The equations of relative motion in the orbital reference frame

The analysis of relative motion of two spacecraft in Earth-bound orbits is usually carried out on the basis of simplifying assumptions. In particular, the reference spacecraft is assumed to follow a circular orbit, in which case the equations of relative motion are governed by the well-known Hill–Clohessy–Wiltshire equations. Circular motion is not, however, a solution when the Earth’s flattening is accounted for, except for equatorial orbits, where in any case the acceleration term is not Newtonian. Several attempts have been made to account for the \(J_2\) effects, either by ingeniously taking advantage of their differential effects, or by cleverly introducing ad-hoc terms in the equations of motion on the basis of geometrical analysis of the \(J_2\) perturbing effects. Analysis of relative motion about an unperturbed elliptical orbit is the next step in complexity. Relative motion about a \(J_2\)-perturbed elliptic reference trajectory is clearly a challenging problem, which has received little attention. All these problems are based on either the Hill–Clohessy–Wiltshire equations for circular reference motion, or the de Vries/Tschauner–Hempel equations for elliptical reference motion, which are both approximate versions of the exact equations of relative motion. The main difference between the exact and approximate forms of these equations consists in the expression for the angular velocity and the angular acceleration of the rotating reference frame with respect to an inertial reference frame. The rotating reference frame is invariably taken as the local orbital frame, i.e., the RTN frame generated by the radial, the transverse, and the normal directions along the primary spacecraft orbit. Some authors have tried to account for the non-constant nature of the angular velocity vector, but have limited their correction to a mean motion value consistent with the \(J_2\) perturbation terms. However, the angular velocity vector is also affected in direction, which causes precession of the node and the argument of perigee, i.e., of the entire orbital plane. Here we provide a derivation of the exact equations of relative motion by expressing the angular velocity of the RTN frame in terms of the state vector of the reference spacecraft. As such, these equations are completely general, in the sense that the orbit of the reference spacecraft need only be known through its ephemeris, and therefore subject to any force field whatever. It is also shown that these equations reduce to either the Hill–Clohessy–Wiltshire, or the Tschauner–Hempel equations, depending on the level of approximation. The explicit form of the equations of relative motion with respect to a \(J_2\)-perturbed reference orbit is also introduced.     

On the period of the periodic orbits of the restricted three body problem

We will show that the period T of a closed orbit of the planar circular restricted three body problem (viewed on rotating coordinates) depends on the region it encloses. Roughly speaking, we show that, \(2 T=k\pi +\int _\Omega g\) where k is an integer, \(\Omega \) is the region enclosed by the periodic orbit and \(g:{\mathbb {R}}^2\rightarrow {\mathbb {R}}\) is a function that only depends on the constant C known as the Jacobian constant; it does not depend on \(\Omega \). This theorem has a Keplerian flavor in the sense that it relates the period with the space “swept” by the orbit. As an application we prove that there is a neighborhood around \(L_4\) such that every periodic solution contained in this neighborhood must move clockwise. The same result holds true for \(L_5\).     

