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1.
Hydrostatic equilibrium and energy conservation determine the conditions in the gravitationally stabilized solar fusion reactor. We assume a matter density distribution varying non-linearly through the central region of the Sun. The analytic solutions of the differential equations of mass conservation, hydrostatic equilibrium, and energy conservation, together with the equation of state of the perfect gas and a nuclear energy generation rate = 0 n T nT m ,are given in terms of Gauss' hypergeometric function. This model for the structure of the Sun gives the run of density, mass, pressure, temperature, and nuclear energy generation through the central region of the Sun. Because of the assumption of a matter density distribution, the conditions of hydrostatic equilibrium and energy conservation are separated from the mode of energy transport in the Sun.  相似文献   

2.
In this note we derive an exact solution of transfer equation in a plane-parallel semiinfinite atmosphere with albedo >1, by the method of Laplace transform and Wiener-Hopf technique. The emergent intensityI(0, ) is obtained in terms of theH 0-functionH 0() (Das Gupta, 1978) for which some good approximations are given. Intensity at any depth is also obtained.I(0, )/I(0, 0) is plotted in graphs against [0,1], and shows a maximum which drops and shifts towards the origin as increases.  相似文献   

3.
We study the classical problem of two-dimensional motion of a particle in the field of a central force proportional to a real power of the distancer. for negative energy and (0, 2), each energy levelI h is foliated by the invariant toriI hc of constant angular momentumc and, by Liouville-Arnold's theorem, the flow on eachI hc is conjugated to a linear flow of rotation number h (c).A well-known result asserts that if we require h (c) to be rational for every value ofh andc, the, must be equal to one (Kepler's problem). In this paper we prove that for almost every (0, 2) h (c) is a non-constant continuous function ofc, for everyh<0. In particular, we deduce that motion under central potentials is generically non-periodic.Partially supported by CIRIT under grant No. EE88/2.  相似文献   

4.
Savopulos  M.  Quenby  J. J. 《Solar physics》1998,180(1-2):479-485
A numerical solution to the Fisk and Lee (1980) equations for the particle intensity upstream of a corotating interplanetary shock is considered for the November 1991 event observed at Ulysses. A numerically derived parallel diffusion coefficient is available for this region (Quenby et al., 1993), based upon in-situ magnetometer data. Fitting the transport equations solution to the upstream energetic particle distribution function, employing a radial diffusion coefficient = 0 r, where r and are, respectively, radial distance from the Sun and particle velocity, and with 0 fixed from the magnetometer derived coefficient yielded a range of statistically acceptable values of (, ). These ran from (0.5, 0.0) to (1.8, 1.6) along a thin strip of — space, hence demonstrating the improbability that the velocity and radial dependence of the particle diffusion can be fixed from such particle and magnetic field data alone.  相似文献   

5.
A map of the Moon at 2 cm wavelength is presented. The angular ( 1arc) and temperature resolution (< 0.1 K) is sufficient to study systematic details of the brightness distribution. In particular, the centre-to-limb variation is considered. An estimate of the dielectric constant is possible (1.4 2.5). The existence of a temperature gradient in the lunar surface layers is used to derive the depth of penetration of electromagnetic waves (L e ), which isL e 8 m for 2 cm wavelength. The parameters derived from the 2 cm map are found to be compatible with those obtained from a former observation at 6 cm wavelength.  相似文献   

