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1.
A two-parameter time transformationdt=r 3/2(α0+α1 r)?1/2 dτ is proposed, where τ is the radial distance while α0 and α1 are, if not constants, at least conservative functions of positions and velocities. In Keplerian systems, the quadrature implied by the transformation may by carried out by elliptic functions. When α0=0, τ is the eccentric anomaly; if α1=0, then τ is the intermediate or elliptic anomaly. Considering several values of α0 and α1, numerical examples of the relation of thegeneralized elliptic anomaly τ with the classical and elliptic anomalies are given. Application of this transformation to some perturbed Kepler problems is briefly outlined.  相似文献   

2.
This paper studies the motion of an infinitesimal body near the out-of-plane equilibrium points, L 6,7, in the perturbed restricted three-body problem. The problem is perturbed in the sense that the primaries of the system are oblate spheroids as well as sources of radiation and small perturbations are give to the Coriolis and centrifugal forces. It locates the positions and examines the stability of L 6,7 with a particular application to the binary system Struve 2398. It is observed that their positions are affected by the radiation, oblateness and a small perturbation in the centrifugal force, but is unaffected by that of the Coriolis force. They are also found to be unstable.  相似文献   

3.
This paper studies the motion of a test particle (infinitesimal mass) in the neighborhood of the triangular point L 4 in the framework of the perturbed relativistic restricted three-body problem (R3BP). The problem is perturbed in the sense that a small perturbation is given to the centrifugal force. It is found that the position and stability of the triangular point are affected by both the relativistic factor and a small perturbation in the centrifugal force.  相似文献   

4.
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5.
Tuominen  Jaakko 《Solar physics》1974,38(1):15-31
Oscillator strengths are calculated for 259 lines of Ti i by taking configuration interaction into account in a somewhat simplified treatment. Radial integrals are obtained by an adaptation of the scaled Thomas-Fermi method. The great majority of selected lines originate from slightly perturbed terms. From a satisfactory comparison of our g \({\text{f}}\) -values with recent experimental data and with solar results in the visible region, we have been able to extend the work for some infrared lines. Very few transition probabilities were known for Ti i in that spectral range. On the basis of our oscillator strengths, a mean value logN ti = 4.88 ± 0.12 (in the usual scale), is proposed for the solar photospheric abundance, in agreement with the meteoritic value.  相似文献   

6.
The motion of a satellite subject to an inverse-square gravitational force of attraction and a perturbation due to the Earth's oblateness as theJ 2 term is analyzed, and a uniform, analytic solution correct to first-order inJ 2, is obtained using a noncanonical approach. The basis for the solution is the transformation and uncoupling of the differential equations for the model. The resulting solution is expressed in terms of elementary functions of the independent variable (the ‘true anomaly’), and is of a compact and simple form. Numerical results are comparable to existing solutions.  相似文献   

7.
In this paper the full recurrent power series solution is established for J 2-gravity perturbed motion in terms of the Eulerian redundant parameters. Applications of the method for the problem of the final state prediction are illustrated by numerical examples of some typical ballistic missiles, a final state of very high accuracy is obtained for each case study.  相似文献   

8.
We investigate the evolution of a spherically symmetric dust-like cloud at the linear and nonlinear stages in the framework of ΛCDM models of the universe with nonzero three-space curvature. The evolution conditions are expressed for any redshift z ≥ 0 in terms of the amplitude δmin of the fluctuation which stops to expand at infinite time, the amplitude δ ta of the fluctuation which stops to expand at a given moment, and the amplitude δ c of the fluctuation which collapses at a given moment. These amplitudes are calculated as functions of cosmological model parameters and redshift. The ratios D nl r /D l of nonlinear amplitude estimates to linear ones and the typical fluctuation scales k nl /k are approximated by a function of the linear amplitude δ z .  相似文献   