Geomagnetic Effects of Corotating Interaction Regions

We present an analysis of the geoeffectiveness of corotating interaction regions (CIRs), employing the data recorded from 25 January to 5 May 2005 and throughout 2008. These two intervals in the declining phase of Solar Cycle 23 are characterised by a particularly low number of interplanetary coronal mass ejections (ICMEs). We study in detail how four geomagnetic-activity parameters (the Dst, Ap, and AE indices, as well as the Dst time derivative, \(\mathrm{dDst}/\mathrm{d}t\)) are related to three CIR-related solar wind parameters (flow speed, \(V\), magnetic field, \(B\), and the convective electric field based on the southward Geocentric solar magnetospheric (GSM) magnetic field component, \(\mathit{VB}_{s}\)) on a three-hour time resolution. In addition, we quantify statistical relationships between the mentioned geomagnetic indices. It is found that Dst is correlated best to \(V\), with a correlation coefficient of \(\mathrm{cc}\approx0.6\), whereas there is no correlation between \(\mathrm{dDst}/\mathrm{d}t\) and \(V\). The Ap and AE indices attain peaks about half a day before the maximum of \(V\), with correlation coefficients ranging from \(\mathrm{cc}\approx0.6\) to \(\mathrm{cc}\approx0.7\), depending on the sample used. The best correlations of Ap and AE are found with \(\mathit{VB}_{s}\) with a delay of 3 h, being characterised by \(\mathrm{cc}\gtrsim 0.6\). The Dst derivative \(\mathrm{dDst}/\mathrm{d}t\) is also correlated with \(\mathit{VB}_{s}\), but the correlation is significantly weaker \(\mathrm{cc}\approx 0.4\)?–?0.5, with a delay of 0?–?3 h, depending on the employed sample. Such low values of correlation coefficients indicate that there are other significant effects that influence the relationship between the considered parameters. The correlation of all studied geomagnetic parameters with \(B\) are characterised by considerably lower correlation coefficients, ranging from \(\mathrm{cc}=0.3\) in the case of \(\mathrm{dDst}/\mathrm{d}t\) up to \(\mathrm{cc}=0.56\) in the case of Ap. It is also shown that peak values of geomagnetic indices depend on the duration of the CIR-related structures. The Dst is closely correlated with Ap and AE (\(\mathrm{cc}=0.7\)), Dst being delayed for about 3 h. On the other hand, \(\mathrm{dDst}/\mathrm{d}t\) peaks simultaneously with Ap and AE, with correlation coefficients of 0.48 and 0.56, respectively. The highest correlation (\(\mathrm{cc}=0.81\)) is found for the relationship between Ap and AE.     

Averaged model to study long-term dynamics of a probe about Mercury

This paper provides a method for finding initial conditions of frozen orbits for a probe around Mercury. Frozen orbits are those whose orbital elements remain constant on average. Thus, at the same point in each orbit, the satellite always passes at the same altitude. This is very interesting for scientific missions that require close inspection of any celestial body. The orbital dynamics of an artificial satellite about Mercury is governed by the potential attraction of the main body. Besides the Keplerian attraction, we consider the inhomogeneities of the potential of the central body. We include secondary terms of Mercury gravity field from \(J_2\) up to \(J_6\), and the tesseral harmonics \(\overline{C}_{22}\) that is of the same magnitude than zonal \(J_2\). In the case of science missions about Mercury, it is also important to consider third-body perturbation (Sun). Circular restricted three body problem can not be applied to Mercury–Sun system due to its non-negligible orbital eccentricity. Besides the harmonics coefficients of Mercury’s gravitational potential, and the Sun gravitational perturbation, our average model also includes Solar acceleration pressure. This simplified model captures the majority of the dynamics of low and high orbits about Mercury. In order to capture the dominant characteristics of the dynamics, short-period terms of the system are removed applying a double-averaging technique. This algorithm is a two-fold process which firstly averages over the period of the satellite, and secondly averages with respect to the period of the third body. This simplified Hamiltonian model is introduced in the Lagrange Planetary equations. Thus, frozen orbits are characterized by a surface depending on three variables: the orbital semimajor axis, eccentricity and inclination. We find frozen orbits for an average altitude of 400 and 1000 km, which are the predicted values for the BepiColombo mission. Finally, the paper delves into the orbital stability of frozen orbits and the temporal evolution of the eccentricity of these orbits.     

Metrics in Keplerian orbits quotient spaces

Quotient spaces of Keplerian orbits are important instruments for the modelling of orbit samples of celestial bodies on a large time span. We suppose that variations of the orbital eccentricities, inclinations and semi-major axes remain sufficiently small, while arbitrary perturbations are allowed for the arguments of pericentres or longitudes of the nodes, or both. The distance between orbits or their images in quotient spaces serves as a numerical criterion for such problems of Celestial Mechanics as search for common origin of meteoroid streams, comets, and asteroids, asteroid families identification, and others. In this paper, we consider quotient sets of the non-rectilinear Keplerian orbits space \(\mathbb H\). Their elements are identified irrespective of the values of pericentre arguments or node longitudes. We prove that distance functions on the quotient sets, introduced in Kholshevnikov et al. (Mon Not R Astron Soc 462:2275–2283, 2016), satisfy metric space axioms and discuss theoretical and practical importance of this result. Isometric embeddings of the quotient spaces into \(\mathbb R^n\), and a space of compact subsets of \(\mathbb H\) with Hausdorff metric are constructed. The Euclidean representations of the orbits spaces find its applications in a problem of orbit averaging and computational algorithms specific to Euclidean space. We also explore completions of \(\mathbb H\) and its quotient spaces with respect to corresponding metrics and establish a relation between elements of the extended spaces and rectilinear trajectories. Distance between an orbit and subsets of elliptic and hyperbolic orbits is calculated. This quantity provides an upper bound for the metric value in a problem of close orbits identification. Finally the invariance of the equivalence relations in \(\mathbb H\) under coordinates change is discussed.     

