scholarly journals Systematic survey of the dynamics of Uranus Trojans

2020 ◽  
Vol 633 ◽  
pp. A153 ◽  
Author(s):  
Lei Zhou ◽  
Li-Yong Zhou ◽  
Rudolf Dvorak ◽  
Jian Li
Keyword(s):  
Solar System ◽  
Stability Region ◽  
Large Fraction ◽  
Instability Strip ◽  
Stability Regions ◽  
Lagrange Points ◽  
Secular Resonances ◽  
Secondary Resonances ◽  
The Stability ◽  
Planetary Migration

Context. The discovered Uranus Trojan (UT) 2011 QF99 and several candidate UTs have been reported to be in unstable orbits. This implies that the stability region around the triangular Lagrange points L4 and L5 of Uranus should be very limited. Aims. In this paper, we aim to locate the stability region for UTs and find out the dynamical mechanisms responsible for the structures in the phase space. The null detection of primordial UTs also needs to be explained. Methods. Using the spectral number as the stability indicator, we constructed the dynamical maps on the (a0, i0) plane. The proper frequencies of UTs were determined precisely with a frequency analysis method that allows us to depict the resonance web via a semi-analytical method. We simulated radial migration by introducing an artificial force acting on planets to mimic the capture of UTs. Results. We find two main stability regions: a low-inclination (0° −14°) and a high-inclination regime (32° −59°). There is also an instability strip in each of these regions at 9° and 51°, respectively. These strips are supposed to be related with g − 2g5 + g7 = 0 and ν8 secular resonances. All stability regions are in the tadpole regime and no stable horseshoe orbits exist for UTs. The lack of moderate-inclined UTs is caused by the ν5 and ν7 secular resonances, which could excite the eccentricity of orbits. The fine structures in the dynamical maps are shaped by high-degree secular resonances and secondary resonances. Surprisingly, the libration centre of UTs changes with the initial inclination, and we prove it is related to the quasi 1:2 mean motion resonance (MMR) between Uranus and Neptune. However, this quasi-resonance has an ignorable influence on the long-term stability of UTs in the current planetary configuration. About 36.3% and 0.4% of the pre-formed orbits survive fast and slow migrations with migrating timescales of 1 and 10 Myr, respectively, most of which are in high inclination. Since low-inclined UTs are more likely to survive the age of the solar system, they make up 77% of all such long-life orbits by the end of the migration, making a total fraction up to 4.06 × 10−3 and 9.07 × 10−5 of the original population for fast and slow migrations, respectively. The chaotic capture, just like depletion, results from secondary resonances when Uranus and Neptune cross their mutual MMRs. However, the captured orbits are too hot to survive until today. Conclusions. About 3.81% UTs are able to survive the age of the solar system, among which 95.5% are on low-inclined orbits with i0 <  7.5°. However, the depletion of planetary migration seems to prevent a large fraction of such orbits, especially for the slow migration model. Based on the widely adopted migration models, a swarm of UTs at the beginning of the smooth outward migration is expected and a fast migration is favoured if any primordial UTs are detected.

scholarly journals The Stability of the Planetary Triangular Lagrange Points

1990 ◽  
Vol 123 ◽  
pp. 533-536
Author(s):  
Seppo Mikkola ◽  
K.A. Innanen
Keyword(s):  
Solar System ◽  
Progress Report ◽  
Scientific Applications ◽  
Medium Term ◽  
Lagrange Points ◽  
The Earth ◽  
Self Consistent ◽  
The Stability

AbstractNumerical, self-consistent, n-body integrations of the solar system show significant indications of medium-term (i.e. several million-year) stability for the various planet-Sun L4,L5 configurations. A progress report of our computations, emphasizing the inner solar system, will be given. There exist interesting possibilities for these locations (including the Earth) as the sites for longer term scientific applications, both pure and applied.

scholarly journals Stability conditions for some distributed systems: buffered random access systems

