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1.
This paper studies the existence and stability of equilibrium points under the influence of small perturbations in the Coriolis and the centrifugal forces, together with the non-sphericity of the primaries. The problem is generalized in the sense that the bigger and smaller primaries are respectively triaxial and oblate spheroidal bodies. It is found that the locations of equilibrium points are affected by the non-sphericity of the bodies and the change in the centrifugal force. It is also seen that the triangular points are stable for 0<μ<μ c and unstable for \(\mu_{c}\le\mu <\frac{1}{2}\), where μ c is the critical mass parameter depending on the above perturbations, triaxiality and oblateness. It is further observed that collinear points remain unstable.  相似文献   

2.
This study explores the effects of small perturbations in the Coriolis and centrifugal forces, radiation pressures and triaxiality of the two stars (primaries) on the position and stability of an infinitesimal mass (third body) in the framework of the planar circular restricted three-body problem (R3BP). it is observed that the positions of the usual five (three collinear and two triangular) equilibrium points are affected by the radiation, triaxiality and a small perturbation in the centrifugal force, but are unaffected by that of the Coriolis force. The collinear points are found to remain unstable, while the triangular points are seen to be stable for 0<μ<μ c and unstable for $\mu_{c} \le\mu\le\frac{1}{2}$ , where μ c is the critical mass ratio influenced by the small perturbations in the Coriolis and centrifugal forces, radiation and triaxiality. It is also noticed that the former one and all the latter three posses stabilizing and destabilizing behavior respectively. Therefore, the overall effect is that the size of the region of stability decreases with increase in the values of the parameters involved.  相似文献   

3.
The existence and stability of a test particle around the equilibrium points in the restricted three-body problem is generalized to include the effect of variations in oblateness of the first primary, small perturbations ϵ and ϵ′ given in the Coriolis and centrifugal forces α and β respectively, and radiation pressure of the second primary; in the case when the primaries vary their masses with time in accordance with the combined Meshcherskii law. For the autonomized system, we use a numerical evidence to compute the positions of the collinear points L 2κ , which exist for 0<κ<∞, where κ is a constant of a particular integral of the Gylden-Meshcherskii problem; oblateness of the first primary; radiation pressure of the second primary; the mass parameter ν and small perturbation in the centrifugal force. Real out of plane equilibrium points exist only for κ>1, provided the abscissae x < \fracn(k-1)b\xi<\frac{\nu(\kappa-1)}{\beta}. In the case of the triangular points, it is seen that these points exist for ϵ′<κ<∞ and are affected by the oblateness term, radiation pressure and the mass parameter. The linear stability of these equilibrium points is examined. It is seen that the collinear points L 2κ are stable for very small κ and the involved parameters, while the out of plane equilibrium points are unstable. The conditional stability of the triangular points depends on all the system parameters. Further, it is seen in the case of the triangular points, that the stabilizing or destabilizing behavior of the oblateness coefficient is controlled by κ, while those of the small perturbations depends on κ and whether these perturbations are positive or negative. However, the destabilizing behavior of the radiation pressure remains unaltered but grows weak or strong with increase or decrease in κ. This study reveals that oblateness coefficient can exhibit a stabilizing tendency in a certain range of κ, as against the findings of the RTBP with constant masses. Interestingly, in the region of stable motion, these parameters are void for k = \frac43\kappa=\frac{4}{3}. The decrease, increase or non existence in the region of stability of the triangular points depends on κ, oblateness of the first primary, small perturbations and the radiation pressure of the second body, as it is seen that the increasing region of stability becomes decreasing, while the decreasing region becomes increasing due to the inclusion of oblateness of the first primary.  相似文献   

