Higher order resonance stability of triangular libration points for radiating primaries in ER3BP

The main aim of this paper is to study the existence of resonance and stability of the triangular equilibrium points in the framework of ER3BP when both the attracting bodies are sources of radiation at w1=w2, w1=2w2, w1=3w2 in both circular and elliptical cases .A practical application of this model could be seen in the case of binary systems ( Achird, Luyten, α Cen- AB, Kruger 60, Xi Bootis). The study is carried out both analytically and numerically by considering various values of radiation pressures and around binary systems .In both cases (CR3BP and ER3BP) it is found that w1=w2 corresponds to the boundary region of the stability for the system, whereas the other two cases w1=2w2, w1=3w2 correspond to the resonant cases. In order to investigate the stability, the Hamiltonian is normalized up to the fourth order by using linear canonical transformation of variables. Then KAM theorem is applied to investigate the stability for different values of radiation pressures in general and around the binary systems in particular. Finally, simulation technique is applied to study the correlation between radiation pressures and mass ratio in circular case; mass ratio and eccentricity in elliptical case. It is found that all the binary systems considered are stable. Also, it is found that except for some values of the radiation pressure parameters and for m<=mc =0.0385209 the triangular equilibrium points are stable.

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Analysis on nonlinear stability of the triangular libration points for radiating and oblate primaries in ER3BP

International Journal of Advanced Astronomy ◽

10.14419/ijaa.v5i1.7104 ◽

2017 ◽

Vol 5 (1) ◽

pp. 50

Author(s):

Nutan Singh ◽

A. Narayan

Keyword(s):

Resonance Frequency ◽

Linear Stability ◽

Radiation Pressure ◽

Nonlinear Stability ◽

Binary Systems ◽

Equilibrium Points ◽

Boundary Region ◽

Resonance Case ◽

Triangular Libration Points ◽

The Stability

In this paper we study the non linear stability of the triangular librations points in ER3BP considering both the primaries as radiating and oblate. The study is carried out near the resonance frequency satisfying the conditions in resonance as well as non resonance case. The study is conducted for various values of radiation pressure and oblateness parameters. It is observed that the case corresponds to the boundary region of the stability for the system Further, it is examined that the system experiences resonance at for different values of radiation pressures and oblateness parameter. In non resonance case, it is observed that the equilibrium points are stable. In resonance case, for and the triangular equilibrium points are unstable. In case, when for some values of radiation pressure and oblateness parameter, it is stable and for some it is unstable. The model is best suited to the binary systems (Achird, Luyten, α Cen AB, Kruger- 60, Xi- Bootis).

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Nonlinear Stability of the Triangular Libration Points for Radiating and Oblate Primaries in CR3BP in Nonresonance Condition

Advances in Astronomy ◽

10.1155/2015/850252 ◽

2015 ◽

Vol 2015 ◽

pp. 1-14 ◽

Cited By ~ 2

Author(s):

Nutan Singh ◽

A. Narayan

Keyword(s):

Resonance Frequency ◽

Radiation Pressure ◽

Nonlinear Stability ◽

Equilibrium Points ◽

Kam Theory ◽

Boundary Region ◽

Libration Points ◽

Triangular Libration Points ◽

Nonresonance Condition ◽

The Stability

This paper investigates the existence of resonance and nonlinear stability of the triangular equilibrium points when both oblate primaries are luminous. The study is carried out near the resonance frequency, satisfying the conditionsω1=ω2, ω1=2ω2, andω1=3ω2in circular cases by the application of Kolmogorov-Arnold-Moser (KAM) theory. The study is carried out for the various values of radiation pressure and oblateness parameters in general. It is noticed that the system experiences resonance atω1=2ω2, ω1=3ω2for different values of radiation pressures and oblateness parameter. The caseω1=ω2corresponds to the boundary region of the stability for the system. It is found that, except for some values of the radiation pressure, and oblateness parameters and forμ≤μc=0.0385209, the triangular equilibrium points are stable.

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Stability analysis of an interacting holographic dark energy model

Modern Physics Letters A ◽

10.1142/s0217732319501475 ◽

2019 ◽

Vol 34 (19) ◽

pp. 1950147

Author(s):

Sudip Mishra ◽

Subenoy Chakraborty

Keyword(s):

Dark Energy ◽

System Analysis ◽

Holographic Dark Energy ◽

Equilibrium Points ◽

Einstein Field Equations ◽

Field Equations ◽

Lyapunov Function Method ◽

Transformation Of Variables ◽

The Stability ◽

Manifold Theory

This work deals with dynamical system analysis of Holographic Dark Energy (HDE) cosmological model with different infra-red (IR)-cutoff. By suitable transformation of variables, the Einstein field equations are converted to an autonomous system. The critical points are determined and the stability of the equilibrium points are examined by Center Manifold Theory and Lyapunov function method. Possible bifurcation scenarios have also been explained.

