Analytical and numerical study of the ground-track resonances of Dawn orbiting Vesta

The aim of Dawn mission is the acquisition of data from orbits around two bodies (4) Vesta and (1) Ceres, the two most massive asteroids.Due to the low thrust propulsion, Dawn will slowly cross and transit through ground-track resonances, where the perturbations on Dawn orbit may be significant. In this context, to safety go the Dawn mission from the approach orbit to the lowest science orbit, it is essential to know the properties of the crossed resonances.This paper analytically investigates the properties of the major ground-track resonances (1:1, 1:2, 2:3 and 3:2) appearing for Vesta orbiters: location of the equilibria, aperture of the resonances and period at the stable equilibria. We develop a general method using an averaged Hamiltonian formulation with a spherical harmonic approximation of the gravity field. If the values of the gravity field coefficient change, our method stays correct and applicable. We also discuss the effect of one uncertainty on the C₂₀ and C₂₂ coefficients on the properties of the 1:1 resonance. These results are checked by numerical tests. We determine that the increase of the eccentricity appearing in the 2:3 resonance is due to the C₂₂ and S₂₂ coefficients.Our method can be easily adapted to missions similar to Dawn because, contrarily to the numerical results, the analytical formalism stays the same and is valid for a wide range of physical parameters of the asteroid (namely the shape and the mass) as well as for different spacecraft orbits.Finally we numerically study the probability of the capture in resonance 1:1. Our paper reproduces, explains and supplements the results of Tricarico and Sykes (2010).     

Analytical and numerical results on the stability of a planetary precessional model

A model for planetary precession is investigated using analytical and numerical techniques. A Hamiltonian function governing the model is derived in terms of action-angle Andoyer-Déprit variables under the assumption of equatorial symmetry. As a first approximation a simplified Hamiltonian with zero-eccentricity is considered and stability estimates are derived using KAM theory. A validation of the analytical results is performed computing Poincaré surfaces of section for the circular and elliptical model. We also investigate the role of the eccentricity and its connection with the appearance of resonances. Special attention is devoted to the particular case of the Earth–Moon system. This revised version was published online in July 2006 with corrections to the Cover Date.     

Analytical and numerical manifolds in a symplectic 4-D map

We study analytically the orbits along the asymptotic manifolds from a complex unstable periodic orbit in a symplectic 4-D Froeschlé map. The orbits are given as convergent series. We compare the analytic results by truncating the series at various orders with the corresponding numerical results and we find agreement along a more extended length, as the order of truncation increases. The agreement is improved when the parameters approach those of the stability domain. Along the manifolds no terms with small divisors appear in the series. The same result is found if we use a parametrization method along the asymptotic curves. In the case of orbits starting close to the manifolds small divisors appear, but the orbits remain close to the manifolds for an extended period of time. If the parameters of the map are close to the stable domain the orbits recede and approach the origin several times and remain confined in a certain volume around the origin for a long time before escaping to large distances. For special sets of parameters we see resonance phenomena and the orbits take particular forms near every resonance.     

Analytical development of the lunisolar disturbing function and the critical inclination secular resonance

We provide a detailed derivation of the analytical expansion of the lunar and solar disturbing functions. Although there exist several papers on this topic, many derivations contain mistakes in the final expansion or rather (just) in the proof, thereby necessitating a recasting and correction of the original derivation. In this work, we provide a self-consistent and definite form of the lunisolar expansion. We start with Kaula’s expansion of the disturbing function in terms of the equatorial elements of both the perturbed and perturbing bodies. Then we give a detailed proof of Lane’s expansion, in which the elements of the Moon are referred to the ecliptic plane. Using this approach the inclination of the Moon becomes nearly constant, while the argument of perihelion, the longitude of the ascending node, and the mean anomaly vary linearly with time. We make a comparison between the different expansions and we profit from such discussion to point out some mistakes in the existing literature, which might compromise the correctness of the results. As an application, we analyze the long-term motion of the highly elliptical and critically-inclined Molniya orbits subject to quadrupolar gravitational interactions. The analytical expansions presented herein are very powerful with respect to dynamical studies based on Cartesian equations, because they quickly allow for a more holistic and intuitively understandable picture of the dynamics.     

