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On the restricted three-body problem with crossing singularities - - PowerPoint PPT Presentation

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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

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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

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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 = −G

  • m⊙

(y − y⊙(t)) |y − y⊙(t)|3 + m⊕ (y − y⊕(t)) |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

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The restricted three–body problem

In heliocentric coordinates ¨ x = −k2 x |x|3 + µ (x − x′) |x − x′|3 − x′ |x′|3

  • where

x = y − y⊙, x′ = y⊕ − y⊙; k2 = Gm⊙, µ = m⊕

m⊙ is a small parameter;

−k2µ (x−x′)

|x−x′|3 is the direct perturbation of the planet on the

asteroid; k2µ 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

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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 G = L √ 1 − e2 Z = G cos I    ℓ = n(t − t0) g = ω 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

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Canonical formulation of the problem

Hamilton’s equations are ˙ Y = J ∇YH , where H = H0 + ǫH1, ǫ = µk2, J = O3 −I3 I3 O3

  • .

H0 = − k4 2L2 (unperturbed part), H1 = −

  • 1

|X − X ′| − X · X ′ |X ′|3

  • (perturbing function).

Here X, X ′ denote x, x′ as functions of Y, Y′.

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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The Keplerian distance function

Let (Ej, vj), j = 1, 2 be the orbital elements of two celestial bodies on Keplerian orbits with a common focus: Ej represents the trajectory of a body, vj is a parameter along it. Set V = (v1, v2). For a given two-orbit configuration E = (E1, E2), we introduce the Keplerian distance function T2 ∋ V → d(E, V) = |X1 − X2| We are interested in the local minimum points of d.

y z

d

x

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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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

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Critical points of d2

Gronchi SISC (2002), CMDA (2005) Apart from the case of two concentric coplanar circles, or two

  • verlapping ellipses, d2 has finitely many critical points.

There exist configurations with 12 critical points, and 4 local minima of d2. 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

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The orbit distance

Let Vh = Vh(E) be a local minimum point of V → d2(E, V). Consider the maps E → dh(E) = d(E, Vh) , E → dmin(E) = min

h dh(E) .

The map E → dmin(E) gives the orbit distance.

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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Singularities of dh and dmin

2 4 1 2 3 4

  • rbital elements

distance dmin 2 4 1 2 3 4

  • rbital elements

distance d1 d2 d2 d1 2 4 1 2 3 4

  • rbital elements

distance d1

(i) dh and dmin are not differentiable where they vanish; (ii) two local minima can exchange their role as absolute minimum thus dmin loses its regularity without vanishing; (iii) when a bifurcation occurs the definition of the maps dh may become ambiguous after the bifurcation point.

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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Smoothing through change of sign

y−axis x−axis y−axis x−axis

Toy problem: f(x, y) =

  • x2 + y2

˜ f(x, y) = −f(x, y) for x > 0 f(x, y) for x < 0 Can we smooth the maps dh(E), dmin(E) through a change of sign?

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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Local smoothing of dh at a crossing singularity

Smoothing dh, the procedure for dmin is the same. Consider the points on the two orbits X (h)

1

= X1(E1, v(h)

1 ) ;

X (h)

2

= X2(E2, v(h)

2 ) .

corresponding to the local minimum point Vh = (v(h)

1 , v(h) 2 ) of d2;

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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Local smoothing of dh at a crossing singularity

introduce the tangent vectors to the trajectories E1, E2 at these points: τ1 = ∂X1 ∂v1 (E1, v(h)

1 ) ,

τ2 = ∂X2 ∂v2 (E2, v(h)

2 ) ,

and their cross product τ3 = τ1 × τ2;

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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Local smoothing of dh at a crossing singularity

define also ∆ = X1 − X2 , ∆h = X (h)

1

− X (h)

2

. The vector ∆h joins the points attaining a local minimum of d2 and |∆h| = dh. Note that ∆h × τ3 = 0

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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Smoothing the crossing singularity

Gronchi and Tommei, DCDS-B (2007) smoothing rule: ˜ dh = sign(τ3 · ∆h)dh E → ˜ dh(E) is an analytic map in a neighborhood of most crossing configurations.

