Almost stable regions for an area-preserving mapping

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).     

The effect of oblateness of the bigger primary on collinear libration points in the restricted problem of three bodies

In the restricted problem of three bodies, the effect of oblateness of the bigger primary appears as an additional term in the potential. As a result, the location of libration points and the roots of the characteristic equation at these points depend not only upon the mass parameter but also on the oblateness termI of the bigger primary. Series solutions are developed in terms of andI which are used for locating the collinear libration points and for determining the mean motions and characteristic exponents at these points.The work is supported by a fellowship awarded to the second author by University Grant Commission, India.     

On solving Kepler's equation for nearly parabolic orbits

We deal here with the efficient starting points for Kepler's equation in the special case of nearly parabolic orbits. Our approach provides with very simple formulas that allow calculating these points on a scientific vest-pocket calculator. Moreover, srtarting with these points in the Newton's method we can calculate a root of Kepler's equation with an accuracy greater than 0.001 in 0–2 iterations. This accuracy holds for the true anomaly || 135° and |e – 1| 0.01. We explain the reason for this effect also.Dedicated to the memory of Professor G.N. Duboshin (1903–1986).     

The structure of chaos in a potential without escapes

We study the structure of chaos in a simple Hamiltonian system that does no have an escape energy. This system has 5 main periodic orbits that are represented on the surface of section by the points (1)O(0,0), (2)C ₁,C ₂(±y _c, 0), (3)B ₁,B ₂(O,±1) and (4) the boundary . The periodic orbits (1) and (4) have infinite transitions from stability (S) to instability (U) and vice-versa; the transition values of are given by simple approximate formulae. At every transitionS U a set of 4 asymptotic curves is formed atO. For larger the size and the oscillations of these curves grow until they destroy the closed invariant curves that surroundO, and they intersect the asymptotic curves of the orbitsC ₁,C ₂ at infinite heteroclinic points. At every transitionU S these asymptotic curves are duplicated and they start at two unstable invariant points bifurcating fromO. At the transition itself the asymptotic curves fromO are tangent to each other. The areas of the lobes fromO increase with ; these lobes increase even afterO becomes stable again. The asymptotic curves of the unstable periodic orbits follow certain rules. Whenever there are heteroclinic points the asymptotic curves of one unstable orbit approach the asymptotic curves of another unstable orbit in a definite way. Finally we study the tangencies and the spirals formed by the asymptotic curves of the orbitsB ₁,B ₂. We find indications that the number of spiral rotations tends to infinity as . Therefore new tangencies between the asymptotic curves appear for arbitrarily large . As a consequence there are infinite new families of stable periodic orbits that appear for arbitrarily large .     

On the coupling of lunisolar resonances for Earth satellite orbits

The resonance C1 occurs when the longitude of the perigee measured from the equinox becomes a slow angle in the doubly averaged equations of motion. This resonance is one of the critical inclination family with I 46°. For prograde Earth satellite orbits, up to five critical points can be identified. Only simple pitchfork bifurcations occur for the single resonance C1. A two degrees of freedom system is studied to check how a coupling of two lunisolar resonances affects the results furnished by the analysis of an isolated resonance case. In the system with two critical angles (g+h and h,+2 , seven types of critical points have been identified. The critical points arise and change their stability through 11 bifurcations. If the initial conditions are selected close to the critical points, the system becomes chaotic as shown in Poincaré maps.     

The Existence and Stability of Equilibrium Points in the Robe’s Restricted Three-Body Problem

The existence of all the equilibrium points, their location and stability in the Robe's (1977) restricted three—body problem have been studied. It is seen that the center of the first primary is always an equilibrium point, whatever be the values of the density parameter K, eccentricity parameter eand mass parameter . The other equilibrium points exist only when K 0 and e = 0 that is when the second primary, a mass point, moves around the first, a spherical shell filled with fluid, in a circular orbit. When K > 1, there is one additional equilibrium point lying on the line joining the center of the first primary and the second primary. When K + = 1, there are infinite number of equilibrium points in the x—yplane lying on a circle of radius one and center as the second primary, provided the points are inside the spherical shell. When K < 0 and K + > 0, there are two more equilibrium points lying in the x—z plane forming triangles with the center of the shell and the second primary. Results of the stability of the equilibrium point (–,0,0), center of the first primary are the same as those given by Robe (1977). Circular points and triangular points are always unstable. The equilibrium point collinear with the center of the shell and the second primary is stable provided and Ksatisfy the inequality