6.
We consider the dynamics of a test particle co-orbital with a satellite of mass m s which revolves around a planet of mass M 0 m s with a mean motion n s and semi-major axis a s. We study the long term evolution of the particle motion under slow variations of (1) the mass of the primary, M 0, (2) the mass of the satellite, m s and (3) the specific angular momentum of the satellite J s. The particle is not restricted to small harmonic oscillations near L 4 or L 5, and may have any libration amplitude on tadpole or horseshoe orbits. In a first step, no torque is applied to the particle, so that its motion is described by a Hamiltonian with slowly varying parameters. We show that the torque applied to the satellite, as measured by s = js/(n s J s) induces an distortion of the phase space which is entirely described by an asymmetry coefficient = s/, where = m s/M. The adiabatic invariance of action implies furthermore that the long term evolution of the particle co-orbital motion depends only on the variation of m s a s with time. Applying a constant torque to the particle, as measured by s = js/(n s J p) is then merely equivalent to replacing = s/ by = (s – p)/. However, if the torque acting on the particle exhibits a radial gradient, then the action is no more conserved and the evolution of the particle orbit is no more controlled by m s a s only. We show that even mild torque gradients can dominate the orbital evolution of the particle, and eventually decide whether the latter will be pulled towards the stable equilibrium points L 4 or L 5, or driven away from them. Finally, we show that when the co-orbital bodies are two satellites with comparable masses m 1 and m 2, we can reduce the problem to that of a test particle co-orbital with a satellite of mass m 1 + m 2. This new problem has then parameters varying at rates which are combinations, with appropriate coefficients, of the changes suffered by each satellite.  相似文献   

7.
Hickson's compact galaxy groups were classified using the statistical criterion which includes the radial velocities of galaxies as well as their relative positions. These groups on the whole and their components are identified as the confident and probable non-chance ones as well as probable and confident chance ones. All confident chance objects have the discordant radial velocities (the differences of radial velocitiesDV [1000; 20000] km s–1). The special class of objects bright discordants is selected. These galaxies have the discordant radial velocities withDV [825; 8440] km s–1 and have a strong tendency to be the brightest components of their groups. The lowest difference of radial velocities for the last class of objects DV = (1.0±0.2) × 103 km s–1 and we accept this value of DV as the lowest value of discordant radial velocities. It is found that the biggest part of Hickson's compact groups consist of non-chance aggregations of galaxies and some of the cases of discordant-redshifts require a special study in order to explain their origin from a dynamic or some other point of view.Present address.  相似文献   

8.
This work contains a transformation of Hill-Brown differential equations for the coordinates of the satellite to a type which can be integrated in a literal form using an analytical programming language. The differential equation for the parallax of the satellite is also established. Its use facilitates the computation of Hill's periodic intermediary orbit of the satellite and provides a good check for the expansion of the coordinates and frequencies. The knowledge of the expansion of the parallax facilitates the formation of differential equations for terms with a given characteristic. These differential equations are put into a form which favors the solution by means of iteration on the computer. As in the classical theory we obtain the expansions of the coordinates and of the parallax in the form of trigonometric series in four arguments and in powers of the constants of integration. We expand the differential operators into series in squares of the constants of integration. Only the terms of order zero in these expansions are employed in the integration of the differential equations. The remaining terms are responsible for producing the cross-effects between the perturbations of different order. By applying the averaging operator to the right sides of the differential equations we deduce the expansion of the frequencies in powers of squares of the constants of integration.Basic Notations f the gravitational constant - E the mass of the planet - M the mass of the satellite - t dynamical time - x, y, z planetocentric coordinates of the satellite - u x+y–1 - s x–y–1 - the planetocentric distance of the satellite - w 1/ - 0 the variational part of - w 0 the variational part ofw, - n the mean daily sidereal motion of the satellite - a the mean semi-major axis of the satellite defined by means of the Kepler relation:a 3 n 2=f(E+M) - a the mean semi-major axis defined as the constant factor attached to the variational solution - e the constant of the eccentricity of the satellite - the sine of one half the orbital inclination of the satellite relative to the orbit of the sun - c(n–n) the anomalistic frequency of the satellite - c 0 the part ofc independent frome,e, and - g(n–n) the draconitic frequency of the satellite, - g 0 the part ofg independent frome,e, and - exp (n–n)t–1 - D d/d - e the eccentricity of the solar planetocentric orbit - a the semi-major axis of the solar orbit - n the mean daily motion of the sun in its orbit around the planet - m n/(n–n) - a/a-the parallactic factor - the disturbing function  相似文献   

9.
We consider the symmetric planar (3 + 1)-body problem with finite masses m 1 = m 2 = 1, m 3 = µ and one small mass m 4 = . We count the number of central configurations of the restricted case = 0, where the finite masses remain in an equilateral triangle configuration, by means of the bifurcation diagram with as the parameter. The diagram shows a folding bifurcation at a value consistent with that found numerically by Meyer [9] and it is shown that for small > 0 the bifurcation diagram persists, thus leading to an exact count of central configurations and a folding bifurcation for small m 4 > 0.  相似文献   