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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The averaging method

The averaging principle is used to study the qualitative behavior

  • f solutions of ODEs in perturbation theory, see Arnold, Kozlov,

Neishtadt (1997). unperturbed ˙ φ = ω(I) ˙ I = 0 φ ∈ Tn, I ∈ Rm perturbed ˙ φ = ω(I) + ǫf(φ, I, ǫ) ˙ I = ǫg(φ, I, ǫ) averaged ˙ J = ǫG(J) , G(J) = 1 (2π)n

  • Tn g(φ, J, 0) dφ1 . . . dφn

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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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 φ1(t), φ2(t), i.e. if h1 ˙ φ1 + h2 ˙ φ2 = 0, with h1, h2 small integers.

φ1 φ2 Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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Averaged equations

Gronchi and Milani, CMDA (1998) Averaged Hamilton’s equations: ˙ Y = ǫ J ∇YH1 , (1) with Y = (G, Z, g, z). If no orbit crossing occurs, (1) are equal to ˙ Y = ǫ J ∇YH1 (2) with H1 = 1 (2π)2

  • T2 H1 dℓ dℓ′ = −

1 (2π)2

  • T2

1 |X − X ′| dℓ dℓ′ The average of the indirect term of H1 is zero.

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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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. H1 = − 1 (2π)2

  • T2

1 |X − X ′| dℓ dℓ′ and

  • X(E1, v(h)

1 ) − X ′(E2, v(h) 2 )

  • = 0 .

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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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

  • f them cross the orbit of the Earth during their evolution.

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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Derivative jumps

Let Ec be a non–degenerate crossing configuration for dh, with

  • nly 1 crossing point.

Given a neighborhood W of Ec, we set W+ = W ∩ {˜ dh > 0} , W− = W ∩ {˜ dh < 0} . The averaged vector field ∇YH1 is not defined on Σ = {dH = 0}.

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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Derivative jumps

Gronchi and Tardioli, DCDS-B (2013) The averaged vector field ∇YH1 can be extended to two Lipschitz–continuous vector fields (∇YH1)±

h on a neighborhood W of Ec.

The components of the extended fields, restricted to W+, W− respectively, correspond to ∂H1

∂yk .

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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Derivative jumps

Moreover the following relations hold: Diffh ∂H1 ∂yk

  • def

= ∂H1 ∂yk −

h −

∂H1 ∂yk +

h =

= − 1 π ∂ ∂yk

  • 1
  • det(Ah)
  • ˜

dh + 1

  • det(Ah)

∂˜ dh ∂yk

  • ,

where yk is a component of Delaunay’s elements Y.

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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Generalized solutions

Figure: Runge-Kutta-Gauss method and continuation of the solutions

  • f equations (1) beyond the singularity.

The averaged solutions are piecewise–smooth

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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Averaged evolution of (1620) Geographos

3000 4000 5000 6000 0.3335 0.334 0.3345 0.335 0.3355 t e 3000 4000 5000 6000 280 285 290 295 300 305 310 315 t ω 3000 4000 5000 6000 13.15 13.2 13.25 13.3 13.35 13.4 13.45 t I 3000 4000 5000 6000 305 310 315 320 325 330 335 t Ω Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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Proper elements for NEAs: (1620) Geographos

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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Proper elements for NEAs: (2102) Tantalus

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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Secular evolution of the orbit distance

Define the secular evolution of the minimal distances dh(t) = ˜ dh(E(t)) , dmin(t) = ˜ dmin(E(t)) in an open interval containing a crossing time tc. Assume tc is a crossing time and Ec = E(tc) is a non-degenerate crossing configuration with only one crossing point, i.e. dh(Ec) = 0. Then there exists an interval (ta, tb), ta < tc < tb such that dh ∈ C1((ta, tb); R).

