This is a preprint of: “Periodic orbits for the perturbed planar circular restricted 3-body problem”, Elbaz I. Abouelmagd, Juan Luis Garcia Guirao, Jaume Llibre, Discrete Contin. Dyn. Syst. Ser. B, vol. 24(3), 2019. PERIODIC ORBITS FOR THE PERTURBED PLANAR CIRCULAR RESTRICTED 3–BODY PROBLEM ELBAZ I. ABOUELMAGD1 , JUAN LUIS GARCÍA GUIRAO2 AND JAUME LLIBRE3 Abstract. We characterize when the classical first and second kind of periodic orbits of the planar circular restricted 3–body problem obtained by Poincaré, can be extended to perturbed planar circular restricted 3–body problems. We put special emphasis when the perturbed forces are due to zonal harmonic or to a solar sail. 1. Introduction The circular restricted 3–body problem is considered the simplest non–integrable dynam- ical system in Celestial Mechanics. However it has many applications not only in the solar system but also in the studies of the stellar dynamics. Therefore many different dynamical astronomical systems can be studied within its frame. Thus, for instance this model is used in some space missions motion of spacecrafts in the Earth–Moon system, or between two planets. Also there are specific applications of the circular restricted 3–body problem in stellar systems for studying exoplanets around either one or both components of a binary star system, (see for instance [13]), and in particular for analyzing the existence of habitable zones in these exoplanets, see [12, 15, 17, 18, 21]. Many researchers pay their attention to present analytical studies on the circular restricted 3–body problem within the frame of some perturbed forces, due to oblateness, zonal harmonic coefficients, triaxial and radiation pressure effects. In this context the existence and linear stability analysis of libration points as well as the periodic orbits around these points are studied by [1, 2, 3, 16, 17, 18, 30]. Furthermore some researchers devoted their work to construct the possible solutions of the same problem with some numerical techniques, if it either includes perturbed forces or not. Some of these techniques are Bulirsch–Stoer method, Runge–Kutta algorithm, Lie series approach, symplectic integrators, hybrid integrators, etc. For more details on these methods see for instance [4, 5, 8, 9, 10, 14, 20]. But within frame of continuation solutions [19] establishes that a symmetric periodic orbit of the circular restricted 3–body problem, when the mass of the third body is small, can be analytically extended into a periodic solution of the full 3–body problem in a synodic reference frame. It also proves that a family of symmetric periodic orbits of this problem can be analytically extended into a family of periodic motion of the full problem, for a fixed value of the small mass of the third body. In [22] the authors develop a new relationship to find a connection between the periodic orbits of the planar 3–body problem and the planar circular restricted 3–body problem. They also state new conditions to prove that the symmetric 2010 Mathematics Subject Classification. Primary: 70F07, 70F15. Key words and phrases. Periodic orbits continuation, circular restricted 3–body problem, perturbed forces, zonal harmonic, solar sail. http://www.gsd.uab.cat 1 2 E. I. ABOUELMAGD, J. L. G. GUIRAO AND J. LLIBRE periodic orbits of the elliptic restricted system can be extended into the full planar 3–body problem. Our objective