Spectroscopic Variability of Supergiant Star HD14134, B3Ia

Profile variations in the \(\hbox {H}\alpha \) and \(\hbox {H}\beta \) lines in the spectra of the star HD14134 are investigated using observations carried out in 2013–2014 and 2016 with the 2-m telescope at the Shamakhy Astrophysical Observatory. The absorption and emission components of the \(\hbox {H}\alpha \) line are found to disappear on some observational days, and two of the spectrograms exhibit inverse P-Cyg profile of \(\hbox {H}\alpha \). It was revealed that when the \(\hbox {H}\alpha \) line disappeared or an inversion of the P-Cyg-type profile is observed in the spectra, the \(\hbox {H}\beta \) line is displaced to the longer wavelengths, but no synchronous variabilities were observed in other spectral lines (CII \( \lambda \) 6578.05 Å, \( \lambda \) 6582.88 Å  and HeI \( \lambda \) 5875.72 Å) formed in deeper layers of the stellar atmosphere. In addition, the profiles of the \(\hbox {H}\alpha \) and \(\hbox {H}\beta \) lines have been analysed, as well as their relations with possible expansion, contraction and mixed conditions of the atmosphere of HD14134. We suggest that the observational evidence for the non-stationary atmosphere of HD14134 can be associated in part with the non-spherical stellar wind.     

Origin and evolution of the atmospheres of early Venus,Earth and Mars

We review the origin and evolution of the atmospheres of Earth, Venus and Mars from the time when their accreting bodies were released from the protoplanetary disk a few million years after the origin of the Sun. If the accreting planetary cores reached masses \(\ge 0.5 M_\mathrm{Earth}\) before the gas in the disk disappeared, primordial atmospheres consisting mainly of H\(_2\) form around the young planetary body, contrary to late-stage planet formation, where terrestrial planets accrete material after the nebula phase of the disk. The differences between these two scenarios are explored by investigating non-radiogenic atmospheric noble gas isotope anomalies observed on the three terrestrial planets. The role of the young Sun’s more efficient EUV radiation and of the plasma environment into the escape of early atmospheres is also addressed. We discuss the catastrophic outgassing of volatiles and the formation and cooling of steam atmospheres after the solidification of magma oceans and we describe the geochemical evidence for additional delivery of volatile-rich chondritic materials during the main stages of terrestrial planet formation. The evolution scenario of early Earth is then compared with the atmospheric evolution of planets where no active plate tectonics emerged like on Venus and Mars. We look at the diversity between early Earth, Venus and Mars, which is found to be related to their differing geochemical, geodynamical and geophysical conditions, including plate tectonics, crust and mantle oxidation processes and their involvement in degassing processes of secondary \(\hbox {N}_2\) atmospheres. The buildup of atmospheric \(\hbox {N}_2\), \(\hbox {O}_2\), and the role of greenhouse gases such as \(\hbox {CO}_2\) and \(\hbox {CH}_4\) to counter the Faint Young Sun Paradox (FYSP), when the earliest life forms on Earth originated until the Great Oxidation Event \(\approx \) 2.3 Gyr ago, are addressed. This review concludes with a discussion on the implications of understanding Earth’s geophysical and related atmospheric evolution in relation to the discovery of potential habitable terrestrial exoplanets.     