Singular points of the polarization tensor

The problem to compute the magnetic field above the chromosphere using data of the vector = B _t/Bt that gives the projected field direction can be solved with different approximations. The field of direction vectors is, however, not the only field accessible to observations. The Stokes parameters, which are components of the radiation tensor, can be measured at each point of the image plane. The directions of the eigenvectors of the radiation tensor define two mutually orthogonal systems of integral curves in the image plane. These families of curves have singular points, which are generally of different type than those of the vector field. When the morphology of H chromospheric fibrils are used to infer the topology of the magnetic field, a similar problem is met, suggesting that singular points should also be present there.     

Force-free equilibrium at magnetic neutral points

It is shown that, at neutral points of force-free magnetic fields, the electric current density must vanish. This property is independent of whether the neutral points are isolated or (e.g.) fill lines or surfaces. One implication is the fact that in a cold pressureless plasma the formation of neutral current sheets cannot be adiabatically slow. The field-line topology in the neighbourhood of neutral points is discussed. At neutral points of force-free magnetic fields in general three constant- surfaces, defined by the equation ×B=B, with the same value of intersect orthogonally. If, during a time-development, the magnetic field gradient matrix B _i/x _j becomes singular at a neutral point, the field topology can change qualitatively — in general connected with the merger of two or more neutral points into one and/or the splitting up of one neutral point into several others. This can be interpreted as implying the transition from a quasi-static evolution to a dynamical state in which magnetic energy is released.     

On the theoretical possibility of the libration cloud

In the present paper, the problem of whether the interplanetary matter has a tendency to accumulate around the Lagrangian libration pointsL ₄ andL ₅, is examined statistically. It is concluded that: (1) If the particles are initially assumed to be distributed uniformly, they keep the uniformity ever after around the libration points. (2) If the particles receive random stochastic perturbations, their distribution tends to become uniform even if initially they have non-uniform distributions. (3) If the particles mutually collide inelastically, they have a tendency to avoid the regions near the libration points. Therefore, the interplanetary matter will not tend to accumulate near the libration points. Even if the observations of the libration cloud so far reported are confirmed, the clouds are likely to be but temporary objects.     

Extrema in Sunspot Cycle Linked to Sun's Motion

Partitions of 178.8-year intervals between instances of retrograde motion in the Sun's oscillation about the center of mass of the solar system seem to provide synchronization points for the timing of minima and maxima in the 11-year sunspot cycle. In the investigated period 1632–1990, the statistical significance of the relationship goes beyond the level P=0.001. The extrapolation of the observed pattern points to sunspot maxima around 2000.6 and 2011.8. If a further connection with long-range variations in sunspot intensity proves reliable, four to five weak sunspot cycles (R0) are to be expected after cycle 23 with medium strength (R100).     

Spectral investigation of chromospheric fine structure

H spectra and effectively simultaneous filtergrams were taken at the Fraunhofer Observatory on Capri with the 35 cm domeless Coudé. The spatial resolution of the 19 best spectra selected for analysis was estimated to be 1–2 arc sec. The comparison of several hundred H line profiles emitted by typical chromospheric structure elements with theoretical prediction yielded strong evidence to suggest that the chromosphere consists of two parts: A lower, rather uniform layer at rest superposed by clouds (condensations of great spatial variability) which constitute the well-known structure pattern of H filtergrams. For most image points the line-of-sight velocity, optical thickness, source function and Doppler broadening of these clouds could be determined. While the values of the latter three quantities were found to be similar to what Beckers (1968) has found for limb spicules the velocity of the bright and of the dark mottles is considerably smaller than to be expected if these features were the spicules as seen on the disk. However, our results do not rule out the possibility that the spicules rise at the centers of rosettes where they are difficult to detect.Mitteilung aus dem Fraunhofer Institut No. 105.     