in the present work is to analyze when the first and second kind of periodic orbits found by Poincaré [26, 27] in the planar circular restricted 3–body problem can be or not extended to periodic orbits of the perturbed planar circular restricted 3–body problem. The orbits of first (respectively second) kind in the planar circular restricted 3–body problem are the ones which come from the circular (respectively elliptic) orbits of the two–body problem or Kepler problem in synodical (or rotating) coordinates. We recall that in the perturbed circular circular restricted 3–body problem the primaries are rotating in circular orbits around their common center of masses. In this setting the infinitesimal third body is affected by the gravitational forces of the primaries as well as small perturbed force. We shall work in synodical coordinates where the primaries are fixed on the x–axis. Thus the model of the circular restricted 3–body problem is perturbed by an extra small force, obtaining the so called the perturbed circular restricted 3–body problem. There are two standard proofs for showing that the circular periodic orbits of the rotating Kepler problem can be extended to the first kind of periodic orbits of the circular restricted 3–body problem. The first proof uses the Poincaré variables, see for instance [29] and the proof of Theorem 2 of this paper, and the second proof uses the multipliers associated to the circular periodic orbits, see for instance [24] or the proof of Theorem 1 of this paper. Inspired in both proofs we shall show that, if the circular restricted 3–body problem with a small perturbation has a first integral, then the periodic solutions of first kind can be extended from the circular restricted 3–body problem to the perturbed circular restricted 3–body problem. But we shall see that if such first integral does not exist then both proofs do not work, and it remains as open problem: “to know when the periodic solutions of first kind can be extended or not when the perturbed problem has not a first integral”. Finally we study in Theorem 3 the continuation of the second kind of periodic solutions from the planar circular restricted 3–body problem to a perturbed planar circular restricted 3–body problem. 2. Equations of motion We assume, without loss of generality, that the separation distance between the primaries, the sum of their masses and the universal constant of gravitation are equal to one. Then the equations of motion can be written in dimensionless synodic coordinate system xyz (see [25]) as (1) r̈ + 2n ∧ ṙ = ∇V where r = [x y z]T , n = [0 0 1]T , (1 − µ) µ 1 V = + + |n ∧ r|2 , r1 r2 2 r1 = [(x − µ) y z]T , r2 = [(x − µ + 1) y z]T , Now we assume that the infinitesimal body will receive an additional force, say perturbed force, with the acceleration in synodic frame a. Using equation (1) the perturbed restricted THE PERTURBED PLANAR RESTRICTED 3–BODY PROBLEM 3 3–body problem can be written as (2) r̈ + 2n ∧ ṙ = ∇V + a, where a = a(r, ṙ, t) being a = [ax (x, y, z, ẋ, ẏ, ż, t) ay (x, y, z, ẋ, ẏ, ż, t) az (x, y, z, ẋ, ẏ, ż, t)]T , and n = [0 0 n]T the mean motion vector. In cartesian coordinates equations (2) have the form ẍ − 2nẏ = n2 x − (x − µ)f (x, y, z) − (x − µ + 1)g(x, y, z) + ax , (3) ÿ + 2nẋ = n2 y − y[f (x, y, z) + g(x, y, z)] + ay , z̈ = −z[f (x, y, z) + g(x, y, z)] + az , where 1−µ µ f (x, y, z) = and g(x, y, z) = . r13 r23 We