4.
The phenomenological nature of a new gravitational type interaction between two different bodies derived from Verlinde’s entropic approach to gravitation in combination with Sorkin’s definition of Universe’s quantum information content, is investigated. Assuming that the energy stored in this entropic gravitational field is dissipated under the form of gravitational waves and that the Heisenberg principle holds for this system, one calculates a possible value for an absolute minimum time scale in nature t = \frac1516 \fracL1/2(h/2p) Gc4 ~ 9.27×10-105\tau=\frac{15}{16} \frac{\Lambda^{1/2}\hbar G}{c^{4}}\sim9.27\times10^{-105} seconds, which is much smaller than the Planck time t P =(ħG/c 5)1/2∼5.38×10−44 seconds. This appears together with an absolute possible maximum value for Newtonian gravitational forces generated by matter Fg=\frac3230\fracc7L (h/2p) G2 ~ 3.84×10165F_{g}=\frac{32}{30}\frac{c^{7}}{\Lambda \hbar G^{2}}\sim 3.84\times 10^{165} Newtons, which is much higher than the gravitational field between two Planck masses separated by the Planck length F gP =c 4/G∼1.21×1044 Newtons.  相似文献   

5.
We have examined the effects of oblateness up to J 4 of the less massive primary and gravitational potential from a circum-binary belt on the linear stability of triangular equilibrium points in the circular restricted three-body problem, when the more massive primary emits electromagnetic radiation impinging on the other bodies of the system. Using analytical and numerical methods, we have found the triangular equilibrium points and examined their linear stability. The triangular equilibrium points move towards the line joining the primaries in the presence of any of these perturbations, except in the presence of oblateness up to J 4 where the points move away from the line joining the primaries. It is observed that the triangular points are stable for 0 < μ < μ c and unstable for \(\mu_{\mathrm{c}} \le \mu \le \frac {1}{2},\) where μ c is the critical mass ratio affected by the oblateness up to J 4 of the less massive primary, electromagnetic radiation of the more massive primary and potential from the belt, all of which have destabilizing tendencies, except the coefficient J4 and the potential from the belt. A practical application of this model could be the study of motion of a dust particle near a radiating star and an oblate body surrounded by a belt.  相似文献   

6.
This paper studies the motion of an infinitesimal mass in the framework of the restricted three-body problem (R3BP) under the assumption that the primaries of the system are radiating-oblate spheroids, enclosed by a circular cluster of material points. It examines the effects of radiation and oblateness up to J 4 of the primaries and the potential created by the circular cluster, on the linear stability of the liberation locations of the infinitesimal mass. The liberation points are found to be stable for 0<μ<μ c and unstable for $\mu_{c}\le\mu\le\frac{1}{2}$ , where μ c is the critical mass value depending on terms which involve parameters that characterize the oblateness, radiation forces and the circular cluster of material points. The oblateness up to J 4 of the primaries and the gravitational potential from the circular cluster of material points have stabilizing propensities, while the radiation of the primaries and the oblateness up to J 2 of the primaries have destabilizing tendencies. The combined effect of these perturbations on the stability of the triangular liberation points is that, it has stabilizing propensity.  相似文献   

7.
We have studied a modified version of the classical restricted three-body problem (CR3BP) where both primaries are considered as oblate spheroids and are surrounded by a homogeneous circular planar cluster of material points centered at the mass center of the system. In this dynamical model we have examined the effects of oblateness of both primaries up to zonal harmonic J 4; together with gravitational potential from the circular cluster of material points on the existence and linear stability of the triangular equilibrium points. It is found that, the triangular points are stable for 0<μ<μ c and unstable for $\mu_{c} \le \mu \le \frac{1}{2}$ , where μ c is the critical mass ratio affected by the oblateness up to J 4 of the primaries and potential from the circular cluster of material points. The coefficient J 4 has stabilizing tendency, while J 2 and the potential from the circular cluster of material points have destabilizing tendency. A practical application of this model could be the study of the motion of a dust particle near oblate bodies surrounded by a circular cluster of material points.  相似文献   