10.
The process of re-escalation of the scalar field as R 3, the energy density as R 3, and the pressurep aspR 3P, lends itself to obtain a reduced equation that represents, for a wide variety of equations of state, the cosmological evolution of an homogeneous and isotropic, flat Universe. A particular solution to this equation is presented.  相似文献   

11.
We prove existence and multiplicity of T-periodic solutions (for any given T) for the N-body problem in m (any m 2) where one of the bodies has mass equal to 1 and the others have masses 2,..., N , small. We find solutions such that the body of mass 1 moves close to x = 0 while the body of mass i moves close to one of the circular solutions of the two body problem of period T/k i, where ki is any odd number. No relation has to be satisfied by k 2,...,k N.  相似文献   

12.
General theory of electrical conductivity of a multicomponent mixture of degenerate fermions in a magnetic fieldB, developed in the preceding article (this volume), is applied to a matter in neutron star interiors at densities 0, where 0 = 2.8×1014 g cm–3 is the standard nuclear matter density. A model of free-particle mixture ofn, p, e is used, with account for appearance of –-hyperons at > c , where c 40. The electric resistivities along and acrossB, and , and the Hall resistivity H are calculated and fitted by simple analytical formulae at c and > c for the cases of normal or superfluid neutrons provided other particles are normal. Charge transport alongB is produced by electrons, due to their Coulombic collisions with other charged particles; is independent ofB and almost independent of the neutron superfluidity. Charge transport acrossB at largeB may be essentially determined by other charged particles. If c , one has = [1 + (B/B 0)2] for the normal neutrons, and for the superfluid neutrons, while H = B/B e for both cases. HereB e 109 T 8 2 G,B 01011 T 8 2 G, andT 8 is temperature in units of 108 K. Accordingly for the normal neutrons atBB 0, the transverse resistivity suffers an enhancement, 1/4 1. When 50 andB varies from 0 toBB p 1013 T 8 2 G, increases by a factor of about 103–104 and H changes sign. WhenBB p , remains constant for the superfluid neutrons, and H B 2 for the normal neutrons, while H B for any neutron state. Strong dependence of resistivity onB, T, and may affect evolution of magnetic fields in neutron star cores. In particular, the enhancement of at highB may noticeably speed up the Ohmic decay of those electric currents which are perpendicular toB.  相似文献   

13.
A static relativistic theory of the stability of the equilibrium of an isentropic spherically-symmetric star is deduced from the properties of a functionu which is solution of a second-order differential equation, and which is related to the model by means of the formulau = m(v, c)/c, wherem is the mass-energy inside the coordinate volumev and c is the central mass-energy density.Work done in the Laboratorio Astrofisico di Frascati, Roma.  相似文献   

14.
By solving the Euler hydrodynamical equations we have obtained closed form solutions for the angular velocities and pressures of a three stratified non-confocal heterogeneous oblate spheroid. Limiting and particular solutions cases, such as a spheroid with N layers, a stratified spheroid with the same eccentricities, as well as confocal layered spheroids are also explicitly written down. As an application, we have numerically estimated planet Earth's outer and inner cores' ellipticities to be o=1/413.318 and i=1/424.616, respectively. These Earth's ellipticities values are in good agreement with those found in the literature.  相似文献   

15.
A comparison is made between the stability criteria of Hill and that of Laplace to determine the stability of outer planetary orbits encircling binary stars. The restricted, analytically determined results of Hill's method by Szebehely and co-workers and the general, numerically integrated results of Laplace's method by Graziani and Black are compared for varying values of the mass parameter =m 2/(m 1+m 2). For 00.15, the closest orbit (lower limit of radius) an outer planet in a binary system can have and still remain stable is determined by Hill's stability criterion. For >0.15, the critical radius is determined by Laplace's stability criterion. It appears that the Graziani-Black stability criterion describes the critical orbit within a few percent for all values of .  相似文献   