Correlation Between the Magnetic Field and Plasma Parameters at 1 AU

The physical parameters of the solar wind observed in-situ near 1 AU have been studied for several decades, and relationships between them, such as the positive correlation between the solar wind plasma temperature, \(T\), and velocity, \(V\), and the negative correlation between density, \(N\), and velocity, \(V\), are well known. However, the magnetic field intensity, \(B\), does not appear to be well correlated with any individual plasma parameter. In this article, we discuss previously under-reported correlations between \(B\) and the combined plasma parameters \(\sqrt{N V^{2}} \) as well as between \(B\) and \(\sqrt{NT}\). These two correlations are strong during periods of corotating interaction regions and high-speed streams, and moderate during intervals of slow solar wind. The results indicate that the magnetic pressure in the solar wind is well correlated both with the plasma dynamic pressure and the thermal pressure.     

Star-formation rate in compact star-forming galaxies

We use the data for the \(\text{H}\beta\) emission-line, far-ultraviolet (FUV) and mid-infrared 22 μm continuum luminosities to estimate star formation rates \(\langle \mbox{SFR} \rangle \) averaged over the galaxy lifetime for a sample of about 14000 bursting compact star-forming galaxies (CSFGs) selected from the Data Release 12 (DR12) of the Sloan Digital Sky Survey (SDSS). The average coefficient linking \(\langle \mbox{SFR} \rangle \) and the star formation rate \(\mbox{SFR}_{0}\) derived from the \(\text{H}\beta\) luminosity at zero starburst age is found to be 0.04. We compare \(\langle \mbox{SFR} \rangle \mbox{s}\) with some commonly used SFRs which are derived adopting a continuous star formation during a period of \({\sim}\,100~\mbox{Myr}\), and find that the latter ones are 2–3 times higher. It is shown that the relations between SFRs derived using a geometric mean of two star-formation indicators in the UV and IR ranges and reduced to zero starburst age have considerably lower dispersion compared to those with single star-formation indicators. We suggest that our relations for \(\langle \mbox{SFR} \rangle \) determination are more appropriate for CSFGs because they take into account a proper temporal evolution of their luminosities. On the other hand, we show that commonly used SFR relations can be applied for approximate estimation within a factor of \({\sim}\,2\) of the \(\langle \mbox{SFR} \rangle \) averaged over the lifetime of the bursting compact galaxy.     

STEREO observations of hybrid stars V775 Tau and V483 Tau

In this study, pulsational and physical characteristics of two \(\delta\) Scuti stars, V775?Tau and V483?Tau, are analysed by using four-year high-precision photometric data of the STEREO satellite. Thus, it is aimed to gain new insights into behaviours of these pulsators and evolution of \(\delta\) Scuti, \(\gamma\) Dor and Am type stars. The data are taken between 2007–2011 and examined with the help of the Lomb–Scargle method. The detection precision in the four-year combined data is around \(10^{-5}\) cd^?1 in frequency and \(10^{-5}\) mag in amplitude. It is revealed that V775?Tau exhibits weak pulsation characteristic which is interpreted as the existence of the interaction between the helium loss in the partial ionization zone and pulsation intensities. It is also considered that the absence of strong pulsations is also related to the evolution status of the star. Further, its periodogram shows low-frequency peaks. If these oscillations are g-modes, V775?Tau can be thought to be one of the rare stars that show all \(\gamma\) Dor, \(\delta\) Scuti and Am type variations. V483?Tau is comparatively more luminous, hotter and has higher rotational velocity. Therefore, although it shares the same region with V775?Tau in the H–R diagram, it is not considered to be an Am star. Yet, it exactly overlaps with the \(\gamma\) Dor stars. These clues as well as g-modes detected in its periodogram indicate that V483?Tau is a hybrid star. Finally, both V775?Tau and V483?Tau display period changes whose rates are between \(10^{-3}\) and \(10^{-4}\) yr^?1. Considering the \(\delta\) Scuti nature, it may be speculated that these changes are non-evolutionary.     

