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1.
In this Letter we propose to consider the four-energy-space whose coordinates are composed as follows: (i) the coordinate 0 refers to the internal energy of the body (it is involved as an unknown function of the rest-energy and the kinetic energy of the body), and (ii) the coordinates 1, 2, 3 relate to the presence of gravitational, electromagnetic, and thermal energy at the location of the body respectively. We involve yet the proper energy interval d2 by analogy to the four-interval ds 2 in general relativity. From such metric field we calculate the Ricci tensor in the simplest case. In addition, we require its form to be the same one as that considered by Schwarzschild. Comparing both solutions we obtain Einstein's relationE=mc 2.  相似文献   

2.
The effect of small perturbations and in the coriolis and the centrifugal forces respectively on the stability of the triangular points in the restricted problem of three bodies with variable mass has been studied. It is found that the range of stability of triangular points increases or decreases depending upon whether the perturbation point (, ) lies in one or the other of the two parts in which the (, ) plane is divided by the line J8–J9=0 where J8 and J9 depend upon , the constant due to the variation in mass governed by Jeans' law.  相似文献   

3.
The location and the stability of the libration points in the restricted problem have been studied when small perturbation and are given to the Coriolis and the centrifugal forces respectively. It is seen that the pointsL 4 andL 5 form nearly equilateral triangles with the primaries and the pointsL 1,L 2,L 3 remain collinear. It is further observed that for the pointsL 4 andL 5, the range of stability increases or decreases depending upon whether the point (, ) lies in one or the other of the two parts in which the (, ) plane is divided by the line 36-19=0 and the stability of the collinear points is not influenced by the perturbations and they remain unstable.  相似文献   

4.
A family of symplectic integrators adapted for the integration of perturbed Hamiltonian systems of the form H=A+B was given in (McLachlan, 1995). We give here a constructive proof that for all integer p, such integrator exists, with only positive steps, and with a remainder of order O(p + 22), where is the stepsize of the integrator. Moreover, we compute the analytical expressions of the leading terms of the remainders at all orders. We show also that for a large class of systems, a corrector step can be performed such that the remainder becomes O(p +42). The performances of these integrators are compared for the simple pendulum and the planetary three-body problem of Sun–Jupiter–Saturn.  相似文献   

5.
A generalization is expressed of the Poisson theorem referring to the invariance of the planetary semi-major axes using the restricted problem model. In particular, it is shown that first and second approximation in terms of a change in the initial states of planets describing closed motions in the solar system remain invariant in modulus after any number of revolutions. But third-order terms contain secular parts and, thus, they undergo a secular change in their orbital motion. Such change would be apparent after -2 Jovian years, where is a constant and is the maximum initial deviation of each planet from its reference orbit.  相似文献   

6.
A model of a first generation intermediate star of 5M , with Z=0 has been considered. The model is at an advanced stage of its evolution and has a double shell burning. It burns helium in the inner shell, and hydrogen, via CNO cycle, in the outer shell. =(log/log) T and T =(log/logT) were computed allowing for the oscillations of the relative mass abundance of the reagents in nuclear reactions. Including =(log/log) T and =(log/logT) of mean molecular weight and the effect of the oscillations of abundances due to nuclear reactions, stability was studied. Contrary to the results of the static calculations, we found that instability due to the excitation mechanism provided by the high temperature sensitivity of energy generation rate propagates up to the surface. Thus the model in question was found to be unstable against radial adiabatic pulsations, in its fundamental mode.  相似文献   

7.
Two hyades giant stars, and Tau, have been studied from an analysis of strong line profiles. We get for Tau,T e =4750K and logg=2.7, and for Tau,T e =4700K and logg=2.8. Hydrogen-to-metal ratio for the two stars is nearly the same as that of solar value.  相似文献   

8.
An expansion of the force function ofn-body dynamical systems, where the equations of motion are expressed in the Jacobian coordinate system, is shown to give rise naturally to a set of (n–1) (n–2) dimensionless parameters ki li {i = 2,...,n;k = 2,...,i – 1 (i 3);l =i + 1,...,n (i n – 1)}, representative of the size of the disturbances on the Keplerian orbits of the various bodies. The expansion is particularized to the casen=3 which involves the consideration of only two parameters 23 and 32. Further, the work of Szebehely and Zare (1977) is reviewed briefly with reference to a sufficient condition for the stability of corotational coplanar three-body systems, in which two of the bodies form a binary system. This condition is sufficient in the sense that it precludes any possibility of an exchange of bodies, i.e. Hill type stability, however, it is not a necessary condition. These two approaches are then combined to yield regions of stability or instability in terms of the parameters 23 and 32 for any system of given masses and orbital characteristics (neglecting eccentricities and inclinations) with the following result: that there is a readily applicable rule to assess the likelihood of stability or instability of any given triple system in terms of 23 and 32.Treating a system ofn bodies as a set of disturbed three-body systems we use existing data from the solar system, known triple systems and numerical experiments in the many-body problem to plot a large number of triple systems in the 23, 32 plane and show the results agree well with the 23, 32 analysis above (eccentricities and inclinations as appropriate to most real systems being negligible). We further deal briefly with the extension of the criteria to many-body systems wheren>4, and discuss several interesting cases of dynamical systems.  相似文献   

