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1.
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   

2.
The classic Lagrange's expansion of the solutionE(e, M) of Kepler's equation in powers of eccentricity is extended to highly eccentric orbits, 0.6627 ... <e<1. The solutionE(e, M) is developed in powers of (e–e*), wheree* is a fixed value of the eccentricity. The coefficients of the expansion are given in terms of the derivatives of the Bessel functionsJ n (ne). The expansion is convergent for values of the eccentricity such that |e–e*|<(e*), where the radius of convergence (e*) is a positive real number, which is calculated numerically.  相似文献   

3.
In this paper we derive some recurrence formulae which can be used to calculate the Fourier expansions of the functions (r/a) n cosmv and (r/a) n sinmv in terms of the eccentric anomalyE or the mean anomalyM. We also establish a recurrence process for computing the series expansions for alln andm when the expansions of two basic series are known. These basic series were given in explicit form in the classical literature. The recurrence formulae are linear in the functions involved and thus make very simple the computation of the series.This work was supported by NASA contract No. NASr 54(06).—The paper was presented at the AIAA/AAS meeting, Princeton University, August 1969.  相似文献   

4.
An efficient algorithm is presented for the solution of Kepler's equationf(E)=E–M–e sinE=0, wheree is the eccentricity,M the mean anomaly andE the eccentric anomaly. This algorithm is based on simple initial approximations that are cubics inM, and an iterative scheme that is a slight generalization of the Newton-Raphson method. Extensive testing of this algorithm has been performed on the UNIVAC 1108 computer. Solutions for 20 000 pairs of values ofe andM show that for single precision (10–8) 42.0% of the cases require one iteration, 57.8% two and 0.2% three. For double precision (10–18) one additional iteration is required. Single- and double-precision FORTRAN subroutines are available from the author.  相似文献   

5.
A qualitative solution is presented of the critical inclination problem in artificial satellite theory for motions in which the orbits are nearly circular. The effects of all the zonal harmonics are taken into account, and bothshallow anddeep resonance regimes are considered. An investigation of the (e sing,e cosg)-plane reveals that six fundamentally different types of phase-plane portraits exist. These portraits illustrate the long-term behaviour of the eccentricity and line of apsides.  相似文献   

6.
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   

7.
New expansions of elliptic motion based on considering the eccentricitye as the modulusk of elliptic functions and introducing the new anomalyw (a sort of elliptic anomaly) defined byw=u/2K–/2,g=amu–/2 (g being the eccentric anomaly) are compared with the classic (e, M), (e, v) and (e, g) expansions in multiples of mean, true and eccentric anomalies, respectively. These (q,w) expansions turn out to be in general more compact than the classical ones. The coefficients of the (e,v) and (e,g) expansions are expressed as the hypergeometric series, which may be reduced to the hypergeometric polynomials. The coefficients of the (q,w) expansions may be presented in closed (rational function) form with respect toq, k, k=(1–k 2)1/2,K andE, q being the Jacobi nome relatedk whileK andE are the complete elliptic integrals of the first and second kind respectively. Recurrence relations to compute these coefficients have been derived.on leave from Institute of Applied Astronomy, St.-Petersburg 197042, Russia  相似文献   

8.
Elemental abundances of the VH group of cosmic radiation have been measured in the energy interval 250–550 MeV nucl–1 in a balloon exposure at Sioux Falls (South Dakota) of a plastic detector LeXAN stack. The so obtained abundances have been extrapolated to the sources in the frame of the homogeneous model correcting for energy loss. After taking into account solar modulation, the best fit to model values has led to a escape mean free path e = 5E –0.4 g cm–2, whereE is the energy in GeV nucl–1, forE>1 GeV nucl–1, and a constant e = 5 g cm–2 forE1 GeV nucl–1. When turning to the diffusion model, also including an energy loss term, a diffusion coefficientD=3×1028 cm2 s–1 has been estimated.  相似文献   

9.
In extending the results of Henon and Petit (1986) an algorithm is suggested for constructing the series representing the general encounter-type solution of the spatial eccentric Hill's problem. The series are arranged in powers of the eccentricity E of Hill's problem and two integration constants e and k characterizing eccentricity and inclination of the relative motion. A particular non-periodic solution of Henon and Petit corresponding to E = e = k = 0 is taken as an intermediary. The perturbations to this solution are constructed similar to the lunar theory of Hill and Brown.  相似文献   

