Why are inner planets not inclined Andrew Clarke Jacques Fejoz Marcel Guardia October 21 2022

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Why are inner planets not inclined?
Andrew Clarke Jacques Fejoz Marcel Guardia
October 21, 2022
Abstract
Poincar´e’s work more than one century ago, or Laskar’s numerical simulations from the 1990’s on,
have irrevocably impaired the long-held belief that the Solar System should be stable. But mathematical
mechanisms explaining this instability have remained mysterious. In 1968, Arnold conjectured the exis-
tence of “Arnold diffusion” in celestial mechanics. We prove Arnold’s conjecture in the planetary spatial
4-body problem as well as in the corresponding hierarchical problem (where the bodies are increasingly
separated), and show that this diffusion leads, on a long time interval, to some large-scale instability.
Along the diffusive orbits, the mutual inclination of the two inner planets is close to π/2, which hints at
why even marginal stability in planetary systems may exist only when inner planets are not inclined.
More precisely, consider the normalised angular momentum of the second planet, obtained by rescaling
the angular momentum by the square root of its semimajor axis and by an adequate mass factor (its
direction and norm give the plane of revolution and the eccentricity of the second planet). It is a vector
of the unit 3-ball. We show that any finite sequence in this ball may be realised, up to an arbitrary
precision, as a sequence of values of the normalised angular momentum in the 4-body problem. For
example, the second planet may flip from prograde nearly horizontal revolutions to retrograde ones. As
a consequence of the proof, the non-recurrent set of any finite-order secular normal form accumulates on
circular motions – a weak form of a celebrated conjecture of Herman.
Contents
1 Introduction 2
1.1 A case for instability in the solar system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Mainresults.............................................. 5
1.3 Main ideas of the proof of Theorem 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Main results in Deprit coordinates 11
2.1 Settinguptheproblem........................................ 12
2.2 The Deprit coordinates and reduction by rotational symmetry . . . . . . . . . . . . . . . . . . 13
2.3 Arnold diffusion in Deprit coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 The averaging procedure and the secular Hamiltonian 17
3.1 The averaging procedure and the secular Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Expansion of the secular Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4 Analysis of the first-order Hamiltonian 24
5 The inner dynamics 26
5.1 The parametrisation of the cylinder and the inner Hamiltonian . . . . . . . . . . . . . . . . . 27
5.2 The Hessian of the averaged inner Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . 33
6 The outer dynamics 36
6.1 Thescatteringmap.......................................... 36
6.2 Computation of the Poincar´e-Melnikov potential . . . . . . . . . . . . . . . . . . . . . . . . . 43
1
arXiv:2210.11311v1 [math.DS] 20 Oct 2022
7 Reduction to a Poincar´e map and the shadowing argument 46
7.1 ThePoincar´emap .......................................... 46
7.2 Constructing transition chains of almost invariant tori . . . . . . . . . . . . . . . . . . . . . . 49
7.3 Application of the shadowing theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
8 Continuation in the planetary regime 53
9 From Deprit coordinates to elliptic elements: proof of Theorem 4 54
A Deprit’s coordinates 55
B The scattering map of a normally hyperbolic invariant manifold 56
C A general shadowing argument 58
D Computation of the phase shift in ˜
ψ161
E Expansion of the inner Hamiltonian 63
F Corrigendum of [38] 63
1 Introduction
1.1 A case for instability in the solar system
Hook’s and Newton’s discovery of universal attraction in the xvii century masterly reconciles two seemingly
contradictory physical principles: the principle of inertia, put forward by Galileo and Descartes in terrestrial
mechanics, and the laws of Kepler, governing the elliptical motion of planets around the Sun [1, 6, 80]. The
unforeseen mathematical consequence of Hook’s and Newton’s discovery was to question the belief that the
solar system be stable: it was no longer obvious that planets kept moving immutably, without collisions or
ejections, because of their mutual (“universal”) attraction. Newton himself, in an additional and staggering
tour de force, estimated the first order effect on Mars of the attraction of other planets. But infinitesimal
calculus was in its infancy and the necessary mathematical apparatus to understand the long-term influence
of mutual attractions did not exist.