Analytical evaluation of the Voigt function using binomial coefficients and incomplete gamma functions

Using the binomial expansion theorem, the simple general analytical expressions are obtained for the Voigt function arising in various fields of physical research. As we will seen, the present formulation yields compact closed-form expressions which enable the ready analytical calculation of the Voigt function. The validity of this approximation is tested by other calculation methods. The series expansion relations established in this work are accurate enough in the whole range of parameters. The convergence rate of the series is estimated and discussed. Some examples of this methodology are presented.     

A numerical investigation of secular terms of the planetary disturbing function

The secular terms of the planetary disturbing function are given, after elimination of short period terms by von Zeipel's transformation. The adequacy of this expansion up to terms of eighth order in the inclination and eccentricity is investigated by numerical processes, as a function of the Keplerian elementsa, e andi. The eccentricityé of the outer planet, is taken equal to zero. It is concluded that for values ofi which are not small the inclusion of additional terms in the expression for the disturbing function, results to drastic changes in its values, while larger values ofe do not have an equaly large effect on the disturbing function.     

Time dependent monochromatic scattering of radiation in one dimensional media: Analytical and numerical solutions

In order to choose a numerical method for solving the time dependent equations of radiative transport, we obtain an exact solution for the time dependent radiation field in a one dimensional infinite medium with monochromatic, isotropic scattering for sources with an arbitrary spatial distribution and an arbitrary time variation of their power. The Lax-Wendroff method seems to be the most suitable. Because it is assumed that radiation delay is caused by the finite speed of light, the following difficulty arises when the numerical method is used: the region of variation of the variables (position τ and time t) is triangular (the inequality τ ≤ t). This difficulty is overcome by expanding the unknown functions in series in terms of small values of the time and position. By comparing the numerical and exact solutions for a point source with a given time dependence for its power and with pure scattering, the number of steps in the variables required to obtain a desired accuracy is estimated. This numerical method can be used to calculate the intensity and polarization of the radiation from sources in the early universe during epochs close to the recombination epoch. __________ Translated from Astrofizika, Vol. 51, No. 1, pp. 109–123 (February 2008).     

Covariance function analysis and the clustering of galaxies

Data on a statistic derived from the angular covariance function show that (contrary to the claim of Peebles that galaxies are distributed continuously with no distinct scales), superclusters and the maximum size of clusters are probably defined at scales of 15 and 2.0h ^–1 Mpc. This suggests some stepped-density profile like the idealized models of de Vaucouleurs and Wertz: consideration is therefore given to a semi-continuous hierarchy in which there are galaxies outside clusters, clusters outside superclusters etc. Theories of the origin of clustering by gravitational clumping and the escape of galaxies from clusters suggests the hypothesis that the average mass (m _g) of galaxies outside clusters is smaller than that of those inside (=fractionf of the total), a hypothesis supported by results on the continuity of the angular and spatial covariance functions. In a semi-continuous hierarchy, the overall packing fractionf _e and the fraction (1-f) of galaxies outside clusters both appear to increase as the distancer from a local origin increases, because a line-of-sight to greater depths intersects systems of the hierarchy of continually greater size (R _i). If the hypothesis is valid thatm _g inside clusters is slightly larger thanm _g outside, the apparent effect is to makem _g systematically distance-dependent from a local origin with and ₁0.3. No direct data on galaxy masses exist to refute such a small trend, but since the absolute magnitudes of galaxies are known to be correlated (very weakly) with their masses, a semi-continuous hierarchy has a location-dependent luminosity function, (M). Within uncertainties as to the steepness of (M) at the bright end, the model is consistent with optical number counts to a limiting photographic magnitudem _pg (isotropic slope,q=0.6; semicontinuous modelq=0.64; observation,q=0.67±0.03, standard error.) this removes the discrepancy between the determinations by de Vaucouleurs and Sandageet al. of the thinning factor (1.7). Predictions of the semi-continuous model are made which are at present observationally feasible to carry out. In particular, it is predicted thatq(20<|M|<22)/q(14<|M|<19)2(±0.2).     

