Critical Mass Parameter Research Articles

The Predictions of Noncollinear Equilibria Positions in ER3BP with Yukawa-Like Corrections

The existence and stability of noncollinear equilibrium points in the elliptic restricted three-body problem under the consideration of Yukawa correction to Newtonian potential are studied in this paper. The effects of various parameters (μ, ê, α, and λ) on the noncollinear equilibrium points are discussed briefly, and it is found that only ordinate of noncollinear equilibria E4,5 is affected by Yukawa correction while abscissa is affected by only mass parameter μ. The noncollinear equilibria was found linearly stable for a critical mass parameter μc. A critical point λ = ½ is also obtained for the critical mass parameter μc, and at this point, the critical mass parameter μc has maximum or minimum values according to α < 0 or α > 0, respectively.

Noncollinear Equilibrium Points in CRTBP with Yukawa-Like Corrections to Newtonian Potential under an Oblate Primary Model

This study is about the effects of Yukawa-like corrections to Newtonian potential on the existence and stability of noncollinear equilibrium points in a circular restricted three-body problem when bigger primary is an oblate spheroid. It is observed that ∂x0/∂λ = 0 = ∂y0/∂λ at λ0 = 1/2, so we have a critical point λ0 = 1/2 at which the maximum and minimum values of x0 and y0 can be obtained, where λ ∈ (0, ∞) is the range of Yukawa force and (x0, y0) are the coordinates of noncollinear equilibrium points. It is found that x0 and y0 are increasing functions in λ in the interval 0 < λ < λ0 and decreasing functions in λ in the interval λ0 < λ < ∞ for all α ∈ α+. On the other hand, x0 and y0 are decreasing functions in λ in the interval 0 < λ < λ0 and increasing functions in λ in the interval λ0 < λ < ∞ for all α ∈ α−, where α ∈ (−1, 1) is the coupling constant of Yukawa force to gravitational force. The noncollinear equilibrium points are found linearly stable for the critical mass parameter β0, and it is noticed that ∂β0/∂λ = 0 at λ ∗ = 1/3; thus, we got another critical point which gives the maximum and minimum values of β0. Also, ∂β0/∂λ > 0 if 0 < λ < λ ∗ and ∂β0/∂λ < 0 if λ ∗ < λ < ∞ for all α ∈ α−, and ∂β0/∂λ < 0 if 0 < λ < λ ∗ and ∂β0/∂λ > 0 if λ ∗ < λ < ∞ for all α ∈ α+. Thus, the local minima for β0 in the interval 0 < λ < λ ∗ can also be obtained.

Stability and velocity sensitivities of libration points in the elliptic restricted synchronous three-body problem under an oblate primary and a dipole secondary

The study investigates the stability and velocity sensitivities of libration points in the elliptic restricted synchronous three-body problem under an oblate primary and a dipole secondary for Luhman-16 and HD188753 systems. We have observed that the position of L4 moves away from the centre of origin for both systems as the oblateness and the half mass dipole distance increases. As the oblateness and the half mass dipole distance increase, there is a shift in the position of L5 closer to the centre of the origin for both systems. The Poincare Surfaces of Section (PSS) for both systems have revealed that the behaviour of the system changes significantly with a bit change in the initial conditions, oblateness and the half mass dipole distance. We have observed that the sensitivity of both systems to change in position and velocities results in either regular orbits or irregular orbits. Hence, the dynamical behaviour of the systems is chaotic. Considering the range of a stable and unstable libration points for the problem under study given as 0<ν<νc and νc≤ν≤14respectively, our study has revealed that, the triangular libration points are stable and unstable for some values of oblateness for the binary systems. In the absence of oblateness, the νc* indicates that triangular points are stable for Luhman-16 system. When the parameters are varied differently with order of commensurability k, the critical mass parameters show that the triangular points are stable for Luhman-16 and unstable for HD188753 system. Using binary systems in our study, the results obtained can be used as springboards for broading the scope of interest in Celestial Mechanics and its investigations have shown significant improvement in the study of this longstanding problem.