Resonance and Capture of Jupiter Comets

A number of Jupiter family comets such as Otermaand Gehrels 3make a rapid transition from heliocentric orbits outside the orbit of Jupiter to heliocentric orbits inside the orbit of Jupiter and vice versa. During this transition, the comet can be captured temporarily by Jupiter for one to several orbits around Jupiter. The interior heliocentric orbit is typically close to the 3:2 resonance while the exterior heliocentric orbit is near the 2:3 resonance. An important feature of the dynamics of these comets is that during the transition, the orbit passes close to the libration points L ₁and L ₂, two of the equilibrium points for the restricted three-body problem for the Sun-Jupiter system. Studying the libration point invariant manifold structures for L ₁and L ₂is a starting point for understanding the capture and resonance transition of these comets. For example, the recently discovered heteroclinic connection between pairs of unstable periodic orbits (one around the L ₁and the other around L ₂) implies a complicated dynamics for comets in a certain energy range. Furthermore, the stable and unstable invariant manifold tubes associated to libration point periodic orbits, of which the heteroclinic connections are a part, are phase space conduits transporting material to and from Jupiter and between the interior and exterior of Jupiter's orbit.     

Periodic orbits and their bifurcations in a 3-D system

We study some simple periodic orbits and their bifurcations in the Hamiltonian . We give the forms of the orbits, the characteristics of the main families, and some existence diagrams and stability diagrams. The existence diagram of the family 1a contains regions that are stable (S), simply unstable (U), doubly unstable (DU) and complex unstable (). In the regionsS andU there are lines of equal rotation numberm/n. Along these lines we have bifurcations of families of periodic orbits of multiplicityn. When these lines reach the boundary of the complex unstable region, they are tangent to it. Inside the region there are linesm/n, along which the orbits 1a, describedn-times, are doubly unstable; however, along these lines there are no bifurcations ofn-ple periodic orbits. The families bifurcating from 1a exist only in certain regions of the parameter space (, ). The limiting lines of these regions join at particular points representing collisions of bifurcations. These collisions of bifurcations produce a nonuniqueness of the various families of periodic orbits. The complicated structure of the various bifurcations can be understood by constructing appropriate stability diagrams.     

Effect of Krein signatures on the stability of relative equilibria

We consider the linear stability of the equilibrium points of the generic rotating potentials U(r), U(r, ), U(r, and U(r, , . The stability analysis is performed using the concept of Krein's signature. This signature is calculated for all eigenvalues of the above potentials. Thereby, the Lagrangian points of the restricted three-body problem and the synchronous satellites of oblate and prolate planets are also studied. We find also the new positions of the eigenvalues for perturbations of the original L ₄ and L ₅ points of Mars, Jupiter, Saturn, Uranus and Neptune. Finally, we study the problem with the mass ratio µ close to the critical value and the stability of geostationary satellites perturbed by the Moon.     

Three-Dimensional p–q Resonant Orbits Close to Second Species Solutions

The purpose of this paper is to study, for small values of , the three-dimensional p–q resonant orbits that are close to periodic second species solutions (SSS) of the restricted three-body problem. The work is based on an analytic study of the in- and out-maps. These maps are associated to follow, under the flow of the problem, initial conditions on a sphere of radius around the small primary, and consider the images of those initial points on the same sphere. The out-map is associated to follow the flow forward in time and the in-map backwards. For both mappings we give analytical expressions in powers of the mass parameter. Once these expressions are obtained, we proceed to the study of the matching equations between both, obtaining initial conditions of orbits that will be 'periodic' with an error of the order ^1–, for some (1/3,1/2). Since, as 0, the inner solution and the outer solution will collide with the small primary, these orbits will be close to SSS.     

Asymmetric librations in exterior resonances

The purpose of this paper is to present a general analysis of the planar circular restricted problem of three bodies in the case of exterior mean-motion resonances. Particularly, our aim is to map the phase space of various commensurabilities and determine the singular solutions of the averaged system, comparing them to the well-known case of interior resonances.In some commensurabilities (e.g. 1/2, 1/3) we show the existence of asymmetric librations; that is, librations in which the stationary value of the critical angle =(p+q)₁–p–q is not equal to either zero or . The origin, stability and morphogenesis of these solutions are discussed and compared to symmetric librations. However, in some other resonances (e.g. 2/3, 3/4), these fixed points of the mean system seem to be absent. Librations in such cases are restricted to =0 mod(). Asymmetric singular solutions of the planar circular problem are unkown in the case of interior resonances and cannot be reproduced by the reduced Andoyer Hamiltonian known as the Second Fundamental Model for Resonance. However, we show that the extended version of this Hamiltonian function, in which harmonics up to order two are considered, can reproduce fairly well the principal topological characteristics of the phase space and thereby constitutes a simple and useful analytical approximation for these resonances.     