assume that the parameter µ and the acceleration a are small. Within this frame we can write µ = εµ1 and a = εa1 , where the parameter ε is very small (i.e. 0 < ε ≪ 1). In what follows for simplicity the subindex 1 in µ1 and a1 will be dropped. Then equations (3) can be written as (4) (1 − µε)(x − µε) µε(x − µε + 1) ẍ = 2nẏ + n2 x − 3/2 − + εax , [(x − µε)2 + y 2 + z 2 ] [(x − µε + 1)2 + y 2 + z 2 ]3/2 [ ] µε 1 − µε ÿ = −2nẋ + n2 y − y + + εay , [(x − µε + 1)2 + y 2 + z 2 ]3/2 [(x − µε)2 + y 2 + z 2 ]3/2 [ ] µε 1 − µε z̈ = −z + + εaz . [(x − µε + 1)2 + y 2 + z 2 ]3/2 [(x − µε)2 + y 2 + z 2 ]3/2 After expanding (4) in powers of ε this system can be written into the form x ẍ = 2nẏ + n2 x − + O(ε), (x2 + y 2 + z 2 )3/2 y (5) ÿ = −2nẋ + n2 y − + O(ε), (x2 + y 2 + z 2 )3/2 z z̈ = − + O(ε). (x2 + y 2 + z 2 )3/2 If z = 0, then ż = 0. So the plane (x, y) is invariant, in the sense that if we choose a point in that plane then the solution which passes through this point with ż = 0 remains always in that plane. We shall study if the periodic orbits of first and second kind persist in the perturbed planar circular restricted 3–body problem (5) with z = 0, i.e. in the differential 4 E. I. ABOUELMAGD, J. L. G. GUIRAO AND J. LLIBRE system x ẍ = 2nẏ + n2 x − + O(ε), (x2 + y 2 )3/2 (6) y ÿ = −2nẋ + n2 y − + O(ε). (x2 + y 2 )3/2 3. Periodic orbits of first kind For ε = 0 system (6) is the Kepler problem in dimensionless synodic coordinates. We do the usual change to polar coordinates with the corresponding momenta as follows (x, y, ẋ, ẏ) → (r, θ, Q, P ) where 1 1 x = r cos θ, ẋ = ny + Q cos θ − P sin θ, y = r sin θ, ẏ = −nx + Q sin θ + P cos θ. r r Then the differential system (6) writes ṙ = f1 (r, θ, Q, P ) = Q, P θ̇ = f2 (r, θ, Q, P ) = − n, r2 (7) P2 1 Q̇ = f3 (r, θ, Q, P ) = 3 − 2 + O(ε), r r Ṗ = f4 (r, θ, Q, P ) = O(ε). Let (r(t), θ(t), Q(t), P (t)) be a T –periodic solution of system (7) for ε = 0. A con- tinuation of this periodic solution is a pair of smooth functions, (r(t, ε), θ(t, ε), Q(t, ε), P (t, ε)) and T (ε), defined for ε > 0 near 0 such that (r(t, 0), θ(t, 0), Q(t, 0), P (t, 0)) = (r(t), θ(t), Q(t), P (t)), T (0) = T , and (r(t, ε), θ(t, ε), Q(t, ε), P (t, ε)) is a T (ε)−periodic solu- tion of system (7). Note that the variational equation associated to the T –periodic solution ϕ(t, ε) = (r(t, ε), θ(t, ε), Q(t, ε), P (t, ε)) is ( ) ∂(f1 , f2 , f3 , f4 ) (8) Ṁ = M, ∂(r, θ, Q, P ) (r,θ,Q,P )=ϕ(t,ε) where M is a 4 × 4 matrix. Of course ∂(f1 , f2 , f3 , f4 )/∂(r, θ, Q, P ) denotes the Jacobian matrix of (f1 , f2 , f3 , f4 ) with respect to the variables (r, θ, Q, P ). The monodromy matrix associated to the T –periodic solution ϕ(t, ε) is the solution M (T, ε) of (8) satisfying that M (0, ε) is the identity matrix. The eigenvalues of the monodromy matrix M (T, 0) associated to the periodic solution ϕ(t, ε) are called the multipliers of the periodic orbit. A periodic solution ϕ(t, ε) always has +1 as an eigenvalue of its associated monodromy matrix M (T, ε), and +1 with multiplicity two for a system with a non–degenerate first integral. Drop one +1 multiplier for a general autonomous differential system, and drop two +1 multipliers from the list of multipliers for an autonomous differential system with a non– degenerate first integral to get the nontrivial multipliers. If the nontrivial multipliers are not equal to one then the periodic solution is called elementary. It