Symmetric and asymmetric librations in extrasolar planetary systems: a global view

We present a global view of the resonant structure of the phase space of a planetary system with two planets, moving in the same plane, as obtained from the set of the families of periodic orbits. An important tool to understand the topology of the phase space is to determine the position and the stability character of the families of periodic orbits. The region of the phase space close to a stable periodic orbit corresponds to stable, quasi periodic librations. In these regions it is possible for an extrasolar planetary system to exist, or to be trapped following a migration process due to dissipative forces. The mean motion resonances are associated with periodic orbits in a rotating frame, which means that the relative configuration is repeated in space. We start the study with the family of symmetric periodic orbits with nearly circular orbits of the two planets. Along this family the ratio of the periods of the two planets varies, and passes through rational values, which correspond to resonances. At these resonant points we have bifurcations of families of resonant elliptic periodic orbits. There are three topologically different resonances: (1) the resonances (n + 1):n, (2:1, 3:2, ...), (2) the resonances (2n + 1):(2n-1), (3:1, 5:3, ...) and (3) all other resonances. The topology at each one of the above three types of resonances is studied, for different values of the sum and of the ratio of the planetary masses. Both symmetric and asymmetric resonant elliptic periodic orbits exist. In general, the symmetric elliptic families bifurcate from the circular family, and the asymmetric elliptic families bifurcate from the symmetric elliptic families. The results are compared with the position of some observed extrasolar planetary systems. In some cases (e.g., Gliese 876) the observed system lies, with a very good accuracy, on the stable part of a family of resonant periodic orbits.     

Periodic orbits generated by Lagrangian solutions of the restricted three body problem when one of the primaries is an oblate body

We have studied periodic orbits generated by Lagrangian solutions of the restricted three body problem when one of the primaries is an oblate body. We have determined the periodic orbits for different values of μ, h and A (h is energy constant, μ is mass ratio of the two primaries and A is an oblateness factor). These orbits have been determined by giving displacements along the tangent and normal to the mobile coordinates as defined by Karimov and Sokolsky (Celest. Mech. 46:335, 1989). These orbits have been drawn by using the predictor-corrector method. We have also studied the effect of oblateness by taking some fixed values of μ, A and h. As starters for our method, we use some known periodic orbits in the classical restricted three body problem.     

Nonlinear Evolution of Axisymmetric Twisted Flux Tubes in the Solar Tachocline

We study the periodicity of twisting motions in sunspot penumbral filaments, which were recently discovered from space (Hinode) and ground-based (SST) observations. A sunspot was well observed for 97 minutes by Hinode/SOT in the G-band (4305 Å) on 12 November 2006. By the use of the time?–?space gradient applied to intensity space?–?time plots, twisting structures can be identified in the penumbral filaments. Consistent with previous findings, we find that the twisting is oriented from the solar limb to disk center. Some of them show a periodicity. The typical period is about ≈?four minutes, and the twisting velocity is roughly 6 km s^?1. However, the penumbral filaments do not always show periodic twisting motions during the time interval of the observations. Such behavior seems to start and stop randomly with various penumbral filaments displaying periodic twisting during different intervals. The maximum number of periodic twists is 20 in our observations. Studying this periodicity can help us to understand the physical nature of the twisting motions. The present results enable us to determine observational constraints on the twisting mechanism.     

The rectilinear three-body problem using symbol sequence II: role of the periodic orbits

We study the change of phase space structure of the rectilinear three-body problem when the mass combination is changed. Generally, periodic orbits bifurcate from the stable Schubart periodic orbit and move radially outward. Among these periodic orbits there are dominant periodic orbits having rotation number (n − 2)/n with n ≥ 3. We find that the number of dominant periodic orbits is two when n is odd and four when n is even. Dominant periodic orbits have large stable regions in and out of the stability region of the Schubart orbit (Schubart region), and so they determine the size of the Schubart region and influence the structure of the Poincaré section out of the Schubart region. Indeed, with the movement of the dominant periodic orbits, part of complicated structure of the Poincaré section follows these orbits. We find stable periodic orbits which do not bifurcate from the Schubart orbit.     