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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Secular evolution of the orbit distance

idea of the proof: lim

t→t−

c

˙ dh(t) − lim

t→t+

c

˙ dh(t) = Diffh

  • ∇YH1
  • · ǫ J2∇Y˜

dh

  • E=Ec

= − ǫ π√det Ah {˜ dh, ˜ dh}Y

  • E=Ec = 0 ,

The secular evolution of ˜ dmin is more regular than that of the

  • rbital elements in a neighborhood of a planet crossing time.

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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Evolution of the orbit distance for 1979 XB

1950 2000 2050 2100 2150 2200 −0.02 −0.015 −0.01 −0.005 0.005 0.01 0.015 0.02 0.025 0.03 time (yr)

  • rbit distance (AU)

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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Transition through a planet crossing for 1979 XB

linearized secular evolution

2000 2050 2100 2150 2200 0.005 0.01 0.015 0.02 0.025 0.03 time (yr)

  • rbit distance (AU)

t1 t2

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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Transition through a planet crossing for 1979 XB

nonlinear secular evolution

2000 2050 2100 2150 2200 0.005 0.01 0.015 0.02 0.025 0.03 time (yr)

  • rbit distance (AU)

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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Mean motion resonances

Mar`

  • and Gronchi, SIADS (2018)

Resonance condition: hn + h′n′ = 0, h, h′ ∈ Z. Extended Hamiltonian: ˜ H = H0 + n′L′ + ǫH1 Resonant normal form to order N: HN(V, L, L′; X) =

  • k∈R,|k|≤N

ˆ Hk(L, L′; X)eik·V. Here V = (ℓ, ℓ′), X are the other (secular) variables, R = {k = (k, k′) ∈ Z2 : ∃n ∈ Z with k = nh}, h = (h, h′), and ˆ Hk(L, L′; X) = 1 (2π)2

  • T2

˜ H(V, L, L′; X)e−ik·VdV. V = (ℓ, ℓ′) when the latter are integration variables.

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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Resonant normal form

Note that HN(V, L, L′; X) = 1 (2π)2

  • T2 DN(h · V − h · V) ˜

H(V, L, L′; X)dV, where DN(x) =

  • |n|≤N

einx = sin((N + 1/2)x) sin(x/2) is the Dirichlet kernel. We introduce a canonical transformation Ψ through the relations σ σ′

  • = A

ℓ ℓ′

  • ,

S S′

  • = A−T

L L′

  • ,

with A =

  • h

h′ 1/h

  • ,

A−T =

  • 1/h

−h′ h

  • .

σ = hℓ + h′ℓ′ is the resonant angle.

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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Resonant normal form

Set X = (G, Z, g, z) and let us define KN(σ, S, T; X) = HN ◦ Ψ−1(σ, τ, S, T; X). Fix Nmax and take the resonant normal form in the new variables KNmax = K0 + ǫ(K1 + KNmax

res ),

(3) with K0(S; S′) = H0

  • hS, h′S + S′

h

  • = −

k4 2(hS)2 + n′ h′S + S′ h

  • ,

K1(S, X; S′) = H1(hS, h′S + S′ h , X) = − 1 (2π)2

  • T2

1 d(ℓ, ℓ′)dℓdℓ′, KNmax

res (S, σ, X; S′) = −

1 (2π)2

  • T2
  • DNmax(h · V − σ) − 1
  • H1
  • V, hS, h′S + S′

h , X

  • dV.

Since KNmax does not depend on σ′, the value of S′ is constant.

