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1.
Using Euler's equation of motion, the equation for disturbed fluid motion against a hydrostatic equilibrium has been derived, and the nonequilibrium dynamical equation of a P-PI nuclear reaction system driven by He3 has been analysed using developed nonequilibrium theory. We find that the system in the solar core is unstable in the layer extending from about 0.2R to 0.4R if the core is disturbed by fluid motion; this instability may be related to thermal diffusion.  相似文献   

2.
It is shown that the equation of motion Du j/Ds = (e/mc 2)F ji u i , a natural generalization to the curved spacetime of the Heaviside-Lorentz law of ponderomotive force, is equivalent to the metric independent and covariant Van Dantzig's equations of motion dx j [jpi] = 0 or L v p i = 0, where p i is the conjugate momentum 4-vector and v a vector determined by the condition p i v i, only with respect to holonomic coordinates. With respect to an anholonomic system, the Heaviside-Lorentz equation is a particular case of the VD equations valid for a privileged class of anholonomic frames, those consisting of orthogonal unit vectors.  相似文献   

3.
The Kepler problem for the resistive force r/r 2 is known to have a conserved vector which is the analogue to Hamilton's vector for the standar Kepler problem. In this note it is shown in a very elementary way that many similar force laws display the same property. The orbit equation can be obtained easily in such cases.  相似文献   

4.
We consider a horizontally stratified isothermal model of the solar atmosphere, with vertical and uniform B 0, and v A 2 v s 2 . The equations of motion are linearized about a background which is in hydrostatic equilibrium. A homogeneous wave equation results for the motions perpendicular to B 0; this wave equation is similar to the equation for the MHD fast mode. On the other hand, the equation for the parallel motions is inhomogeneous, containing driving terms which arise from the presence of the fast mode; the homogeneous form of this equation is identical to the equation describing vertically-propagating gravity-modified acoustic waves. We demonstrate that a resonance can exist between the (driving) fast wave and the (driven) gravity-modified acoustic wave, in such a way that very large parallel velocities can be driven by small perpendicular velocities. Applications of this resonance to solar spicules, jets, and other phenomena are discussed.The National Center for Atmospheric Research is sponsored by the National Science Foundation.  相似文献   

5.
Interaction between orbital motion and attitude libration dynamics of an arbitrary rigid body moving in a central Newtonian field is considered to second order. Advantage is taken of the decoupling between inplane-pitch and roll-yaw out-of-plane motion to restrict the motion to the orbital plane by an appropriate choice ofinitial conditions. An averaged solution to the nonlinear inplane-pitch equations whose accuracy is determined by ignoring terms of order {·G32/a 2, 2,2,G32/a 2} and higher is presented. The results show that the near-resonant motion is characterized by a periodic interchange of energy between the attitude and orbital motion.Associate Professor, Department of Aeronautics and Astronautics.  相似文献   

6.
The Kustaanheimo-Stiefel (KS) transformation is shown to follow naturally from the general solution of the two-body motion if half-arguments are introduced. Application to collision orbits and to the exact triangular solutions of Lagrange (vide E. Stiefel and G. Scheifele: 1971,Linear and Regular Celestial Mechanics, Springer, Berlin-Heidelberg-New York, p. 23–35).Notations x Position vector (x, y, z) - r=|x| Distance from the origin - 1/2h Energy constant or Kepler motion - c Angular momentum vector of Kepler motion - t physical time ()·=d/dt () - new independent variable ()=d/d () Note by editor: This is the well-known Three-dimensional regularization, published in 1965 by P. Kustaanheimo and E. Stiefel, Perturbation Theory of Kepler Motion Based on Spinor Regularization,J. reine angewandte Mathematik 218, 204. The present article was written during Professor Volk's stay at the Zurich Technische Hochschule in 1972, when he also celebrated his 80th birthday.  相似文献   

