A family of stacked central configurations in the planar five-body problem

We study planar central configurations of the five-body problem where three bodies, \(m_1, m_2\) and \(m_3\), are collinear and ordered from left to right, while the other two, \(m_4\) and \(m_5\), are placed symmetrically with respect to the line containing the three collinear bodies. We prove that when the collinear bodies form an Euler central configuration of the three-body problem with \(m_1=m_3\), there exists a new family, missed by Gidea and Llibre (Celest Mech Dyn Astron 106:89–107, 2010), of stacked five-body central configuration where the segments \(m_4m_5\) and \(m_1m_3\) do not intersect.     

On the photo-gravitational restricted four-body problem with variable mass

This paper deals with the photo-gravitational restricted four-body problem (PR4BP) with variable mass. Following the procedure given by Gascheau (C. R. 16:393–394, 1843) and Routh (Proc. Lond. Math. Soc. 6:86–97, 1875), the conditions of linear stability of Lagrange triangle solution in the PR4BP are determined. The three radiating primaries having masses \(m_{1}\), \(m_{2}\) and \(m_{3}\) in an equilateral triangle with \(m_{2}=m_{3}\) will be stable as long as they satisfy the linear stability condition of the Lagrangian triangle solution. We have derived the equations of motion of the mentioned problem and observed that there exist eight libration points for a fixed value of parameters \(\gamma (\frac{m \ \text{at time} \ t}{m \ \text{at initial time}}, 0<\gamma\leq1 )\), \(\alpha\) (the proportionality constant in Jeans’ law (Astronomy and Cosmogony, Cambridge University Press, Cambridge, 1928), \(0\leq\alpha\leq2.2\)), the mass parameter \(\mu=0.005\) and radiation parameters \(q_{i}, (0< q_{i}\leq1, i=1, 2, 3)\). All the libration points are non-collinear if \(q_{2}\neq q_{3}\). It has been observed that the collinear and out-of-plane libration points also exist for \(q_{2}=q_{3}\). In all the cases, each libration point is found to be unstable. Further, zero velocity curves (ZVCs) and Newton–Raphson basins of attraction are also discussed.     

Dynamics of “jumping” Trojans: a perturbative treatment

The term “jumping” Trojan was introduced by Tsiganis et al. (Astron Astrophys 354:1091–1100, 2000) in their studies of long-term dynamics exhibited by the asteroid (1868) Thersites, which had been observed to jump from librations around \(L_4\) to librations around \(L_5\). Another example of a “jumping” Trojan was found by Connors et al. (Nature 475:481–483, 2011): librations of the asteroid 2010 TK7 around the Earth’s libration point \(L_4\) preceded by its librations around \(L_5\). We explore the dynamics of “jumping” Trojans under the scope of the restricted planar elliptical three-body problem. Via double numerical averaging we construct evolutionary equations, which allow analyzing transitions between different regimes of orbital motion.     

15-digit accuracy calculations of Ambartsumian-Chandrasekhar’s $$-functions for four-term phase functions with the double-exponential formula

We have established an iterative scheme to calculate with 15-digit accuracy the numerical values of Ambartsumian-Chandrasekhar’s \(H\)-functions for anisotropic scattering characterized by the four-term phase function: the method incorporates some advantageous features of the iterative procedure of Kawabata (Astrophys. Space Sci. 358:32, 2015) and the double-exponential integration formula (DE-formula) of Takahashi and Mori (Publ. Res. Inst. Math. Sci. Kyoto Univ. 9:721, 1974), which proved highly effective in Kawabata (Astrophys. Space Sci. 361:373, 2016). Actual calculations of the \(H\)-functions have been carried out employing 27 selected cases of the phase function, 56 values of the single scattering albedo \(\varpi_{0}\), and 36 values of an angular variable \(\mu(= \cos\theta)\), with \(\theta\) being the zenith angle specifying the direction of incidence and/or emergence of radiation. Partial results obtained for conservative isotropic scattering, Rayleigh scattering, and anisotropic scattering due to a full four-term phase function are presented. They indicate that it is important to simultaneously verify accuracy of the numerical values of the \(H\)-functions for \(\mu<0.05\), the domain often neglected in tabulation. As a sample application of the isotropic scattering \(H\)-function, an attempt is made in Appendix to simulate by iteratively solving the Ambartsumian equation the values of the plane and spherical albedos of a semi-infinite, homogeneous atmosphere calculated by Rogovtsov and Borovik (J. Quant. Spectrosc. Radiat. Transf. 183:128, 2016), who employed their analytical representations for these quantities and the single-term and two-term Henyey-Greenstein phase functions of appreciably high degrees of anisotropy. While our results are in satisfactory agreement with theirs, our procedure is in need of a faster algorithm to routinely deal with problems involving highly anisotropic phase functions giving rise to near-conservative scattering.     