is known that an elementary THE PERTURBED PLANAR RESTRICTED 3–BODY PROBLEM 5 periodic solution for ε = 0 can be extended to ε > 0 sufficiently small, see Proposition 9.1.1 of [24]. System (7) for ε = 0 becomes P P2 1 ṙ = Q, θ̇ =− n, Q̇ = − , Ṗ = 0. r2 r3 r2 Clearly the angular momentum P is a first integral. Taking P = c for a fixed constant c3 ̸= 1/n, we get the circular periodic solution r = c2 , Q = 0 of period |2πc3 /(1 − nc3 )|. Computing the multipliers of this circular periodic solution we obtain +1 with multiplicity two and exp(±i2π/(1 − nc3 )) ̸= +1. If the perturbed problem has a first integral, by Proposition 9.1.1 of [24], this periodic orbits can be extended, and take the following result: Theorem 1. Consider a circular restricted 3–body problem with a small perturbation having a first integral. If c3 ̸= 1/n and 1/(1 − nc3 ) is an integer, then the perturbed circular orbits of the rotating Kepler problem with angular momentum c can be extended to this problem. For instance this theorem can be applied to the models of the perturbed circular restricted 3–body problem by zonal harmonic coefficients, oblateness, triaxial, and radiation pressure effects because these models have a first integral. If the circular restricted 3–body problem with a small perturbation has not a first integral the multipliers of the circular periodic orbits cannot be used for knowing if such orbits can be extended, this is the case of planar circular restricted 3–body problem with a solar sail [28]. We introduce the Delaunay variables (ℓ, g, L, G) instead of the variables (r, θ, Q, P ) through √ G2 2 1 t Q = − 2 + − 2 , P = G, ℓ = 3 , g = θ − f, r r L L where t is the time and f the true anomaly. For ε = 0 we have the Kepler problem √ in rotating coordinates and the Delaunay variables have √ the following meaning: L = a where a is the semi–major axis of the elliptic orbits, G = L 1 − e2 where e is the eccentricity of the elliptic orbits, g is the angle of the pericenter of the elliptic orbits, and ℓ is the mean anomaly (which is the angular variable measured from the pericenter). For more details on the Delaunay variables see page 164 of [24] and its neighborhood. In terms of the Delaunay variables the differential system (7) writes 1 (9) ℓ̇ = + O(ε), L̇ = O(ε), ġ = −n + O(ε), Ġ = O(ε). L3 We note that the Delaunay variables are valid only in a neighborhood of the phase space where we have the elliptic orbits of the Kepler problem in rotating coordinates, and now we introduce the Poincaré variables (λ, Λ, ξ, η) given by √ √ λ = ℓ + g, Λ = L, ξ = 2(L − G) cos ℓ, η = 2(L − G) sin ℓ, for studying the orbits in a neighborhood of circular periodic orbits of the rotating Kepler problem. In the Poincaré variables the differential system (9) becomes (10) λ̇ = Γ − n + O(ε), Λ̇ = O(ε), ξ˙ = −Γη + O(ε), η̇ = Γξ + O(ε), where Γ = 1/Λ3 . 6 E. I. ABOUELMAGD, J. L. G. GUIRAO AND J. LLIBRE The periodic solutions of system (10) for ε = 0 are (11) λ = (Γ0 − n) t + λ0 , Λ = Λ0 , ξ = ξ0 cos(Γ0 t) − η0 sin(Γ0 t), η = ξ0 sin(Γ0 t) + η0 cos(Γ0 t), where Λ0 is a non–zero constant and Γ0 = 1/Λ30 . The orbit (11) is periodic if and only if 2πl/(Γ0 − n) = 2πl/Γ0 , i.e. Γ0 must be k/(k − l) with l < k positive integers. Note that k/(k − l) can be any positive rational number. The circular periodic solutions of system (10) for ε = 0 are λ = (Γ0 − n) t + λ0 , Λ = Λ0 , ξ = 0, η = 0. √ Note that ξ = η = 0 if and only if L = G, and since G = L 1 − e2 , we have that ξ = η = 0 if and only if the eccentricity e = 0. Now applying the Poincaré continuation method (see for more details on this method [11]), we shall study which of these circular periodic solutions can be extended to periodic solutions of system (10) with ε > 0 sufficiently small. We denote by (12) (ξ(t; ε, ξ0 , η0 , λ0 , Λ0 ), η(t; ε, ξ0 , η0 , λ0 , Λ0 ), λ(t; ε, ξ0 , η0 , λ0 , Λ0 ), Λ(t; ε, ξ0 , η0 , λ0 , Λ0 )) the solution of system (10) such that ξ(0; ε, ξ0 , η0 , λ0 , Λ0 ) = ξ0 , η(0; ε, ξ0 , η0 , λ0 , Λ0 ) = η0 , λ(0; ε, ξ0 , η0 , λ0 , Λ0 = λ0 , Λ(0; ε, ξ0 , η0 , λ0 , Λ0 ) = Λ0 . We want to study the periodic solutions of period T = T0 + τ of system (10) which are continuation of the circular periodic solutions of (10) with ε = 0, i.e. the solutions (12) satisfying ξ(T ; ε, ξ0 , η0 , λ0 , Λ0 ) = ξ(0; ε, ξ0 , η0 , λ0 , Λ0 ), η(T ; ε, ξ0 , η0 , λ0 , Λ0 ) = η(0; ε, ξ0 , η0 , λ0 , Λ0 ), (13) λ(T ; ε, ξ0 , η0 , λ0 , Λ0 ) = λ(0; ε, ξ0 , η0 , λ0 , Λ0 ) + 2πl, Λ(T ; ε, ξ0 , η0 , λ0 , Λ0 ) = Λ(0; ε, ξ0 , η0 , λ0 , Λ0 ), and ξ(t; 0, ξ0 , η0 , λ0 , Λ0 ) = 0, η(t; 0, ξ0 , η0 , λ0 , Λ0 ) = 0, λ(t; 0, ξ0 , η0 , λ0 , Λ0 = (Γ0 − n) t + λ0 , Λ(t; 0, ξ0 , η0 , λ0 , Λ0 ) = Λ0 . Thus we expect to find initial conditions (ξ0 (ε), η0 (ε), λ0 (ε), Λ0 (ε)) for a periodic solution of period T0 of system (10), such that for ε = 0 we have ξ0 (0) = η0 (0) = 0, Λ0 (0) = Λ0 and (Γ0 (0) − n)T0 = 2πl. where Γ0 (0) = 1/Λ0 (0), therefore ( )1/3 2πl + nT0 T0 Γ0 (0) = and Λ0 (0) = , T0 2πl + nT0 THE PERTURBED PLANAR RESTRICTED 3–BODY PROBLEM 7 with λ0 (0) arbitrary. We want to solve system (13) with T = T0 +τ , more precisely to find the zeros (ξ0 , η0 , Λ0 , τ ) = (ξ0 (ε), η0 (ε), Λ0 (ε), τ (ε)) of the map (ψ1 , ψ2 , ψ3 , ψ4 ) given by ψ1 (ε, ξ0 , η0 , λ0 , Λ0 , τ ) = ξ(T0 + τ ; ε, ξ0 , η0 , λ0 , Λ0 ) − ξ(0; ε, ξ0 , η0 , λ0 , Λ0 ), ψ2 (ε, ξ0 , η0 , λ0 , Λ0 , τ ) = η(T0 + τ ; ε, ξ0 , η0 , λ0 , Λ0 ) − η(0; ε, ξ0 , η0 , λ0 , Λ0 ), (14) ψ3 (ε, ξ0 , η0 , λ0 , Λ0 , τ ) = λ(T0 + τ ; ε, ξ0 , η0 , λ0 , Λ0 ) − λ(0; ε, ξ0 , η0 , λ0 , Λ0 ) − 2πl, ψ4 (ε, ξ0 , η0 , λ0 , Λ0 , τ ) = Λ(T0 + τ ; ε, ξ0 , η0 , λ0 , Λ0 ) − Λ(0; ε, ξ0 , η0 , λ0 , Λ0 ). The Jacobian matrix of (ψ1 , ψ2 , ψ3 , ψ4 ) with respect to ξ0 , η0 , Λ0 and τ0 evaluated at (ξ0 , η0 , Λ0 , τ ) = (0, 0, Λ̄0 , 0) and ε = 0 is   ∂ψ1 ∂ψ1 ∂ψ1 ∂ψ1  ∂ξ0 ∂η0 ∂Λ0 ∂τ         ∂ψ2 ∂ψ2 ∂ψ2 ∂ψ2     ∂ξ0 ∂η0 ∂Λ0 ∂τ    D=  .  ∂ψ ∂ψ ∂ψ ∂ψ   3 3 3 3    ∂ξ0 ∂η0 ∂Λ0 ∂τ       ∂ψ4 ∂ψ4 ∂ψ4 ∂ψ4  ∂ξ0 ∂η0 ∂Λ0 ∂τ (ξ0 ,η0 ,Λ0 ,τ,ε)=(0,0,Λ̄0 ,0,0) Since ψ1 (ε, ξ0 , η0 , λ0 , Λ0 , τ ) = ξ0 cos[Γ0 (T0 + τ )] − η0 sin[Γ0 (T0 + τ )] − ξ0 + O(ε), ψ2 (ε, ξ0 , η0 , λ0 , Λ0 , τ ) = ξ0 sin[Γ0 (T0 + τ )] + η0 cos[Γ0 (T0 + τ )] − η0 + O(ε), 1 ψ3 (ε, ξ0 , η0 , λ0 , Λ0 , τ ) = ( 3 − n)τ + O(ε), Λ0 ψ4 (ε, ξ0 , η0 , λ0 , Λ0 , τ ) = O(ε). then   cos(Γ0 T0 ) − 1 − sin(Γ0 T0 ) ∗ ∗      sin(Γ0 T0 ) cos(Γ0 T0 ) − 1 ∗ ∗      D= = 0.  2πl    0 0 0   T0    0 0 0 0 Since the determinant D = 0 ,the Poincaré variables which allow to continue the circular periodic orbits of the rotating Kepler problem to the circular restricted 3–body problem, do not work for continuing periodic orbits to the circular restricted 3–body problem with some small perturbation without first integrals. But if this perturbed problem has a first integral, using this first integral we can compute in each level of the fixed first integral the variable Λ 8 E. I. ABOUELMAGD, J. L. G. GUIRAO AND J. LLIBRE as a function of the variables (ξ, η, λ), and we can omit in system (14) the equation ψ4 = 0. Thus the determinant becomes   cos(Γ0 T0 ) − 1 − sin(Γ0 T0 ) ∗     8πl 2 1  sin(Γ0 T0 ) cos(Γ0 T0 ) − 1 ∗  D= =− sin Γ0 T0 .   