Triangular Equilibria in R3BP under the Consideration of Yukawa Correction to Newtonian Potential

We study the triangular equilibrium points in the framework of Yukawa correction to Newtonian potential in the circular restricted three-body problem. The effects of α and λ on the mean-motion of the primaries and on the existence and stability of triangular equilibrium points are analyzed, where α ∈ − 1 , 1 is the coupling constant of Yukawa force to gravitational force, and λ ∈ 0 , ∞ is the range of Yukawa force. It is observed that as λ ⟶ ∞ , the mean-motion of the primaries n ⟶ 1 + α 1 / 2 and as λ ⟶ 0 , n ⟶ 1 . Further, it is observed that the mean-motion is unity, i.e., n = 1 for α = 0 , n &gt; 1 if α &gt; 0 and n &lt; 1 when α &lt; 0 . The triangular equilibria are not affected by α and λ and remain the same as in the classical case of restricted three-body problem. But, α and λ affect the stability of these triangular equilibria in linear sense. It is found that the triangular equilibria are stable for a critical mass parameter μ c = μ 0 + f α , λ , where μ 0 = 0.0385209 ⋯ is the value of critical mass parameter in the classical case of restricted three-body problem. It is also observed that μ c = μ 0 either for α = 0 or λ = 0.618034 , and the critical mass parameter μ c possesses maximum ( μ c max ) and minimum ( μ c min ) values in the intervals − 1 &lt; α &lt; 0 and 0 &lt; α &lt; 1 , respectively, for λ = 1 / 3 .

Exact nonsingular black holes and thermodynamics

We show that regular black holes given by Culetu [14] can be obtained by coupling Einstein's gravity with the nonlinear electrodynamics source. The non-singular black hole has a mass function m(r)=Me−k/r, k is the deviation parameter, and it interpolates between the Schwarzschild black hole (k=0) and the Reissner-Nordstrom black hole (r≫k). Interestingly, there exists a critical mass parameter M=Mc, which corresponds to an extremal black hole when Cauchy and event horizons coincide. For M>Mc, it describes a nonextremal black hole with two horizons and no black hole for M<Mc. The Hawking temperature of the nonsingular black hole is maximum, where the specific heat diverges and changes its sign at the value of mass Mc2>Mc1, and the second-order phase transition occurs at that point. The smaller nonsingular black holes are always stable due to positive heat capacity and negative free energy. A discussion on the quasinormal modes of scalar field perturbations on nonsingular black holes background is included.

Effect of slip boundary condition and non-newtonian rheology of lubricants on the dynamic characteristics of finite hydrodynamic journal bearing

In the present study, the influence of various slip zone locations on the dynamic stability of finite hydrodynamic journal bearing lubricated with non-Newtonian and Newtonian lubricants has been investigated. Linearized equation of motion with free vibration of rigid rotor has been used to find the optimum location of the slip region with maximum stability margin limit. It has been observed that bearing with interface of slip and no-slip region near the upstream side of minimum film-thickness location is effective in improving the direct and cross stiffness coefficient, critical mass parameter, and critical whirling speed. The magnitude of dynamic performance parameters with slip effect is highly dependent on the rheology of lubricant. Shear-thinning lubricants combined with slip boundary condition shows higher dynamic stability as compared to the Newtonian lubricants under the conventional boundary condition. For all considered rheology of lubricants, the dynamic stability of bearing with slip effect is improving by increasing the eccentricity ratio.