Periodic orbits and bifurcations in the Sitnikov four-body problem when all primaries are oblate

We study the motions of an infinitesimal mass in the Sitnikov four-body problem in which three equal oblate spheroids (called primaries) symmetrical in all respect, are placed at the vertices of an equilateral triangle. These primaries are moving in circular orbits around their common center of mass. The fourth infinitesimal mass is moving along a line perpendicular to the plane of motion of the primaries and passing through the center of mass of the primaries. A relation between the oblateness-parameter ‘A’ and the increased sides ‘ε’ of the equilateral triangle during the motion is established. We confine our attention to one particular value of oblateness-parameter A=0.003. Only one stability region and 12 critical periodic orbits are found from which new three-dimensional families of symmetric periodic orbits bifurcate. 3-D families of symmetric periodic orbits, bifurcating from the 12 corresponding critical periodic orbits are determined. For A=0.005, observation shows that the stability region is wider than for A=0.003.     

Asymmetric Periodic Solutions in the Restricted Problem of Three Bodies

A systematic numerical exploration of the families of asymmetric periodic orbits of the restricted three-body problem when a) the primary bodies are equal and b) for the Earth-Moon mass ratio, is presented. Decades families of asymmetric periodic solutions were found and three of the simplest ones, in the first case, and ten of the second one are illustrated. All of these families consist of periodic orbits which are asymmetric with respect to x-axis while are simple symmetric periodic orbits with respect to y-axis (i.e. the orbit has only one perpendicular intersection at half period with y-axis). Many asymmetric periodic orbits, members of these families, are calculated and plotted. We studied the stability of all the asymmetric periodic orbits we found. These families consist, mainly, of unstable periodic solutions but there exist very small, with respect to x, intervals where these families have stable periodic orbits. We also found, using appropriate Poincaré surface of sections, that a relatively large region of phase space extended around all these stable asymmetric periodic orbits shows chaotic motion.     

Effects of the Polarity States of the Heliospheric Magnetic Field and Particle Drifts in Cosmic Radiation

Forbush decrease (FD) events recorded at the ground-based neutron monitors (NMs) during the period 1961 – 1999, have been selected and recovery characteristic of these events have been analyzed. The average profile of FDs observed during different polarity states of the heliosphere is obtained by superposed epoch analysis separately for the periods 1961 – 1969 (A < 0), 1971 – 1979 (A > 0), 1981 – 1989 (A < 0) and 1991 – 1999 (A > 0). Hourly count rate of neutron monitors of different cut-off rigidities have been utilized. The results are compared with model predictions including drifts. No marked difference is observed in the amplitudes of FDs during A < 0 and A > 0. Rigidity spectrum fitted with a power law yields the values of spectral exponent that are closer to values predicted by two-dimensional models including drifts. The recovery rate of FDs varies with the polarity of HMF and the rate is higher (recovery time smaller) during A > 0 than during A < 0 epoch, consistent with the model predictions including the drift effects in the HMF. This difference in recovery time of FDs during A > 0 and A < 0 polarity conditions provides experimental evidence that drift plays an important role in cosmic ray modulation.     

Characteristics of resonant periodic orbits

We study the families of periodic orbits in a time-independent two-dimensional potential field symmetric with respect to both axes. By numerical calculations we find characteristic curves of several families of periodic orbits when the ratio of the unperturbed frequencies isA ^1/2/B ^1/2=2/1. There are two groups of characteristic curves: (a) The basic characteristic and the characteristics which bifurcate from it. (b) The characteristics which start from the boundary line and the axisx=0.     

A photometric study of the nova‐like variable TT Arietis with the MOST satellite

Variability on all time scales between seconds and decades is typical for cataclysmic variables (CVs). One of the brightest and best studied CVs is TT Ari, a nova‐like variable which belongs to the VY Scl subclass, characterized by occasional low states in their light curves. It is also known as a permanent superhumper at high state, revealing “positive” (P_S > P₀) as well as “negative” (P_S < P₀) superhumps, where PS is the period of the superhump and P₀ the orbital period. TT Ari was observed by the Canadian space telescope MOST for about 230 hours nearly continuously in 2007, with a time resolution of 48 seconds. Here we analyze these data, obtaining a dominant “negative” superhump signal with a period P_S = 0.1331 days and a mean amplitude of 0.09 mag. Strong flickering with amplitudes up to 0.2 mag and peak‐to‐peak time scales of 15–20 minutes is superimposed on the periodic variations. We found no indications for significant quasi‐periodic oscillations with periods around 15 minutes, reported by other authors. We discuss the known superhump behaviour of TTAri during the last five decades and conclude that our period value is at the upper limit of all hitherto determined “negative” superhump periods of TTAri, before and after the MOST run. (© 2013 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Basic Functions of the Light Curve Analysis of Eclipsing Variables in the Frequency Domain

