The Adiabatic Invariant in Classical Mechanics

Context. It is well known that asteroids and comets fall into the Sun. Metal pollution of white dwarfs and transient spectroscopic signatures of young stars like β-Pic provide growing evidence that extra solar planetesimals can attain extreme orbital eccentricities and fall into their parent stars. Aims. We aim to develop a general, implementable, semi-analytical theory of secular eccentricity excitation of small bodies (planetesimals) in mean motion resonances with an eccentric planet valid for arbitrary values of the eccentricities and including the short-range force due to General Relativity. Methods. Our semi-analytic model for the restricted planar three-body problem does not make use of series expansion and therefore is valid for any eccentricity value and semi-major axis ratio. The model is based on the application of the adiabatic principle, which is valid when the precession period of the longitude of pericentre of the planetesimal is much longer than the libration period in the mean motion resonance. In resonances of order larger than 1 this is true except for vanishingly small eccentricities. We provide prospective users with a Mathematica notebook with implementation of the model allowing direct use. Results. We confirm that the 4:1 mean motion resonance with a moderately eccentric (e < ∼ 0.1) planet is the most powerful one to lift the eccentricity of planetesimals from nearly circular orbits to star-grazing ones. However, if the planet is too eccentric, we find that this resonance is unable to pump the planetesimal's eccentricity to a very high value. The inclusion of the General Relativity effect imposes a condition on the mass of the planet to drive the planetesimals into star-grazing orbits. For a planetesimal at ∼1 AU around a solar mass star (or white dwarf), we find a threshold planetary mass of about 17 Earth masses. We finally derive an analytical formula for this critical mass. Conclusions. Planetesimals can easily fall into the central star even in the presence of a single moderately eccentric planet, but only from the vicinity of the 4:1 mean motion resonance. For sufficiently high planetary masses the General Relativity effect does not prevent the achievement of star-grazing orbits.

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“…which is the adiabatic invariant of the dynamics (Henrard 1993). We now explain this procedure in more detail.…”

Section: Studying the Averaged Hamiltonianmentioning

confidence: 99%

Extreme secular excitation of eccentricity inside mean motion resonance

Pichierri

Morbidelli

Lai

2017

A&A

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“…This was generalized by Neishtadt (1986) and Tennyson et al (1986). Henrard (1993) summarized, justified, and improved various estimates and illustrated many significant applications. Haberman (1990,1994) showed that the nonlinear oscillations obtained by averaging and the nearly homoclinic orbits could be connected by the method of matched asymptotic expansions.…”

Section: Introductionmentioning

confidence: 94%

“…Examples include double-well potentials and the strongly nonlinear Hamiltonian system describing generic weakly nonlinear 1-1 primary resonance [see, for example, Guckenheimer and Holmes (1983) and Kevorkian and Cole (1981)]. Other examples are described by Neishtadt (1991) and Henrard (1993). With a small dissipative perturbation, the method of averaging is used to describe the long time behavior of weakly damped strongly nonlinear oscillations.…”

Section: Introductionmentioning

confidence: 99%

Higher-order change of the Hamiltonian (energy) for nearly homoclinic orbits

Haberman¹,

Ho²

1994

Methods and Applications of Analysis

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ABSTRACT. Conservative dynamical systems with a homoclinic orbit are analyzed under a small dissipative perturbation. The asymptotic expansions of a nearly homoclinic orbit are shown to fail as the surrounding saddle regions are approached. A weakly nonlinear analysis of the surrounding saddle regions is derived assuming the dissipation can be approximated by linear damping in the neighborhood of the saddle point. By matching (to sufficiently high order) the asymptotic expansion of the nearly homoclinic orbit to the solutions in the surrounding saddle regions, the change in the Hamiltonian (the energy dissipation) over a complete nearly homoclinic orbit is obtained. We show there is a new higher-order logarithmic correction to the well-known Melnikov integral for the change of the Hamiltonian from one saddle approach to the next. Numerical computations of this change in the Hamiltonian for a double-well potential are shown to compare favorably with the asymptotic result.

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“…As shown in [18], in case of positive circulation the spin slows approaching the resonance; in the librational regime the trajectory tends to the exact resonance; for negative circulation there are two possible behaviors: if 8 √ α > 2π γ β the guiding trajectory tends to the resonance, while if 8 √ α < 2π γ β the motion can evolve toward an invariant curve attractor. To provide a concrete example, let us follow the trajectory with initial conditions x = 0, y = −0.2; the set of parameters (α, β, γ) = (0.0061, 0.01, 0.001) satisfies the condition 8 √ α < 2π γ β and the corresponding dynamics is attracted by an invariant curve (see Figure 7, left panel); on the contrary, such condition is not fulfilled by (α, β, γ) = (0.0063, 0.01, 0.001) and consistently we find that the corresponding trajectory is attracted by a resonance as shown in Figure 7, right panel.…”

Section: Adiabatic Invariants Of the Pendulummentioning

confidence: 97%

“…According to [18] the adiabatic invariant Y ≡ 1 2π y dx slowly changes for a small variation of the dissipation factor β according to Y (t) = e −βt Y (0). The phase-space area enclosed by a guiding trajectory is provided by the formula…”

Section: Adiabatic Invariants Of the Pendulummentioning

confidence: 99%

Weakly Dissipative Systems in Celestial Mechanics

Celletti

Lecture Notes in Physics

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Summary. We investigate the dynamics associated to nearly-integrable dissipative systems, with particular reference to some models of Celestial Mechanics which can be described in a weakly dissipative framework. We start by studying some paradigmatic models provided by the dissipative standard maps in 2 and 4 dimensions. The dynamical investigation is performed applying frequency analysis and computing the differential fast Lyapunov indicators. After recalling a few properties of adiabatic invariants, we provide some examples of nearly-integrable dissipative systems borrowed from Celestial Mechanics, and precisely the spin-orbit coupling and the 3-body problem. We conclude with a discussion on the existence of periodic orbits in dissipative autonomous and non-autonomous systems.

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The Adiabatic Invariant in Classical Mechanics

Cited by 97 publications

References 83 publications

Extreme secular excitation of eccentricity inside mean motion resonance

Extreme secular excitation of eccentricity inside mean motion resonance

Higher-order change of the Hamiltonian (energy) for nearly homoclinic orbits

Weakly Dissipative Systems in Celestial Mechanics

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