Perturbation due to a Circumbinary Disc around &lt;i&gt;L&lt;sub&gt;4,5&lt;/sub&gt;&lt;/i&gt; in the Photogravitational Elliptic Restricted Three-Body Problem

The motion is investigated of dust/gas particles in the elliptic restricted three-body problem (ER3BP) in which the less massive primary is an oblate spheroid and the more massive a luminous body surrounded by a circumbinary disk. The paper has investigated both analytically and numerically the effects of oblateness and radiation pressure of the primaries respectively together with the gravitational potential from a disk on the triangular equilibrium L4,5 of the system, all in the elliptic framework of the restricted problem of three bodies. The important result obtained therein is a move towards the line joining the primaries in the presence of any /all perturbation(s). A significant shift away from the origin as the radiation pressure factor decreases and oblateness of the smaller primary increase is also observed. It is also seen that, all aforementioned parameters in the region of stability have destabilizing tendencies resulting in a decrease in the size of the region of stability except the gravitational potential from the disc. The binary system Ruchbah in the constellation Cassiopeiae is an excellent model for the problem, using the analytic results obtained, the locations of the triangular points and the critical mass parameter are computed numerically.

Perturbation due to a Circumbinary Disc around &lt;i&gt;L&lt;sub&gt;4,5&lt;/sub&gt;&lt;/i&gt; in the Photogravitational Elliptic Restricted Three-Body Problem

The motion is investigated of dust/gas particles in the elliptic restricted three-body problem (ER3BP) in which the less massive primary is an oblate spheroid and the more massive a luminous body surrounded by a circumbinary disk. The paper has investigated both analytically and numerically the effects of oblateness and radiation pressure of the primaries respectively together with the gravitational potential from a disk on the triangular equilibrium L4,5 of the system, all in the elliptic framework of the restricted problem of three bodies. The important result obtained therein is a move towards the line joining the primaries in the presence of any /all perturbation(s). A significant shift away from the origin as the radiation pressure factor decreases and oblateness of the smaller primary increase is also observed. It is also seen that, all aforementioned parameters in the region of stability have destabilizing tendencies resulting in a decrease in the size of the region of stability except the gravitational potential from the disc. The binary system Ruchbah in the constellation Cassiopeiae is an excellent model for the problem, using the analytic results obtained, the locations of the triangular points and the critical mass parameter are computed numerically.

Control of self-excited rotor disturbances in three-lobe journal bearing space using a couple stress lubricant

In this study, the dynamic behavior of three-lobe noncircular hydrodynamic journal bearings lubricated with couple stress fluid is presented. Currently, application of journal bearings with different geometries such as lobed bearings and using non-Newtonian lubricants, including various additives, are the common suggested solutions by tribology researchers to improve the performance of rotor-journal bearing systems. Due to the incompatibility of new lubricant behaviors including polymer chains or other types of suspended particles with considerable size from the classic Navier–Stokes equations, different models such as couple stress fluid theory have been proposed to evaluate their performance. To investigate the self-excited rotor disturbances caused by various factors, like operating in critical speeds, simultaneous solution of governing Reynolds equation of oil lubricant and the rotor motion equations has been done. The result of dynamic analyses using linear and nonlinear approaches indicates the possibility of enhancing the stiffness and damping ability of the three-lobe bearings by strengthening the couple stress properties of lubricant. It is seen from the results that by increasing the couple stress parameter of oil, the type of rotor perturbations in the bearing clearance space changes from divergent oscillations, as a leading cause of collision between the rotor surface and the bearing shell, to limit cycle motion around the rotor stable position or convergent fluctuations to the static equilibrium point. This improvement in the stability range of three-lobe journal bearings using a couple stress lubricant appears in the results of a linear analysis model in terms of increasing the critical mass parameter and decreasing the whirl frequency ratio. In addition, bifurcation diagram results show that using noncircular three-lobe bearings instead of circular types and replacing the Newtonian lubricant with the couple stress fluid improve the dynamic stability of rotor- bearing supports.