8.
In this paper, we study the existence of libration points and their linear stability when the three participating bodies are axisymmetric and the primaries are radiating, we found that the collinear points remain unstable, it is further seen that the triangular points are stable for 0<μ<μ c , and unstable for where , it is also observed that for these points the range of stability will decrease. In addition to this we have studied periodic orbits around these points in the range 0<μ<μ c , we found that these orbits are elliptical; the frequencies of long and short orbits of the periodic motion are affected by the terms which involve parameters that characterize the oblateness and radiation repulsive forces. The implication is that the period of long periodic orbits adjusts with the change in its frequency while the period of short periodic orbit will decrease.  相似文献   

9.
This paper investigates the motion of an infinitesimal body in the generalized restricted three-body problem. It is generalized in the sense that both primaries are radiating, oblate bodies, together with the effect of gravitational potential from a belt. It derives equations of the motion, locates positions of the equilibrium points and examines their linear stability. It has been found that, in addition to the usual five equilibrium points, there appear two new collinear points L n1, L n2 due to the potential from the belt, and in the presence of all these perturbations, the equilibrium points L 1, L 3 come nearer to the primaries; while L 2, L 4, L 5, L n1 move towards the less massive primary and L n2 moves away from it. The collinear equilibrium points remain unstable, while the triangular points are stable for 0<μ<μ c and unstable for $\mu_{c} \le\mu\le\frac{1}{2}$ , where μ c is the critical mass ratio influenced by the oblateness and radiation of the primaries and potential from the belt, all of which have destabilizing tendency. A practical application of this model could be the study of the motion of a dust particle near the oblate, radiating binary stars systems surrounded by a belt.  相似文献   

10.
We have investigated an improved version of the classic restricted three-body problem where both primaries are considered oblate and are enclosed by a homogeneous circular planar cluster of material points centered at the mass center of the system. In this dynamical model we have examined the effect on the number and on the linear stability of the equilibrium locations of the small particle due to both, the primaries’ oblateness and the potential created by the circular cluster. We have drawn the zero-velocity surfaces and we have found that in addition to the usual five Lagrangian equilibrium points of the classic restricted three-body problem, there exist two new collinear points L n1,L n2 due to the potential from the circular cluster of material points. Numerical investigations reveal that with the increase in the mass of the circular cluster of material points, L n2 comes nearer to the more massive primary, while L n1 moves away from it. Owing to oblateness of the bodies, L n1 comes nearer to the more massive primary, while L n2 moves towards the less massive primary. The collinear equilibrium points remain unstable, while the triangular points are stable for 0<μ<μ c and unstable for $\mu_{c} \le \mu \le \frac{1}{2}$ , where μ c is the critical mass ratio influenced by oblateness of the primaries and the potential from the circular cluster of material points. The oblateness and the circular cluster of material points have destabilizing tendency.  相似文献   

11.
It is surprising that we hardly know only 4% of the universe. Rest of the universe is made up of 73% of dark-energy and 23% of dark-matter. Dark-energy is responsible for acceleration of the expanding universe; whereas dark-matter is said to be necessary as extra-mass of bizarre-properties to explain the anomalous rotational-velocity of galaxy. Though the existence of dark-energy has gradually been accepted in scientific community, but the candidates for dark-matter have not been found as yet and are too crazy to be accepted. Thus, it is obvious to look for an alternative theory in place of dark-matter. Milgrom (Astrophys. J. 270:365, 1983a; 270:371, 1983b) has suggested a ‘Modified Newtonian Dynamics (MOND)’ which appears to be highly successful for explaining the anomalous rotational-velocity. But unfortunately MOND lacks theoretical support. The MOND, in-fact, is (empirical) modification of Newtonian-Dynamics through modification in the kinematical acceleration term ‘a’ (which is normally taken as a=\fracv2ra=\frac{v^{2}}{r}) as effective kinematic acceleration aeffective = a m(\fracaa0)a_{\mathit{effective}} = a \mu(\frac{a}{a_{0}}), wherein the μ-function is 1 for usual-values of accelerations but equals to \fracaa0 ( << 1)\frac{a}{a_{0}} (\ll1) if the acceleration ‘a’ is extremely-low lower than a critical value a 0(10−10 m/s2). In the present paper, a novel variant of MOND is proposed with theoretical backing; wherein with the consideration of universe’s acceleration a d due to dark-energy, a new type of μ-function on theoretical-basis emerges out leading to aeffective = a(1 -K \fraca0a)a_{\mathit{effective}} = a(1 -K \frac{a_{0}}{a}). The proposed theoretical-MOND model too is able to fairly explain ‘qualitatively’ the more-or-less ‘flat’ velocity-curve of galaxy-rotation, and is also able to predict a dip (minimum) on the curve.  相似文献   