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Modeling the evection resonance for Trojan satellites: application to the Saturn system

Astronomy and Astrophysics ◽

10.1051/0004-6361/201833735 ◽

2018 ◽

Vol 620 ◽

pp. A90 ◽

Cited By ~ 1

Author(s):

C. A. Giuppone ◽

F. Roig ◽

X. Saad-Olivera

Keyword(s):

Phase Space ◽

Equilibrium Points ◽

Dissipation Factor ◽

The Other ◽

Orbital Dynamics ◽

Mass Ratio ◽

Tidal Dissipation ◽

The Past ◽

The Stability ◽

Averaged Hamiltonian

Context. The stability of satellites in the solar system is affected by the so-called evection resonance. The moons of Saturn, in particular, exhibit a complex dynamical architecture in which co-orbital configurations occur, especially close to the planet where this resonance is present. Aims. We address the dynamics of the evection resonance, with particular focus on the Saturn system, and compare the known behavior of the resonance for a single moon with that of a pair of moons in co-orbital Trojan configuration. Methods. We developed an analytic expansion of the averaged Hamiltonian of a Trojan pair of bodies, including the perturbation from a distant massive body. The analysis of the corresponding equilibrium points was restricted to the asymmetric apsidal corotation solution of the co-orbital dynamics. We also performed numerical N-body simulations to construct dynamical maps of the stability of the evection resonance in the Saturn system, and to study the effects of this resonance under the migration of Trojan moons caused by tidal dissipation. Results. The structure of the phase space of the evection resonance for Trojan satellites is similar to that of a single satellite, differing in that the libration centers are displaced from their standard positions by an angle that depends on the periastron difference ϖ2 −ϖ1 and on the mass ratio m2∕m1 of the Trojan pair. In the Saturn system, the inner evection resonance, located at ~8 RS, may capture a pair of Trojan moons by migration; the stability of the captured system depends on the assumed values of the dissipation factor Q of the moons. On the other hand, the outer evection resonance, located at >0.4 RHill, cannot exist at all for Trojan moons, because Trojan configurations are strongly unstable at distances from Saturn longer than ~0.15 RHill. Conclusions. The interaction with the inner evection resonance may have been relevant during the early evolution of the Saturn moons Tethys, Dione, and Rhea. In particular, Rhea may have had Trojan companions in the past that were lost when it crossed the evection resonance, while Tethys and Dione may either have retained their Trojans or have never crossed the evection. This may help to constrain the dynamical processes that led to the migration of these satellites and to the evection itself.

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Pulsating Different Curves of Zero Velocity around Triangular Equilibrium Points in Elliptical Restricted Three-Body Problem

Journal of Mathematics ◽

10.1155/2013/936859 ◽

2013 ◽

Vol 2013 ◽

pp. 1-8 ◽

Cited By ~ 4

Author(s):

A. Narayan ◽

Amit Shrivastava

Keyword(s):

Binary Systems ◽

Body Problem ◽

Equilibrium Points ◽

Restricted Three Body Problem ◽

Three Body Problem ◽

Triangular Points ◽

Zero Velocity ◽

Simulation Techniques ◽

The Stability ◽

Three Body

The oblateness and the photogravitational effects of both the primaries on the location and the stability of the triangular equilibrium points in the elliptical restricted three-body problem have been discussed. The stability of the triangular points under the photogravitational and oblateness effects of both the primaries around the binary systems Achird, Lyeten, Alpha Cen-AB, Kruger 60, and Xi-Bootis, has been studied using simulation techniques by drawing different curves of zero velocity.

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Generalized Out-of-Plane Equilibrium Points in the Frame of Elliptic Restricted Three-Body Problem: Impact of Oblate Primary and Luminous-Triaxial Secondary

Advances in Astronomy ◽

10.1155/2019/3278946 ◽

2019 ◽

Vol 2019 ◽

pp. 1-12

Author(s):

Aminu Abubakar Hussain ◽

Aishetu Umar

Keyword(s):

Binary Systems ◽

Equilibrium Points ◽

Semimajor Axis ◽

Three Body Problem ◽

Out Of Plane ◽

Elliptic Restricted Problem ◽

The Stability ◽

Zero Velocity Curves ◽

Three Body ◽

Plane Equilibrium

This paper studies the motion of a third body near the 1st family of the out-of-plane equilibrium points, L6,7, in the elliptic restricted problem of three bodies under an oblate primary and a radiating-triaxial secondary. It is seen that the pair of points (ξ0,0,±ζ0) which correspond to the positions of the 1st family of the out-of-plane equilibrium points, L6,7, are affected by the oblateness of the primary, radiation pressure and triaxiality of the secondary, semimajor axis, and eccentricity of the orbits of the principal bodies. But the point ±ζ0 is unaffected by the semimajor axis and eccentricity of the orbits of the principal bodies. The effects of the parameters involved in this problem are shown on the topologies of the zero-velocity curves for the binary systems PSR 1903+0327 and DP-Leonis. An investigation of the stability of the out-of-plane equilibrium points, L6,7 numerically, shows that they can be stable for 0.32≤μ≤0.5 and for very low eccentricity. L6,7 of PSR 1903+0327 and DP-Leonis are however linearly unstable.