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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Resonant normal form

Equations for the motion of the asteroid: ˙ Y = J∇YKNmax, where Y = (S, G, Z, σ, g, z), or, in components, ˙ S = −∂KNmax ∂σ = −ǫ∂KNmax

res

∂σ , ˙ G = −∂KNmax ∂g = −ǫ ∂KNmax

res

∂g + ∂K1 ∂g

  • ,

˙ Z = −∂KNmax ∂z = −ǫ ∂KNmax

res

∂z + ∂K1 ∂z

  • ,

˙ σ = ∂KNmax ∂S = hk4 (hS)3 + n′h′ + ǫ ∂KNmax

res

∂S + ∂K1 ∂S

  • ,

˙ g = ∂KNmax ∂G = ǫ ∂KNmax

res

∂G + ∂K1 ∂G

  • ,

˙ z = ∂KNmax ∂Z = ǫ ∂KNmax

res

∂Z + ∂K1 ∂Z

  • .

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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Mean motion resonances

If Ec corresponds to a crossing configuration with Jupiter, then the following relation holds in a neighborhood W Diffh ∂KNmax ∂yi

  • = ǫ

∂K1 ∂yi −

h −

∂K1 ∂yi +

h +

∂KNmax

res

∂yi −

h −

∂KNmax

res

∂yi +

h

  • = − ǫ

π DNmax(σ − h · Vh)

  • ∂

∂yi

  • 1
  • det(Ah)
  • ˜

dh + 1

  • det(Ah)

∂˜ dh ∂yi

  • .

Let σc = h · Vh. We observe that lim

Nmax→∞ DNmax(σ − σc) = δσc,

that is, for Nmax → ∞, the Dirichlet kernel converges in the sense of distributions to the Dirac delta centered in σc.

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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Recent work (in progress)

Joint work (in progress) with M. Fenucci Open questions: Can we prove that the averaged solutions are a good approximation of the solutions of the full equations? What happens in case of close approaches with some planet? In case of mean motion resonances, can we prove that the solutions of Hamilton’s equations for the resonant normal form are a good approximation of the solutions of the full equations?

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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Comparison for (1620) Geographos: 64 clones

2500 3000 3500 4000 4500 5000 −0.33 −0.328 −0.326 −0.324 −0.322 −0.32 −0.318 t (yr) h 2500 3000 3500 4000 4500 5000 −0.09 −0.085 −0.08 −0.075 −0.07 t (yr) k 2500 3000 3500 4000 4500 5000 −0.08 −0.075 −0.07 −0.065 −0.06 −0.055 −0.05 t (yr) p 2500 3000 3500 4000 4500 5000 0.08 0.085 0.09 0.095 0.1 0.105 t (yr) q Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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Comparison in the resonant case: 64 clones, Nmax = 15

2500 3000 3500 4000 95 100 105 110 t (yr) ω (deg) 2500 3000 3500 4000 250 255 260 265 t (yr) Ω (deg) 2500 3000 3500 4000 0.4915 0.492 0.4925 0.493 0.4935 0.494 t (yr) e 2500 3000 3500 4000 9.4 9.5 9.6 9.7 9.8 t (yr) I (deg) 2500 3000 3500 4000 1.367 1.3675 1.368 1.3685 1.369 t (yr) a (AU) 2500 3000 3500 4000 50 100 150 200 250 t (yr) σ (deg)

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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The arithmetic mean

Let V = (ℓ, ℓ′) and I be the other variables. Consider the arithmetic mean ˆ IN(t) = 1 N

N

  • j1,j2=1

Ij1,j2(t) where Ij1,j2(t) = I(t; I(0), Vj1,j2(0)), Vj1,j2(0) = 2π N (j1, j2), j1, j2 = 1, . . . , N. The solutions Ij1,j2(t) are computed through Kustaanheimo-Stiefel regularization of binary collisions. Consider also the standard deviation of the solutions: stdI(t) =

  • N
  • j1,j2=1

(Ij1,j2(t) − ˆ IN(t))2 N2 − 1 1/2 Then compare ˆ IN(t) with the solutions of Hamilton’s equations for the normal form, for different values of N.