12.
Recently, Bijalwan (Astrophys. Space Sci., doi:, 2011a) discussed charged fluid spheres with pressure while Bijalwan and Gupta (Astrophys. Space Sci. 317, 251–260, 2008) suggested using a monotonically decreasing function f to generate all possible physically viable charged analogues of Schwarzschild interior solutions analytically. They discussed some previously known and new solutions for Schwarzschild parameter u( = \fracGMc2a ) ￡ 0.142u( =\frac{GM}{c^{2}a} ) \le 0.142, a being radius of star. In this paper we investigate wide range of u by generating a class of solutions that are well behaved and suitable for modeling Neutron star charge matter. We have exploited the range u≤0.142 by considering pressure p=p(ω) and f = ( f0(1 - \fracR2(1 - w)a2) +fa\fracR2(1 - w)a2 )f = ( f_{0}(1 - \frac{R^{2}(1 - \omega )}{a^{2}}) +f_{a}\frac{R^{2}(1 - \omega )}{a^{2}} ), where w = 1 -\fracr2R2\omega = 1 -\frac{r^{2}}{R^{2}} to explore new class of solutions. Hence, class of charged analogues of Schwarzschild interior is found for barotropic equation of state relating the radial pressure to the energy density. The analytical models thus found are well behaved with surface red shift z s ≤0.181, central red shift z c ≤0.282, mass to radius ratio M/a≤0.149, total charge to total mass ratio e/M≤0.807 and satisfy Andreasson’s (Commun. Math. Phys. 288, 715–730, 2009) stability condition. Red-shift, velocity of sound and p/c 2 ρ are monotonically decreasing towards the surface while adiabatic index is monotonically increasing. The maximum mass found to be 1.512 M Θ with linear dimension 14.964 km. Class of charged analogues of Schwarzschild interior discussed in this paper doesn’t have neutral counter part. These solutions completely describe interior of a stable Neutron star charge matter since at centre the charge distribution is zero, e/M≤0.807 and a typical neutral Neutron star has mass between 1.35 and about 2.1 solar mass, with a corresponding radius of about 12 km (Kiziltan et al., [astro-ph.GA], 2010).  相似文献   

13.
The Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI) gives us a chance to investigate the theoretical Neupert effect using the correlation between the thermal-energy derivative and the nonthermal energy, or the thermal energy and the integral nonthermal energy. Based on this concept, we analyze four M-class RHESSI flares on 13 November 2003, 4 November 2004, 3 and 25 August 2005. According to the evolution of the temperature [T], emission measure [EM], and thermal energy [E th], each event is divided into three phases during the nonthermal-energy input [ \frac dEnthdt\frac {\mathrm{d}E_{\mathrm{nth}}}{\mathrm{d}t} in the units of erg s−1]. Phase 1 is identified as the interval before the temperature maximum, while after the thermal-energy maximum is phase 3, between them is phase 2. We find that these four flares show the Neupert effect in phase 1, but not in phase 3. The Neupert effect still works well in the second phase, although the cooling becomes slightly important. We define the parameter μ in the relation of \fracdEthdt=m\fracdEnth(t)dt\frac{\mathrm {d}E_{\mathrm{th}}}{\mathrm{d}t}=\mu\frac{\mathrm{d}E_{\mathrm {nth}}(t)}{\mathrm{d}t} or Eth(t0)=mò0t0\fracdEnth(t)dt dtE_{\mathrm{th}}(t_{0})=\mu\int_{0}^{t_{0}}\frac{\mathrm{d}E_{\mathrm{nth}}(t)}{\mathrm{d}t}\,\mathrm{d}t when the cooling is ignored in phase 1. Considering the uncertainties in estimating the energy from the observations, it is not possible to precisely determine the fraction of the known energy in the nonthermal electrons transformed into the thermal energy of the hottest plasma observed by RHESSI. After a rough estimate of the flare volume and the assumption of the filling factor, we investigate the parameter μ in these four events. Its value ranges from 0.02 to 0.20, indicating that a small fraction (2% – 20%) of the nonthermal energy can be efficiently transformed into thermal energy, which is traced by the soft X-ray emission, and the bulk of the energy is lost possibly due to cooling.  相似文献   