9.
In this paper, the classical and generalized Sundman time transformations are used to establish new generating set of differential equations of motion in terms of the Eulerian redundant parameters. The implementation of this set on digital computers for the commonly used independent variables is developed once and for all. Motion prediction algorithms based on these equations are developed in a recursive manner for the motions in the Earth's gravitational field with axial symmetry whatever the number of the zonal harmonic terms may be. Applications for the two types of short and long term predictions are considered for the perturbed motion in the Earth's gravitational field with axial symmetry with zonal harmonic terms up to J 36 . Numerical results proved the very high efficiency and flexibility of the developed equations.  相似文献   

10.
In this paper, the connections between orbit dynamics and rigid body dynamics are established throughout the Eulerian redundant parameters, the perturbation equations for any conic motion of artificial satellites are derived in terms of these parameters. A general recursive and stable computational algorithm is also established for the initial-value problem of the Eulerian parameters for satellites prediction in the Earth's gravitational field with axial symmetry. Applications of the algorithm are considered for the two cases of short and long term predictions. For the short-term prediction, we consider the problem of the final state prediction of some typical ballistic missiles in the geopotential model with zonal harmonic terms up to J 36, while for the long-term prediction, we consider the perturbed J 2 motion of Explorer 28 over 100 revolutions.  相似文献   

11.
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   

12.
For an Oort cloud comet to be seen as a new comet, its perihelion must be moved from a point exterior to the loss cylinder boundary to a point interior to observable limits in a single orbit. The galactic tide can do this continuously, in a non-impulsive manner. Near-parabolic comets, with specific angular momentum , will most easily be made observable. Therefore, to reduce the perihelion distance H must decrease. Since weakly perturbed comets are, in general, more numerous than strongly perturbed comets, we can anticipate that new comets made observable by a weak tidal torque will more likely be first observed when their slowly changing perihelion distances are approaching their minimum osculating values under the action of the tide, rather than receding from their minimum values. That is, defining ΔHtide as the vector change due to the galactic tidal torque during the prior orbit, and Hobs as the observed vector, the sign S≡Sign(Hobs·ΔHtide) will more likely be −1 than +1 if a weak galactic tidal perturbation indeed dominates in making comets observable. Using comet data of the highest quality class (1A) for new comets (a>10,000 AU), we find that 49 comets have S=−1 and 22 have S=+1. The binomial probability that as many or more would exhibit this characteristic if in fact S=?1 were equally likely is only 0.0009. This characteristic also persists in other long-period comet populations, lending support to the notion that they are dominated by comets recently arrived from the outer Oort cloud. The preponderance of S=−1 also correlates with weakly perturbed (i.e., smaller semimajor axis) new comets in a statistically significant manner. This is strong evidence that the data are of sufficiently high quality and sufficiently free of observational selection effects to detect this unique imprint of the tide.  相似文献   

13.
While solutions for bounded orbits about oblate spheroidal planets have been presented before, similar solutions for unbounded motion are scarce. This paper develops solutions for unbounded motion in the equatorial plane of an oblate spheroidal planet, while taking into account only the J 2 harmonic in the gravitational potential. Two cases are distinguished: A pseudo-parabolic motion, obtained for zero total specific energy, and a pseudo-hyperbolic motion, characterized by positive total specific energy. The solutions to the equations of motion are expressed using elliptic integrals. The pseudo-parabolic motion unveils a new orbit, termed herein the fish orbit, which has not been observed thus far in the perturbed two-body problem. The pseudo-hyperbolic solutions show that significant differences exist between the Keplerian flyby and the flyby performed under the the J 2 zonal harmonic. Numerical simulations are used to quantify these differences.  相似文献   

