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1.
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   

2.
The pulsed plasma probe technique has been expanded to include simultaneous determinations of absolute electron density, density fluctuations, electron temperature, and mean-ion-mass with resolution limited only by probe geometry, sheath size, and telemetry. The technique has been designed to test for coupling of electron density variations and ion composition irregularities in multi-component plasmas by the comparison of electron density fluctuation power spectraP N(k) and a newly-developed diagnostic parameter, the mean-ion-mass fluctuation spectraM i/M iP M(k). In addition, the experiment extends satellite-borne irregularity spectral analyses down to the 5–20 m range while attempting to identify F-region plasma instability processes on the basis of characteristics inN e,T e, N e,P N,M i, andP M. Initial results demonstrate the expanded diagnostic capability for high spatial resolution measurements of mean-ion-mass and provide experimental evidence for the role of ion composition in multi-stepped plasma instability processes. Specific results include a spectral indexX n inP N=A nf–X n of 1.6–2.9 over the wavelength range from 1 km to 6 m under conditions identified with an unstable equatorial nighttime ionosphere. Simultaneous measurements ofM i/M i(P M=A M f –X m) andN e/N e(P N=A n f –X n) have shown a general behavior tending to lower power (A m<A n) and softer spectra (X m<X n) in ion mass fluctuations when compared with fluctuations in total plasma density. Limited analyses of the two power spectral elements raise hopes for the differentiation between plasma mechanisms that can lead to similar indices inP N.Paper originally submitted to the journalSpace Science Instrumentation.  相似文献   

3.
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4.
We consider the problem of finding the generalized potential function V = U i(q 1, q 2,..., q n)q i + U(q 1, q 2,...;q n) compatible with prescribed dynamical trajectories of a holonomic system. We obtain conditions necessary for the existence of solutions to the problem: these can be cast into a system of n – 1 first order nonlinear partial differential equations in the unknown functions U 1, U 2,...;, U n, U. In particular we study dynamical systems with two degrees of freedom. Using adapted coordinates on the configuration manifold M 2 we obtain, for potential function U(q 1, q 2), a classic first kind of Abel ordinary differential equation. Moreover, we show that, in special cases of dynamical interest, such an equation can be solved by quadrature. In particular we establish, for ordinary potential functions, a classical formula obtained in different way by Joukowsky for a particle moving on a surface.Work performed with the support of the Gruppo Nazionale di Fisica Matematica (G.N.F.M.) of the Italian National Research Council.  相似文献   

5.
The number of equivalence classes of central configurations (abbr. c.c.) in the planar 4-body problem with three arbitrary and a fourth small mass is investigated. These c.c. are derived according to their generic origin in the 3-body problem. It is shown that each 3-body collinear c.c. generates exactly 2 non-collinear c.c. (besides 4 collinear ones) of 4 bodies with smallm 40; and that any 3-body equilateral triangle c.c. generates exactly 8 or 9 or 10 (depending onm 1,m 2,m 3) planar 4-body c.c. withm 4=0. Further, every one of these c.c. can be continued uniquely to sufficiently smallm 4>0 except when there are just 9; then exactly one of them is degenerate, and we conjecture that it is not continuable tom 4>0.Paper presented at the 1981 Oberwolfach Conference on Mathematical Methods in Celestial Mechanics.  相似文献   

6.
In this article we study the conditions for obtaining canonical transformationsy=f(x) of the phase space, wherey(y 1,y 2,...,y 2n ) andx(x 1,x 2,...,x 2m ) in such a way that the number of variables is increased. In particular, this study is applied to the rotational motion in functions of the Eulerian parameters (q 0,q 1,q 2,q 3) and their conjugate momenta (Q 0,Q 1,Q 2,Q 3) or in functions of complex variables (z 1,z 2,z 3,z 4) and their conjugate momenta (Z 1,Z 2,Z 3,Z 4) defined by means of the previous variables. Finally, our article include some properties on the rotational motion of a rigid body moving about a fixed point.  相似文献   

7.
General integral transform of the exponential integralsE n is considered and will be denoted asB (k) n (). Different expressions and the equations satisfied byB (k) n are developed. Two-term recurrence formula forB (k) n (0) and three-term recurrence formula forB (k) n (); 0 will be established for a givenk1 andn=2,3, ...,N. The computational algorithms based on these formulae are also constructed for the casesk=1,2,3, andn2. Finally the numerical results fork=2,3 andn=2(1)25 are presented to 15-digit accuracy  相似文献   