Anisotropic model with decaying cosmological term

A spatially homogeneous and anisotropic locally rotationally symmetric Bianchi type-I spacetime with cosmological term \(\varLambda \) in \(f(R,T) \) theory has been studied. The exact solution of the field equations is obtained under a variation law of the Hubble parameter \((H) \) which yields a time dependent deceleration parameter (Banerjee and Das in Gen. Relativ. Gravit. 37:10, 2005). The model presents a cosmological scenario which describes early deceleration and late time acceleration. The physical parameters of the model have been analysed.     

Cosmic strings in a five dimensional spherically symmetric background in $$ gravity

We study the physical behavior of a five dimensional non-static spherically symmetric cosmological models in the presence of massive strings in the framework of \(f(R,T)\) gravity proposed by Harko et al. (Phys. Rev. D 84:024020, 2011). Here \(R\) is the Ricci scalar and \(T\) is the trace of the stress energy tensor and the fifth dimension is not observed because it is compact. We solve the field equations (i) using a relation between the scale factors given by Samantha and Dhal (Int. J. Theor. Phys. 52:1334, 2013) and (ii) equations of state for string models. The models obtained correspond to \(p\)-string, geometric string and massive string models in this modified theory in five dimensions. Cosmological parameters of the models are determined and their dynamical properties are discussed.     

Theoretical scaling law of coronal magnetic field and electron power-law index in solar microwave burst sources

It is first proposed a theoretical scaling law respectively for the coronal magnetic field strength \(B\) and electron power-law index \(\delta\) versus frequency and coronal height in solar microwave burst sources. Based on the non-thermal gyro-synchrotron radiation model (Ramaty in Astrophys. J. 158:753, 1969), \(B\) and \(\delta\) are uniquely solved by the observable optically-thin spectral index and turnover (peak) frequency, the other parameters (plasma density, temperature, view angle, low and high energy cutoffs, etc.) are relatively insensitive to the calculations, thus taken as some typical values. Both of \(B\) and \(\delta\) increase with increasing of radio frequency but with decreasing of coronal height above photosphere, and well satisfy a square or cubic logarithmic fitting.     

An analysis of the convergence of Newton iterations for solving elliptic Kepler’s equation

In this note a study of the convergence properties of some starters \( E_0 = E_0(e,M)\) in the eccentricity–mean anomaly variables for solving the elliptic Kepler’s equation (KE) by Newton’s method is presented. By using a Wang Xinghua’s theorem (Xinghua in Math Comput 68(225):169–186, 1999) on best possible error bounds in the solution of nonlinear equations by Newton’s method, we obtain for each starter \( E_0(e,M)\) a set of values \( (e,M) \in [0, 1) \times [0, \pi ]\) that lead to the q-convergence in the sense that Newton’s sequence \( (E_n)_{n \ge 0}\) generated from \( E_0 = E_0(e,M)\) is well defined, converges to the exact solution \(E^* = E^*(e,M)\) of KE and further \( \vert E_n - E^* \vert \le q^{2^n -1}\; \vert E_0 - E^* \vert \) holds for all \( n \ge 0\). This study completes in some sense the results derived by Avendaño et al. (Celest Mech Dyn Astron 119:27–44, 2014) by using Smale’s \(\alpha \)-test with \(q=1/2\). Also since in KE the convergence rate of Newton’s method tends to zero as \( e \rightarrow 0\), we show that the error estimates given in the Wang Xinghua’s theorem for KE can also be used to determine sets of q-convergence with \( q = e^k \; \widetilde{q} \) for all \( e \in [0,1)\) and a fixed \( \widetilde{q} \le 1\). Some remarks on the use of this theorem to derive a priori estimates of the error \( \vert E_n - E^* \vert \) after n Kepler’s iterations are given. Finally, a posteriori bounds of this error that can be used to a dynamical estimation of the error are also obtained.     