T0 2  2πl  0 0 T0 and D ̸= 0 if Γ0 T0 ̸= 2πk with k a positive integer. Consequently we have the following result: Theorem 2. Let T0 = 2πr with r an arbitrary positive rational number. Consider a planar circular restricted 3–body problem with a small perturbation having a first integral. If T0 ̸= 2πk with k being a positive integer and ε > 0 sufficiently small, then there exists a periodic orbit for this perturbed problem which tends to a circular periodic orbit of the rotating Kepler problem with period T0 when ε → 0. Clearly Theorem 1 and 2 are related because both provide sufficient conditions in order that the circular periodic orbits of the rotating Kepler problem can be extended to the perturbed planar circular restricted 3–body problem. 4. Applications of the periodic orbits of first kind In this section the previous results of Theorems 1 and 2 will be applied to two perturbed planar circular restricted 3–body problems. In the first application the perturbed problem has a first integral [2], and the second one does not have a first integral [28]. 4.1. Planar circular restricted 3–body problem with a zonal harmonic effect. The equations of motion of the planar circular restricted 3–body problem under the effect of zonal harmonic coefficients up to J2 , for the more massive primary, within dimensionless synodic coordinate system xyz can be written as in [2] ẍ − 2ẏ = x − (x − µ)f (x, y, z) − (x − µ + 1)g(x, y, z) + ax , (15) ÿ + 2ẋ = y − y[f (x, y, z) + g(x, y, z)] + ay , where ax = A(x − µ)h(x, y, z), ay = Ayh(x, y, z), being A the zonal harmonic parameter and the function h(x, y, z) = 3(1 − µ)/(2r15 ). System (15) admits a Hamiltonian formulation with the Hamiltonian 1 2 1−µ H(x, y, p1 , p2 ) = (p1 + p22 ) + n(yp1 − xp2 ) − 2 [(x − µ)2 + y 2 ]1/2 (16) µ (1 − µ)A − 1/2 − . [(x − µ + 1)2 + y 2 ] [(x − µ)2 + y 2 ]3/2 Consider µ = εµ1 and A = εA1 where ε is a small parameter. In what follows for simplicity the subindex 1 in µ1 and A1 will be dropped. Then H(x, y, p1 , p2 ) = Hk (x, y, p1 , p2 ) + εHp (x, y, p1 , p2 ) + O(ε2 ), THE PERTURBED PLANAR RESTRICTED 3–BODY PROBLEM 9 where Hk is the Hamiltonian of the rotating Kepler problem and Hp comes from the per- turbed force, i.e. 1 2 1 Hk (x, y, p1 , p2 ) = (p1 + p22 ) + n(yp1 − xp2 ) − , 2 (x + y 2 )1/2 2 µ A µ[x(x − 1) + y 2 ] Hp (x, y, p1 , p2 ) = − − + . [(x + 1)2 + y 2 ]1/2 2 [(x + 1)2 + y 2 ]3/2 (x2 + y 2 )3/2 Now we change the cartesian variables (x, y, p1 , p2 ) to the polar variables (r, θ, Q, P ), and the Hamiltonian becomes H(r, θ, Q, P ) = Hk (r, θ, Q, P ) + εHp (r, θ, Q, P ) + O(ε2 ). where 1 2 P2 1 Hk (r, θ, Q, P ) = (Q + 2 ) − nP − , 2 r r µ A µ(r − cos θ) Hp (r, θ, Q, P ) = − − + . (r2 + 2r cos θ + 1)1/2 2 (r2 + 2r cos θ + 1)3/2 r2 and system (15) takes the form (17) ṙ = Q, P θ̇ = − n, r2 [ ] P2 1 µ(r − 2 cos θ) µ(r + cos θ) 6A(r + cos θ) Q̇ = − 2 +ε − 3/2 − 5/2 + O(ε2 ), r3 r r3 2 (r + 2r cos θ + 1) 2 (r + 2r cos θ + 1) [ ] µ µr sin θ 3Ar sin θ Ṗ = −ε 2 sin θ − 3/2 − 5/2 + O(ε2 ). r 2 (r + 2r cos θ + 1) 2 (r + 2r cos θ + 1) Since system (17) is a particular case of the general system (7) having a first integral, then Theorems 1 and 2 hold to the the planar circular restricted 3–body problem with the zonal harmonic coefficient perturbation. 