7.
Using a method previously applied to the treatment of the Mathieu differential equation, we solve the Hill's differential equation of lunar theory through the way of operational calculus, which avoids the cumbersome infinite determinants of the classical procedure. The one-sided Laplace transformation changes it into a difference equation with an infinite number of terms and variable coefficients. When its first member is divided by a suitable factor, this difference equation is the image of an integral equation of the Volterra type which is equivalent to the initial Hill's differential equation. Solution of this Volterra integral equation is unique and it is the general solution of the Hill's differential equation. This solution is a series in the powers of a small dimensionless parameter 2 which appears as a factor in the second member of the Hill's differential equation. We reduce it to the sum of its terms of degree 12 with respect to which is the precision usually required in a lunar theory and we write down effectively the coefficients of the terms in 2, ( 2)2 and the coefficient of the term in ( 2)3 which depends upon the initial valuey(0) of the Hill's differential equation.  相似文献   

8.
Expansions of the functions (r/a)cos jv and (r/a)m sin jv of the elliptic motion are extended to highly eccentric orbits, 0.6627 ... <e<1. The new expansions are developed in powers of (e–e*), wheree* is a fixed value of the eccentricity. The coefficients of these expansions are expressed in terms of the derivatives of Hansen's coefficients with respect to the eccentricity. The new expansions are convergent for values of the eccentricity such that |e–e*|<(e*), where the radius of convergence (e*) is the same of the extended solution of Kepler's equation. The new expansions are intrinsically related to Lagrange's series.  相似文献   

9.
Letx 0 (t),x 0 4 be a homothetic solution of the planar three-body problem with total energyh, described in relative coordinates with respect to one body. It is shown that the variational equation of the problem atx 0 (t) can be solved explicitly in terms of hypergeometric functions. This is done by using the scaled true anomaly of the one-dimensional Kepler motion as the independent variable.The classical theorems about hypergeometric functions allow a simple calculation of all the values needed in applications. By means of this theory the past of a homothetic triple close encounter may be described in a linearized approximation.Proceedings of the Sixth Conference on Mathematical Methods in Celestial Mechanics held at Oberwolfach (West Germany) from 14 to 19 August, 1978.  相似文献   

10.
The motion and radiation of relativistic particles with radiation reaction in a strong magnetic field has been considered. The kinetic equation determining the relaxation of the distribution function with radiation reaction has been investigated. The universal one-dimensional distribution function is found to which any isotropic ultrarelativistic distribution in a strong magnetic field is relaxed. It is of power type –3 for ultrarelativistic energies mc 2. Estimations are made which indicate that under the pulsar conditions the one-dimensional electron distribution function is likely formed due to radiation losses while for ions the one-dimensionalization is associated with the conservation of the adiabatic invariant.  相似文献   

11.
The Schwarzschild field of a central massM is used to derive the general relativistic motion of a particle in a bounded orbit aroundM. A quadrature gives the central angle as a quasi-periodic function (f) of an effective true anomalyf. The linear term is an infinite series, whose second term yields the usual rate of advance of pericenter. For an artificial satellite this may be as large as 17 of arc per year. The periodic part is a sine series, with coefficients containing the small parameter 2GM/c 2 p, wherep is closely approximated by the classical semi-latus rectum. The radius vectorr is a Kepler-like function off.The essentially new features of the calculation are the appropriate factoring of a certain cubic polynomialF(p/r), the use of the above effective true anomalyf, and the introduction of an effective eccentric anomalyE. The latter serves to reduce the differential equation forf as a function of the timet, obtained by combining the solution for (f) with the relativistic integrals of motion, to a Kepler equation forE.Knowing the constants of the motion, one can then solve successively forE(t), f(t), r(t), and (t). This is best done as a variational calculation, comparing the relativistic orbiter with a classical orbiter having the same initial conditions. The resulting variations agree with those of Lass and Solloway, but the present method is quite different from theirs and results in a simpler algorithm. The results show that the radial and transverse corrections, r andr , arising from the Schwarzschild field, may be of the order of a kilometer for 1000 revolutions of an Earth satellite of orbital eccentricitye 00.6.For bounded motion, the cubic polynomialF(p/r) has three positive real zeros, the two smaller ones corresponding to apocenter and pericenter. The third and apparently non-physical one occurs forrSchwarzschild radius. It may thus correspond to the incipient fall of the orbiter into the central body, if the latter is a black hole.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 27–September 2, 1972.Research sponsored by NASA Goddard Space Flight Center under Contract No. NAS 5-11909.  相似文献   