16.
Using the flux-transport equation in the absence of sources, we study the relation between a highly peaked polar magnetic field and the poleward meridional flow that concentrates it. If the maximum flow speed m greatly exceeds the effective diffusion speed /R, then the field has a quasi-equilibrium configuration in which the poleward convection of flux via meridional flow approximately balances the equatorward spreading via supergranular diffusion. In this case, the flow speed () and the magnetic field B() are related by the steady-state approximation () (/R)B()/B() over a wide range of colatitudes from the poles to midlatitudes. In particular, a general flow profile of the form sin p cos q which peaks near the equator (q p) will correspond to a cos n magnetic field at high latitudes only if p = 1 and m = n /R. Recent measurements of n 8 and 600 km2 s–1 would then give m 7 m s–1.  相似文献   

17.
The concept of velocity correlation functions is introduced and applied to the calculation of cosmic ray spatial diffusion coefficients. It is assumed that the pitch angle scattering coefficient is already known from some other theory, and is reasonably well-behaved. Previous results for the coefficient for diffusion parallel to the mean field are recovered when the velocity-changing mechanism is artificially restricted to pitch angle scattering. The velocity correlation method is then applied to the more general case where there are fluctuations in the local mean field. It is found that the parallel diffusion coefficient is reduced in proportion to the amplitude of the field fluctuations, and that the ratio of the perpendicular to parallel diffusion coefficients cannot be greater than B x 2 /B 0 2 . It is shown in the appendix that the Liouville form of the scattering equation implies that the Fokker-Planck coefficients 2/t=2D and /t=D /, and that all higher-order coefficients are identically zero.  相似文献   

18.
We study some aspects of the restricted three-body problem when the mass parameter is sufficiently small. First, we describe the global flow of the two-body rotating problem, =0, and we use it for the analysis of the collision and parabolic orbits when 0. Also we show that for any fixed value of the Jacobian constant and for any >0, there exists a 0>0 such that if the mass parameter [0,0], then the set of bounded orbits which are not contained in the closure of the set of symmetric periodic orbits has Lebesgue measure less than .Paper presented at the 1981 Oberwolfach Conference on Mathematical Methods in Celestial Mechanics.  相似文献   

19.
Analytic structure of high-density steady isothermal spheres is discussed using the TOV equation of hydrostatic equilibrium which satisfies an equation of state of the kind:P = K g , = g c 2.Approximate analytical solutions to the Tolman-Oppenheimer-Volkoff (TOV) equations of hydrostatic equilibrium in (, ), (,U) and (u, v) phase planes in concise and simple form useful for short computer programmes or on small calculator, have been given. In Figures 1, 2, and 3, respectively, we display the qualitative behaviours of the ratio of gas density g to the central density gc , g / gc ; pressureP to the gc ,P/ gc ; and the metric componente –, for three representative general relativistic (GR) isothermal configurations =0.1, 0.2, and 0.3. Figure 4 shows the solution curve (, ) for =0.1, 0.2, and 0.3 (=0 represents the classical (Newtonian) curve). Numerical values of physical quantitiesv (=4r 2 P *(r)), in steps ofu (=M(r)/r)=0.03, and the mass functionU, in steps of =0.2 (dimensionless radial distance), are given, respectively, in Tables I and II. Other interesting features of the configurations, such as ratio of gravitational radius 2GM/c 2 to the coordinate radiusR, mass distributionM(r)/M, pressure (or density) distributionP/P c , binding energy (B.E.), etc., have also been incorporated in the text. It has further been shown that velocity of sound inside the configurations is always less than the velocity of light.Part of the work done at Azerbaijan State University, Baku, U.S.S.R., and Mosul University, Mosul, Iraq, 1985-1986  相似文献   

20.
Consider a Hamiltonian system of two degrees of freedom at an equilibrium. Suppose that the linearized vectorfield has eigenvaluesi,i,–i,–i ( , >0) and is not semisimple. In this paper we discuss the real normalization of the Hamiltonian function of such a system. We normalize the Hamiltonian up to 4th order and show how to compute its coefficients. For the planar restricted three body problem atL 4 the coefficient that plays an important role in the investigation of the qualitative behaviour of periodic solutions near the equilibrium is explicitly calculated.  相似文献   

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