In their study of Jupiter’s and Saturn’s motions, Lagrange, Laplace, Poisson and others laid the foun-
dations of the Hamiltonian theory of the variation of constants. They managed to compute the secular
dynamics, i.e. the slow deformations of Keplerian ellipses, at the first order with respect to the masses,
eccentricities and inclinations of the planets. This level of approximation is still integrable, and Keplerian
ellipses have slow but non vanishing precession and rotation frequencies, thus departing from the dynamical
degeneracy decribed by Bertrand’s theorem. Besides, the analysis of the spectrum of the linearised vector
field entailed a resounding stability theorem for the solar system: the observed variations of adiabatic in-
variants in the motion of Jupiter and Saturn come from resonant terms of large amplitude and long period,
but with zero average ([60, p. 164], [62]). Yet it is a mistake, which Laplace made, to infer the topological
stability of the non-truncated planetary system.
In the xviii and xix centuries, mathematicians spent an inordinate amount of energy trying to prove the
stability of the Solar system... until Poincar´e discovered a remarkable set of arguments strongly speaking
against stability: generic divergence of perturbation series, non-integrability of the three-body problem, and
entanglement of the stable and unstable manifolds of the Lagrange relative equilibrium in the restricted
3-body problem [83].
In the mid xx century, Siegel and Kolmogorov still proved that, respectively for the linearisation problem
of a one-dimensional complex map and for the perturbation of an invariant torus of fixed frequency in a
Hamiltonian system, perturbation series do converge, albeit non uniformly, under some arithmetic assump-
tion of Diophantine type, ensuring that the frequencies of the motion are far from low order resonances, in a
quantitative way. The obtained solutions are quasiperiodic and densely fill Lagrangian invariant tori. They
form a large set in the measure theoretic sense, but a small set from the topological viewpoint. Besides,
2
starting from dimension 6, invariant tori do not separate energy levels and thus do not confine neighboring
motions, so, outside invariant tori, nothing prevents adiabatic invariants to drift. Kolmogorov’s theorem was
successfully adapted to the planetary system, despite the numerous degeneracies of the latter, and assuming
that the masses of the planets are very small [3, 21, 35, 84]. The obtained solutions are small perturbations
of (Diophantine) Laplace-Lagrange motions.
Soon afterward, Arnold imagined an example of dynamical instability in a near-integrable Hamiltonian
system with many degrees of freedom, where action variables may drift (for some well chosen orbits), by an
amount uniform with respect to the smallness of the perturbation [4]. Of course, the drifting time tends
to infinity as the size of the perturbation tends to 0, consistently with the continuity of the time-tmap of
the flow with respect to parameters. Drifting orbits shadow the stable and unstable manifolds of a chain
of hyperbolic invariant tori (“transition chain”). This phenomenon has been called Arnold diffusion, since
Chirikov coined the phrase, referring to the (in part conjectural) stochastic properties of such a dynamics [22].
In fact, in this seminal paper, Arnold conjectured the following.
Conjecture 1 (Arnold [4]).The mechanism of transition chains [...] is also applicable to the case of general
Hamiltonian systems (for example, to the problem of three bodies).
Arnold’s example has proved difficult to generalise because of the so-called large gap problem: usually
the transition chain is a (totally disconnected) Cantor set of hyperbolic tori and it is not obvious whether
there exist orbits shadowing these tori. A better strategy has emerged, consisting in shadowing normally
hyperbolic cylinders (whether they contain invariant tori or not). Nearly integrable Hamiltonian systems
are usually classified as a priori unstable and a priori stable [19]. A priori unstable models are those whose
integrable approximation presents some hyperbolicity (the paradigmatic example being a pendulum weakly
coupled with several rotators). In this case, the unperturbed model has a normally hyperbolic invariant
manifold with attached invariant manifolds that one can use, for the perturbed model, as a “highway”
for diffusing orbits. The existence of Arnold diffusion generically in these models is nowadays rather well
understood, at least for two and a half degrees of freedom (see [72, 18, 86, 27, 49, 8, 31], or [87, 29, 50] for
results in higher dimension).