An analytically tractable method of transforming the problem of light curve analysis of eclipsing binaries from the time domain into the frequency domain was introduced by Kopal (1975, 1979, 1990). This method uses a new general formulation of eclipse functions α, the so-called moments A _2m , and their combinations as g _2m = A _2m+2/(A _2m A _2m+4) functions for the basic spherical model. In this paper, I will review the use of these functions in the light curve analysis of eclipsing binaries.     

Long-Term Variations in the Solar Differential Rotation

Using Greenwich data (1879–1976) and SOON/NOAA data (1977–2002) on sunspot groups we found the following results: (i) The Sun's mean (over all the concerned cycles during 1879–1975) equatorial rotation rate (A) is significantly larger (≈0.1%) in the odd-numbered sunspot cycles (ONSCs) than in the even-numbered sunspot cycles (ENSCs). The mean rotation is significantly (≈10%) more differential in the ONSCs than in the ENSCs. North–south difference in the mean equatorial rotation rate is larger in the ONSCs than in the ENSCs. North–south difference in the mean latitude gradient of the rotation is significant in the ENSCs and insignificant in the ONSCs. (ii) The known very large decrease in A from cycle 13 to cycle 14 is confirmed. The amount of this decrease in the mean A was about 0.017 μrad s⁻¹. Also, we find that A decreased from cycle 17 to cycle 18 by about 0.008 μrad s⁻¹ and from cycle 21 to cycle 22 by about 0.016 μrad s⁻¹. From cycle 13 to cycle 14 the decrease in A was more in the northern hemisphere than in the southern hemisphere, it is opposite in the later two epochs. The time gap between the consecutive drops in A is about 44 years, suggesting the existence of a `44-yr' cycle or `double Hale cycle' in A. The time gap between the two large drops, viz., from cycle 13 to cycle 14 and from cycle 21 to cycle 22, is about 90 years (Gleissberg cycle). We predict that the next drop (moderate) in A will be occurring from cycle 25 to cycle 26 and will be followed by a relatively large-amplitude `double Hale cycle' of sunspot activity. (iii) Existence of a 90-yr cycle is seen in the cycle-to-cycle variation of the latitude gradient (B). A weak 22-yr modulation in B seems to be superposed on the relatively strong 90-yr modulation. (iv) The coefficient A varies significantly only during ONSCs and the variation has maximum amplitude in the order of 0.01 μrad s<sup>−1</sup> around activity minima. (v) There exists a good anticorrelation between the mean variation of B during the ONSCs and that during the ENSCs, suggesting the existence of a `22-yr' periodicity in B. The maximum amplitude of the variation of B is of the order of 0.05 μrad s⁻¹ around the activity minima. (vi) It seems that the well-known Gnevyshev and Ohl rule of solar activity is applicable also to the cycle-to-cycle amplitude modulation of B from cycle 13 to cycle 20, but the cycles 12 (in the northern hemisphere, Greenwich data) and 21 (in both hemispheres, SOON/NOAA data) seem to violate this rule in B. And (vii) All the aforesaid statistically significant variations in A and B seem to be related to the approximate 179-yr cycle, 1811–1989, of variation in the Sun's motion about the center of mass of the solar system.     

Influence of short-duration matter temperature perturbations on the cosmological hydrogen recombination spectrum

A perturbation in the ratio of the matter temperature to the radiation temperature in the form of a Gaussian with amplitude A and width σ (in units of the redshift z) centered at some redshift z _c is considered, with some “standard” temperature ratio obtained from a simultaneous solution of the cosmological recombination kinetics and energy equations being taken as the initial (unperturbed) one. Comparatively small (A = ± 0.01), fast (σ = 17) perturbations are shown to give rise to distinct narrow absorption (for A > 0) or emission (for A < 0) quasi-lines in each of the subordinate continua. The positions of these quasi-lines correlate with the position of the perturbation center, while their intensities are very sensitive to the perturbation amplitude. At the same time, the manifestation of the perturbation is much less clear in hydrogen lines (subordinate ones and the Ly-α line) and two-photon emission. As a result, the full perturbed spectrum is characterized by the presence of the narrow quasi-lines mentioned above and by a general decrease (for A > 0) or increase (for A < 0) in intensity with increasing wavelength.     