10.
We describe a method for the analysis of magnetic data taken daily at the Vacuum Telescope at Kitt Peak. In this technique, accurate position differences of very small magnetic features on the solar surface outside active regions are determined from one day to the next by a cross-correlation analysis. In order to minimize systematic errors, a number of corrections are applied to the data for effects originating in the instrument and in the Earth's atmosphere. The resulting maps of solar latitude vs central meridian distance are cross-correlated from one day to the next to determine daily motions in longitude and latitude. Some examples of rotation and meridional motion results are presented. For the months of May 1988 and October–November 1987, we find rotation coefficients A = 2.894 ± 0.011, B = - 0.428 ± 0.070, and C = -0.370 ± 0.077 in rad s–1 from the expansion = A + B sin2 + C sin4, where is the latitude. The differential rotation curve for this interval is essentially flat within 20 deg of the equator in these intervals. For the same intervals we find a poleward meridional motion a = 16.0 ± 2.8 m sec -1 from the relation v = a sin, where v is the line-of-sight velocity.Operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.  相似文献   

11.
An outline of a new method is given to obtain the expansion of –S by the method of W. M. Smart, using Taylor theorem and neglecting powers higher than the fourth in the eccentricity and inclination.  相似文献   

12.
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   

13.
We emphasize the sharp distinctions between different one-body gravitational trajectories made by the ratio of time averagesR(t)–E kin/E pot.R is calculated as a function of the eccentricity (e) and of the energy (E). Whent, independently ofe andE, R1/2 for closed orbits (this clearly illustrates the fulfillment of the virial theorem in classical mechanics); whereasR1, at any time, for open orbits.  相似文献   

14.
Colliding comets in the Solar System may be an important source of gamma ray bursts. The spherical gamma ray comet cloud required by the results of the Venera Satellites (Mazets and Golenetskii, 1987) and the BATSE detector on the Compton Satellite (Meeganet al., 1992a, b) is neither the Oort Cloud nor the Kuiper Belt. To satisfy observations ofN(>P max) vsP max for the maximum gamma ray fluxes,P max > 10–5 erg cm–2 s–1 (about 30 bursts yr–1), the comet density,n, should increase asn a 1 from about 40 to 100 AU wherea is the comet heliocentric distance. The turnover above 100 AU requiresn a –1/2 to 200 AU to fit the Venera results andn a 1/4 to 400 AU to fit the BATSE data. Then the masses of comets in the 3 regions are from: 40–100 AU, about 9 earth masses,m E; 100–200 AU about 25m E; and 100–400 AU, about 900m E. The flux of 10–5 erg cm–2 s–1 corresponds to a luminosity at 100 AU of 3 × 1026 erg s–1. Two colliding spherical comets at a distance of 100 AU, each with nucleus of radiusR of 5 km, density of 0.5 g cm–3 and Keplerian velocity 3 km s–1 have a combined kinetic energy of 3 × 1028 erg, a factor of about 100 greater than required by the burst maximum fluxes that last for one second. Betatron acceleration in the compressed magnetic fields between the colliding comets could accelerate electrons to energies sufficient to produce the observed high energy gamma rays. Many of the additional observed features of gamma ray bursts can be explained by the solar comet collision source.  相似文献   

15.
We identified the family of the binary asteroid 423 Diotima consisting of 411 members in the phase space of orbital elements—semimajor axes a (or mean motions n), eccentricities e, and inclinations i—by using an electronic version of the ephemerides of minor planets EMP-2003 containing osculating orbital elements for 34992 asteroids of the main belt. The 9/4 resonance with Jupiter clearly separates the family of 423 Diotima from the family of Eos, which, according to EMP for 2003, contains 1204 asteroids.  相似文献   

16.
The sidereal daily rotation of the Sun, (), depends on the data used. From an appropriate selection of the data — sunspots with regular motion — it is found that ()=14.31–2.70 sin2 , where denotes the heliographic latitude. Moreover, it seems that there is a variation, of the order of 3%, with the solar activity.  相似文献   