14.
  1. The short-period terms of a second-order general planetary theory are removed through the Hori's method based on a development of the HamiltonianF in a Lie series which involves a determining functionS not depending upon mixed canonical variables as in the Von Zeipel's method but upon all the canonical variables resulting from the elimination of the short period terms ofF. Canonical variables adopted are the slow Delaunay variables. Eccentricitiese j and sines γj of the semi inclinations are respectively replaced by the Jacques Henrard variablesE j ,J j which lead to formulas remarkably simple.F is reduced to the sumF 0+F 1 of its terms of degrees 0,1 in small parameter ε of the order of the masses. Only one disturbing planet is considered.F 1 is not calculated beyond its terms of degree 3 inE j ,E j ,J j , the determining functionS 2 of degree 2 in ε not being therefore calculated beyond its terms of degree 2 inE′ j ,E j ,J j and the expressions of slow Delaunay canonical variables of the disturbed planetP 1 and the disturbing planetP 2 in terms of the new slow Delaunay canonical variables ofP 1 andP 2 which result from the elimination of the short period terms ofF 1 being therefore reduced to their terms of degree <1 in theE′ j ,E′ j ,J′ j . Calculation of the principal partF 1m ofF 1 is carried out through Laplace coefficients and operatorD=α(d/dα) applied to Laplace coefficients, α ratio of the semi major axis ofP 1 andP 2. Eccentricitye 2 of the disturbed planetP 2 is assumed to be zero, such an assumption not restricting our aim which is to investigate the mechanism of the elimination of short period terms in a second order general planetary theory carried out through the Hori's method, not to perform the elimination of those terms for a complete second order general planetary theory. Expressions of the slow Delaunay canonical variables in terms of the new ones resulting from the elimination of the short period terms ofF 1 are written down only for the disturbed planetP 1.
  2. Small divisors in 1/E′ 1 and 1/E′ 1 2 appear in the longitude ?1 of perihelia ofP 1. No small divisors appear in the other five slow Delaunay variables ofP 1. The only Jacques Henrard variables which appear in the longitude Ω1 of the ascending node ofP 1 are the J j′ j=1, 2 and no Jacques Henrard variables appear in the slow Delaunay canonical variablesX 1,Y 1,Z 1, λ1. The solving of the ten canonical equations ofP 1 andP 2 in the slow Delaunay canonical variablesX′ j ,Y′ 1,Z′ j ,λ′ j ,ω′ j ,Ω′ j resulting from the elimination of the short period terms ofF 1 reduces to that of four canonical equations inZ′ j ,©′ j and to six quadratures three of them expressing theX′ j ,Y′ 1 are constants and the three others expressingλ′ j ,?′ j as functions of timet. Solving of the four canonical equations inZ′ j ,Ω′ j reduces to that of a first order non linear differential equation and to two quadratures. Sinceγ′ 1 is then constant, so is the Jacques Henrard variableE′ 1. If the eccentricitye 2 ofP 2 is no more assumed to be zero, additive small divisors inE′ 2/E′ 2 1 appear in longitude ?′1 of perihelia ofP 1 and the solving of the twelve canonical equations ofP 1 andP 2 inX′ j ,Y′ j ,Z′ j ,λ′ j ,?′ j ,Ω′ j is reduced to that of eight canonical equations inY′ j ,?′ j ,Z′ j ,Ω′ j and to four quadratures expressingX′ j are constants andλ′ j as functions oft. Those eight canonical equations split into two systems of four canonical equations, one of them inY′ j ,?′ j and the other one inZ′ j ,Ω′ j . Each of those two systems is identical to the system inZ′ j ,Ω′ j corresponding toe 2=0 and its solving reduces to that of a first order non linear differential equation and to two quadratures identical to those of the casee 2=0.
  3. Expressions ofX 1,Y 1,Z 1,λ 1,? 1,Ω 1 as functions ofX′ j ,Y′ 1,Z′ j ,λ′ j ,?′ 1,Ω′ j ;j=1, 2 are sums of sines and cosines of the multiples ofλ′ j ,?′ 1,Ω′ j for the terms arising from the indirect partF 1j ofF 1, Fourier series in those sines and cosines or products of two such Fourier series for the terms arising from the principal partF 1m ofF 1, coefficients of those sums and Fourier series having one of the eight forms: $$A,{\text{ }}\frac{B}{{E'}},{\text{ }}\frac{C}{{E'^2 }},{\text{ }}D\frac{{j'^{2_1 } }}{{E'^{2_1 } }},{\text{ }}E\frac{{j'^{2_2 } }}{{E'^{2_1 } }},{\text{ }}F\frac{{j'^{_1 } j'^2 }}{{E'^{2_1 } }},{\text{ }}G\frac{{j'^2 }}{{j'^{_1 } }},{\text{ }}H\frac{{j'^{22} }}{{j'^{2_1 } }}{\text{.}}$$ A,..., H being constants which depend upon ratio α. Numerical calculation of the constantsA,..., H arising from the terms ofF 1j is easily carried out; that of theA,..., H arising from the terms ofF 1m require more manipulations, Fourier series in sines and cosines of the multiples ofλ′ j ,?′ j ,Ω ij and products of two such Fourier series having then to be reduced to sums of a finite number of terms and treated through the methods of harmonic analysis. Divisors inp+qα3/2;p, q relative integers, or products of such divisors appear inA,..., H.
  4. the method extends to the case whenF 1 is calculated beyond its terms of degree 3 in the Jacques Henrard variables.F 1 being calculated up to its terms of degree 8 in the Jacques Henrard variables which is the precision required to eliminate the short period terms of a complete second order general planetary theory,S 2 has to be calculated up to its terms of degree 7 and the expression of the slow Delaunay canonical variables ofP 1 andP 2 in terms of the slow Delaunay canonical variables ofP 1 andP 2 resulting from the elimination of the short period terms ofF 1 have, therefore, to be calculated up to their terms of degree 5 in the Jacques Henrard variables.
  相似文献   