Effects of new micro-pocketed bore surface topographies on the performance behaviours of aerodynamic journal bearing

This paper presents the exploration for improving the static and dynamic performance behaviours of a self-acting rigid gas journal bearing employing some new conceived micro-depth pocketed surface topographies at the bore. The conceived micro-pockets comprise of relatively large size rectangular pocket having micro-depth at bearing bore in the converging zone followed by placing of different designs of sub-pockets at the trailing edge of the previous one in the direction of journal rotation. The form of equation achieved from Patir and Cheng’s model for the case of hydrodynamic lubrication regime and the related dynamic pressure equations have been solved using the finite volume method discretisation scheme followed by the solution of the algebraic equations using the Gauss-Seidel iterative method. The minimum film thickness, frictional force, side leakage, bearing dynamic coefficients, effective stiffness, effective damping, and critical mass parameters have been investigated with each new bore surface topography and compared with the performances of conventional aerodynamic journal bearing. Substantial improvements in both static and dynamic performances have been found with the new micro-pocketed bore surface topographies as compared to conventional one. Moreover, the established best bore design has produced significant increase (21%) in minimum film thickness, substantial reduction (12%) in coefficient of friction, and excellent improvement (170%) in the stability parameter (critical mass) as compared to the conventional case.

Stationary scalar clouds supported by rapidly-rotating acoustic black holes in a photon-fluid model

It has recently been proved that, in the presence of vortex flows, the fluctuation dynamics of a rotating photon-fluid model is governed by the Klein-Gordon equation of an effective massive scalar field in a $(2+1)$-dimensional acoustic black-hole spacetime. Interestingly, it has been demonstrated numerically that the rotating acoustic black hole, like the familiar Kerr black-hole spacetime, may support spatially regular stationary density fluctuations (linearized acoustic scalar `clouds') in its exterior regions. In particular, it has been shown that the composed rotating-acoustic-black-hole-stationary-scalar-field configurations of the photon-fluid model exist in the narrow dimensionless regime $\alpha\equiv\Omega_0/m\Omega_{\text{H}}\in(1,\alpha_{\text{max}})$ with $\alpha_{\text{max}}\simeq 1.08$ [here $\Omega_{\text{H}}$ is the angular velocity of the black-hole horizon and $\{\Omega_0,m\}$ are respectively the effective proper mass and the azimuthal harmonic index of the acoustic scalar field]. In the present paper we use analytical techniques in order to explore the physical and mathematical properties of the acoustic scalar clouds of the photon-fluid model in the regime $\Omega_{\text{H}}r_{\text{H}}\gg1$ of rapidly-spinning central supporting acoustic black holes. In particular, we derive a remarkably compact analytical formula for the discrete resonance spectrum $\{\Omega_0(\Omega_{\text{H}},m;n)\}$ which characterizes the stationary bound-state acoustic scalar clouds of the photon-fluid model. Interestingly, it is proved that the critical (maximal) mass parameter $\alpha_{\text{max}}$, which determines the regime of existence of the composed acoustic-black-hole-stationary-bound-state-massive-scalar-field configurations, is given by the exact dimensionless relation $\alpha_{\text{max}}=\sqrt{{{32}\over{27}}}$.

Effect of Perturbations in Coriolis and Centrifugal Forces on Libration Points in the Restricted Six-Body Problem

This paper deals with the existence and stability of libration points in linear sense in central square configuration of restricted six-body problem when small perturbations υ and ρ are given to the coriolis and the centrifugal forces, respectively. This is observed that there exist twelve libration points out of which four libration points are collinear and eight are non-collinear and all libration points lie on the concentric circles C1, C2 and C3 centered at origin. This is also observed that the eight libration points are on the axes and four are off the axes, i.e., L1, L2, L3 and L4 are on x-axis, L5, L6, L7 and L8 are on y-axis and rest are off the axes. Also, the locations of all the libration points are only influenced by a small change in the centrifugal force. It is further observed that the stability of libration points lying on circles C1 and C3 is not influenced by the perturbations and they remain unstable for all values of mass parameter μ while the libration points lying on circle C2 are influenced by the perturbations and stable for the critical mass parameter μc.

