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.     

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.     

The Effect of Magnetic and Density Differences on the Fast Kink Oscillations of Neighboring Coronal Loops

It is reasonable that neighboring coronal loops may obtain similar momentum during a flare. The fast kink oscillations (FKOs) between them are thus mainly influenced by their physical differences. We discuss the dependencies of FKO on the physical properties of coronal loops in a low-\(\beta \) thin-tube approximation. From the analysis, we obtain the analytic relationship between the density [\(\rho _{\mathrm{i}}\)] and magnetic field [\(B\)] of loops and the corresponding period [\(\tau \)] and amplitude [\(A\)] of FKO, which may provide us with a powerful tool to diagnose the physical differences between neighboring loops.     

First integrals for the Kepler problem with linear drag

In this work we consider the Kepler problem with linear drag, and prove the existence of a continuous vector-valued first integral, obtained taking the limit as \(t\rightarrow +\infty \) of the Runge–Lenz vector. The norm of this first integral can be interpreted as an asymptotic eccentricity \(e_{\infty }\) with \(0\le e_{\infty } \le 1\). The orbits satisfying \(e_{\infty } <1\) approach the singularity by an elliptic spiral and the corresponding solutions \(x(t)=r(t)e^{i\theta (t)}\) have a norm r(t) that goes to zero like a negative exponential and an argument \(\theta (t)\) that goes to infinity like a positive exponential. In particular, the difference between consecutive times of passage through the pericenter, say \(T_{n+1} -T_n\), goes to zero as \(\frac{1}{n}\).     

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.     

Spatial collinear restricted four-body problem with repulsive Manev potential

We outline some aspects of the dynamics of an infinitesimal mass under the Newtonian attraction of three point masses in a symmetric collinear relative equilibria configuration when a repulsive Manev potential (\(-1/r +e/r^{2}\)), \(e>0\), is applied to the central mass. We investigate the relative equilibria of the infinitesimal mass and their linear stability as a function of the mass parameter \(\beta \), the ratio of mass of the central body to the mass of one of two remaining bodies, and e. We also prove the nonexistence of binary collisions between the central body and the infinitesimal mass.     

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.     

Motions about a fixed point by hypergeometric functions: new non-complex analytical solutions and integration of the herpolhode

The present study highlights the dynamics of a body moving about a fixed point and provides analytical closed form solutions. Firstly, for the symmetrical heavy body, that is the Lagrange–Poisson case, we compute the second (precession, \(\psi \)) and third (spin, \(\varphi \)) Euler angles in explicit and real form by means of multiple hypergeometric (Lauricella) functions. Secondly, releasing the weight assumption but adding the complication of the asymmetry, by means of elliptic integrals of third kind, we provide the precession angle \(\psi \) completing the treatment of the Euler–Poinsot case. Thirdly, by integrating the relevant differential equation, we reach the finite polar equation of a special motion trajectory named the herpolhode. Finally, we keep the symmetry of the first problem, but without weight, and take into account a viscous dissipation. The use of motion first integrals—adopted for the first two problems—is no longer practicable in this situation; therefore, the Euler equations, faced directly, are driving to particular occurrences of Bessel functions of order \(-\,1/2\).     

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.     

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.     

Statistical approach to triple systems in three-dimensional motion

This paper considers disruption of triple close approaches with low initial velocities and equal masses in the framework of statistical escape theory in a three-dimensional space. The statistical escape theory is based on the assumption that the phase trajectory of a triple system is quasi-ergodic. This system is described by allowing for both energy and angular momentum conservation in the phase space. In this paper, “possibility of escape” is derived with the formation of a binary on the basis of relative distances of the participating bodies. The complete statistical solutions (i.e. the semi-major axis \(a\), the distributions of eccentricity \(e\) of the binary, binary energy \({E}_{{b}}\), escape energy \({E}_{{s}}\) of escaper, and its escape velocity \({v}_{{s}}\)) of the system are derived from the allowable phase space volumes and are in good agreement with the numerical results in the range of perturbing velocities \({v}_{{i}}\)(\(10^{ - 1} \le {v}_{{i}} \le 10^{ - 10}\)) and directions of \({v}_{{i}}(0 \le \alpha _{{i}},\beta _{{i}},\gamma _{{i}} \le \pi )\), \({i} = 1,2,3\). In this paper, the double limit process has been applied to approximate the escape probability. Through this process, it is observed that the perturbing velocity \({v}_{{i}} \to 0^{ +} \), as the product of the semi-major axis \(a\) of the final binary and the square of the escape velocity \({v}_{{s}}\) approach 2/3, i.e. \({a} {v}_{{s}}^{2} \to 2 / 3\), whatever direction of \(\mathbf{v}_{{i}}\) may be.     

The rectilinear three-body problem as a basis for studying highly eccentric systems

The rectilinear elliptic restricted three-body problem (TBP) is the limiting case of the elliptic restricted TBP when the motion of the primaries is described by a Keplerian ellipse with eccentricity \(e'=1\), but the collision of the primaries is assumed to be a non-singular point. The rectilinear model has been proposed as a starting model for studying the dynamics of motion around highly eccentric binary systems. Broucke (AIAA J 7:1003–1009, 1969) explored the rectilinear problem and obtained isolated periodic orbits for mass parameter \(\mu =0.5\) (equal masses of the primaries). We found that all orbits obtained by Broucke are linearly unstable. We extend Broucke’s computations by using a finer search for symmetric periodic orbits and computing their linear stability. We found a large number of periodic orbits, but only eight of them were found to be linearly stable and are associated with particular mean motion resonances. These stable orbits are used as generating orbits for continuation with respect to \(\mu \) and \(e'<1\). Also, continuation of periodic solutions with respect to the mass of the small body can be applied by using the general TBP. FLI maps of dynamical stability show that stable periodic orbits are surrounded in phase space with regions of regular orbits indicating that systems of very highly eccentric orbits can be found in stable resonant configurations. As an application we present a stability study for the planetary system HD7449.     