15.
《Icarus》1987,70(2):269-288
We simulate the Oort comet cloud to study the rate and properties of new comets and the intensity and frequency of comet showers. An ensemble of ∼106 comets is perturbed at random times by a population of main sequence stars and white dwarfs that is described by the Bahcall-Soneira Galaxy model. A cloning procedure allows us to model a large ensemble of comets efficiently, without wasting computer time following a large number of low eccentricity orbits. For comets at semimajor axis a = 20,000 AU, about every 100 myr a star with mass in the range 1M⊙−2M⊙ passes within ∼10,000 AU of the Sun and triggers a shower that enhances the flux of new comets by more than a factor of 10. The time-integrated flux is dominated by the showers for comets with semimajor axes less than ∼30,000 AU. For semimajor axes greater than ∼30,000 AU the comet loss rate is roughly constant and strong showers do not occur. In some of our simulations, comets are also perturbed by the Galactic tidal field. The inclusion of tidal effects increases the loss rate of comets with semimajor axes between 10,000 and 20,000 AU by about a factor of 4. Thus the Galactic tide, rather than individual stellar perturbations, is the dominant mechanism which drives the evolution of the Oort cloud.  相似文献   

16.
A new non-singular analytical theory for the motion of near-Earth satellite orbits with the air drag effect is developed in terms of uniformly regular KS canonical elements. Diurnally varying oblate atmosphere is considered with variation in density scale height dependent on altitude. The series expansion method is utilized to generate the analytical solutions and terms up to fourth-order terms in eccentricity and c (a small parameter dependent on the flattening of the atmosphere) are retained. Only two of the nine equations are solved analytically to compute the state vector and change in energy at the end of each revolution, due to symmetry in the equations of motion. The important drag perturbed orbital parameters: semi-major axis and eccentricity are obtained up to 500 revolutions, with the present analytical theory and by numerical integration over a wide range of perigee height, eccentricity and inclination. The differences between the two are found to be very less. A comparison between the theories generated with terms up to third- and fourth-order terms in c and e shows an improvement in the computation of the orbital parameters semi-major axis and eccentricity, up to 9%. The theory can be effectively used for the re-entry of the near-Earth objects, which mainly decay due to atmospheric drag.  相似文献   