A priori stable systems are those whose integrable approximation is foliated by quasiperiodic invariant
Lagrangian tori. Since the unperturbed Hamiltonian does not possess hyperbolic invariant objects, in order
to construct the diffusing “highway” one has to rely on a first perturbation and face involved singular
perturbation problems. One of the difficulties is that one cannot avoid double resonances, where the system
is intrinsically non-integrable. Arnold conjecture refers to these models. The work of Mather on minimizing
measures has been deeply influential. In the finite smoothness category, the papers [9, 17, 58] show the
typicality (in the cusped residual sense as defined by J. Mather) of Arnold diffusion in a priori stable
Hamiltonian systems of 3 degrees of freedom. Yet, many questions remain unsolved. In particular, the
original Arnold conjecture on the typicality of Arnold diffusion for analytic non degenerate nearly integrable
Hamiltonian systems of 3 or more degrees of freedom remains open (see however [44, 45]).1
In the 1990s, with extensive numerical computations Laskar showed that over the physical life span of
the Sun, or even over a few hundred million years, collisions and ejections of inner planets occur with some
probability [61, 64].2Our solar system is now believed only marginally stable. This has been corroborated
by abundant numerical evidence, as overviewed in Morbidelli’s book [77]. In particular, the effect of mean
motion (Keplerian) resonances in the asteroid belt has been described by [67]. Numerical evidence has
also been suggesting that secular resonances are a major source of chaos in the Solar system [65, 63, 46].
For example, astronomers have established that Mercury’s eccentricity is chaotic and can increase so much
that collisions with Venus or the Sun become possible, as a result from an intricate network of secular
resonances [13]. On the other hand, that Uranus’s obliquity (97o) is essentially stable, is explained, to a
1One can also consider the so called a priori chaotic case, where the unperturbed Hamiltonian presents “local non-
integrability”. In particular, it has a first integral and a periodic orbit with transverse homoclinics at each energy level.
Examples of such settings are certain geodesic flows with a time dependent potential, see [11, 24, 25, 26, 47, 48].
2Such long term computations are checked to pass various consistency tests (e.g. the preservation of first integrals). But
due to the exponential divergence of solutions, they are statistical in nature: an uncertainty of a few centimeters on the initial
position of the Earth leads to an uncertainty of the size of the Solar System after a few hundred millions years. But one likes
to believe that such Hamiltonian systems have good shadowing properties, i.e. that any finite-time pseudo-orbit (as computed
numerically) is shadowed by orbits.
3
large extent, by the absence of any low-order secular resonance [12, 65]. The effects of secular resonances of
the inner planets have later been studied systematically using both computer algebra and numerics and the
main “sources of chaos” in the inner solar system have been identified [7, 74].
The mathematical theory of instability remains in its infancy and the Astronomers’ instability mechanisms
are still mysterious. A matter of discontent with Arnold diffusion is that the time needed for actions to drift
looks larger than in other, far from integrable, instability mechanisms that astronomers observe. Resonance
overlapping, a phenomenon described by Chirikov [22], would be a fantastic competing mechanism. But to
our knowledge it lacks mathematical explanation (see however a simple example in [39]).
Regarding “the oldest problem in dynamical systems”, in his ICM lecture [55] Herman formulated the
following two precise conjectures. Consider the N-body problem in space, with N≥3. Assume that the
center of mass is fixed at the origin and that on the energy surface of level ewe C∞-reparametrise the flow
by a C∞function ϕesuch that the collisions now occur only in infinite time (ϕe>0 is a Cωfunction outside
collisions).3
Conjecture 2 (Global instability).Is for every ethe non-wandering set of the Hamiltonian flow of Heon
H−1
e(0) nowhere dense in H−1
e(0)?