8.
In this paper, we consider the inverse problem of central configurations of n-body problem. For a given \({q=(q_1, q_2, \ldots, q_n)\in ({\bf R}^d)^n}\), let S(q) be the admissible set of masses denoted \({ S(q)=\{ m=(m_1,m_2, \ldots, m_n)| m_i \in {\bf R}^+, q}\) is a central configuration for m}. For a given \({m\in S(q)}\), let S m (q) be the permutational admissible set about m = (m 1, m 2, . . . , m n ) denoted
$S_m(q)=\{m^\prime | m^\prime\in S(q),m^\prime \not=m \, {\rm and} \, m^\prime\,{\rm is\, a\, permutation\, of }\, m \}.$
The main discovery in this paper is the existence of a singular curve \({\bar{\Gamma}_{31}}\) on which S m (q) is a nonempty set for some m in the collinear four-body problem. \({\bar{\Gamma}_{31}}\) is explicitly constructed by a polynomial in two variables. We proved:
  1. (1)
    If \({m\in S(q)}\), then either # S m (q) = 0 or # S m (q) = 1.
     
  2. (2)
    #S m (q) = 1 only in the following cases:
    1. (i)
      If s = t, then S m (q) = {(m 4, m 3, m 2, m 1)}.
       
    2. (ii)
      If \({(s,t)\in \bar{\Gamma}_{31}\setminus \{(\bar{s},\bar{s})\}}\), then either S m (q) = {(m 2, m 4, m 1, m 3)} or S m (q) = {(m 3, m 1, m 4, m 2)}.
       
     
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9.
New stacked central configurations for the planar 5-body problem   总被引：1，自引：0，他引：1  
A stacked central configuration in the n-body problem is one that has a proper subset of the n-bodies forming a central configuration. In this paper we study the case where three bodies with masses m 1, m 2, m 3 (bodies 1, 2, 3) form an equilateral central configuration, and the other two with masses m 4, m 5 are symmetric with respect to the mediatrix of the segment joining 1 and 2, and they are above the triangle generated by {1, 2, 3}. We show the existence and non-existence of this kind of stacked central configurations for the planar 5-body problem.  相似文献   

10.
Bianchi Type I string dust cosmological models in presence and absence of magnetic field following the techniques used by Letelier and Stachel, are investigated. To get the deterministic solution, we have assumed that σ 11 is proportional to the expansion (θ) where σ 11 is the eigen value of shear tensor (σ i j ) and which leads to A=N(BC)−n , n>0 where A,B,C are metric potentials and , N and ℓ are constants. The behaviour of the models in presence and absence of magnetic field are discussed. The other physical and geometrical aspects of the model are also discussed.  相似文献   

11.
We analyze the hypothesis of quantization in bands for the angular momenta of binary systems and for the maount of actionA c in stable and pulsating stars. This parameter isA c=Mv eff R eff, where the effective velocity corresponds to the kinetic energy in the stellar interior and the effective radius corresponds to the potential energyGM 2/R eff. Analogous parameters can be defined for a pulsating star withm=M where is the rate of the massm participating in the oscillation to the total massM andv osc,R osc the effective velocity and oscillation radius.From an elementary dimensional analysis one has thetA c (energy x time) (period)1/3 independently ifA c corresponds to the angular momentum in a binary system, or to the oscillation in a pulsating star or the inner energy and its time-scaleP eff in a stable star.From evolving stellar models one has that P effP eff(solar)1.22 hr a near-invariant for the Main Sequence and for the range of masses 0.6M <M<1.6M .With this one can give scalesn k=kn 1 withk integers andn 1=(P/P 1)1/3 withP 1=P eff1.22 hr. In these scales proportional toA c, one sees that the periods in binary and pulsating stars are clustered in discrete unitsn 1,n 2,n 3, etc.This can be seen in pulsating Scuti, Cephei, RR Lyrae, W Virginis, Cephei, semi-regular variables, and Miras and in binary stars as cataclysmic binaries, W Ursa Majoris, Algols, and Lyrae with the corresponding subgroups in all these materials. Phase functions (n k) in RR Lyrae and Cephei are also associated with discrete levelsn k.the suggested scenario is that the potential energies and the amounts of actionE p(t), Ac(t) are indeed time-dependent, but the stars remain more time in determinated most proble states. The Main Sequence itself is an example of this. These most probable states in binary systems, or pulsating or stable stars, must be associated with velocities sub-multiplesc/ F , given by the velocity of light and the fine structure constant.Additional tests for such a hypothesis are suggested when the sufficient amount of observational data are available. They can made with oscillation velocities in pulsating stars and velocity differences of pairs of galaxies.  相似文献   