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Maximum mass ratio of am CVn-type binary systems and maximum white dwarf mass in ultra-compact x-ray binaries (addendum - Serb. Astron. J. No. 183 (2011), 63)

Serbian Astronomical Journal ◽

10.2298/saj1284105a ◽

2012 ◽

pp. 105-107

Author(s):

B. Arbutina

Keyword(s):

Mass Transfer ◽

White Dwarf ◽

Binary Systems ◽

Accretion Rate ◽

Practical Relevance ◽

Maximum Mass ◽

Mass Ratio ◽

X Ray ◽

The Stability ◽

X Ray Binaries

We recalculated the maximum white dwarf mass in ultra-compact X-ray binaries obtained in an earlier paper (Arbutina 2011), by taking the effects of super-Eddington accretion rate on the stability of mass transfer into account. It is found that, although the value formally remains the same (under the assumed approximations), for white dwarf masses M2 >~0.1MCh mass ratios are extremely low, implying that the result for Mmax is likely to have little if any practical relevance.

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Non- linear stability of triangular librations points in circular restricted three body under radiating and oblate primaries in presence of resonance

International Journal of Advanced Astronomy ◽

10.14419/ijaa.v3i2.4772 ◽

2015 ◽

Vol 3 (2) ◽

pp. 58

Author(s):

Ashutosh Narayan ◽

Nutan Singh

Keyword(s):

Radiation Pressure ◽

Fourth Order ◽

Nonlinear Stability ◽

Binary Systems ◽

Normal Forms ◽

Boundary Region ◽

Third Order ◽

Order Resonance ◽

The Stability ◽

Three Body

The nonlinear stability of the triangular librations points is studied in the presence resonance considering both the primaries as radiating and oblate. The study is carried out for various values of radiation pressure and oblateness parameter in general and binary systems in particular. It is found that the normal forms of the Hamiltonian contains both the resonance cases; ω1= 2ω2 and ω1= 3ω2. The case ω1= ω2 corresponds to the boundary region of the stability for the system.It is investigated that for the motion is unstable for third order resonance but stable for fourth order resonance.

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Perturbed locations and linear stability of collinear Lagrangian points in the elliptic restricted three body problem with triaxial primaries

Open Astronomy ◽

10.1515/astro-2019-0013 ◽

2019 ◽

Vol 28 (1) ◽

pp. 145-153

Author(s):

Walid Ali Rahoma ◽

Akram Masoud ◽

Fawzy Ahmed Abd El-Salam ◽

Elamira Hend Khattab

Keyword(s):

Linear Stability ◽

Body Problem ◽

Equilibrium Points ◽

Classical Case ◽

Restricted Three Body Problem ◽

Mass Ratio ◽

Three Body Problem ◽

The Difference ◽

The Stability ◽

Three Body

Abstract This paper aims to study the effect of the triaxiality and the oblateness as a special case of primaries on the locations and stability of the collinear equilibrium points of the elliptic restricted three body problem (in brief ERTBP). The locations of the perturbed collinear equilibrium points are first determined in terms of mass ratio of the problem (the smallest mass divided by the total mass of the system) and different concerned perturbing factors. The difference between the locations of collinear points in the classical case of circular restricted three body problem and those in the perturbed case is represented versus mass ratio over its range. The linear stability of the collinear points is discussed. It is observed that the stability regions for our model depend mainly on the eccentricity of the orbits in addition to the considered perturbations.

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Trajectories of the infinitesimal mass around the triangular equilibrium points in elliptical restricted three bodies problem under oblate and radiating primaries for the binary systems

International Journal of Advanced Astronomy ◽

10.14419/ijaa.v3i2.5304 ◽

2015 ◽

Vol 3 (2) ◽

pp. 107

Author(s):

Nutan Singh ◽

Ashutosh Narayan ◽

B. Ishwar

Keyword(s):

Radiation Pressure ◽

Binary Systems ◽

Equilibrium Points ◽

Simulation Technique ◽

Triangular Points ◽

The Stability ◽

Infinitesimal Mass

This paper describes the trajectory of the infinitesimal mass around L4 of the triangular equilibrium points for the binary systems in the elliptical restricted three body’s problem (ERTBP), where both oblate primaries are radiating. The solutions for the perturbed motion in the vicinity of L4 is given by u(f) and v(f) function .The stability of the infinitesimal mass around the triangular points is also studied by plotting u(f) and v(f) curve. It is found that radiation pressure, oblateness and eccentricity show a significant effect on the trajectory and stability of the infinitesimal mass around the triangular equilibrium points. Simulation technique has been used to design the trajectory of the binary systems (Achird, Luyten, α Cen AB, Kruger-60 and Xi-Bootis).

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