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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Crossing case: 64 clones

2000 4000 6000 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 t e 2000 4000 6000 43 44 45 46 47 48 t i 2000 4000 6000 7 7.5 8 8.5 9 9.5 10 10.5 t Ω 2000 4000 6000 68 70 72 74 76 t ω a=1.8, e=0.75, I=45,ω=75, Ω=10, tend=5000, µ=0.0001,64 clones

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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Crossing case: 625 clones

2000 4000 6000 0.72 0.73 0.74 0.75 0.76 0.77 0.78 t e 2000 4000 6000 43 44 45 46 47 48 t i 2000 4000 6000 6 7 8 9 10 11 t Ω 2000 4000 6000 66 68 70 72 74 76 78 t ω a=1.8, e=0.75, I=45,ω=75, Ω=10, tend=5000, µ=0.0001,625 clones

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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Crossing case: 3600 clones

2000 4000 6000 0.7 0.72 0.74 0.76 0.78 t e 2000 4000 6000 43 44 45 46 47 48 t i 2000 4000 6000 2 4 6 8 10 12 14 t Ω 2000 4000 6000 65 70 75 80 t ω a=1.8, e=0.75, I=45,ω=75, Ω=10, tend=5000, µ=0.0001,3600 clones

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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non-crossing case: 625 clones

1000 2000 3000 4000 0.1496 0.1498 0.15 0.1502 0.1504 0.1506 0.1508 0.151 t e 1000 2000 3000 4000 44.99 44.995 45 45.005 45.01 45.015 t i 1000 2000 3000 4000 8.5 9 9.5 10 10.5 t Ω 1000 2000 3000 4000 59.8 60 60.2 60.4 60.6 60.8 61 61.2 t ω a=1.8, e=0.15, I=45,ω=60, Ω=10, tend=3000, µ=0.0001,625 clones

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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non-crossing case: 8100 clones

1000 2000 3000 4000 0.1496 0.1498 0.15 0.1502 0.1504 0.1506 0.1508 0.151 t e 1000 2000 3000 4000 44.99 44.995 45 45.005 45.01 45.015 t i 1000 2000 3000 4000 8.5 9 9.5 10 10.5 t Ω 1000 2000 3000 4000 59.8 60 60.2 60.4 60.6 60.8 61 61.2 t ω a=1.8, e=0.15, I=45,ω=60, Ω=10, tend=3000, µ=0.0001,8100 clones

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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Conclusions and future work

We can compute the secular evolution of planet crossing asteroids, by averaging over the fast angles: the solutions are piecewise–smooth; the orbit distance along the averaged evolution is more regular than the orbital elements; We can compute the long term evolution of planet crossing asteroids also in case of mean motion resonances; the arithmetic mean of the solutions seems to be close to the solution of Hamilton’s equations for the normal form of the Hamiltonian, both in the non-resonant and in the resonant case.

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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Thanks for your attention!

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences

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References

[1] Gronchi and Milani: ‘Averaging on Earth-crossing orbits’, Cel.

  • Mech. Dyn. Ast., 71/2, 109–136 (1998)

[2] Gronchi: ‘On the stationary points of the squared distance between two ellipses with a common focus’, SIAM Journ. Sci. Comp. 24/1, 61–80 (2002) [3] Gronchi and Tommei: ‘On the uncertainty of the minimal distance between two confocal Keplerian orbits’, DCDS-B 7/4, 755–778 (2007) [4] Gronchi and Tardioli: ‘The evolution of the orbit distance in the double averaged restricted 3-body problem with crossing singularities’, DCDS-B 8/5, 1323–1344 (2013) [5] Mar`

  • and Gronchi: ‘Long Term Dynamics for the Restricted

N-Body Problem with Mean Motion Resonances and Crossing Singularities’, SIAM Journ. Appl. Dyn. Sys. 17/2, 1786-1815 (2018)

Giovanni F. Gronchi Mathematical Models and Methods in Earth and Space Sciences