The formation of a gas giant planet in a viscously evolved protoplanetary disk within the core accretion model

The gas giant planets’ formation processes in a viscously evolved protoplanetary disk are studied in the context of the core accretion model. In this paper, we follow the entire formation process of the core accretion model (the three stages). We find that the gas giant planets’ final masses and formation regions have strong dependence on the molecular cloud core’s properties (angular velocity \(\omega \) and mass \(M _{c d}\)) and the \(\alpha _{ \mathit{min} }\) parameter. We find and build the relationship between gas giant planets’ properties and molecular cloud core’s properties. In contrast to the previous works, we find that the formation process can be finished within the protoplanetary disk’s lifetime (4×10⁶ yr) in our disk model. This is because the mass influx produced by the molecular cloud core can provide enough material to the protoplanetary disk. We also find that the gas giant planets’ final masses increase generally with the viscosity coefficient \(\alpha \). This is because most of the gas giant planet’s mass is captured during the rapid gas accretion phase (the third stage of the core accretion model), and furthermore the accretion of gas in this phase is dominated by the “gap limiting case”. And our numerical results can also be compared with the observed data of exoplanet systems.     

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.     

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.     

De Sitter universe described by a binary mixture with a variable cosmological constant $\lambda $

We have constructed a self-consistent system of Bianchi Type VI₀ cosmology, and mingling of perfect fluid and dark energy in five dimensions. The usual equation of state \(p = \gamma \rho \) with \(\gamma \in [0, 1]\) is chosen by the perfect fluid. The dark energy assumed to be chosen is taken into consideration to be either the quintessence or Chaplygin gas. The same solutions pertaining to the corresponding the field equations of Einstein are obtained as a quadrature. State parameter’s equations for dark energy \(\omega \) is found to be consistent enough with the recent observations of SNe Ia data (SNe Ia data with CMBR anisotropy) and galaxy clustering statistics. Here our models predict that the rate of expansion of Universe would increase with passage of time. The cosmological constant \(\varLambda \) is traced as a declining function of time and it gets nearer to a small positive value later on which is supported by the results from the current supernovae Ia observations. Also a detail discussion is made on the physical and geometrical aspects of the models.     

Minimal energy configurations of gravitationally interacting rigid bodies

Consider a collection of n rigid, massive bodies interacting according to their mutual gravitational attraction. A relative equilibrium motion is one where the entire configuration rotates rigidly and uniformly about a fixed axis in \(\mathbb {R}^3\). Such a motion is possible only for special positions and orientations of the bodies. A minimal energy motion is one which has the minimum possible energy in its fixed angular momentum level. While every minimal energy motion is a relative equilibrium motion, the main result here is that a relative equilibrium motion of \(n\ge 3\) disjoint rigid bodies is never an energy minimizer. This generalizes a known result about point masses to the case of rigid bodies.     

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.     

A novel look at the pulsar force-free magnetosphere

The stationary axisymmetric force-free magnetosphere of a pulsar is considered. We present an exact dipolar solution of the pulsar equation, construct the magnetospheric model on its basis and examine its observational support. The new model has toroidal rather than common cylindrical geometry, in line with that of the plasma outflow observed directly as the pulsar wind nebula at much larger spatial scale. In its new configuration, the axisymmetric magnetosphere consumes the neutron star rotational energy much more efficiently, implying re-estimation of the stellar magnetic field, \(B_{\mathrm{new}}^{0}=3.3\times10^{-4}B/P\), where \(P\) is the pulsar period. Then the 7-order scatter of the magnetic field derived from the rotational characteristics of the pulsars observed appears consistent with the \(\cot\chi\)-law, where \(\chi\) is a random quantity uniformly distributed in the interval \([0,\pi/2]\). Our result is suggestive of a unique actual magnetic field strength of the neutron stars along with a random angle between the magnetic and rotational axes and gives insight into the neutron star unification on the geometrical basis.     

Photospheric Shear Flows in Solar Active Regions and Their Relation to Flare Occurrence

Solar active regions (ARs) that produce major flares typically exhibit strong plasma shear flows around photospheric magnetic polarity inversion lines (MPILs). It is therefore important to quantitatively measure such photospheric shear flows in ARs for a better understanding of their relation to flare occurrence. Photospheric flow fields were determined by applying the Differential Affine Velocity Estimator for Vector Magnetograms (DAVE4VM) method to a large data set of 2548 coaligned pairs of AR vector magnetograms with 12-min separation over the period 2012?–?2016. From each AR flow-field map, three shear-flow parameters were derived corresponding to the mean (\(\langle S\rangle \)), maximum (\(S_{\mathrm{max}}\)) and integral (\(S_{\mathrm{sum}}\)) shear-flow speeds along strong-gradient, strong-field MPIL segments. We calculated flaring rates within 24 h as a function of each shear-flow parameter and we investigated the relation between the parameters and the waiting time (\(\tau \)) until the next major flare (class M1.0 or above) after the parameter observation. In general, it is found that the larger \(S_{\mathrm{sum}}\) an AR has, the more likely it is for the AR to produce flares within 24 h. It is also found that among ARs which produce major flares, if one has a larger value of \(S_{\mathrm{sum}}\) then \(\tau \) generally gets shorter. These results suggest that large ARs with widespread and/or strong shear flows along MPILs tend to not only be more flare productive, but also produce major flares within 24 h or less.