New holographic dark energy model with constant bulk viscosity in modified $$ gravity theory

The aim of this paper is to study new holographic dark energy (HDE) model in modified \(f(R,T)\) gravity theory within the framework of a flat Friedmann-Robertson-Walker model with bulk viscous matter content. It is thought that the negative pressure caused by the bulk viscosity can play the role of dark energy component, and drive the accelerating expansion of the universe. This is the motive of this paper to observe such phenomena with bulk viscosity. In the specific model \(f(R,T)=R+\lambda T\), where \(R\) is the Ricci scalar, \(T\) the trace of the energy-momentum tensor and \(\lambda \) is a constant, we find the solution for non-viscous and viscous new HDE models. We analyze new HDE model with constant bulk viscosity, \(\zeta =\zeta _{0}= \text{const.}\) to explain the present accelerated expansion of the universe. We classify all possible scenarios (deceleration, acceleration and their transition) with possible positive and negative ranges of \(\lambda \) over the constraint on \(\zeta _{0}\) to analyze the evolution of the universe. We obtain the solutions of scale factor and deceleration parameter, and discuss the evolution of the universe. We observe the future finite-time singularities of type I and III at a finite time under certain constraints on \(\lambda \). We also investigate the statefinder and \(\mathit{Om}\) diagnostics of the viscous new HDE model to discriminate with other existing dark energy models. In late time the viscous new HDE model approaches to \(\varLambda \mathit{CDM}\) model. We also discuss the thermodynamics and entropy of the model and find that it satisfies the second law of thermodynamics.     

Relative equilibria of the restricted three-body problem in curved spaces

We use a formulation of the N-body problem in spaces of constant Gaussian curvature, \({\kappa }\in \mathbb {R}\), as widely used by A. Borisov, F. Diacu and their coworkers. We consider the restricted three-body problem in \(\mathbb {S}^2\) with arbitrary \({\kappa }>0\) (resp. \(\mathbb {H}^2\) with arbitrary \({\kappa }<0\)) in a formulation also valid for the case \({\kappa }=0\). For concreteness when \({\kappa }>0\) we restrict the study to the case of the three bodies at the upper hemisphere, to be denoted as \(\mathbb {S}^2_+\). The main goal is to obtain the totality of relative equilibria as depending on the parameters \({\kappa }\) and the mass ratio \(\mu \). Several general results concerning relative equilibria and its stability properties are proved analytically. The study is completed numerically using continuation from the \({\kappa }=0\) case and from other limit cases. In particular both bifurcations and spectral stability are also studied. The \(\mathbb {H}^2\) case is similar, in some sense, to the planar one, but in the \(\mathbb {S}^2_+\) case many differences have been found. Some surprising phenomena, like the coexistence of many triangular-like solutions for some values \(({\kappa },\mu )\) and many stability changes will be discussed.     

The Sitnikov problem for several primary bodies configurations

In this paper we address an \(n+1\)-body gravitational problem governed by the Newton’s laws, where n primary bodies orbit on a plane \(\varPi \) and an additional massless particle moves on the perpendicular line to \(\varPi \) passing through the center of mass of the primary bodies. We find a condition for the described configuration to be possible. In the case when the primaries are in a rigid motion, we classify all the motions of the massless particle. We study the situation when the massless particle has a periodic motion with the same minimal period as the primary bodies. We show that this fact is related to the existence of a certain pyramidal central configuration.     

Role of primordial black holes in the direct collapse scenario of supermassive black hole formation at high redshifts

In this paper, we explore the possibility of accreting primordial black holes as the source of heating for the collapsing gas in the context of the direct collapse black hole scenario for the formation of super-massive black holes (SMBHs) at high redshifts, \(z\sim \) 6–7. One of the essential requirements for the direct collapse model to work is to maintain the temperature of the in-falling gas at \(\approx \)10\(^4\) K. We show that even under the existing abundance limits, the primordial black holes of masses \(\gtrsim \)10\(^{-2}M_\odot \), can heat the collapsing gas to an extent that the \(\mathrm{H}_2\) formation is inhibited. The collapsing gas can maintain its temperature at \(10^4\) K till the gas reaches a critical density \(n_{{c}} \,{\approx }\, 10^3~\hbox {cm}^{-3}\), at which the roto-vibrational states of \(\mathrm{H}_2\) approaches local thermodynamic equilibrium and \(\mathrm{H}_2\) cooling becomes inefficient. In the absence of \(\mathrm{H}_2\) cooling, the temperature of the collapsing gas stays at \(\approx \)10\(^4\) K even as it collapses further. We discuss scenarios of subsequent angular momentum removal and the route to find collapse through either a supermassive star or a supermassive disk.     