Dynamics of Large-Scale Solar-Wind Streams Obtained by the Double Superposed Epoch Analysis: 3. Deflection of the Velocity Vector

This work is a continuation of our previous articles (Yermolaev et al. in J. Geophys. Res.120, 7094, 2015 and Yermolaev et al. in Solar Phys.292, 193, 2017), which describe the average temporal profiles of interplanetary plasma and field parameters in large-scale solar-wind (SW) streams: corotating interaction regions (CIRs), interplanetary coronal mass ejections (ICMEs, including both magnetic clouds (MCs) and ejecta), and sheaths as well as interplanetary shocks (ISs). Changes in the longitude angle, \(\varphi\), in CIRs from ?2 to \(2^{\circ}\) agree with earlier results (e.g. Gosling and Pizzo, 1999). We have also analyzed the average temporal profiles of the bulk velocity angles in sheaths and ICMEs. We have found that the angle \(\varphi\) in ICMEs changes from 2 to \(-2^{\circ}\), while in sheaths it changes from ?2 to \(2^{\circ}\) (similar to the change in CIRs), i.e. the angle in CIRs and sheaths deflects in the opposite sense to ICMEs. When averaging the latitude angle \(\vartheta\) on all the intervals of the chosen SW types, the angle \(\vartheta\) is almost constant at \({\sim}\,1^{\circ}\). We made for the first time a selection of SW events with increasing and decreasing \(\vartheta\) and found that the average \(\vartheta\) temporal profiles in the selected events have the same “integral-like” shape as for \(\varphi\). The difference in \(\varphi\) and \(\vartheta\) average profiles is explained by the fact that most events have increasing profiles for the angle in the ecliptic plane as a result of solar rotation, while for the angle in the meridional plane, the numbers of events with increasing and decreasing profiles are equal.     

Transition between general relativity and quantum gravity using quark and strange quark matter with some kinematical test

Contemporary piece of writing devotes to the investigation of plane symmetric cosmological model with quark and strange quark matter in the deformations of the Einstein’s theory of General Relativity (GR). At small or large scales (ultraviolet or infrared gravity), deformations of the Einstein’s theory could provide a better handling of cosmic acceleration without magnetism (along with singularities). In particular, a proper deformation of GR in the ultraviolet regime could play the role of describing the transition between GR and quantum gravity. As a matter of fact, although with a different purpose in mind, it was Einstein himself who proposed in the 30’s the reformulation of GR by taking the field of orthonormal frames or tetrads as the dynamical variable instead of the metric tensor (Einstein, Phys. Math. Kl 217, 401, 1928). As per the observation, pressure and energy density of the model approaches the bag constant in negative and positive ways at \(t\rightarrow \infty \), i.e. \(p\rightarrow -B_c \) and \(\rho \rightarrow B_c \), the negative pressure due to the Dark Energy (DE) in the context of accelerated expansion of the universe. So the strange quark matter gives an idea of existence of dark energy in the universe and supports the observations of the SNe-I (Riess et al., Astron. J. 116,1009, 1998; Perlmutter et al., Astrophys. J. 517, 565, 1999). Also these results agree with the study of Aktas and Aygun (Chinese J. Phys. 55, 71, 2017) and Sahoo et al. (New. Astron. 60, 80, 2018).     

Average Magnetic Field Magnitude Profiles of <Emphasis Type="Italic">Wind</Emphasis> Magnetic Clouds as a Function of Closest Approach to the Clouds’ Axes and Comparison to Model