This would imply that bounded orbits are nowhere dense and no topological stability occurs.4The
conjecture is wide open, and we are still at the stage of looking for non-recurrent orbits having negative
energy (due to the Lagrange-Jacobi identity, non-wandering orbits have negative energy) [37]. A growing
number of such orbits are known to exist, most often associated with some kind of symbolic dynamics: close
to parabolic motions [2, 32, 51, 52, 53, 70, 78, 69, 85], close to triple collisions [71, 73], close to Poincar´e’s
periodic orbits of second species [10], close to the Euler-Lagrange points [14], on the submanifold of zero
angular momentum of the 3-body problem [75, 76], and so on. It is part of the richness of the N-body
problem to include such diverse kinds of behavior.
Yet, scarce mathematical mechanisms have been described regarding more astronomical regimes, which
would be plausible for subsystems of solar or extra-solar systems. Within the domain of negative energy, of
special astronomical relevance is the planetary problem, where planets with small masses revolve around the
Sun.5Another well-known problem is the hierarchical problem, an extension to N-bodies of the so-called
Hill problem or lunar problem, where one body (the Sun) revolves far away around the other two (the Earth
and the Moon).6These problems are rendered difficult by the proximity to a degenerate (“super-integrable”)
integrable system of two uncoupled Kepler problems. As mentioned above, the recurrent set has positive
Lebesgue measure due to Arnold-Herman’s invariant tori theorem, while the existence of unstable orbits relies
on the so-called Arnold diffusion. Herman further argues that What seems not an unreasonable question to
ask (and possibly prove in a finite time with a lot of technical details) is that:
Conjecture 3 (Planetary instability).If one of the masses is fixed (m0= 1) and the other masses mj=ρemj,
1≤j≤n−1,emj>0,ρ > 0, then in any neighbourhod of fixed different circular orbits around m0moving
in the same direction in a plane, when ρis small, there are wandering domains.
Even if Arnold in his seminal paper conjectured that his Arnold diffusion mechanism via transition chains
should be present in the 3 body problem, even nowadays the results in this direction are rather scarce. Indeed,
as far as the authors know, the only complete analytical proof of Arnold diffusion in Celestial Mechanics,
prior to the present work, is contained in an article by Delshams, Kaloshin, de la Rosa and Seara [32]. In this
paper, the authors consider the Restricted Planar Elliptic 3 Body Problem and construct orbits with large
drift in angular momentum, assuming the mass ratio and the eccentricities of the primaries are sufficiently
small. A key point of this paper is that the body of zero mass is close to the so called parabolic motions.
That is, it relies on the invariant manifolds of infinity which are already present in the two body problem.
They then perform a delicate perturbative analysis of the model.
3Here indeed it is natural to count the non-wandering set without collision orbits, since positions do not necessarily go to
infinity at collisions. Herman furthermore claims that this reparametrised flow is complete. This is not clear due to the potential
presence of non-collision singularities. But conjecture 2 remains relevant with a possibly incomplete flow.
4This incidentally contradicts a conjecture of Poincar´e, that periodic orbits are dense [83, End of section 36].
5Jupiter, the largest planet of the solar system, weighs roughly 1/1000 the mass of the Sun.
6The lunar distance is roughly 1/400 the distance from Earth to Sun.
4
Some works have uncovered instability mechanisms in celestial mechanics related to Arnold diffusion,
relying on computer assisted computations (see [16, 40] and also [15] which relies on computer assisted
proofs) or conditionally to a plausible transversality hypothesis [88]. The article [38] proves the existence of
some hyperbolic invariant set with symbolic dynamics in the hierarchical truncated secular spatial 3-body
problem, with no hope of finding Arnold diffusion in the full 3-body problem because of the lack of additional
slow degrees of freedom.