GONG p-Mode Parameters Through Two Solar Cycles

We investigate the parameters of global solar p-mode oscillations, namely damping width \(\Gamma\), amplitude \(A\), mean squared velocity \(\langle v^{2}\rangle\), energy \(E\), and energy supply rate \(\mathrm{d}E/\mathrm{d}t\), derived from two solar cycles’ worth (1996?–?2018) of Global Oscillation Network Group (GONG) time series for harmonic degrees \(l=0\,\mbox{--}\,150\). We correct for the effect of fill factor, apparent solar radius, and spurious jumps in the mode amplitudes. We find that the amplitude of the activity-related changes of \(\Gamma\) and \(A\) depends on both frequency and harmonic degree of the modes, with the largest variations of \(\Gamma\) for modes with \(2400~\upmu\mbox{Hz}\le\nu\le3300~\upmu\mbox{Hz}\) and \(31\le l \le60\) with a minimum-to-maximum variation of \(26.6\pm0.3\%\) and of \(A\) for modes with \(2400~\upmu\mbox{Hz}\le\nu\le 3300~\upmu\mbox{Hz}\) and \(61\le l \le100\) with a minimum-to-maximum variation of \(27.4\pm0.4\%\). The level of correlation between the solar radio flux \(F_{10.7}\) and mode parameters also depends on mode frequency and harmonic degree. As a function of mode frequency, the mode amplitudes are found to follow an asymmetric Voigt profile with \(\nu_{\text{max}}=3073.59\pm0.18~\upmu\mbox{Hz}\). From the mode parameters, we calculate physical mode quantities and average them over specific mode frequency ranges. In this way, we find that the mean squared velocities \(\langle v^{2}\rangle\) and energies \(E\) of p modes are anticorrelated with the level of activity, varying by \(14.7\pm0.3\%\) and \(18.4\pm0.3\%\), respectively, and that the mode energy supply rates show no significant correlation with activity. With this study we expand previously published results on the temporal variation of solar p-mode parameters. Our results will be helpful to future studies of the excitation and damping of p modes, i.e., the interplay between convection, magnetic field, and resonant acoustic oscillations.     

Landau damping of dust acoustic waves in the presence of hybrid nonthermal nonextensive electrons

Based on the kinetic theory, Landau damping of dust acoustic waves (DAWs) propagating in a dusty plasma composed of hybrid nonthermal nonextensive distributed electrons, Maxwellian distributed ions and negatively charged dust grains is investigated using Vlasov-Poisson’s equations. The characteristics of the DAWs Landau damping are discussed. It is found that the wave frequency increases by decreasing (increasing) the value of nonextensive (nonthermal) parameter, \(q\) (\(\alpha \)). It is recognized that \(\alpha \) plays a significant role in observing damping or growing DAW oscillations. For small values of \(\alpha \), damping modes have been observed until reaching a certain value of \(\alpha \) at which \(\omega _{i}\) vanishes, then a growing mode appears in the case of superextensive electrons. However, only damping DAW modes are observed in case of subextensive electrons. The present study is useful in the space situations where such distribution exists.     

Supernova neutrinos: Flavor conversion independent of their mass

In extremely dense neutrino environments like in supernova core, the neutrino-neutrino refraction may give rise to self-induced flavor conversion. These neutrino flavor oscillations are well understood from the idea of the exponentially growing modes of the interacting oscillators in the flavor space. Until recently, the growth rates of these modes were found to be of the order of the vacuum oscillation frequency \(\Delta m^2/2E\) [\(\mathcal {O}(1~\mathrm{km}^{-1})\)] and were considered slow growing. However, in the last couple of years it was found that if the system was allowed to have different zenith-angle distributions for the emitted \(\nu _e\) and \(\bar{\nu }_e\) beams then the fastest growing modes of the interacting oscillators grew at the order of \(\mu =\sqrt{2} G_\mathrm{F}n_{\nu }\), a typical \(\nu \)–\(\nu \) interaction energy [\(\mathcal {O}(10^5~\mathrm{km}^{-1})\)]. Thus the growth rates are very large in comparison to the so-called ‘slow oscillations’ and can result in neutrino flavor conversion on a much faster scale. In fact, the point that the growth rates are no longer dependent on the vacuum oscillation frequency \(\Delta m^2/2E\), makes these ‘fast flavor conversions’ independent of \(\Delta m^2\) (thus mass) and energy. This is a surprising result as neutrino flavor conversions are considered to be the ultimate proof of massive neutrinos. However, the importance of this effect in the realistic astrophysical scenarios still remains to be understood.     