1994 ◽  
Vol 26 (02) ◽  
pp. 498-515 ◽  
Author(s):  
Wojciech Szpankowski
Keyword(s):  
Stability Region ◽  
Sufficient Conditions ◽  
Random Access ◽  
Symmetric System ◽  
Slotted Aloha ◽  
Stability Regions ◽  
Novel Technique ◽  
Token Passing ◽  
The Stability ◽  
Necessary And Sufficient

We consider the standard slotted ALOHA system with a finite number of buffered users. Stability analysis of such a system was initiated by Tsybakov and Mikhailov (1979). Since then several bounds on the stability region have been established; however, the exact stability region is known only for the symmetric system and two-user ALOHA. This paper proves necessary and sufficient conditions for stability of the ALOHA system. We accomplish this by means of a novel technique based on three simple observations: applying mathematical induction to a smaller copy of the system, isolating a single queue for which Loynes' stability criteria is adopted, and finally using stochastic dominance to verify the required stationarity assumptions in the Loynes criterion. We also point out that our technique can be used to assess stability regions for other multidimensional systems. We illustrate it by deriving stability regions for buffered systems with conflict resolution algorithms (see also Georgiadis and Szpankowski (1992) for similar approach applied to stability of token-passing rings).

INVESTIGATION OF THE GEOMETRY OF THE D-PARTITION OF ONE-DIMENSIONAL PLANE OF PARAMETER OF THE CHARACTERISTIC EQUATION OF A CONTINUOUS SYSTEM

2021 ◽  
Vol 4 ◽  
pp. 125-136
Author(s):  
Leonid Movchan ◽  
◽  
Sergey Movchan ◽  
Keyword(s):  
Imaginary Part ◽  
Characteristic Equation ◽  
Stability Region ◽  
Practical Interest ◽  
Stability Regions ◽  
Odd Order ◽  
Stability Intervals ◽  
Even Order ◽  
The Stability ◽  
Two Parameters

The paper considers two types of boundaries of the D-partition in the plane of one parameter of linear continuous systems given by the characteristic equation with real coefficients. The number of segments and intervals of stability of the X-partition curve is estimated. The maximum number of stability intervals is determined for different orders of polynomials of the equation of the boundary of the D-partition of the first kind (even order, odd order, one of even order, and the other of odd order). It is proved that the maximum number of stability intervals of a one-parameter family is different for all cases and depends on the ratio of the degrees of the polynomials of the equation of the D-partition curve. The derivative of the imaginary part of the expression of the investigated parameter at the initial point of the D-partition curve is obtained in an analytical form, the sign of which depends on the ratio of the coefficients of the characteristic equation and establishes the stability of the first interval of the real axis of the parameter plane. It is shown that for another type of the boundary of the D-partition in the plane of one parameter, there is only one interval of stability, the location of which, as for the previous type of the boundary of the stability region (BSR), is determined by the sign of the first derivative of the imaginary part of the expression of the parameter under study. Consider an example that illustrates the effectiveness of the proposed approach for constructing a BSR in a space of two parameters without using «Neimark hatching» and constructing special lines. In this case, a machine implementation of the construction of the stability region is provided. Considering that the problem of constructing the boundary of the stability region in the plane of two parameters is reduced to the problem of determining the BSR in the plane of one parameter, then the given estimates of the maximum number of stability regions in the plane of one parameter allow us to conclude about the number of maximum stability regions in the plane of two parameters, which are of practical interest. In this case, one of the parameters can enter nonlinearly into the coefficients of the characteristic equation.

scholarly journals Convergence and Stability in Collocation Methods of Equationu′(t)=au(t)+bu([t])

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Han Yan ◽  
Shufang Ma ◽  
Yanbin Liu ◽  
Hongquan Sun
Keyword(s):  
Numerical Stability ◽  
Numerical Experiments ◽  
Stability Region ◽  
Collocation Methods ◽  
Stability Regions ◽  
Optimal Convergence ◽  
Analytic Stability ◽  
The Stability ◽  
Local Superconvergence ◽  
Global Superconvergence

