On the restricted three-body problem with crossing singularities Giovanni Federico Gronchi Dipartimento di Matematica, Universit` a di Pisa e-mail: gronchi@dm.unipi.it Mathematical Models and Methods in Earth and Space Science Universit` a di Roma 2 ‘Tor Vergata’ March 19-22, 2019 Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences
The restricted three-body problem Three-body problem: Sun, Earth, asteroid. Restricted problem: the asteroid does not influence the motion of the two larger bodies. Equations of motion of the asteroid: � ( y − y ⊙ ( t )) ( y − y ⊕ ( t )) � y = − G ¨ | y − y ⊙ ( t ) | 3 + m ⊕ m ⊙ | y − y ⊕ ( t ) | 3 y is the unknown position of the asteroid; y ⊙ ( t ) , y ⊕ ( t ) are known functions of time, solutions of the two-body problem Sun-Earth. Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences
The restricted three–body problem In heliocentric coordinates � ( x − x ′ ) � x x ′ �� x = − k 2 ¨ | x | 3 + µ | x − x ′ | 3 − | x ′ | 3 where x = y − y ⊙ , x ′ = y ⊕ − y ⊙ ; k 2 = Gm ⊙ , µ = m ⊕ m ⊙ is a small parameter; − k 2 µ ( x − x ′ ) | x − x ′ | 3 is the direct perturbation of the planet on the asteroid; k 2 µ x ′ | x ′ | 3 is the indirect perturbation, due to the interaction Sun-planet. Hint! We can model the dynamics of an asteroid in the solar system by summing up the contribution of each planet to the perturbation. Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences
Canonical formulation of the problem Use Delaunay’s variables Y = ( L , G , Z , ℓ, g , z ) for the motion of the asteroid: L = k √ a ℓ = n ( t − t 0 ) √ G = L 1 − e 2 g = ω Z = G cos I z = Ω These are canonical variables, representing the osculating orbit, solution of the 2-body problem Sun-asteroid. Denote by Y ′ = ( L ′ , G ′ , Z ′ , ℓ ′ , g ′ , z ′ ) Delaunay’s variables for the planet. Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences
Canonical formulation of the problem Hamilton’s equations are ˙ Y = J ∇ Y H , where � O 3 � − I 3 ǫ = µ k 2 , H = H 0 + ǫ H 1 , J = . I 3 O 3 H 0 = − k 4 ( unperturbed part ) , 2 L 2 |X − X ′ | − X · X ′ � � 1 H 1 = − ( perturbing function ) . |X ′ | 3 Here X , X ′ denote x , x ′ as functions of Y , Y ′ . Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences
The Keplerian distance function Let ( E j , v j ) , j = 1 , 2 be the orbital elements of two celestial bodies on Keplerian orbits with a common focus: E j represents the trajectory of a body, v j is a parameter along it. Set V = ( v 1 , v 2 ) . z For a given two-orbit configuration E = ( E 1 , E 2 ) , we introduce the Keplerian distance function T 2 ∋ V �→ d ( E , V ) = |X 1 − X 2 | y We are interested in the local x minimum points of d . d Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences
Geometry of two confocal Keplerian orbits Is there still something that we do not know about distance of points on conic sections? ἐθεώρουν σε σπεύδοντα μετασχεῖν τῶν πεπραγμένων ἡμῖν κωνικῶν ( 1 ) (Apollonius of Perga, Conics , Book I) ( 1 ) I observed you were quite eager to be kept informed of the work I was doing in conics. Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences
Critical points of d 2 Gronchi SISC (2002), CMDA (2005) Apart from the case of two concentric coplanar circles, or two overlapping ellipses, d 2 has finitely many critical points. There exist configurations with 12 critical points, and 4 local minima of d 2 . This is thought to be the maximum possible, but a proof is not known yet. ( 1 ) A simple computation shows that, for non-overlapping trajectories, the number of crossing points is at most two. ( 1 ) Albouy, Cabral and Santos, ‘Some problems on the classical n-body problem’ CMDA 113/4 , 369-375 (2012) Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences
The orbit distance Let V h = V h ( E ) be a local minimum point of V �→ d 2 ( E , V ) . Consider the maps E �→ d h ( E ) = d ( E , V h ) , E �→ d min ( E ) = min h d h ( E ) . The map E �→ d min ( E ) gives the orbit distance. Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences
Singularities of d h and d min 4 4 4 3 3 3 distance distance distance d min 2 2 2 d 2 d 1 1 1 1 d 1 d 2 d 1 0 0 0 0 2 4 0 2 4 0 2 4 orbital elements orbital elements orbital elements (i) d h and d min are not differentiable where they vanish; (ii) two local minima can exchange their role as absolute minimum thus d min loses its regularity without vanishing; (iii) when a bifurcation occurs the definition of the maps d h may become ambiguous after the bifurcation point. Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences
Smoothing through change of sign x−axis y−axis x−axis y−axis Toy problem: � − f ( x , y ) for x > 0 x 2 + y 2 � ˜ f ( x , y ) = f ( x , y ) = f ( x , y ) for x < 0 Can we smooth the maps d h ( E ) , d min ( E ) through a change of sign? Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences
Local smoothing of d h at a crossing singularity Smoothing d h , the procedure for d min is the same. Consider the points on the two orbits X ( h ) = X 1 ( E 1 , v ( h ) X ( h ) = X 2 ( E 2 , v ( h ) 1 ) ; 2 ) . 1 2 corresponding to the local minimum point V h = ( v ( h ) 1 , v ( h ) 2 ) of d 2 ; Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences
Local smoothing of d h at a crossing singularity introduce the tangent vectors to the trajectories E 1 , E 2 at these points: τ 1 = ∂ X 1 τ 2 = ∂ X 2 ( E 1 , v ( h ) ( E 2 , v ( h ) 1 ) , 2 ) , ∂ v 1 ∂ v 2 and their cross product τ 3 = τ 1 × τ 2 ; Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences
Local smoothing of d h at a crossing singularity define also ∆ h = X ( h ) − X ( h ) ∆ = X 1 − X 2 , . 1 2 The vector ∆ h joins the points attaining a local minimum of d 2 and | ∆ h | = d h . Note that ∆ h × τ 3 = 0 Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences
Smoothing the crossing singularity Gronchi and Tommei, DCDS-B (2007) smoothing rule: ˜ d h = sign ( τ 3 · ∆ h ) d h E �→ ˜ d h ( E ) is an analytic map in a neighborhood of most crossing configurations. Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences
The averaging method The averaging principle is used to study the qualitative behavior of solutions of ODEs in perturbation theory, see Arnold, Kozlov, Neishtadt (1997). � ˙ φ = ω ( I ) φ ∈ T n , I ∈ R m unperturbed ˙ I = 0 � ˙ φ = ω ( I ) + ǫ f ( φ, I , ǫ ) perturbed ˙ I = ǫ g ( φ, I , ǫ ) 1 � ˙ averaged J = ǫ G ( J ) , G ( J ) = T n g ( φ, J , 0 ) d φ 1 . . . d φ n ( 2 π ) n Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences
Averaging over 2 angular variables Using the averaged equations corresponds to substituting the time average with the space average. Case of 2 angles: a problem occurs if there are resonant relations of low order between the motions φ 1 ( t ) , φ 2 ( t ) , i.e. if h 1 ˙ φ 1 + h 2 ˙ φ 2 = 0 , with h 1 , h 2 small integers. φ 2 φ 1 Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences
Averaged equations Gronchi and Milani, CMDA (1998) Averaged Hamilton’s equations: ˙ Y = ǫ J ∇ Y H 1 , (1) with Y = ( G , Z , g , z ) . If no orbit crossing occurs, (1) are equal to ˙ Y = ǫ J ∇ Y H 1 (2) with � � 1 1 1 T 2 H 1 d ℓ d ℓ ′ = − |X − X ′ | d ℓ d ℓ ′ H 1 = ( 2 π ) 2 ( 2 π ) 2 T 2 The average of the indirect term of H 1 is zero. Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences
Crossing singularities If there is an orbit crossing, then averaging on the fast angles ℓ, ℓ ′ produces a singularity in the averaged equations: we take into account every possible position on the orbits, thus also the collision configurations. 1 � 1 |X − X ′ | d ℓ d ℓ ′ H 1 = − ( 2 π ) 2 T 2 and � X ( E 1 , v ( h ) 1 ) − X ′ ( E 2 , v ( h ) � = 0 . � � 2 ) Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences
Near-Earth asteroids and crossing orbits (433) Eros: the first near-Earth asteroid (NEA, with q = a ( 1 − e ) ≤ 1 . 3 au), discovered in 1898; it crosses the trajectory of Mars. from NEAR mission (NASA) Today (March 19, 2019) we know about 19800 NEAs: several of them cross the orbit of the Earth during their evolution. Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences
Derivative jumps Let E c be a non–degenerate crossing configuration for d h , with only 1 crossing point. Given a neighborhood W of E c , we set W + = W ∩ { ˜ d h > 0 } , W − = W ∩ { ˜ d h < 0 } . The averaged vector field ∇ Y H 1 is not defined on Σ = { d H = 0 } . Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences
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