Distribution of lifetimes for coronal soft X-ray bright points

We have measured the lifetimes of all compact emission features visible on three sets of high time resolution soft X-ray images. The spectrum of lifetimes is found to be heavily weighted toward short lifetimes. The number of features present on the disk which live 2–48 hours is at least ten times as great as the number living more than 48 hours. The distribution of lifetimes can be fit in all three cases by a four-parameter function N(t) = N _s exp(-t/ _s ) + N _L exp(-t/ _L ), with _s = 8.7±0.2, _L = 35±4 and N _s 10N _L . Features living two days or less have a very broad latitude distribution (Golub et al., 1974, 1975) whereas nearly all longer-lived features are found within 30° of the equator. The growth rates of long-lived vs short-lived points are the same to within 20%, the major difference being that long-lived points continue to grow and generally reach larger sizes.Harvard College Observatory/Smithsonian Astrophysical Observatory.     

Power spectrum of geomagneticK p-index detected at Misallat magnetic station

TheK _p-indices measured at Misallat station [ø = 29°45 N; = 30°54 E] during the period 1958–1989, have been compared with the data of two geomagnetic stations; one of them has similar longitude as Misallat and the second has almost similar latitude. The spectra of daily and hourly data of the three stations are compared together to define the latitude and longitude-effects on the detected periodicities. Daily periodicities ofK _p-index activities at the values 32.7, 21.2, 12.9, 11.6 and 9.2 days have been obtained to be common in the spectra of the three stations. The existence of 5-days periodicity in these stations has also been detected.     

Explosive hydrogen burning in supernovae

Rapid proton capture is supposed to be responsible for the synthesis of a number of proton-rich nuclei. This process of hydrogen burning is considered here for mass elements, the atomic numbers of which range fromZ=10 toZ=20. The possible site for this process is assumed to be the outer envelope of the supernova at a proton number density (n _p )ranging fromn _p =10²² cm^–3 ton _p =10²⁸ cm^–3 at temperatures in the range ofT=2–3×10⁹ K.The capture path is determined by considering that a dynamical equilibrium between (p, ) and (,p) reactions exists between the reacting nuclei. In this situation, the abundances of elements become proportional to the lifetime of ⁺ decaying nuclei at the waiting points.It is suggested that these rapid proton-capture reactions are responsible for the production of a number of nuclei in the rangeA40 during supernova outbursts.     

The size dependence of contrasts and numbers of small magnetic flux tubes in an active region

Intensity contrasts and number densities of bright points, knots and pores ranging in size between 0.15 and 4 are studied using high resolution pictures in Mg b₁ of a young active region. On the average, the contrast in the wing of the line increases very strongly with decreasing size, while the continuum contrast increases more slowly. The ratio of contrast in the line to contrast in the continuum increases rapidly with decreasing size. The possibility is explored of using this contrast ratio as an indicator of size. The distribution of the contrast ratio in a part of the active region is used in this way to derive a size distribution of facular points. The resulting distribution has a limited accuracy, but is free from systematic distortion due to selection effects. Validity checks on the method are presented. We measure the size distribution of the pores in the same area, and combine the result with that for the facular points. The combined distribution shows that the surface area covered by magnetic elements with diameter has a maximum near = 0.8. It increases roughly proportional to for < 0.3 and falls off as for > 1.5. It is inferred that elements with 0.5 < < 1.6, which show no conspicuous contrast in the line wing or in the continuum, occupy as much area as the pores, and twice as much as the facular points. We suggest that the changing appearance of a facular area with increasing height of formation reflects at least as much the increasing weight of the small elements in the contrast as a real change in intrinsic properties (such as the diameter) of individual elements. A spatial resolution better than 0.1 may be needed to resolve the individual elements in plages and the chromospheric network.The observed variation of continuum contrast of facular points with size agrees with predictions based on magnetostatic flux tube models if a field strength of about 2000 G is assumed.