4.2. Planar circular restricted 3–body problem with solar sail perturbation. In this case the equations of motion with reflecting the solar sail reflector (see [23] and [28]) are (18) r̈ + 2n ∧ ṙ = ∇V + a, where β(1 − µ) a= (r̂1 . s)2 s, r12 1 s= R, R R = r1 + r2 , √ R= [2(x − µ) + 1]2 + 4(y 2 + z 2 ) . and 10 E. I. ABOUELMAGD, J. L. G. GUIRAO AND J. LLIBRE • r is the position vector of the solar reflector, • n is the angular velocity vector of a rotating frame with unity value, • a is the acceleration on a perfectly reflecting solar sail, • V is the effective potential function, • r1 is the position vector of the space reflector with respect to the Sun, • r̂1 is the unit vector in the direction of r1 , • r2 is the position vector of the space reflector with respect to the Earth, • s is the attitude unit normal vector of the reflector, • µ mass ratio parameter, • β sail lightness number, and • The dot denotes derivative with respect to the time t. In cartesian coordinates equations (18) are ẍ − 2ẏ = x − (x − µ)f (x, y, z) − (x − µ + 1)g(x, y, z) + ax , (19) ÿ + 2ẋ = y − y[f (x, y, z) + g(x, y, z)] + ay , z̈ = −z[f (x, y, z) + g(x, y, z)] + az , where ax = β[2(x − µ) + 1]h(x, y, z), ay = 2βyh(x, y, z), az = 2βzh(x, y, z), being 1−µ µ 1−µ[ 2 2 2 ] f (x, y, z) = , g(x, y, z) = , h(x, y, z) = (x − µ)[2(x − µ) + 1] + 2(y + z ) . r13 r23 r14 R3 We assume that both parameters µ and β are small. Within this frame we can write µ = εµ1 and β = εβ1 , where the parameter ε is very small (i.e. 0 < ε ≪ 1), and µ1 and β1 are positive constants. In what follows for simplicity the subindex 1 in µ1 and β1 will be dropped. Then equations (19) can be written as (1 − µε)(x − µε) µε(x − µε + 1) ẍ = 2ẏ + x − − + [(x − µε)2 + y2 + z 2 ]3/2 [(x − µε + 1)2 + y 2 + z 2 ]3/2 2 βε(1 − µε)[2(x + µε) − 1] [(x + µε)[2(x + µε) − 1] + 2y 2 + 2z 2 ] , [(x − µε)2 + y 2 + z 2 ]2 [(2(x + µε) − 1)2 + 4 (y 2 + z 2 )]3/2 [ ] µε 1 − µε ÿ = −2ẋ + y − y + + [(x − µε + 1)2 + y 2 + z 2 ]3/2 [(x − µε)2 + y 2 + z 2 ]3/2 (20) 2 2βyε(1 − µε) [(x + µε)[2(x + µε) − 1] + 2y 2 + 2z 2 ] , [(x − µε)2 + y 2 + z 2 ]2 [(2(x + µε) − 1)2 + 4 (y 2 + z 2 )]3/2 [ ] µε 1 − µε z̈ = −z + + [(x − µε + 1)2 + y 2 + z 2 ]3/2 [(x − µε)2 + y 2 + z 2 ]3/2 2 2βzε(1 − µε) [(x + µε)[2(x + µε) − 1] + 2y 2 + 2z 2 ] . [(x − µε)2 + y 2 + z 2 ]2 [[2(x + µε) − 1]2 + 4 (y 2 + z 2 )]3/2 After expanding (20) in powers of ε this system can be written into the form (5), and we can study its periodic orbits in the plane z = 0 using the system (6). But since now the THE PERTURBED PLANAR RESTRICTED 3–BODY PROBLEM 11 planar circular restricted 3–body problem with solar sail perturbation has no first integrals, the results of Theorems 1 and 2 cannot be applied, and it remains an open problem to know if the periodic orbits of first kind can be extended or not to this perturbed problem. But in the case of radial solar sail, that is when the sail is oriented along the Sun–Line (i.e. along the normal vector n = r̂1 ), then system (18) has a first integral see [31], and in this case the results of Theorems 1 and 2 will apply. 