12.
We study a theory for the ninth satellite of Saturn, Phoebe, based on the literal solution we have obtained in the main problem of the lunar theory.These series were computed by solving, by successive approximations, the Lagrange's equations expressed in variables, functions of the elliptic elements.We may consider the case of Phoebe simpler than a lunar case because we seek less precision (1/10 geocentric) than in the Lunar case, although the eccentricity of Phoebe is stronger.Main problem: our series are computed to the complete seventh order and a great part of the perturbations of the eighth and ninth order, where we have attributed to the small lunar parameters the order 1 tom 0=n/n 0,e 0,e, sin (i 0/2), the order 2 to 0=(a 0/a)((M 1–)/(M 1+M)) and the order 4 toµ 0(a 0/a)M 1 M/M 1 2 –M 2.In the case of Phoebe,µ 0 equal zero and ±0 is the ratioa 0/a.We study the further development of these series by using, instead of parameterm 0, the quantity m 0=n/n 0–m 1 wherem 1 is an approached value ofm 0, in order to accelerate the convergence of the series with respect tom 0.Comparison with a numerical integration we are adjusting a numerical integration to the observations. We have already more than 100 observations, for the period 1900–1957.At first, we compare the series of the main problem to a numerical integration of the Keplerian problem.

Proceedings of the Conference on Analytical Methods and Ephemerides: Theory and Observations of the Moon and Planets. Facultés universitaires Notre Dame de la Paix. Namur, Belgium, 28–31 July, 1980.  相似文献   

13.
T. Takakura 《Solar physics》1984,91(2):311-324
In some gradual hard X-ray bursts with high intensity, hard X-ray source (15–40 keV) is steadily located in the corona along with softer X-ray source (5–10 keV).Two stationary models, high density and high temperature models, are proposed to solve the difficult problem of confinement of hot (or nonthermal) plasma in the direction of the magnetic field along the loops in the corona. In both models, an essential point is that the effective X-ray source is composed of fine dense filamentary loops imbeded in a larger rarefied coronal loop, and the electron number density in the filaments is so high as 1011–1012 cm-3. If the density is so high heat conduction can be as reasonably small as of the order of 1027 erg s -1 for the given emission measures of observed X-rays, since the required cross-sectional area is small and also classical conduction is valid. Collisional confinement of thermal tail, and nonthermal electrons if any, up to 50–60 keV in the filaments is also possible, so that the hard X-ray images can be loop like structure instead of double source (foot points).High density model is applicable to the coronal filamentary loops with temperature T m < 5 × 107 K at the loop summit. The heat flow from the summit downwards is lost almost completely by the radiation from the loop during the conduction to the foot points. A continuous energy release is assumed near the summit to maintain the stationary temperature T m, and pressure balance is maintained along the loop. In this model, the number density at the summit is given by n m - 106 T m 2 /sm, where s m is the length of the loop from the summit to the foot point, and the distribution of temperature and density along the loop are given by T = T m(s/sm)1/3 and n = n m(s/sm)-1/3, respectively.High temperature model is applicable to the filamentary loops with higher temperature up to about 108.5 K and comparatively lower number density as 1011 cm-3 for the requirement of magnetic confinement of the hot plasma in radial direction. The radiation from the loop is negligibly small in this model so that the heat flux is nearly conserved down to the foot points. In this case, temperature gradient is smaller than that of the high density model, depending on the tapering of the magnetic bottle.In both models, the differential emission measure is maximum at the highest temperature T m and the brightness distribution along the loop shows a maximum around the summit of the loop if some magnetic tapering is taken into account.  相似文献   

14.
Letn2 mass points with arbitrary masses move circularly on a rotating straight-line central-configuration; i.e. on a particular solution of relative equilibrium of then-body problem. Replacing one of the mass points by a close pair of mass points (with mass conservation) we show that the resultingN-body problem (N=n+1) has solutions, which are periodic in a rotating coordinate system and describe precessing nearlyelliptic motion of the binary and nearlycircular collinear motion of its center of mass and the other bodies; assuming that also the mass ratio of the binary is small.  相似文献   