Regarding Conjecture 3, the construction of wandering domains using Arnold diffusion-like mechanisms
is a difficult problem. As far as we know, the only positive result in this direction is in [66], where wandering
domains for Gevrey nearly integrable symplectic maps are constructed. The methods used in that paper to
construct such domains do not admit an immediate extension to the analytic category.
The present work
•proves Arnold’s Conjecture 1 in the planetary spatial 4-body problem (see Theorem 4 below)
•and proves a weak local version of Herman’s Conjecture 3, dealing with non-recurrent orbits instead
of wandering orbits (Theorem 6), for the corresponding secular dynamics.
1.2 Main results
Consider the 4-body problem, that is 4 bodies numbered from 0 to 3 moving in 3-dimensional space according
to the Newtonian gravitational law,
¨xj=X
0≤i≤3
i6=j
mi
xi−xj
kxi−xjk3,
where xj∈R3is the position and mj>0 is the mass of body jfor j= 0,1,2,3.
For the sake of simplicity, let us first focus on the “hierarchical regime” where body 2 revolves around
and far away from bodies 0 and 1, while body 3 revolves around and even farther away from bodies 0, 1 and
2. Each body thus primarily undergoes the attraction of one other body: bodies 0 and 1 are close to being
isolated, body 2 primarily undergoes the attraction of a fictitious body located at the center of mass of 0
and 1, and body 3 primarily undergoes the attraction of a fictitious body located at the center of mass of 0,
1 and 2. We think of body 0 as the Sun and of the three other bodies as planets. The Jacobi coordinates are
well suited for this regime (Figure 2), but we defer their definition to a later stage. Assuming that the center
of mass is fixed, the small displacements of the Sun may be recovered from the positions of the planets.
The fast dynamics consists in those planets moving along Keplerian ellipses according to the above
approximation, with elliptical elements as first integrals, in addition to the total energy and the total angular
momentum C=C1+C2+C3, where Ciis the angular momentum of planet i. Let a1,a2and a3be the
semimajor axes. In our regime,
a1a2a3
(we will later make a more specific hypothesis on how the semimajor axis ratios ai/ai+1 compare to each
other). Let also e1,e2and e3be the eccentricities. The angular momentum Ci, seen as a vector in space, is
normal to the plane of ellipse iand its length is pai(1 −e2
i), up to a mass factor (see (1) below). Let θij
be the mutual inclinations, i.e. the oriented angle αCi×Cj(Ci, Cj) between the angular momenta of planets
iand j, the orientation being defined by the normal vector Ci×Cj(here assumed non-zero).
The so-called secular dynamics describes the slow evolution of the three Keplerian ellipses. At the first
order of approximation, it is governed by the vector field obtained by averaging out the mean anomalies, thus
defining a dynamical system on the “secular space” of triples of Keplerian ellipses with fixed semimajor axes.
In the hierarchical regime, the dominating term is what is usually called the “quadrupolar” Hamiltonian F12
quad
of the two inner planets. It was introduced in various particular cases by Lidov and then Kozai [59, 68, 90]. It
may come as a surprise that F12
quad is integrable (defined in (26) in Section 3), as noticed by Harrington [54],
due to the fact that it does not depend on the argument of the outer pericenter g2. This dynamics was later
studied more globally in the secular space by Lidov, Kozai and others (see a review in [79]).
Our analysis follows from a higher order, non-integrable approximation of the system (it will rely also on
the quadrupolar Hamiltonian of planets 2 and 3, and the octupolar term of planets 1 and 2, as introduced
later). Precisely understanding the various time scales within the secular dynamics will be key to our analysis.
At this stage, let us only loosely describe the different roles played by the three planets:
5
摘要：

Whyareinnerplanetsnotinclined?AndrewClarkeJacquesFejozMarcelGuardiaOctober21,2022AbstractPoincare'sworkmorethanonecenturyago,orLaskar'snumericalsimulationsfromthe1990'son,haveirrevocablyimpairedthelong-heldbeliefthattheSolarSystemshouldbestable.Butmathematicalmechanismsexplainingthisinstabilityhave...

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