We examine the average magnetic field magnitude (\(| \boldsymbol{B} | \equiv B\)) within magnetic clouds (MCs) observed by the Wind spacecraft from 1995 to July 2015 to understand the difference between this \(B\) and the ideal \(B\)-profiles expected from using the static, constant-\(\alpha\), force-free, cylindrically symmetric model for MCs of Lepping, Jones, and Burlaga (J. Geophys. Res. 95, 11957, 1990, denoted here as the LJB model). We classify all MCs according to an assigned quality, \(Q_{0}\) (\(= 1, 2, 3\), for excellent, good, and poor). There are a total of 209 MCs and 124 when only \(Q_{0} = 1\), 2 cases are considered. The average normalized field with respect to the closest approach (\(\mathit{CA}\)) is stressed, where we separate cases into four \(\mathit{CA}\) sets centered at 12.5 %, 37.5 %, 62.5 %, and 87.5 % of the average radius; the averaging is done on a percentage-duration basis to treat all cases the same. Normalized \(B\) means that before averaging, the \(B\) for each MC at each point is divided by the LJB model-estimated \(B\) for the MC axis, \(B_{0}\). The actual averages for the 209 and 124 MC sets are compared to the LJB model, after an adjustment for MC expansion (e.g. Lepping et al. in Ann. Geophys. 26, 1919, 2008). This provides four separate difference-relationships, each fitted with a quadratic (Quad) curve of very small \(\sigma\). Interpreting these Quad formulae should provide a comprehensive view of the variation in normalized \(B\) throughout the average MC, where we expect external front and rear compression to be part of its explanation. These formulae are also being considered for modifying the LJB model. This modification will be used in a scheme for forecasting the timing and magnitude of magnetic storms caused by MCs. Extensive testing of the Quad formulae shows that the formulae are quite useful in correcting individual MC \(B\)-profiles, especially for the first \({\approx\,}1/3\) of these MCs. However, the use of this type of \(B\) correction constitutes a (slight) violation of the force-free assumption used in the original LJB MC model.     

Anisotropic new holographic dark energy model in Saez–Ballester theory of gravitation

This work is devoted to the investigation of new holographic dark energy (infrared cutoff is the Hubble radius) in locally rotationally symmetric Bianchi type-\(I\) universe within the framework of Saez–Ballester (Phys. Lett. A 113:467, 1986) scalar–tensor theory of gravitation. We construct interacting and non-interacting dark energy models by solving the field equations using a relationship between the metric potentials. This leads to a variable deceleration parameter model which exhibits a transition of the universe from deceleration to acceleration. We evaluate various cosmological parameters of our models. We have observed that the energy density parameters, equation of state and important cosmological planes (\(\omega _{\mathit{de}} - \omega _{\mathit{de}}'\) and \(r - s\)) yield the results compatible with the modern observational data. We have, also, discussed the stability analysis of our models.     

Satellite de-orbiting via controlled solar radiation pressure

We analyze the families of central configurations of the spatial 5-body problem with four masses equal to 1 when the fifth mass m varies from 0 to \(+\infty \). In particular we continue numerically, taking m as a parameter, the central configurations (which all are symmetric) of the restricted spatial (\(4+1\))-body problem with four equal masses and \(m=0\) to the spatial 5-body problem with equal masses (i.e. \(m=1\)), and viceversa we continue the symmetric central configurations of the spatial 5-body problem with five equal masses to the restricted (\(4+1\))-body problem with four equal masses. Additionally we continue numerically the symmetric central configurations of the spatial 5-body problem with four equal masses starting with \(m=1\) and ending in \(m=+\infty \), improving the results of Alvarez-Ramírez et al. (Discrete Contin Dyn Syst Ser S 1: 505–518, 2008). We find four bifurcation values of m where the number of central configuration changes. We note that the central configurations of all continued families varying m from 0 to \(+\infty \) are symmetric.     