A Study of the Elliptic Restricted Three-Body Problem with Triaxial and Radiating Primaries Surrounded by a Belt

This paper examines the effects of radiation pressure and triaxiality of two stars (primaries) surrounded by a belt (circumbinarydisc)onthepositionsandstabilityofathirdbodyofaninfinitesimalmassintheframeworkoftheEllipticrestricted three body problem (ER3BP). We have obtained analytical solutions to the triangular equilibrium points and their stability and have also investigated these solutions numerically and graphically using radiating binary system (Xi- Bootis and Kruger 60). It is observed that their positions and stability are affected by semi-major axis, eccentricity of the orbit, triaxiality, radiation pressure of the primaries and potential from the belt. The perturbed parameters show the destabilizing tendency by decreasing the range of stability. The triangular points are found to bestable for 0 < <i>µ</i> < <i>µ_c</i> where <i>µ_c</i> is the critical mass parameter. The stability analysis for the binary system yielded a stable outcome when we consider the range of mass parameterµ in the region of the Routhonian critical mass ratio (0.03852) when the effect of circumbinary disc is dominant. We found triaxiality and radition factors inducing instability even within this range.

Nonlinear transient analysis of rigid rotor mounted on externally pressurized double-layered porous gas journal bearings accounting velocity slip

The present work aims at estimating the journal center trajectories of a rigid rotor mounted on a hybrid, finite gas lubricated journal bearings with double-layered porous bushing using nonlinear transient analysis. To consider velocity slip in the film at the interface between film and porous region, Reynolds equation is modified using the Beavers–Joseph boundary condition. The governing equations of flow at clearance, porous regions are discretized using finite volume method. The variable values at cell-face centers are obtained using interpolation scheme of third order. Those discretized equations coupled with equations of rigid rotor motion are solved by third-order total variation diminishing Runge–Kutta scheme. Influence of velocity slip and design parameters on critical mass parameter values were explored. The observations are presented in the form of graphs that serves as a reference during the design of such bearings.

Non-linear stability analysis of two-layered porous journal bearings with velocity slip and percolation effect of additives of coupled-stress lubricant

In this paper a non-linear stability analysis of the two-layered porous journal bearing under coupled-stress lubricant has been presented with velocity slip phenomenon and additive’s percolation effect. In this non-linear transient analysis, system stability is determined by tracing the locus of the journal center and various trajectories of journal center locus have been represented in graphical form for different operating conditions. Furthermore, stability characteristics in respect of critical mass parameter and whirl ratio have been studied under various parametric conditions and a comparison between the linear and non-linear stability analysis have been demonstrated. To acquire the non-dimensional pressure values, non-dimensional transient Reynolds equation has been solved and with these pressure values, bearing load carrying capacity are derived. Fourth order Runge-Kutta method is used to solve the second order equations of motion for journal bearing system to obtain the stability characteristics. Results of this analysis may be helpful for designing such bearings.

Dynamic stability analysis of noncircular two-lobe journal bearings with couple stress lubricant regime

In this research, the steady state and dynamic performances of two-lobe noncircular journal bearings with couple stress lubricant are presented. The lubricating oil, containing additives and contaminants, is modeled as the couple stress fluid. The modified Reynolds equation is obtained using the couple stress lubrication theory and is then solved by finite element method as an efficient numerical technique. The steady-state characteristics of bearings, including the load carrying capacity and attitude angle, are determined for various values of the couple stress parameter. The results show that applying the couple stress fluid improves the efficiency of two-lobe bearings in terms of an increased load carrying capacity and reduced attitude angle. Also, the stability performance of the investigated bearings is studied using rotor motion equations based on linear and nonlinear dynamic methods. The results indicate that any increase in the lubricant couple stress parameter enhances the bearing ability to damp the rotor perturbations. In other words, by varying the lubricant from Newtonian oil to the couple stress type and upgrading its properties, the curves of the critical mass parameter and whirl frequency ratio have an increasing and decreasing trend, respectively. Based on the fourth-order Runge–Kutta method results, the dynamic trajectory of the rotor center in the bearing space changes with increasing the couple stress parameter from diverging disturbances and limits the cycle perturbations around the bearing center to converging oscillations to the static equilibrium point. Moreover, the effect of changing lubricant properties on the two-lobe bearing’s performance is more pronounced at higher values of the couple stress parameter, especially with an increase in the noncircularity of bearings’ geometry.