5. Periodic orbits of second kind Birkhoff [7] observed that if the equations of system (6) are invariant with respect to the change variables (x, y, t) → (x, −y, −t), then an orbit which crosses the x−axis perpendicular at t = 0 and again at t = T /2 must be periodic of period T . So a way of finding periodic orbits is to look for orbits having two perpendicular crossings with the x−axis. Barrar [6] used ideas of Birkhoff for extending the elliptical orbits of the rotating Kepler problem to the planar circular restricted 3–body problem. Now we will see that the work of Barrar goes beyond the planar circular restricted 3–body problem, i.e. that it extends to the perturbed planar circular restricted 3–body problems. In Delaunay variables two perpendicular crossing with the x−axis correspond to g(t) = mπ, l(t) = nπ. Indeed g(t) = mπ says that the major axis of the ellipse coincides with the x−axis, and l(t) = nπ says that the infinitesimal body is on the x−axis, either at the apocenter or at the pericenter. Since instantaneous velocity of the infinitesimal body is the same than its motion along the ellipse and rotation of the ellipse, and both motions are perpendicular to the x−axis this implies a perpendicular crossing with the x−axis. For ε = 0 we consider the elliptic orbit with its apocenter at the positive x−axis when t = 0, i.e. (21) g(0) = −π , l(0) = π, from system (9) the equations of motion of this ellipse are 1 ℓ̇ = , L̇ = 0, ġ = −n, Ġ = 0. L3 So its solution satisfying conditions (21) is given by t (22) l(t) = + π, L(t) = L0 , g(t) = −nt − π, G(t) = G0 , L30 where L0 and G0 are constants. The period of this elliptic orbit is 2πL30 and we want to consider the case when 2πL30 = 2πp/q with p and q are coprime. Thus we want to continue this periodic orbit to ε > 0 sufficiently small with a period T = 2πp = 2πqL30 . Therefore g(T /2) = −(1 + p)π, l(T /2) = (1 + q)π, and a such orbit will have two perpendicular crossings with the x−axis. Of course we must avoid the collision with the small body localized at the point (1, 0). 12 E. I. ABOUELMAGD, J. L. G. GUIRAO AND J. LLIBRE For ε > 0 sufficiently small we look for solutions (l(t, ε, L0 , G0 ), g(t, ε, L0 , G0 ), L(t, ε, L0 , G0 ), G(t, ε, L0 , G0 )) of system (9) with initial conditions g(0, ε, L0 , G0 ) = −π , l(0, ε, L0 , G0 ) = π, such that ψ1 (t, ε, L0 , G0 ) = g(t, ε, L0 , G0 ) + (1 + p)π, ψ2 (t, ε, L0 , G0 ) = l(t, ε, L0 , G0 ) − (1 + q)π, with L0 near (p/q)1/3 and t near T /2. From the Implicit Function Theorem such a solution will exist if ( ) ∂(ψ1 , ψ2 ) det ̸= 0. ∂(t, L0 ) ε=0 on the elliptic orbit. Since ∂g ∂l ∂(ψ1 , ψ2 ) ∂t ∂t = ∂(t, L0 ) ε=0 ∂g ∂l ∂L0 ∂L0 ε=0 then, from (22), we get ∂(ψ1 , ψ2 ) 3nT = ̸= 0. ∂(t, L0 ) ε=0 2L40 So we have the following result: Theorem 3. Let p and q be relative prime positive integers and T = 2πp. Then the elliptical periodic solution of period T of the Kepler rotating problem satisfying g(0) = −π , l(0) = π , L3 (0) = p/q, and do not going through the point (1, 0) can be extended to the perturbed planar circular restricted 3–body problem with µ > 0 and ε > 0 small such that its equations of motion (9) come from the equations (6) invariant by the symmetry (x, y, t) → (x, −y, −t). 6. Conclusion We have proved that the periodic orbits of first kind can be extended from the planar circular restricted 3–body problem to a perturbed planar circular restricted 3–body problem if this perturbed problem has a first integral. It is an open problem to know when these periodic orbits can be extended if the perturbed problem has not a first integral. 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