17.
The author's previous studies concerning the Ideal Resonance Problem are enlarged upon in this article. The one-degree-of-freedom Hamiltonian system investigated here has the form $$\begin{array}{*{20}c} { - F = B(x) + 2\mu ^2 A(x)\sin ^2 y + \mu ^2 f(x,y),} \\ {\dot x = - F_y ,\dot y = F_x .} \\ \end{array}$$ The canonically conjugate variablesx andy are respectively the momentum and the coordinate, andμ 2 is a small positive constant parameter. The perturbationf is o (A) and is represented by a Fourier series iny. The vanishing of ?B/?x≡B (1) atx=x 0 characterizes the resonant nature of the problem. With a suitable choice of variables, it is shown how a formal solution to this perturbed form of the Ideal Resonance Problem can be constructed, using the method of ‘parallel’ perturbations. Explicit formulae forx andy are obtained, as functions of time, which include the complete first-order contributions from the perturbing functionf. The solution is restricted to the region of deep resonance, but those motions in the neighbourhood of the separatrix are excluded.  相似文献   

18.
The Delaunay-Similar elements of Scheifele are applied to the problem of an Earth satellite that is perturbed by the Sun, Moon andJ 2. All three effects are assumed to be the same order of magnitude. Since the external body terms depend explicitly on time, the time element appears as an additional angle variable. Also, the eccentric anomaly is used as a noncanonical auxiliary variable. A solution to the first Von Zeipel equation allows simultaneous elimination of short and intermediate period terms. The canonical transformation to mean elements is defined by a generating function that is a series involving Bessel coefficients.  相似文献   

19.
Meteorite impacts onto a small satellite lead to the ejection of a regolith mass, which is much greater than the impactor mass, into cosmic space. Assume that an isotropic ejection with velocities smaller than the maximum possible velocity b took place at the time moment t 0. Since the orbital periods are unequal, the particle trajectories will densely fill a certain domain D. The same domain will be filled after an explosion of an artificial satellite moving in a high orbit. One to three months later, the node and pericenter longitudes will be distributed over the entire circle and the domain D will become a body of revolution, a topological solid torus. We examine the domain of possible particle motion and its boundary S immediately after the impact event (an unperturbed case) and the same domain under the assumption that the initial longitudes of nodes and pericenters were already a result of considerable changes (a perturbed case). In both cases, we managed to construct the domain D and its boundary S analytically: parametric equations containing only relatively simple functions were obtained for S. The basic topologic and differential-geometric properties of S were studied completely.  相似文献   

20.
From the equations of classical tidal theory with Newtonian cooling (Chapman and Lindzen, Atmospheric Tides: thermal and gravitational, Reidel, 1970), formulae are obtained for wind, temperature and pressure oscillations generated by thermal, gravitational and lower-boundary excitations of given frequency. The analysis is an extension of that of Butler and Small (Proc. R. Soc. Lond.A274, 91, 1963) who formulated solutions of the vertical structure equation in terms of two independent solutions of the homogeneous equation and derived expressions for surface pressure oscillations. A comprehensive formulation is presented for wind, temperature and pressure oscillations as functions of height with the above-mentioned sources of excitation and an upper-boundary radiation condition. The formulae obtained are applied at the surface leading to evaluations of the surface oscillation weighting function Wp(z) which weights the thermal excitation at height z according to its differential contribution to the surface oscillation. The formulae are shown to simplify at heights above a region of excitation and evaluations are undertaken of the thermal response weighting function Wt(z) which weights the thermal excitation at height z according to its differential contribution to the oscillation at any height above the region of thermal excitation. Computational procedures are described for obtaining two independent solutions of the homogeneous equation and results are presented for an adopted profile of atmospheric scale height. The problem of deriving the surface pressure oscillation due to a tidal potential is briefly reviewed and results are presented as an example of the application of formulae that have been derived.  相似文献   

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