This paper is concerned with the convergence, global superconvergence, local superconvergence, and stability of collocation methods foru′(t)=au(t)+bu([t]). The optimal convergence order and superconvergence order are obtained, and the stability regions for the collocation methods are determined. The conditions that the analytic stability region is contained in the numerical stability region are obtained, and some numerical experiments are given.

scholarly journals Orbital stability of Earth Trojans

2019 ◽  
Vol 622 ◽  
pp. A97 ◽  
Author(s):  
Lei Zhou ◽  
Yang-Bo Xu ◽  
Li-Yong Zhou ◽  
Rudolf Dvorak ◽  
Jian Li
Keyword(s):  
Orbital Stability ◽  
Dynamical Behaviour ◽  
Secular Resonance ◽  
Lagrange Points ◽  
Secular Resonances ◽  
Frequency Space ◽  
Yarkovsky Effect ◽  
Horseshoe Orbits ◽  
The Stability ◽  
Primordial Earth

The only discovery of Earth Trojan 2010 TK7 and the subsequent launch of OSIRIS-REx have motived us to investigate the stability around the triangular Lagrange points of the Earth, L4 and L5. In this paper we present detailed dynamical maps on the (a0, i0) plane with the spectral number (SN) indicating the stability. Two main stability regions, separated by a chaotic region arising from the ν3 and ν4 secular resonances, are found at low (i0 ≤ 15°) and moderate (24 ° ≤i0 ≤ 37°) inclinations, respectively. The most stable orbits reside below i0 = 10° and they can survive the age of the solar system. The nodal secular resonance ν13 could vary the inclinations from 0° to ∼10° according to their initial values, while ν14 could pump up the inclinations to ∼20° and upwards. The fine structures in the dynamical maps are related to higher degree secular resonances, of which different types dominate different areas. The dynamical behaviour of the tadpole and horseshoe orbits, reflected in their secular precession, show great differences in the frequency space. The secular resonances involving the tadpole orbits are more sensitive to the frequency drift of the inner planets, thus the instabilities could sweep across the phase space, leading to the clearance of tadpole orbits. We are more likely to find terrestrial companions on horseshoe orbits. The Yarkovsky effect could destabilize Earth Trojans in varying degrees. We numerically obtain the formula describing the stabilities affected by the Yarkovsky effect and find the asymmetry between the prograde and retrograde rotating Earth Trojans. The existence of small primordial Earth Trojans that avoid being detected but survive the Yarkovsky effect for 4.5 Gyr is substantially ruled out.

scholarly journals ORDER OF THE RUNGE-KUTTA METHOD AND EVOLUTION OF THE STABILITY REGION

2019 ◽  
Vol 5 (2) ◽  
pp. 64
Author(s):  
Hippolyte Séka ◽  
Kouassi Richard Assui
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Absolute Stability ◽  
Stability Region ◽  
Kutta Method ◽  
Stability Regions ◽  
Runge Kutta Method ◽  
The Absolute ◽  
Runge Kutta ◽  
The Stability

In this article, we demonstrate through specific examples that the evolution of the size of the absolute stability regions of Runge–Kutta methods for ordinary differential equation does not depend on the order of methods.

scholarly journals Stability conditions for some distributed systems: buffered random access systems

1994 ◽  
Vol 26 (2) ◽  
pp. 498-515 ◽  
Author(s):  
Wojciech Szpankowski
Keyword(s):  
Stability Region ◽  
Sufficient Conditions ◽  
Random Access ◽  
Symmetric System ◽  
Slotted Aloha ◽  
Stability Regions ◽  
Novel Technique ◽  
Token Passing ◽  
The Stability ◽  
Necessary And Sufficient