15.
In this paper we find a class of new degenerate central configurations and bifurcations in the Newtonian n-body problem. In particular we analyze the Rosette central configurations, namely a coplanar configuration where n particles of mass m1 lie at the vertices of a regular n-gon, n particles of mass m2 lie at the vertices of another n-gon concentric with the first, but rotated of an angle π /n, and an additional particle of mass m0 lies at the center of mass of the system. This system admits two mass parameters μ = m0/m1 and ε = m2/m1. We show that, as μ varies, if n > 3, there is a degenerate central configuration and a bifurcation for every ε > 0, while if n = 3 there is a bifurcation only for some values of ε.  相似文献   

16.
The adiabatic theory of interaction between high and low frequency waves has been studied for the case of electron plasma oscillations and ion acoustic waves and the results are applied to the solar wind. The modified dispersion relation for ion acoustic waves has been derived, taking a Gaussian distribution for plasmons. Two limiting cases of the spectrum are studied. For a broad spectrum, the plasma turbulence has a destabilising effect by introducing a growth rate denoted by turbulence, which is positive for k 0 > (m e/ m i )1/2 De –1 , k 0 being the central wave numger of the spectrum, De the electron Debye length. Also, even for v d(drift velocity between electrons and ions) < c s, we arrive at unstable ion acoustic modes. For narrow spectrum, the plasma turbulence has a stabilising effect.  相似文献   

17.
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18.
Morozhenko  N. N. 《Solar physics》1974,34(2):313-322
Observations of the lines He i 3888 and H8 in 80 quiescent prominences by the author, and in other prominences by Kubota et al. (1972) and Morozhenko (1971), have been used to derive the dependences of I(3888)/I(H8) on I(H8), N 2 3 s on 0 (H), and N + n e on 0(H) (Figures 1, 2, 3 and 4). The equations of ionization equilibrium and triplet system steady state for a helium atom (27 levels and continuum were considered) were solved together with the radiation transfer equation in the helium Lyman continuum. As given n e () distribution with depth and T e =7500 K were assumed. The 23 S level population N2 3 s, helium emission measure N + n e and the intensity ratios of the He i 3888 and H8 lines were calculated and compared with observation (Figures 2, 3 and 4, solid lines). The figures show that in bright prominences the observed values of N 2 3 s and N + n e are systematically higher than the calculated ones. These deviations cannot be eliminated by decreasing n e . One can make the calculations and observations agree for bright prominences by increasing the UV radiation which penetrates into the prominence.  相似文献   

19.
The method of obtaining the estimates of the maximalt-interval ( –, +) on which the solution of theN-body problem exists and which is such that some fixed mutual distance (e. g. 12) exceeds some fixed non-negative lower bound, for allt contained in ( –, +), is considered. For given masses and initial data, the increasing sequences of the numbers k , each of which provides the estimate + > k , are constructed. It appears that if + = +, then .  相似文献   

20.
Equations for the chemical evolution of the Galaxy are derived, accounting for (i) the dynamical evolution of the Galaxy (i.e. the collapse of the proto-galaxy), and (ii) either a variable mass-spectrum in the birth-rate stellar function of the type (m, t)=(t)(m, t), or a constant mass-spectrum with variable lower mass limit for star birth:m mf=mmf(Z). Simple equations are adopted for the collapse of the proto-galaxy, accounting for the experimental data (i.e. axial ratio and major semi-axis) relative to the halo and to the disk, and best fitted for a rapid collapse; gas density is assumed to be always uniform. Numerical computations of several cases show that there is qualitative agreement with the experimental data relative to theZ(t) function when: (i) the mass-spectrum is nearly constant in time: (m, t)(m)=m –2.35; (ii) the efficiency (t) is sufficiently high; moreover, the super metallic effect (SME) takes place for greater than a given value (1.5); (iii) the shorter the collapse timeT c, the more rapid is the initial increase of metallicity, the asymptotic value being left nearly unaltered. The theoretical present-day values of gas density and metallicity so obtained differ from the experimental values by a factor of 2 or 3. Leaving aside other possible explanations, such a discrepancy is within the range of the uncertainties concerning the amount of gas returned back into space by the decay of the stars. Our theoretical results are not in complete agreement with the observed data bearing on theN n(Z) function (N n is the number of stars whose Main-Sequence lifetime is not less than the age of the Galaxy), while a hypothesis of star formation with different efficiencies in different zones of the Galaxy, and successive stellar mixing from zone to zone, is not inconsistent with such data.  相似文献   

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