On the variational approach to the periodic n-body problem

This expository paper gathers some of the results obtained by the author in recent works in collaboration with Davide Ferrario and Vivina Barutello, focusing on the periodic n-body problem from the perspective of the calculus of variations and minimax theory. These researches were aimed at developing a systematic variational approach to the equivariant periodic n-body problem in the two and three-dimensional space. The purpose of this paper is to expose the main problems and achievements of this approach. The material here was exposed in the talk that given at the Meeting CELMEC IV promoted by SIMCA (Società italiana di Meccanica Celeste).     

Period study of the binary star SW Cygni

The O−C curve of SW Cyg between 1880 and 1977 is presented and discussed. It is found that the orbital period undergoes a systematic change, becoming greater with time. In addition, a periodic oscillation of amplitude 0 _. ^d 015 with period of 43.8 years is superimposed on this general trend. It is concluded that the increase in the period is due to a transfer of mass from the secondary star to the primary and the periodic oscillation is due to the light time effect of the third body of mass functionf(m)=0.006M _⊙.     

The elliptic restricted problem at the 3 : 1 resonance

Four 3 : 1 resonant families of periodic orbits of the planar elliptic restricted three-body problem, in the Sun-Jupiter-asteroid system, have been computed. These families bifurcate from known families of the circular problem, which are also presented. Two of them, I _c , II _c bifurcate from the unstable region of the family of periodic orbits of the first kind (circular orbits of the asteroid) and are unstable and the other two, I _e , II _e , from the stable resonant 3 : 1 family of periodic orbits of the second kind (elliptic orbits of the asteroid). One of them is stable and the other is unstable. All the families of periodic orbits of the circular and the elliptic problem are compared with the corresponding fixed points of the averaged model used by several authors. The coincidence is good for the fixed points of the circular averaged model and the two families of the fixed points of the elliptic model corresponding to the families I _c , II _c , but is poor for the families I _e , II _e . A simple correction term to the averaged Hamiltonian of the elliptic model is proposed in this latter case, which makes the coincidence good. This, in fact, is equivalent to the construction of a new dynamical system, very close to the original one, which is simple and whose phase space has all the basic features of the elliptic restricted three-body problem.     

Periodic orbits and bifurcations in the Sitnikov four-body problem

We study the existence, linear stability and bifurcations of what we call the Sitnikov family of straight line periodic orbits in the case of the restricted four-body problem, where the three equal mass primary bodies are rotating on a circle and the fourth (small body) is moving in the direction vertical to the center mass of the other three. In contrast to the restricted three-body Sitnikov problem, where the Sitnikov family has infinitely many stability intervals (hence infinitely many Sitnikov critical orbits), as the “family parameter” ż₀ varies within a finite interval (while z ₀ tends to infinity), in the four-body problem this family has only one stability interval and only twelve 3-dimensional (3D) families of symmetric periodic orbits exist which bifurcate from twelve corresponding critical Sitnikov periodic orbits. We also calculate the evolution of the characteristic curves of these 3D branch-families and determine their stability. More importantly, we study the phase space dynamics in the vicinity of these orbits in two ways: First, we use the SALI index to investigate the extent of bounded motion of the small particle off the z-axis along its interval of stable Sitnikov orbits, and secondly, through suitably chosen Poincaré maps, we chart the motion near one of the 3D families of plane-symmetric periodic orbits. Our study reveals in both cases a fascinating structure of ordered motion surrounded by “sticky” and chaotic orbits as well as orbits which rapidly escape to infinity.     

Families of Periodic Orbits Emanating From Homoclinic Orbits in the Restricted Problem of Three Bodies

We describe and comment the results of a numerical exploration on the evolution of the families of periodic orbits associated with homoclinic orbits emanating from the equilateral equilibria of the restricted three body problem for values of the mass ratio larger than μ ₁. This exploration is, in some sense, a continuation of the work reported in Henrard [Celes. Mech. Dyn. Astr. 2002, 83, 291]. Indeed it shows how, for values of μ. larger than μ ₁, the Trojan web described there is transformed into families of periodic orbits associated with homoclinic orbits. Also we describe how families of periodic orbits associated with homoclinic orbits can attach (or detach) themselves to (or from) the best known families of symmetric periodic orbits. This revised version was published online in July 2006 with corrections to the Cover Date.     