17.
The negative powers of the mutual distance – between two bodies are developed into series converging at any moment but that of collision. On the base of these expansions the series have been constructed representing – in the perturbation theory of celestial mechanics. In the general case, including intersecting orbits, the terms are quasi-periodic functions of the time. In the case of non-intersecting orbits the expansion is a double Fourier series in the mean anomalies. All the expansions have a literal form with respect to osculating elements.  相似文献   

18.
The area preserving mapping x = x + a(y – y 3), y = y – a(x – x3), for 0.3 a 2.0 has been studied to locate approximately the x-axis points bounding almost stable regions. For each value of a, these are fixed points with variational trace just greater than 2.0. Transition to chaos can occur rapidly as a increases (with n/k fixed).  相似文献   

19.
In the ordinary restricted problem of three bodies, the first-order stability of planar periodic orbits may be determined by means of their characteristic exponents, as derived from the condition of a vanishing determinant for the coefficients of an infinite system of homogenous linear equations associated with the exponential series solutionu, v representing any initially small oscillations about the periodic solutionx, y. In the elliptic restricted problem, periodic solutions are possible only for periods which are equal to, or integral multiples of, the periodP of the elliptic motion of the two primary masses. It is shown that the infinite determinant approach to the determination of the characteristic exponents can be extended to the treatment of superposed free oscillations in the elliptic problem, and that in generaltwo exponents appear in any complete solutionu, v for eachone existing in the corresponding ordinary restricted problem. The value of each exponent depends on a series proceeding in even powers of the eccentricitye of the relative orbit of the two primaries, in addition to its basic dependence on the mass ratio . For stable periodic orbits, the oscillation frequenciesn 1 (,e 2),n 2 (,e 2) associated with these two exponents tend, withe0, to certain limiting valuesn 1 (),n 2(), which differ from each other by the amount of the frequencyN=2/P of the orbital motion of the primaries. One of the two frequencies, sayn 1(), is identical with the frequency of the corresponding oscillations in the ordinary restricted problem, while the second one gives rise to oscillations only in the elliptic restricted problem, withe0.The method will be described in more detail, together with its application to two families of small periodie librations about the equilateral points of the elliptic restricted problem (E. Rabe: Two new Classes of Periodic Trojan Librations in the Elliptic Restricted Problem and their Stabilities) in theProceedings of the Symposium on Periodic Orbits, Stability and Resonances, held at the University of São Paulo, Brasil, 4–12 September, 1969.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 17–23, 1969.  相似文献   

20.
Intensity, polarization, and cooling rate of the two-photon annihilation radiation are studied in detail in the case of one-dimensional power-law distributions of electrons and positrons, assuming that they occupy the ground Landau level in a strong magnetic fieldB1010–1012 G. Simple analytical expressions for limiting cases are obtained and results of numerical calculations of radiation characteristics are presented. Power-lawe ± distributions ± ± –k are shown to generate power-law spectra of the annihilation radiation atEmc 2 andEmc 2, with indices depending on the direction of radiation. The annihilation spectra at =0 show the largest blue-shifts of their maxima and the hardest high-energy tailsI(Emc 2, =0)E –(k–1). The blue-shifts reduce, and the hard tials steepen, with increasing . At >(2mc 2/E)1/2 the slopes of the high-energy tails rapidly transform to that at =2,I(Emc 2, =/2)E –(2k+3). The direction-integrated spectraS(E) also display the power-law tials at low and high energies,S(Emc 2)E –(k+1). The total annihilation rate and energy losses decrease with decreasingk, being higher than for the isotropice ± power-law distributions at the samek. The radiation is linearly polarized in the plane formed by the magnetic field and wave-vector. The polarization degreeP is maximum atEmc 2:P max0.6 for =/2. Annihilation features and power-law-like hard tails observed in many gamma-ray burst spectra may be associated with the annihilation radiation of the magnetized power-law distributed plasma near neutron stars. Comparison of the observed and theoretical spectra allows one to estimate the power-law index of thee – e +-distribution and the gravitational redshift factor in the radiating region.  相似文献   

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