12.
As an outcome of our previous notes [13, 14] on the quaternion regularization of the classical Kepler problem and pre-quantization of the Kepler manifold we show, first, that both the cross product of two quaternions and the cross product of their anti-involutes are susceptible of a simple geometrical representation in the ordinary 3-dimensional euclidean spaceR 3 and, secondly, that they satisfy anSO(4)-invariant relation that implies projection of curves from the quaternion space onto the spaceR 3. ThisSO(4)-invariance allows—in the particular case of orthogonal quaternions of equal norm—a straight derivation: (i) of the correspondence between the free motion on the surface of a sphereS 3 and the physical elliptical Kepler motion (collisions included) on a plane denoted by w ; (ii) of the celebrated Kepler equation and (iii) of the Levi-Civita regularizing time transformation. With (i) and (ii) we recover some of Györgyi's [3] results. The aforesaid orbital plane w and the orbital plane *, arrived at independently by exploiting the Kustaanheimo-Stiefel regularizing transformation, are shown to be inclined exactly at an angle characterizing the ratio of the semi-axes of the elliptical orbits and intimately related to the cross product representation. Thus the eventual superimposition of the two planes confirms the intimate connection between the various regularization procedures—transforming the classical Kepler problem into the geodesic flow onS 3—and the Fock's procedure for the quantum theoretical Kepler problem of the hydrogen atom (accidental degeneracy).This research was supported by the Consiglio Nazionale delle Ricerche of Italy (C.N.R.-G.N.F.M.).  相似文献   

13.
A direct approach of the dynamical equation for the evolution of the two-point density correlation function is given in an expanding flat Friedmann Universe in the Newtonian approximation. If the third and higher moments are neglected, a wave-like equation of third-order for the two-point density correlation function is found. The exact solution of this equation shows, in the large time limit, the usual Jeans instability t 4/3. It is suggested that the highern-point correlation function of the density grow liket 2n/3 in the same approximation. This indicates that every truncation procedure of the hierarchy of the equations is inapplicable at least for large timest.  相似文献   

14.
Starting from the Vlasov equation the steady state and stability properties of the electron sheet in the Cowley neutral sheet model of the geomagnetic tail are considered. Electrostatic ion plasma oscillations propagating from dusk to dawn are found to be unstable provided the thermal spread normal to the current is sufficiently large. Assuming the population of the neutral sheet to be supplied by the polar wind it is shown how a localisation of the cross tail electric field could lead to the instability first appearing around midnight. It is conjectured that the localisation of the cross tail electric field could continually feed the instability, so leading to enough turbulence to give enhanced reconnection of the magnetic field.List of symbols f distribution function - B magnetic field strength far from the neutral sheet - a sheet half thickness - total potential drop across the tail which is localised to the dusk end of the tail in Cowley's model - potential for the steady state electric field normal to the electron current sheet. This potential exists in that region of the tail that excludes the localised region of cross tail electric field - average velocity across the tail of electrons in the current sheet - v average velocity of the electrons normal to the current sheet - p canonical momentum of a particle - energy of a particle - KT electron energy normal to the sheet (1/2m e v 2 ) - KT i ion energy (1/2m i V 2 ) - electron gyrofrequency far from the neutral sheet - i ion gyrofrequency far from the neutral sheet - Ay steady state vector potential for the magnetic field - A –Ay/aB 0 (normalised vector potential) When perturbing the steady state, dashes have been used to denote the time dependent first order quantities. Where no confusion could arise the dashes are dropped, e.g.Ey=Ey since there is no zero orderEy in the region considered in the stability analysis.  相似文献   