The curious case of the greenwich faculae

We analyze the record of facular areas compiled by the Royal Greenwich Observatory (RGO) from daily white-light observations between 1874 and 1976. Curiously, the relative amplitudes of the three largest sunspot cycles 17, 18, and 19 in this record are reversed when they are ranked by facular area. We show that this negative correlation arises from a general decrease of the ratioA _F/A _S, of facular to sunspot area, with increasingA _S. Within a given cycle,A _F/A _Sdecreases in active regions of largeA _S, butA _F/A _Sis also lower at allA _S, in cycles of higher peak amplitude inA _S. This decrease ofA _F/A _Sin large spot groups is consistent with its decrease in younger, more active solar-mass stars, and it may explain why stars only slightly more magnetically active than the Sun tend to exhibit much greater variability in broad-band photometry. We suggest that the physical explanation is an increased spatial filling factor of magnetic flux, favoring formation of sunspots over faculae. We also explain why the decrease inA _F/A_Sis not seen in the disc-integrated Ca K plage areas, nor in theF10.7 microwave index, both of which exhibit remarkable linearity when plotted against smoothed sunspot area. This explanation suggests how complementary data on faculae and plages from RGO and Mt. Wilson could be used to improve empirical models of total irradiance variation, extending back to 1874.     

Classification of cometary orbits based on the concept of orbital mean temperature

The orbital mean temperature (T_m) for periodic cometary orbits has been calculated as a function of eccentricity e for 0 ≤ e ≤ 0.97, the semimajor axis a for 1 ≤ a ≤ 100 AU, and for an albedo A = 0. If water ice is the major constituent of comets, it can be concluded from these results that d'Arrest, Encke, Oterma, Schwassmann-Wachmann 2, and Tempel 2 contain crystalline ice. In the past, they must have experienced a short period of high activity due to the phase transition from amorphous to cubic ice. The actual low activity of those comets is not necessarily due to depletion of volatiles. Schwassmann-Wachmann 1 and Halley are expected to contain amorphous ice in the inner part of the nucleus. Schwassmann-Wachmann 1 will conserve its erratic activity for a long time until all its ice is in the cubic state. Halley will have an important activity at small heliocentric distance until final depletion of volatiles as no thick crystalline crust can be built up. The present statements are also valid for albedos A ≤ 0.3. THe results are generalized to other periodic orbits.     

Stable Orbits of Planets of a Binary Star System in the Three-Dimensional Restricted Problem

The present research was motivated by the recent discovery of planets around binary stars. Our initial intention was thus to investigate the 3-dimensional nearly circular periodic orbits of the circular restricted problem of three bodies; more precisely Stromgren's class L, (direct) and class m, (retrograde). We started by extending several of Hénon's vertical critical orbits of these 2 classes to three dimensions, looking especially for orbits which are near circular and have stable characteristic exponents.We discovered early on that the periodic orbits with the above two qualifications are fairly rare and we decided thus to undertake a systematic exploration, limiting ourselves to symmetric periodic orbits. However, we examined all 16 possible symmetry cases, trying 10000 sets of initial values for periodicity in each case, thus 160000 integrations, all with z _o or _o equal to 0.1 This gave us a preliminary collection of 171 periodic orbits, all fairly near the xy-plane, thus with rather low inclinations. Next, we integrated a second similar set of 160000 cases with z _o or _o equal to 0.5, in order to get a better representation of the large inclinations. This time, we found 167 periodic orbits, but it was later discovered that at least 152 of them belong to the same families as the first set with 0.1Our paper quickly describes the definition of the problem, with special emphasis on the symmetry properties, especially for the case of masses with equal primaries. We also allow a section to describe our approach to stability and characteristic exponents, following our paper on this subject, (Broucke, 1969). Then we describe our numerical results, as much as space permits in the present paper.We found basically only about a dozen families with sizeable segments of simple stable periodic orbits. Some of them are around one of the two stars only but we do not describe them here because of a lack of space. We extended about 170 periodic orbits to families of up to 500 members, (by steps of 0.005 in the parameter), although, in many cases, we do not know the real end of the families. We also give an overview of the different types of periodic orbits that are most often encountered. We describe some of the rather strange orbits, (some of which are actually stable).