9.
The clear sky emissivity 0 and the ground emissivity g in Bahrain is studied. The study reveals that the annual value of 0 is 0.88 ± 0.039 relating to the maximum and the minimum values in August and February, respectively. Meanwhile, the annual value of g is 0.338 ± 0.228, where the maximum and the minimum values are in July and January, respectively. These two parameters are related to the transmittance factor .  相似文献   

10.
In the theory of supergravity (N=1), the supersymmetric version of general relativity, and for the Kasner cosmological model (Bianchi type I) we find a non-trivial solution (for the metric and spinor-vector) under the most simple assumption =11 + 22; 12+21=0 and for a special choosed gaugeN=1,N j=0, 0=0. This method could be also applied to other cosmological metrics and extended to enlarged Grassmann basis.O. Obregón was partially supported by the Alexander von Humboldt Stiftung.  相似文献   

11.
The equilibrium point O of an autonomous Hamiltonian system of two degrees of freedom is considered for small-oscillation frequencies related as 2=21+. If under the precise resonance (=0) the equilibrium is unstable, the inner diameter () of the domain of stability containing the point O is estimated. It is shown that for the normalized variables ()/b where b is the corresponding resonance coefficient. The estimates () for other main resonances are reported.  相似文献   

12.
The nonlinear evolution of waves in a low-density plasma in a strong magnetic field is investigated on the basis of the Chew-Goldberger-Low approximation. The nonlinear effects are found to be essentially different for the magneto-acoustic and Alfvén modes. For the magnetic-acoustic mode, waveform distortion occurs at order 2 (where is a measure of the linear wave amplitude) and shock formation occurs over a time-scale of order –1. For the Alfvén wave, modulation occurs at order 3 and shock formation over a time-scale of order –2. The nature of the waveform distortion is qualitatively different for the two modes.  相似文献   

13.
A small particle moves in the vicinity of two masses, forming a close binary, in orbit about a distant mass. Unique, uniformly valid solutions of this four-body problem are found for motion near both equilateral triangle points of the binary system in terms of a small parameter , where the primaries move in accordance with a uniformly-valid three-body solution. Accuracy is maintained within a constant errorO(8), and the solutions are uniformly valid as tends to zero for time intervalsO(–3). Orbital position errors nearL 4 andL 5 of the Earth-Moon system are found to be less than 5% when numerically-generated periodic solutions are used as a standard of comparison.  相似文献   

14.
In this paper we consider almost integrable systems for which we show that there is a direct connection between symplectic methods and conventional numerical integration schemes. This enables us to construct several symplectic schemes of varying order. We further show that the symplectic correctors, which formally remove all errors of first order in the perturbation, are directly related to the Euler—McLaurin summation formula. Thus we can construct correctors for these higher order symplectic schemes. Using this formalism we derive the Wisdom—Holman midpoint scheme with corrector and correctors for higher order schemes. We then show that for the same amount of computation we can devise a scheme which is of order O(h 6)+(2 h 2), where is the order of perturbation and h the stepsize. Inclusion of a modified potential further reduces the error to O(h 6)+(2 h 4).This revised version was published online in October 2005 with corrections to the Cover Date.  相似文献   

15.
The catalog of positions of 1007 stars (792 FK4 and FK4S stars, 57 OB stars, 49 NPZT stars, and 109 SAO stars) is presented. They were observed during the period from December 1984 to September 1985 with the Tokyo Photoelectric Meridian Circle (Tokyo PMC). The positions in the catalog are referred to the equinox and equator of J2000, and are based on the FK4 system. The internal errors of a single observation were estimated to be ( cos, )=(0.16, 0.19), whereas the mean internal errors of the catalog positions were (0.08, 0.08) for FK4 stars and (0.09, 0.11) for FK4S stars. A comparison of the positions of the FK4 stars in the present catalog with those of the FK4 catalog shows significant differences cos and in some declination zones. Some of those differences are commonly found in other recent catalogs. Thus they may be considered to be real systematic errors in the FK4 system. Neither significant magnitude nor color equations exist in the Tokyo PMC 85 catalog.  相似文献   

16.
In the present paper we prove a theorem giving rigorous estimates in the problem of bringing to normal form a nearly integrable Hamiltonian system, using methods of classical perturbation theory, i.e. series expansions in the small parameter . For any order of normalization, we give a lower bound * r for the convergence radius of the normalized Hamiltonian, and a greater bound for the remainder, i.e. the non normalized part of the Hamiltonian. As an application, we consider the case of weakly coupled harmonic oscillators with highly nonresonant frequencies and show how, by optimizing, for fixed , the orderr of normalization, one gets for the remainder a greater bound of the formAe –(*1/) a , with positive constantsA,a and 1 * exponential estimate of Nekhoroshev's type.  相似文献   