Density variation effect on multi-ions with kinetic Alfven wave around cusp region—a kinetic approach

The kinetic Alfven waves in the presence of homogeneous magnetic field plasma with multi-ions effect are investigated. The dispersion relation and normalised damping rate are derived for low-\(\beta\) plasma using kinetic theory. The effect of density variation of \(\text{H}^{+}\), \(\text{He}^{+}\) and \(\text{O}^{+}\) ions is observed on frequency and damping rate of the wave. The variation of frequency (\(\omega\)) and normalised damping rate (\(\gamma / \varOmega_{H^{ +}} \)) of the wave are studied with respect to \(k_{ \bot} \rho_{j}\), where \(k_{ \bot} \) is the perpendicular wave number, \(\rho_{j}\) is the ion gyroradius and \(j \) denotes \(\text{H}^{+}\), \(\text{He}^{+}\) and \(\text{O}^{+}\) ions. The variation with \(k_{ \bot} \rho_{j}\) is considered over wide range. The parameters appropriate to cusp region are used for the explanation of results. It is found that with hydrogen and helium ions gyration, the frequency of wave is influenced by the density variation of \(\text{H}^{+}\) and \(\text{He}^{+}\) ions but remains insensitive to the change in density of \(\text{O}^{+}\) ions. For oxygen ion gyration, the frequency of wave varies over a short range only for \(\text{O}^{+}\) ion density variation. The wave shows damping at lower altitude due to variation in density of lighter \(\text{H}^{+}\) and \(\text{He}^{+}\) ions whereas at higher altitude only heavy \(\text{O}^{+}\) ions contribute in wave damping. The damping of wave may be due to landau damping or energy transfer from wave to particles. The present study signifies that the both lighter and heavier ions dominate differently to change the characteristics of kinetic Alfven wave and density variation is also an important parameter to understand wave phenomena in cusp region.     

The radio and $\gamma$-ray variability analysis of S5 0716+714

We have studied the variability of S5 0716+714 at radio 15 GHz and \(\gamma\)-ray band using three different methods. A possible periodicity of \(P_{15~\text{GHz}}=266.0\pm11.5\) and \(P_{\gamma}=344.0 \pm16.4\) days are obtained for radio 15 GHz and \(\gamma\)-ray light curves, respectively. The variability may be related to the intrinsically emission mechanism. The difference between the variability timescales of radio 15 GHz and \(\gamma \)-ray may be due to that the emission of radio 15 GHz is produced via the synchrotron process, while the \(\gamma\)-ray is produced by both the SSC and EC processes.     

Effect of electron temperature on small-amplitude electron acoustic solitary waves in non-planar geometry

Effects of electron temperature on the propagation of electron acoustic solitary waves in plasma with stationary ions, cold and superthermal hot electrons is investigated in non-planar geometry employing reductive perturbation method. Modified Korteweg–de Vries equation is derived in the small amplitude approximation limit. The analytical and numerical calculations of the KdV equation reveal that the phase velocity of the electron acoustic waves increases as one goes from planar to non planar geometry. It is shown that the electron temperature ratio changes the width and amplitude of the solitary waves and when electron temperature is not taken into account,our results completely agree with the results of Javidan & Pakzad (2012). It is found that at small values of \(\tau \), solitary wave structures behave differently in cylindrical (\(\text {m} = 1\)), spherical (\(\text {m} = 2\)) and planar geometry (\(\text {m} = 0\)) but looks similar at large values of \(\tau \). These results may be useful to understand the solitary wave characteristics in laboratory and space environments where the plasma have multiple temperature electrons.