Analytical theories of satellite motion constructed with a new package of formula manipulation and compared with numerical integration

Specialized to the Lie series based perturbation method of Kirchgraber and Stiefel (1978) a new computer algebra package called ANALYTOS has been developed for constructing analytical orbital theories either in noncanonical or canonical form. We present results on the (extended) Main Problem of orbital theory of artificial earth satellites and related issues. The order of the solutions achieved is generally one order higher than those known from literature. Moreover, the analytical orbits have been checked succesfully against precise numerical ephemerides. This revised version was published online in July 2006 with corrections to the Cover Date.     

The quasar population density as a result of the formation rate and the evolutionary pathes. I. Analytical formulas and numerical simulations

Wir verallgemeinern die Kontinuitätsgleichung von CAVALIERE et al. (1971) für den Fall, daß die Leuchtkraftentwicklung nicht nur von der aktuellen Leuchtkraft abhängt. Wir zeigen, daß einfache Fälle (monomiale Entstehungsraten) integrierbar sind und vergleichen einige interessante Merkmale. Testverfahren und die Einbeziehung der Beobachtung sind in Vorbereitung.     

Analytical study of the powered Swing-By maneuver for elliptical systems and analysis of its efficiency

Analytical equations describing the velocity and energy variation of a spacecraft in a Powered Swing-By maneuver in an elliptic system are presented. The spacecraft motion is limited to the orbital plane of the primaries. In addition to gravity, the spacecraft suffers the effect of an impulsive maneuver applied when it passes by the periapsis of its orbit around the secondary body of the system. This impulsive maneuver is defined by its magnitude \(\delta V\) and the angle that defines the direction of the impulse with respect to the velocity of the spacecraft (\(\alpha\)). The maneuver occurs in a system of main bodies that are in elliptical orbits, where the velocity of the secondary body varies according to its position in the orbit following the rules of an elliptical orbit. The equations are dependent on this velocity. The study is done using the “patched-conics approximation”, which is a method of simplifying the calculations of the trajectory of a spacecraft traveling around more than one celestial body. Solutions for the velocity and energy variations as a function of the parameters that define the maneuver are presented. An analysis of the efficiency of the powered Swing-By maneuver is also made, comparing it with the pure gravity Swing-by maneuver with the addition of an impulse applied outside the sphere of influence of the secondary body. After a general study, the techniques developed here are applied to the systems Sun-Mercury and Sun-Mars, which are real and important systems with large eccentricity. This problem is highly nonlinear and the dynamics very complex, but very reach in applications.     

Designing weak lensing surveys: a generalized eigenmode analysis

We study the estimators of various second-order weak lensing statistics such as the shear correlation functions  ξ_±  and the aperture mass dispersion  〈 M ²_ap〉  which can directly be constructed from weak lensing shear maps. We compare the efficiency with which these estimators can be used to constrain cosmological parameters. To this end we introduce the Karhunen–Loève (KL) eigenmode analysis techniques for weak lensing surveys. These tools are shown to be very effective as a diagnostics for optimizing survey strategies. The usefulness of these tools to study the effect of angular binning, the depth and width of the survey and noise contributions due to intrinsic ellipticities and number density of source galaxies on the estimation of cosmological parameters is demonstrated. Results from independent analysis of various parameters and joint estimations are compared. We also study how degeneracies among various cosmological and survey parameters affect the eigenmodes associated with these parameters.     

Detailed analysis of fan-shaped jets in three dimensional numerical simulation

We performed three dimensional resistive magnetohydrodynamic simulations to study the magnetic reconnection using an initially shearing magnetic field configuration(force free field with a current sheet in the middle of the computational box).It is shown that there are two types of reconnection jets:the ordinary reconnection jets and fan-shaped jets,which are formed along the guide magnetic field.The fan-shaped jets are significantly different from the ordinary reconnection jets which are ejected by magneti...     

On the symmetric difference quotient and its application to the correction of orbits (II). A numerical analysis

In this paper we present, among other things, the numerical analysis of behaviour of ordinary and symmetric difference quotients for some elementary functions, for rectangular coordinates (velocity components) in relation to orbital elements and initial values of coordinates (velocity components) in Keplerian motion.     