Short timescale UV variability study in NGC 4151 using IUE data

IUE has made very successful long term and intense short time-scale monitoring spectroscopic study of NGC 4151, a Seyfert 1 galaxy for over nearly 18 years from its launch in 1978 to 1996. The long-term observations have been useful in understanding the complex relation between UV continuum and emission line variability Seyfert galaxies. In this paper, we present the results of our studies on the short-timescale intense monitoring campaign of NGC 4151 undertaken during December 1–15, 1993. A most intense monitoring observation of NGC 4151 was carried out by IUE in 1993, when the source was at its historical high flux state with a shortest interval of 70 min between two successive observations. We present our results on emission line and continuum variability amplitudes characterized by \(F_{\mathrm{var}}\) method. We found highest variability of nearly 8.3% at 1325 \(\AA \) continuum with a smallest amplitude of 4% at 2725 \(\AA \). The relative variability amplitudes (\(R_\mathrm{max}\)) have been found to be 1.372, 1.319, 1.302 and 1.182 at 1325, 1475, 1655 and 2725 \(\AA \) continuum respectively. The continuum and emission line variability characteristics obtained in the present analysis are in very good agreement with the results obtained by Edelson et al. (1996) and Crenshaw et al. (1996) from the analysis of the same observational spectral data. The large amplitude rapid variability characteristics obtained in our study have been attributed to the continuum reprocessing of X-rays absorbed by the material in the accretion disk as proposed by Shakura and Sunyaev (1973). The continuum and emission light curves have shown four distinct high amplitude events of flux maxima during the intense monitoring campaign of 15 days, providing a good limit on the amplitude of UV variability and the BLR size in low luminosity Seyfert galaxies and are useful for constraining the continuum emission models. The decreasing \(F_{\mathrm{var}}\) amplitude of UV continuum with respect to increasing wavelength obtained in the present study and consistent with similar observations by Edelson et al. (1996) and Crenshaw et al. (1996) is a significant result of the intense monitoring observations.     

New Hamiltonian expansions adapted to the Trojan problem

A number of studies, referring to the observed Trojan asteroids of various planets in our Solar System, or to hypothetical Trojan bodies in extrasolar planetary systems, have emphasized the importance of so-called secondary resonances in the problem of the long term stability of Trojan motions. Such resonances describe commensurabilities between the fast, synodic, and secular frequency of the Trojan body, and, possibly, additional slow frequencies produced by more than one perturbing bodies. The presence of secondary resonances sculpts the dynamical structure of the phase space. Hence, identifying their location is a relevant task for theoretical studies. In the present paper we combine the methods introduced in two recent papers (Páez and Efthymiopoulos in Celest Mech Dyn Astron 121(2):139, 2015; Páez and Locatelli in MNRAS 453(2):2177, 2015) in order to analytically predict the location of secondary resonances in the Trojan problem. In Páez and Efthymiopoulos (2015), the motion of a Trojan body was studied in the context of the planar Elliptic Restricted Three Body or the planar Restricted Multi-Planet Problem. It was shown that the Hamiltonian admits a generic decomposition \(H=H_b+H_{sec}\). The term \(H_b\), called the basic Hamiltonian, is a model of two degrees of freedom characterizing the short-period and synodic motions of a Trojan body. Also, it yields a constant ‘proper eccentricity’ allowing to define a third secular frequency connected to the body’s perihelion precession. \(H_{sec}\) contains all remaining secular perturbations due to the primary or to additional perturbing bodies. Here, we first investigate up to what extent the decomposition \(H=H_b+H_{sec}\) provides a meaningful model. To this end, we produce numerical examples of surfaces of section under \(H_b\) and compare with those of the full model. We also discuss how secular perturbations alter the dynamics under \(H_b\). Secondly, we explore the normal form approach introduced in Páez and Locatelli (2015) in order to find an ‘averaged over the fast angle’ model derived from \(H_b\), circumventing the problem of the series’ limited convergence due to the collision singularity at the 1:1 MMR. Finally, using this averaged model, we compute semi-analytically the position of the most important secondary resonances and compare the results with those found by numerical stability maps in specific examples. We find a very good agreement between semi-analytical and numerical results in a domain whose border coincides with the transition to large-scale chaotic Trojan motions.     

Relating Solar Energetic Particle Event Fluences to Peak Intensities

Recently we (Kahler and Ling, Solar Phys.292, 59, 2017: KL) have shown that time–intensity profiles [\(I(t)\)] of 14 large solar energetic particle (SEP) events can be fitted with a simple two-parameter fit, the modified Weibull function, which is characterized by shape and scaling parameters [\(\alpha\) and \(\beta\)]. We now look for a simple correlation between an event peak energy intensity [\(I_{\mathrm{p}}\)] and the time integral of \(I(t)\) over the event duration: the fluence [\(F\)]. We first ask how the ratio of \(F/I_{\mathrm{p}}\) varies for the fits of the 14 KL events and then examine that ratio for three separate published statistical studies of SEP events in which both \(F\) and \(I_{\mathrm{p}}\) were measured for comparisons of those parameters with various solar-flare and coronal mass ejection (CME) parameters. The three studies included SEP energies from a 4?–?13 MeV band to \(E > 100~\mbox{MeV}\). Within each group of SEP events, we find a very robust correlation (\(\mathrm{CC} > 0.90\)) in log–log plots of \(F\)versus\(I_{\mathrm{p}}\) over four decades of \(I_{\mathrm{p}}\). The ratio increases from western to eastern longitudes. From the value of \(I_{\mathrm{p}}\) for a given event, \(F\) can be estimated to within a standard deviation of a factor of \({\leq}\,2\). Log–log plots of two studies are consistent with slopes of unity, but the third study shows plot slopes of \({<}\,1\) and decreasing with increasing energy for their four energy ranges from \(E > 10~\mbox{MeV}\) to \({>}\,100~\mbox{MeV}\). This difference is not explained.     