14.
This paper deals with the existence of libration points and their linear stability when the more massive primary is radiating and the smaller is an oblate spheroid. Our study includes the effects of oblateness of $\bar{J}_{2i}$ (i=1,2) with respect to the smaller primary in the restricted three-body problem. Under combining the perturbed forces that were mentioned before, the collinear points remain unstable and the triangular points are stable for 0<μ<μ c , and unstable in the range $\mu_{c} \le\mu\le\frac{1}{2}$ , where $\mu_{c} \in(0,\frac{1}{2})$ , it is also observed that for these points the range of stability will decrease. The relations for periodic orbits around five libration points with their semimajor, semiminor axes, eccentricities, the frequencies of orbits and periods are found, furthermore for the orbits around the triangular points the orientation and the coefficients of long and short periodic terms also are found in the range 0<μ<μ c .  相似文献   

15.
The aim of this paper is to determine the flux emergence rate due to small-scale magnetic features in the quiet Sun using high-resolution Hinode SOT NFI data. Small-scale magnetic features are identified in the data using two different feature identification methods (clumping and downhill); then three methods are applied to detect flux emergence events. The distribution of the intranetwork peak emerged fluxes is determined. When combined with previous emergence results, from ephemeral regions to sunspots, the distribution of all fluxes are found to follow a power-law distribution which spans nearly seven orders of magnitude in flux (1016 – 1023 Mx) and 18 orders of magnitude in frequency. The power-law fit to all these data is of the form
\fracdNdY = \fracn0Y0\fracYY0-2.7,\frac{\mathrm{d}N}{\mathrm{d}\Psi} = \frac{n_0}{\Psi_0}\frac{\Psi}{\Psi _0}^{-2.7},  相似文献   

16.
Detailed analyses by independent research groups over several decades reveal a significant discrepancy between the observed rate of periastron advance in the detached eclipsing binary star systems DI Herculis and V541 Cygni and the values theoretically predicted from the combined classical and general relativistic effects. A modification to Newton’s gravitational theory is proposed in this investigation to account for these discrepancies, and is represented by
F = - \fracGm1m2r3r - \fracGom1m2r2r\mathbf{F} = - \frac{Gm_{1}m_{2}}{r^{3}}\boldsymbol{r} - \frac{G_{o}m_{1}m_{2}}{r^{2}}\boldsymbol{r}  相似文献   

17.
Kinetic Alfven waves are important in a wide variety of areas like astrophysical, space and laboratory plasmas. In cometary environments, waves in the hydromagnetic range of frequencies are excited predominantly by heavy ions. We, therefore, study the stability of the kinetic Alfven wave in a plasma of hydrogen ions, positively and negatively charged oxygen ions and electrons. Each species was modeled by drifting ring distributions in the direction parallel to the magnetic field; in the perpendicular direction the distribution was simulated with a loss cone type distribution obtained through the subtraction of two Maxwellian distributions with different temperatures. We find that for frequencies w* < wcH +\omega^{*} < \omega_{c\mathrm{H}^{ +}} (ω ∗ and wcH +\omega_{c\mathrm{H}^{ +}} being respectively the Doppler shifted and hydrogen ion gyro-frequencies), the growth rate increases with increasing negatively charged oxygen ion densities while decreasing with increasing propagation angles, negative ion temperatures and negative ion mass.  相似文献   