We consider the standard slotted ALOHA system with a finite number of buffered users. Stability analysis of such a system was initiated by Tsybakov and Mikhailov (1979). Since then several bounds on the stability region have been established; however, the exact stability region is known only for the symmetric system and two-user ALOHA. This paper proves necessary and sufficient conditions for stability of the ALOHA system. We accomplish this by means of a novel technique based on three simple observations: applying mathematical induction to a smaller copy of the system, isolating a single queue for which Loynes' stability criteria is adopted, and finally using stochastic dominance to verify the required stationarity assumptions in the Loynes criterion. We also point out that our technique can be used to assess stability regions for other multidimensional systems. We illustrate it by deriving stability regions for buffered systems with conflict resolution algorithms (see also Georgiadis and Szpankowski (1992) for similar approach applied to stability of token-passing rings).

scholarly journals Stability of Jovian Trojans and their collisional families

2020 ◽  
Vol 495 (4) ◽  
pp. 4085-4097 ◽  
Author(s):  
Timothy R Holt ◽  
David Nesvorný ◽  
Jonathan Horner ◽  
Rachel King ◽  
Raphael Marschall ◽  
...  
Keyword(s):  
Solar System ◽  
Age Estimation ◽  
Escape Rate ◽  
Linear Rate ◽  
Lagrange Points ◽  
Short Period Comet ◽  
Short Period ◽  
Population Sizes ◽  
The Stability ◽  
Period Comet

ABSTRACT The Jovian Trojans are two swarms of objects located around the L4 and L5 Lagrange points. The population is thought to have been captured by Jupiter during the Solar system’s youth. Within the swarms, six collisional families have been identified in previous work, with four in the L4 swarm, and two in the L5. Our aim is to investigate the stability of the two Trojan swarms, with a particular focus on these collisional families. We find that the members of Trojan swarms escape the population at a linear rate, with the primordial L4 (23.35 per cent escape) and L5 (24.89 per cent escape) population sizes likely 1.31 and 1.35 times larger than today. Given that the escape rates were approximately equal between the two Trojan swarms, our results do not explain the observed asymmetry between the two groups, suggesting that the numerical differences are primordial in nature, supporting previous studies. Upon leaving the Trojan population, the escaped objects move on to orbits that resemble those of the Centaur and short-period comet populations. Within the Trojan collisional families, the 1996 RJ and 2001 UV209 families are found to be dynamically stable over the lifetime of the Solar system, whilst the Hektor, Arkesilos and Ennomos families exhibit various degrees of instability. The larger Eurybates family shows 18.81 per cent of simulated members escaping the Trojan population. Unlike the L4 swarm, the escape rate from the Eurybates family is found to increase as a function of time, allowing an age estimation of approximately 1.045 ± 0.364 × 109 yr.

scholarly journals A New Eighth Order Runge-Kutta Family Method

2019 ◽  
Vol 11 (2) ◽  
pp. 190
Author(s):  
Séka Hippolyte ◽  
Assui Kouassi Richard
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Stability Region ◽  
Eighth Order ◽  
Stability Regions ◽  
New Family ◽  
Runge Kutta ◽  
The Stability

In this article, a new family of Runge-Kutta methods of 8^th order for solving ordinary differential equations is discovered and depends on the parameters b_8 and a_10;5. For b8 = 49/180 and a10;5 = 1/9, we find the Cooper-Verner method [1]. We show that the stability region depends only on coefficient a_10;5. We compare the stability regions according to the values of a_10;5 with respect to the stability region of Cooper-Verner.

The influence of the Stokes and Archimedes forces on the stability of a two-phase flow

2003 ◽  
Vol 3 ◽  
pp. 266-270
Author(s):  
B.H. Khudjuyerov ◽  
I.A. Chuliev
Keyword(s):  
Spectral Method ◽  
Stability Region ◽  
Gas Flow ◽  
Two Phase Flow ◽  
Solid Phase ◽  
Stability Characteristic ◽  
Phase Flow ◽  
Two Phase ◽  
The Stability

The problem of the stability of a two-phase flow is considered. The solution of the stability equations is performed by the spectral method using polynomials of Chebyshev. A decrease in the stability region gas flow with the addition of particles of the solid phase. The analysis influence on the stability characteristic of Stokes and Archimedes forces.

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