Analytical Solution of the Kozai Resonance and its Application

When Kozai (1962) studied the secular resonance of asteroids, he found the so-called Kozai resonance and expressed the analytical solution with the use of Weierstrass ℘. Here we discuss the case where the disturber is outside a disturbed body and give the analytical solution of the eccentricity, the inclination and the argument of pericenter with the use of the Jacobi elliptic functions, which are more familiar than the Weierstrass ℘. Then we derive the Fourier expansion of the longitude of node and the mean anomaly. The analytical expressions obtained here can be used for any value of the eccentricity and the inclination. Finally we applied these analytical expressions to several dynamical systems – Nereid, that is a highly eccentric satellite of Neptune, and newly discovered retrograde satellites of Uranus. This revised version was published online in July 2006 with corrections to the Cover Date.     

Analytical theory of the libration of the Moon

This paper presents a new theory of the libration of the Moon, completely analytical with respect to the harmonic coefficients of the lunar gravity field. This field is represented through its third degree harmonics for the torque due to the Earth (second degree for the torque due to the Sun).The orbital motion of the Moon is described by the ELP 2000 solution (Chapront-Touzé, 1980) of the main problem of lunar theory.the physical libration variables are obtained as Poisson series and comparisons with the results of Eckhardt (Eckhardt, 1981) and Migus (Migus, 1980) are presented.     

Cosmic electric currents and the generalized Bennett relation

A generalized form of the Bennett pinch is studied in both cylindrical geometry and plane-parallel geometry. In this kind of pinch electromagnetic forces, kinetic pressure gradient forces, centrifugal forces, and gravitational forces may act. For each of the two geometries considered a generalized Bennett relation is derived. By means of these relations it is possible to describe among other things the pure Bennett pinch, Jean's criterion in one and two dimensions, force-free magnetic fields, gravitationally balanced magnetic pressures, and continuous transitions between these states. The theory is applied to electric currents in the magnetosphere, in the solar atmosphere, and in the interstellar medium. It is pointed out that the currents in the solar atmosphere and in the interstellar medium may lead to pinches that are of vital importance to the phenomena of solar flares and star formation, respectively.Paper dedicated to Professor Hannes Alfvén on the occasion of his 80th birthday, 30 May 1988.     

Equivalence of the generalized Lie-Hori method and the method of averaging

In this investigation, a comparison is made of two methods for developing perturbation theories for non-canonical dynamical systems. The methods compared are the generalized Lie-Hori method and the method of averaging. In the comparison presented here, the equivalence of the methods up to the second order in the small parameter is shown. However, the approach used can be extended to demonstrate the equivalence for higher orders. To illustrate the equivalence Duffing's equation, the van der Pol equation and the oscillator with quadratic damping problem are solved using each method.     

Notions of analytic vs numerical stability as applied to the numerical calculation of orbits

This paper deals with the implications of stability as applied to the numerical calculation of orbits. The study was motivated by the recent appearance of several proposed transformations of the classical Newtonian equations of motion which analytically stabilize Cowell's method. This report analyzes the basic properties of such stabilizing transformations and shows the removal of the period as a parameter is the key to these transformations and, that although such transformations do not yield global numerical error bounds, the error propagation properties are more favorable-linear vs quadratic growth.This paper was presented at the AIAA/AAS Astrodynamics Conference Vail, Colorado/July 16–18, 1973.     

Analytical Approximant of the Universal Line Shape

The pressure broadening of spectral lines based in the Unified Franc–Condon (UCF) treatment, leads to universal line shape function L(z) which is a highly complicated integral of the Airy function and exponentialsL(z) = ^-2[Ai (-z)]² exp (-^-3) d.An approximated analytical expression to this function has been found in terms of elementary functions, which is more precise than any previous one, giving also enough accuracy for most of the applications in molecular and plasma spectroscopy. This approximant has been obtained using the recently developed technique of two-point quasifractional approximation. A unique approximant has been found with better accuracy for both sides of the variable. The method used here is an improvement of previous procedures leaving enough free parameters in order to determine exactly the value of the maximum. On the other hand, since there are two different asymptotic forms for large negative and positive values of z, a new way to impose this additional condition has been found.