18.
Stochastic temperatures and turbulence are characterized by average velocities u th and < u turb  > ≡ u 0 and fluctuations u￠th {u'_{th}} and u′ (<u′ > = 0). Thus, the Doppler width of a line also has a fluctuating component Dl￠D \Delta {\lambda '_D} . Observed spectra correspond to the radiative flux averaged over time and over a star’s surface, <Hλ>. Usually, only the average velocities u th and u 0 are taken into account in photospheric models and these yield the Doppler width DlD(0) \Delta \lambda_D^{(0)} of a line in the customary way. The fluctuations Dl￠D \Delta {\lambda '_D} mean that near a line center the average absorption coefficient < αλ > is larger than the usual αλ, which depends only on the average velocities u th and u 0. This enhances the absorption line near the center and is not explained by the photospheric models. This new statistical effect depends on the wavelength of the line. A comparison of observed lines with model profiles yields an estimate for the average level of fluctuations in the Doppler width, h = á | Dl￠D | ñ
/ DlD(0) \eta = {{{\left\langle {\left| {\Delta {{\lambda '}_D}} \right|} \right\rangle }} \left/ {{\Delta \lambda_D^{(0)}}} \right.} , which characterizes the average stochasticity of a photosphere and is important for understanding the physics of photospheres. The depths of lines in synthetic spectra of stars are often greater than the observed values. The observed disagreement between the theoretical and actually observed depths of lines can be corrected by introducing an additional parameter, the fluctuation level η. Then it is possible to obtain estimates of η for a number of stars.  相似文献   

19.
We investigate the late-time dynamics of a four-dimensional universe based on modified scalar field gravity in which the standard Einstein-Hilbert action R is replaced by f(φ)R+f(R) where f(φ)=φ 2 and f(R)=AR 2+BR μν R μν,(A,B)∈ℝ. We discussed two independent cases: in the first model, the scalar field potential is quartic and for this special form it was shown that the universe is dominated by dark energy with equation of state parameter w≈−0.2 and is accelerated in time with a scale factor evolving like a(t)∝t 5/3 and B+3A≈0.036. When, B+3A→∞ which corresponds for the purely quadratic theory, the scale factor evolves like a(t)∝t 1/2 whereas when B+3A→0 which corresponds for the purely scalar tensor theory we found when a(t)∝t 1.98. In the second model, we choose an exponential potential and we conjecture that the scalar curvature and the Hubble parameter vary respectively like R=hH[(f)\dot]/f,h ? \mathbbRR=\eta H\dot{\phi}/\phi,\eta\in\mathbb{R} and H=g[(f)\dot]c,(g,c) ? \mathbbRH=\gamma\dot{\phi}^{\chi},(\gamma,\chi)\in\mathbb{R}. It was shown that for some special values of  χ, the universe is free from the initial singularity, accelerated in time, dominated by dark or phantom energy whereas the model is independent of the quadratic gravity corrections. Additional consequences are discussed.  相似文献   

20.
In this research paper, we have derived the formula for both the changes in energy (δE) and entropy (δS) and thereafter calculated the change in entropy (δS) with corresponding change in energy (δE) taking account the first law of the black hole mechanics relating the change in mass M, angular momentum J, horizon area A and charge Q, of a stationary black hole, when it is perturbed, given by formula satisfying in the vacuum as dM = \frack8p dA + WdJ - udQ\delta M = \frac{k}{8\pi} \delta A + \Omega\delta J - \upsilon\delta Q, specially for Non-spinning black holes.  相似文献   

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