Viscoelastic tides: models for use in Celestial Mechanics

In this note, by using Smale’s \(\alpha \)-theorem on the convergence of Newton’s method, the \(\alpha \)-sets of convergence of some starters of solving the elliptic Kepler’s equation are derived. For each starter we compute the exact \(\alpha \)-set in the eccentricity-main anomaly \((e,M)\in [0,1)\times [0,\pi ]\), showing that these sets are larger than those derived by Avendaño et al. (Celest Mech Dyn Astron 119:27–44, 2014). Further, new convergence tests based on the Newton–Kantorowitch theorem are given comparing with the derived from Smale’s \(\alpha \)-test.     

Anisotropic charged generalized polytropic models

In this paper, we found some new anisotropic charged models admitting generalized polytropic equation of state with spherically symmetry. An analytic solution of the Einstein–Maxwell field equations is obtained through the transformation introduced by Durgapal and Banerji (Phys. Rev. D 27:328, 1983). The physical viability of solutions corresponding to polytropic index \(\eta =1/2\), \(2/3\), 1, 2 is analyzed graphically. For this, we plot physical quantities such as radial and tangential pressure, anisotropy, speed of sound which demonstrated that these models achieve all the considerable physical conditions required for a relativistic star. Further, it is mentioned here that previous results for anisotropic charged matter with linear, quadratic and polytropic equation of state can be retrieved.     

Fast numerics for the spin orbit equation with realistic tidal dissipation and constant eccentricity

We present an algorithm for the rapid numerical integration of a time-periodic ODE with a small dissipation term that is \(C^1\) in the velocity. Such an ODE arises as a model of spin–orbit coupling in a star/planet system, and the motivation for devising a fast algorithm for its solution comes from the desire to estimate probability of capture in various solutions, via Monte Carlo simulation: the integration times are very long, since we are interested in phenomena occurring on timescales of the order of \(10^6\)–\(10^7\) years. The proposed algorithm is based on the high-order Euler method which was described in Bartuccelli et al. (Celest Mech Dyn Astron 121(3):233–260, 2015), and it requires computer algebra to set up the code for its implementation. The payoff is an overall increase in speed by a factor of about 7.5 compared to standard numerical methods. Means for accelerating the purely numerical computation are also discussed.     

Secular resonances between bodies on close orbits II: prograde and retrograde orbits for irregular satellites

In extending the analysis of the four secular resonances between close orbits in Li and Christou (Celest Mech Dyn Astron 125:133–160, 2016) (Paper I), we generalise the semianalytical model so that it applies to both prograde and retrograde orbits with a one-to-one map between the resonances in the two regimes. We propose the general form of the critical angle to be a linear combination of apsidal and nodal differences between the two orbits \( b_1 \Delta \varpi + b_2 \Delta \varOmega \), forming a collection of secular resonances in which the ones studied in Paper I are among the strongest. Test of the model in the orbital vicinity of massive satellites with physical and orbital parameters similar to those of the irregular satellites Himalia at Jupiter and Phoebe at Saturn shows that \({>}20\) and \({>}40\%\) of phase space is affected by these resonances, respectively. The survivability of the resonances is confirmed using numerical integration of the full Newtonian equations of motion. We observe that the lowest order resonances with \(b_1+|b_2|\le 3\) persist, while even higher-order resonances, up to \(b_1+|b_2|\ge 7\), survive. Depending on the mass, between 10 and 60% of the integrated test particles are captured in these secular resonances, in agreement with the phase space analysis in the semianalytical model.