by assuming spatial orbits of small inclination. These results have been verified also by other13able 2: Critical mass-ratios ρ ∗ = m m for various values of m m ρ ∗ g ( F , H a ). For all orbitsit is θ = 0 ◦ , ∆ ω = 0 ◦ , ∆Ω = 180 ◦ ρ a /a e e i ( ◦ )0.001 0.99807216 0.0022 0.7587 15.230.99856609 0.0123 0.8051 20.000.99869836 0.0474 0.8042 15.100.01 0.99804105 0.0197 0.7435 10.000.99833646 0.0289 0.7665 12.610.99860354 0.0558 0.7825 10.070.018 0.99807499 0.0350 0.7361 3.510.99829821 0.0443 0.7518 5.860.99848330 0.0591 0.7633 3.47studies (e.g. Mikkola et al., 2006; Sidorenko et al., 2014; Pousse et al., 2017). We determinedtwo critical orbits along family f , called B cs and B ce . The critical orbit B cs separates thefamily f in two segments, one vertically stable and one vertically unstable. Therefore, B cs is av.c.o., which is continued in the spatial model by adding inclination to the asteroid, and hence,we obtain the family F of spatial periodic QS orbits. The B ce orbit belongs to the segment ofvertically unstable orbits of f . It has period T = 2 π and is continued in the elliptic model andgenerates two families of planar periodic orbits, E a and E p . Both critical orbits and all orbitsof the above mentioned generated families are highly eccentric orbits for the massless body.Also, along family F the inclination reaches the value of 27 ◦ .Family E a contains the critical orbit B es which is a v.c.o. Namely, at B es the family E a turns from vertically unstable to vertically stable and a new stable family, H a is derived bycontinuation in the spatial model. The spatial family F contains a critical orbit B cse , which hasperiod T = 2 π and is continued in the spatial elliptic model generating two new families. Ourcomputations showed that one of the families, which arises from B cse coincides with H a (it isthe same family), which emanates from the planar v.c.o. B es . So, the family H a of the spatialelliptic RTBP forms a bridge between the families E a and F of the elliptic planar and thespatial circular RTBP, respectively. The second generated family, H p is unstable and extendsup to very high eccentricity values. This structure of periodic solutions in phase space holdsat least in the mass range m ⊕ ≤ µ ≤ m J .Apart from the isolated critical orbits B ce and B cse all other orbits of the above mentionedplanar and spatial families are continued by adding mass, m , to the massless body, i.e. bypassing from the RTBP to GTBP. We showed that for m (cid:54) = 0 two families of inclined orbits areformed, called g ( F , H a ) and g ( F , H p ). Family g ( F , H a ) is stable and forms a bridge between14igure 10: a The maximum mutual inclination, ∆ i max , observed in families g ( F , H a ) and g ( F , H p ), when m equals to 0 . m J (green), 1 . m J (blue) and 10 m J (red) as the mass-ratio, ρ , varies. b The eccentricity values of the orbits presented in panel a two orbits of the QS family of the planar GTBP (Giuppone et al., 2010; Hadjidemetriou andVoyatzis, 2011) and reaches a maximum inclination value that depends mainly on the mass-ratio ρ = m /m . This “bridge” becomes lower and lower as ρ increases and disappears for ρ ≈ .
02. The family g ( F , H p ) is located at higher inclinations than those of the “bridge” andit consists mainly of two segments, one stable and one unstable. From a qualitative point ofview, the above structure of periodic QS motion is almost unaltered for m ⊕ ≤ µ ≤ m J .The QS periodic orbits studied in this paper consist the exact 1:1 resonant solutions. Par-ticularly, stable periodic solutions should form islands in phase space, where the resonant angle θ = λ − λ librates regularly. Considering a particular TBP model and a stable periodic orbitof it, we obtain in its vicinity librations for the resonant angle ∆ (cid:36) = (cid:36) − (cid:36) , too. For theplanar models, such librations have been indicated by the studies cited along the paper. Theexistence of inclined librations close to spatial periodic orbits has been checked and verified bynumerical integrations, though they are not presented in this paper. The width of the area ofinclined librations, for both asteroid and planetary QS orbits, requires further studies. Acknowledgements.
The work of KIA was supported by the University of Namur.
Conflict of Interest.
The authors declare that they have no conflict of interest.
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