17.
A three-body problem is considered in which two masses, forming a close binary, orbit a comparatively distant mass. An asymptotic solution of this problem is presented, where the small parameter is related to the distance separating the binary and the remaining mass. Accepting certain model constraints, this solution is accurate within a constant errorO(11) and uniformly valid for time intervalsO(–3). Two specific examples are chosen to verify the literal solution: one relating to the Sun-Earth-Moon configuration of the solar system, the other to an idealized stellar system where the three masses are in the ratio 20:1:1. In both cases close agreement is found when the analytical solution is compared with an equivalent numerically-generated orbit.  相似文献   

18.
Infinite series expansions are obtained for the doubly averaged effects of the Moon and Sun on a high altitude Earth satellite, and the results used to interpret numerically integrated examples. New in this paper are: (1) both sublunar and translunar satellites are considered; (2) analytic expansions include all powers in the satellite and perturbing body semi-major axes; (3) the fact that retrograde orbits have more benign eccentricity behavior than direct orbits should be exploited for high altitude satellite systems; and (4) near circular orbits can be maintained with small expenditures of fuel in the face of an exponential driving force one forI ab, whereI b=180°–I a andI a is somewhat less than 39.2° for sublunar orbits and somewhat greater than 39.2° for translunar orbits.Nomenclature a semi-major axis - A lk coefficient defined in Equation (11) - B lk coefficient defined in Equation (24) - C km coefficient defined in Equation (25) - D, E, F coefficients in Equations (38), (39) - e eccentricity - H k expression defined in Equation (34) - expression defined in Equation (35) - I inclination of satellite orbit on lunar (or solar) ring plane - J 2 coefficient of second harmonic of Earth's gravitational potential (1082.637×10–6 R E 2 ) - K k, Lk, Mk expressions in Section 4 - expressions in Section 4 - p=a(1–e 2) semi-latus rectum - P l Legendre polynomial of degreel - q argument of Legendre polynomial - radial distance of satellite - R E Earth equatorial radius (6378.16 km) - R, S, W perturbing accelerations in the radial, tangential and orbit normal directions - syn synchronous orbit radius (42 164.2 km=6.6107R E) - t time - T satellite orbital period - T orbital period of perturbing body (Moon) - T e period of long periodic oscillations ine for |I|<I a - T s synodic period - U gravitational potential of lunar (or solar) ring - x, y, z Cartesian coordinates of a satellite with (x, y) being the ring plane - coefficient defined in Equation (20) - average change in orbital element over one orbit (=a, e, I, , ) - 1,23 unit vectors in thex, y, z coordinate directions - r , s , w unit vectors in the radial, tangential and orbit normal directions - =+ angle along the orbital plane from the ascending node on the ring plane to the true position of the satellite - angle around the ring - gravitational constant times mass of Earth (3.986 013×105 km s–2) - gravitational constant times mass of Moon (or Sun) - m gravitational constant times mass of Moon (/81.301) - s gravitational constant time mass of Sun (332 946 ) - ratio of the circumference of a circle to its diameter - radius of lunar (or solar) ring - m radius of lunar ring (60.2665R E) - s radius of solar ring (23455R E) - true anomaly - argument of perigee - 0 initial value of - i critical value of in quadranti(i=1, 2, 3, 4) - longitude of ascending node on ring plane This work was sponsored by the Department of the Air Force.  相似文献   

19.
This investigation presents the orbital elements of a satellite moving in a circular ring potential. The ring is considered to be of infinitesimal thickness and of unit radius. The components of the perturbing accelerations due to the ring potential have been substituded into the Gauss form of Lagrange's planetary equations to yield the first-order approximations. The elements of the orbit have been expressed by means of Hansen coefficients. The results include the effects produced by the 2nd, 4th, 6th, and 8th spherical harmonics. Due to their importance we present separately the secular terms from the periodic ones. The general expressions for the orbital elements can be easily extended to include the effects produced by any other higher harmonic.List of Symbols semi-major axis - C jK n (u, ) cosine functions ofu and - e eccentricity of the orbit - f sin2 - inclination of the orbit - M mean anomaly - n mean motion - p semi-latus rectum of the orbit - R, S, andW components of the perturbing acceleration - r magnitude of position vector - S jK n (u, ) sine functions ofu and - T time of periapse passage - u argument of latitude - U gravitational potential - V perturbing potential - G(M r +m) (gravitational constant times the sum of the masses of ring and satellite) - n, k coefficients ofR component of disturbing acceleration (functions off) - n, k coefficients ofS andW components of disturbing acceleration (functions off) - mean anomaly at timet=0 - X 0 n, m zero-order Hansen coefficients - argument of periapse - longitude of the ascending node  相似文献   

20.
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   

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