The Motion of Out-of-Plane Equilibrium Points in the Elliptic Restricted Three-Body Problem at J4

We have investigated the motion of the out-of-plane equilibrium points within the framework of the Elliptic Restricted Three-Body Problem (ER3BP) at J4 of the smaller primary in the field of stellar binary systems: Xi- Bootis and Sirius around their common center of mass in elliptic orbits. The positions and stability of the out-of-plane equilibrium points are greatly affected on the premise of the oblateness at J4 of the smaller primary, semi-major axis and the eccentricity of their orbits. The positions L6, 7 of the infinitesimal body lie in the xz-plane almost directly above and below the center of each oblate primary. Numerically, we have computed the positions and stability of L6, 7 for the aforementioned binary systems and found that their positions are affected by the oblateness of the primaries, the semi-major axis and eccentricity of their orbits. It is observed that, for each set of values, there exist at least one complex root with positive real part and hence in Lyapunov sense, the stability of the out-of-plane equilibrium points are unstable.

On the Dependence of the Relativistic Angular Momentum of a Uniform Ball on the Radius and Angular Velocity of Rotation

In the framework of the special theory of relativity, elementary formulas are used to derive the formula for determining the relativistic angular momentum of a rotating ideal uniform ball. The moment of inertia of such a ball turns out to be a nonlinear function of the angular velocity of rotation. Application of this formula to the neutron star PSR J1614-2230 shows that due to relativistic corrections the angular momentum of the star increases tenfold as compared to the nonrelativistic formula. For the proton and neutron star PSR J1748-2446ad the velocities of their surface’s motion are calculated, which reach the values of the order of 30% and 19% of the speed of light, respectively. Using the formula for the relativistic angular momentum of a uniform ball, it is easy to obtain the formula for the angular momentum of a thin spherical shell depending on its thickness, radius, mass density, and angular velocity of rotation. As a result, considering a spherical body consisting of a set of such shells it becomes possible to accurately determine its angular momentum as the sum of the angular momenta of all the body’s shells. Two expressions are provided for the maximum possible angular momentum of the ball based on the rotation of the ball’s surface at the speed of light and based on the condition of integrity of the gravitationally bound body at the balance of the gravitational and centripetal forces. Comparison with the results of the general theory of relativity shows the difference in angular momentum of the order of 25% for an extremal Kerr black hole.

The Unsteady Flow of a Fluid of Finite Depth with an Oscillating Bottom

In this paper, the unsteady flow of a fluid of finite depth with an oscillating bottom is examined. The flow is assumed in the absence of viscous dissipation. The governing equations of the flow are decoupled in the velocity and temperature fields. The velocity and temperature fields have been obtained analytically. The effects of various material parameters on these fields have been discussed with the help of graphical illustrations. It is noticed that the upward thrust (ρfy) vanishes when Reiner Rivlin coefficient of viscosity (μc) is zero and the transverse force (ρfz) perpendicular to the flow direction vanishes for thermo-viscosity coefficient (α8) is zero. The external forces generated perpendicular to the flow direction is a special feature of thermo-viscous fluid when compared to the other type of fluids.

On Solution of Singular Integral Equations by Operator Method

In this paper, we study the exact solution of singular integral equations using two methods, including Adomian Decomposition Method and Elzaki Transform Method. We propose an analytical method for solving singular integral equations and system of singular integral equations, and have some goals in our paper related to suggested technique for solving singular integral equations. The primary goal is for giving analytical solutions of such equations with simple steps, another goal is to compare the suggested method with other methods used in this study.

The Generalized Poynting Theorem for the General Field and Solution of the 4/3 Problem

The generalized Poynting theorem is applied to the equilibrium system of particles, both inside and outside the system. The particles are bound to each other by means of the electromagnetic and gravitational fields, acceleration field and pressure field. As a result, the correlation is found between the acceleration field coefficient, the pressure field coefficient, the gravitational constant and the vacuum permittivity. This correlation also depends on the ratio of the charge density to the mass density of the particles inside the sphere. Due to the correlation between the given field coefficients the 4/3 problem is solved and the expression for the relativistic energy of the system is refined.

Fixed Points and Stability Analysis in the Motion of Human Heart Valve Leaflet

This work was set out to gain further insight into the kinetics of the human heart valve leaflet. The Korakianitis and Shi lumped parameter model was adopted for this study. The fixed points were determined, and then, their stability properties were assessed by evaluating eigenvalues of the Jacobian matrices. Normal physiological parameters for the valve model were simulated; based on which, a local bifurcation diagram was generated. Phase portraits were plotted from simulated responses, and were used to observe the qualitative properties of the valve leaflet motion. The evaluated fixed points were found to be dependent on pressure and flow effects, and independent on friction or damping effect. Observed switching of stability between the two fixed points indicated that the leaflet motion undergoes transcritical bifurcation. Of the two fixed points, one is always either a stable spiral or generative node while the other is a saddle. Numerical simulations were carried out to verify the analytical solutions.

Investigation of the Stability of a Test Particle in the Vicinity of Collinear Points with the Additional Influence of an Oblate Primary and a Triaxial-Stellar Companion in the Frame of ER3BP

We investigate in the elliptic framework of the restricted three-body problem, the motion around the collinear points of an infinitesimal particle in the vicinity of an oblate primary and a triaxial stellar companion. The locations of the collinear points are affected by the eccentricity of the orbits, oblateness of the primary body and the triaxiality and luminosity of the secondary. A numerical analysis of the effects of the parameters on the positions of collinear points of CEN X-4 and PSR J1903+0327 reveals a general shift away from the smaller primary with increase in eccentricity and triaxiality factors and a shift towards the smaller primary with increase in the semi-major axis and oblateness of the primary on L1. The collinear points remain unstable in spite of the introduction of these parameters.

On Motion around the Collinear Equilibrium Points in the Relativistic R3BP with a Smaller Triaxial Primary

This paper studies the motion of an infinitesimal mass near the collinear equilibrium points in the framework of relativistic restricted three-body problem (R3BP) when the smaller primary is a triaxial body. It is observed that the positions of the collinear points are affected by the relativistic and triaxiality factors. The collinear points are found to remain unstable. Numerical studies in this connection, with the Sun-Earth, Sun-Pluto and Earth-Moon systems have been carried out to show the relativistic and triaxiality effects.

Stability Analysis of a Biological Model of a Marine Resources Allowing Density Dependent Migration

Biology of a marine resources is a descriptive science. The description is the first step towards understanding a system. However, the main objective is to present a rigorous mathematical analysis and numerical simulation of these spatio temporal models. In the present paper, we consider a two species food chain, i.e. a prey and predator populations modeled in a two-patch environment, one of which is a free fishing zone and the other one is protected zone. We study the qualitative analysis of solutions and we establish sufficient conditions under which the endemic and trivial equilibria are asymptotically stable.The asymptotic stability corresponding to the equilibria is graphically shown.

Motion around the Triangular Equilibrium Points in the Circular Restricted Three-Body Problem under Triaxial Luminous Primaries with Poynting-Robertson Drag

This paper explores the motion of an infinitesimal body around the triangular equilibrium points in the framework of circular restricted three-body problem (CR3BP) with the postulation that the primaries are triaxial rigid bodies, radiating in nature and are also under the influence of Poynting–Robertson (P-R) drag. We study the linear stability of these triangular points and for the numerical application, the binary stars Kruger 60 (AB) and Archird have been considered. These triangular points are not only perceived to move towards the line joining the primaries in the direction of the bigger primary with increasing triaxiality, they are also unstable owing to the destabilizing influence of P-R drag.