Elliptic restricted synchronous three-body problem (ERS3BP) with a mass dipole model

The goal of the present manuscript is to determine the points of libration and their linear stability in the elliptic restricted synchronous three-body problem with a mass dipole model. The dipole system consists of two-point masses connected with a massless rod in a constant characteristic distance. To execute this study, we acquire the equations of motion of a spacecraft having negligible mass travelling in the system in which more massive primary is assumed to have a spherical shape, and the lesser one is irregular shaped and considered as a rotating mass dipole. The locations of the libration points are analyzed for various values of e, where e is the eccentricity of the orbit of one massive primary around other. Also, the stability of the libration points is investigated in linear sense, and it is observed that the collinear libration points are unstable while the non-collinear libration points are stable for the critical mass parameter μc. Further, we investigated the regions of motion around libration points.

Periodic orbits in the planar restricted photo-gravitational problem when the smaller primary is an oblate spheroid

Background/Objectives: This study deals wit h the stationary solutions of the planar circular restricted three-body problem when the more massive primary is a source of radiation and the smaller primary is an oblate spheroid with its equatorial plane coincident with the plane of motion. The objective is to study the location of the Lagrangian points and to find the values of critical mass. Also, to study the periodic orbits around the Lagrangian points. Methods: A new mean motion expression by including the secular perturbation due to oblateness utilized by(1,2) is used in the present studies. The characteristic roots are obtained by linearizing the equation of the motion around the Lagrangian points. Findings: The critical mass parameter µcrit(3,4) , which decreases radiation force, whereas it increases with oblateness when we consider the value of new mean motion. Through special choice of initial conditions, retrograde elliptical periodic orbits exist for the case µ = µcrit, whose eccentricity increases with oblateness and decreases with radiation force for non-zero oblateness, although there is slight variation in L2 location. Keywords: Restricted three body problem; Lagrangian points; Eccentricity; Oblateness; Critical mass; Radiation force; Mean motion.

Central-body square configuration of restricted six-body problem

This paper deals with the existence and stability of libration points in linear sense in the central-body square configuration of restricted six-body problem. It is found that there exist twelve libration points, four collinear and eight non-collinear. All libration points lie on the concentric circles C1, C2 and C3 centered at origin. The libration points L1,3,5,7 lie on circle C1, L9,10,11,12 on C2 and L2,4,6,8 on C3. This is also observed that the eight libration points are on the axes and four are off the axes, i.e., L1,2,3,4 are on x-axis, L5,6,7,8 on y-axis and rest are off the axes. The libration points on circles C1 and C3 are unstable for all values of mass parameter µ while the libration points on circle C2 are stable for the critical mass parameter µc = 0.00910065.

Granular Effects on Depositional Processes of Debris Avalanches

Debris avalanches generally have massive mobility and extensive deposited area, which have strong relation to the granular effect. The granular effect is an outcome from the interaction (based on frictional mechanism) among particles. Discrepancy of grain composition, for instance the difference of particle size and their proportion of particles, is a key factor to influence granular effect. A series of laboratory tests have been conducted to research the granular effects on depositional processes of debris avalanches containing fine- and coarse-grained particles. The velocity of debris front, the morphology characteristics and runout distance of the debris avalanche are considered in detail, and the mass parameter and the size parameter are introduced in analysis for the research of granular effect. The experimental results indicate that the granular effects can decrease the dissipation of momentum and increase the accumulation region, leading to further runout distance and more extensive deposited areas. Further, the velocity, the range of depositions of debris avalanches are larger than that in other conditions when the proportion of fine-grained particles is 30%, especially the materials consist of heterogeneous granular. It means there have a critical mass parameter in experiments which lead to the high efficiency of granular effects, and the granular effect is weakening in homogeneous granular composition.