15.
The precise numerical integration of Cowell's equations of satellite motion is frequently performed with an independent variables defined by an equation of the form dt=cr n ds, wheret represents time,r the radial distance from the center of attraction,c is a constant, andn is a parameter. This has been primarily motivated by the uniformizing effects of such a transformation resulting in desirable analytic stepsize control for elliptical orbits. This report discusses the proper choice of the parametern defining the independent variables for various types of orbits and perturbation models, and develops a criterion for its selection.  相似文献   

16.
Jun-Ichi Sakai 《Solar physics》1989,120(1):117-124
We report on the results of plasma jet and shock formation during the current loop coalescence in solar flares. It is shown by a theoretical model based on the ideal MHD equation that the spiral, two-sided plasma jet can be explosively driven by the plasma rotational motion induced during the two current loop coalescence process. The maximum velocity of the jet can exceed the Alfvén velocity, depending on the plasma (= c s 2 v A 2 ) ratio. The acceleration time getting to the maximum jet velocity is quite short and le than 1 s. The rebound following the plasma collapse driven by magnetic pinch effect can strongly induce super-Alfvénic flow. We present the condition of the shock formation. We briefly discuss the high-energy particle acceleration during the plasma collapse as well as by the shocks.  相似文献   

17.
By use of the dispersion equation given by Song, Wu, and Dryer (1987) for a cylinder plasma with mass motion and gravity included, we investigate the linear current instabilities developed in loop prominences. The results indicate that the mode of linear instability depends mainly on whetherv s 2 > or not, wherev s is the sonic velocity at heightz, =GM/(R +z) is the gravity potential,G the gravitational constant,M andR the mass and the radius of the Sun respectively. Ifv s 2 > , then the sausage instability will be dominant. Otherwise, the kink instability will be more important. A possible explanation of knot structure, which appears sometimes in solar loop prominences has been given.  相似文献   

18.
For a given family of orbits f(x,y) = c * which can be traced by a material point of unit in an inertial frame it is known that all potentials V(x,y) giving rise to this family satisfy a homogeneous, linear in V(x,y), second order partial differential equation (Bozis,1984). The present paper offers an analogous equation in a synodic system Oxy, rotating with angular velocity . The new equation, which relates the synodic potential function (x,y), = –V(x, y) + % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSqaaSqaai% aaigdaaeaacaaIYaaaaaaa!3780!\[\tfrac{1}{2}\]2(x 2 + y 2) to the given family f(x,y) = c *, is again of the second order in (x,y) but nonlinear.As an application, some simple compatible pairs of functions (x,y) and f(x, y) are found, for appropriate values of , by adequately determining coefficients both in and f.  相似文献   

19.
We consider the spatially flat Friedmann model For a ≈ tp, especially, if p ≥ 1, this is called power-law inflation. For the Lagrangian L = Rm with p = − (m − 1) (2m − 1)/(m − 2) power-law inflation is an exact solution, as it is for Einstein gravity with a minimally coupled scalar field ϕ in an exponential potential V(ϕ) = exp (μϕ) and also for the higher-dimensional Einstein equation with a special Kaluza-Klein ansatz. The synchronized coordinates are not adapted to allow a closed-form solution, so we write The general solutions reads Q(a) = (ab + C)f/b with free integration constant C (C = 0 gives exact power-law inflation) and m-dependent values b and f: f = −2 + 1/p, b = (4m − 5)/(m − 1). Finally, special solutions for the closed and open Friedmann model are found.  相似文献   

20.
The analysis of a homogeneous sample of 108 Abell clusters has led to an average peculiar velocity for the center of mass motion of these clusters of 610±750 km s–1. From this result, an upper limit for the average mass of the Abell clusters of (1.6±2.4)×1015 M was obtained under the assumption that the peculiar motion is due to the excess of neighbours with respect to an uniform background. A lower limit of (2.42.9) x 1014 h -10.4 M was derived if one assumes that the peculiar velocity results from the mutual acceleration with the nearest neighbour.  相似文献   

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