Stability of resonance rotation of a satellite with respect to its center of mass in the orbit plane

Sadov, Sergey

doi:10.1134/s0010952506020080

“…Since their introduction by Hansen (1885), these coefficients have been earnestly studied in the literature (see, e.g., Hagihara 1970;Hughes 1981;Sadov 2008), and the general consensus holds that closed-form expressions for coefficients with j ≠ 0 do not exist. Nevertheless, Sadov (2006) has demonstrated that in the double limit of e → 1 − and j → ∞ the specific coefficient -X j 3,2 (which Sadov 2006 calls a Chernousko function with index j − 2) approaches the asymptotic form…”

Section: Computation Of Hansen Coefficientsmentioning

confidence: 99%

The Stability Boundary of the Distant Scattered Disk

Batygin

¹

,

Mardling

²

,

Nesvorný

³

2021

ApJ

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The distant scattered disk is a vast population of trans-Neptunian minor bodies that orbit the Sun on highly elongated, long-period orbits. The orbital stability of scattered-disk objects (SDOs) is primarily controlled by a single parameter—their perihelion distance. While the existence of a perihelion boundary that separates chaotic and regular motion of long-period orbits is well established through numerical experiments, its theoretical basis as well as its semimajor axis dependence remain poorly understood. In this work, we outline an analytical model for the dynamics of distant trans-Neptunian objects and show that the orbital architecture of the scattered disk is shaped by an infinite chain of exterior 2:j resonances with Neptune. The widths of these resonances increase as the perihelion distance approaches Neptune’s semimajor axis, and their overlap drives chaotic motion. Within the context of this theoretical picture, we derive an analytic criterion for instability of long-period orbits, and demonstrate that rapid dynamical chaos ensues when the perihelion drops below a critical value, given by q crit = a N ln ( ( 24 2 / 5 ) ( m N / M ⊙ ) a / a N 5 / 2 ) 1 / 2 . This expression constitutes an analytic boundary between the “detached” and actively “scattering” subpopulations of distant trans-Neptunian minor bodies. Additionally, we find that within the stochastic layer, the Lyapunov time of SDOs approaches the orbital period, and show that the semimajor axis diffusion coefficient is approximated by  a ∼ ( 8 / ( 5 π ) ) ( m N / M ⊙ )  M ⊙ a N exp − q / a N 2 / 2 . We confirm our results with direct N-body simulations and highlight the connections between scattered-disk dynamics and the Chirikov Standard Map. Implications of our results for the long-term evolution of minor bodies in the distant solar system are discussed.

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“…Since their introduction by Hansen (1885), these coefficients have been earnestly studied in the literature (see e.g., Hagihara 1970;Hughes 1981;Sadov 2008), and the general consensus holds that closed-form expressions for coefficients with j = 0 do not exist. Nevertheless, Sadov (2006) has demonstrated that in the double limit of e → 1 − and j → ∞ the specific coefficient X −3,2 j (which Sadov 2006 calls a Chernousko function with index j − 2) approaches the asymptotic form:…”

Section: Computation Of Hansen Coefficientsmentioning

confidence: 99%

The Stability Boundary of the Distant Scattered Disk

Batygin,

Mardling,

Nesvorny

2021

Preprint

0

View full text Add to dashboard Cite

The distant scattered disk is a vast population of trans-Neptunian minor bodies that orbit the sun on highly elongated, long-period orbits. The orbital stability of scattered disk objects is primarily controlled by a single parameter -their perihelion distance. While the existence of a perihelion boundary that separates chaotic and regular motion of long-period orbits is well established through numerical experiments, its theoretical basis as well as its semi-major axis dependence remain poorly understood. In this work, we outline an analytical model for the dynamics of distant trans-Neptunian objects and show that the orbital architecture of the scattered disk is shaped by an infinite chain of exterior 2 : j resonances with Neptune. The widths of these resonances increase as the perihelion distance approaches Neptune's semi-major axis, and their overlap drives chaotic motion. Within the context of this theoretical picture, we derive an analytic criterion for instability of long-period orbits, and demonstrate that rapid dynamical chaos ensues when the perihelion drops below a critical value, given by q crit = a N ln((24 2 /5) (m N /M ) (a/a N ) 5/2 ) 1/2 . This expression constitutes an analytic boundary between the "detached" and actively "scattering" sub-populations of distant trans-Neptunian minor bodies. Additionally, we find that within the stochastic layer, the Lyapunov time of scattered disk objects approaches the orbital period, and show that the semi-major axis diffusion coefficient is approximated by D a ∼ (8/(5 π)) (m N /M ) G M a N exp −(q/a N ) 2 /2 . We confirm our results with direct N −body simulations and highlight the connections between scattered disk dynamics and the Chirikov Standard Map. Implications of our results for the long-term evolution of minor bodies in the distant solar system are discussed.

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“…This discovery leads to the numerous studies of the spin-orbital resonance dynamics [2]. The studies of resonances occurring in a planar motion of a satellite (a rigid body) in an elliptic orbit are described in [3][4][5][6][7].…”

Section: Introductionmentioning

confidence: 99%

Mathematical Model of Satellite Rotation near Spin-Orbit Resonance 3:2

Shatina¹,

Djioeva²,

Osipova³

2022

Nelin. Dinam.

0

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This paper considers the rotational motion of a satellite equipped with flexible viscoelastic rods in an elliptic orbit. The satellite is modeled as a symmetric rigid body with a pair of flexible viscoelastic rods rigidly attached to it along the axis of symmetry. A planar case is studied, i. e., it is assumed that the satellite’s center of mass moves in a Keplerian elliptic orbit lying in a stationary plane and the satellite’s axis of rotation is orthogonal to this plane. When the rods are not deformed, the satellite’s principal central moments of inertia are equal to each other. The linear bending theory for thin inextensible rods is used to describe the deformations. The functionals of elastic and dissipative forces are introduced according to this model. The asymptotic method of motions separation is used to derive the equations of rotational motion reflecting the influence of the fluctuations, caused by the deformations of the rods. The method of motion separation is based on the assumption that the period of the autonomous oscillations of a point belonging to the rod is much smaller than the characteristic time of these oscillations’ decay, which, in its turn, is much smaller than the characteristic time of the system’s motion as a whole. That is why only the oscillations induced by the external and inertial forces are taken into account when deriving the equations of the rotational motion. The perturbed equations are described by a third-order system of ordinary differential equations in the dimensionless variable equal to the ratio of the satellite’s absolute value of angular velocity to the mean motion of the satellite’s center of mass, the angle between the satellite’s axis of symmetry and a fixed axis and the mean anomaly. The right-hand sides of the equation depend on the mean anomaly implicitly through the true anomaly. A new slow angular variable is introduced in order to perform the averaging for the perturbed system near the 3:2 resonance, and the averaging is

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Stability of resonance rotation of a satellite with respect to its center of mass in the orbit plane

Cited by 7 publications

References 7 publications

The Stability Boundary of the Distant Scattered Disk

The Stability Boundary of the Distant Scattered Disk

The Stability Boundary of the Distant Scattered Disk

Mathematical Model of Satellite Rotation near Spin-Orbit Resonance 3:2

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