Relation Between Coronal Hole Areas and Solar Wind Speeds Derived from Interplanetary Scintillation Measurements

We investigate the relation between coronal hole (CH) areas and solar wind speeds during 1995?–?2011 using the potential field (PF) model analysis of magnetograph observations and interplanetary scintillation (IPS) observations by the Institute for Space-Earth Environmental Research (formerly Solar-Terrestrial Environment Laboratory) of Nagoya University. We obtained a significant positive correlation between the CH areas (\(A\)) derived from the PF model calculations and solar wind speeds (\(V\)) derived from the IPS observations. The correlation coefficients between them are usually high, but they drop significantly in solar maxima. The slopes of the \(A\)?–?\(V\) relation are roughly constant except for the period around solar maximum, when flatter or steeper slopes are observed. The excursion of the correlation coefficients and slopes at solar maxima is ascribed partly to the effect of rapid structural changes in the coronal magnetic field and solar wind, and partly to the predominance of small CHs. It is also demonstrated that \(V\) is inversely related to the flux expansion factor (\(f\)) and that \(f\) is closely related to \(A^{-1/2}\); hence, \(V \propto A^{1/2}\). A better correlation coefficient is obtained from the \(A^{1/2}\)?–?\(V\) relation, and this fact is useful for improving space weather predictions. We compare the CH areas derived from the PF model calculations with He i 1083 nm observations and show that the PF model calculations provide reliable estimates of the CH area, particularly for large \(A\).     

Modeling the Global Coronal Field with Simulated Synoptic Magnetograms from Earth and the Lagrange Points $L_{3}$, L4$L_{4}$, and L5$L_{5}$

The solar photospheric magnetic flux distribution is key to structuring the global solar corona and heliosphere. Regular full-disk photospheric magnetogram data are therefore essential to our ability to model and forecast heliospheric phenomena such as space weather. However, our spatio-temporal coverage of the photospheric field is currently limited by our single vantage point at/near Earth. In particular, the polar fields play a leading role in structuring the large-scale corona and heliosphere, but each pole is unobservable for \({>}\,6\) months per year. Here we model the possible effect of full-disk magnetogram data from the Lagrange points \(L_{4}\) and \(L_{5}\), each extending longitude coverage by \(60^{\circ}\). Adding data also from the more distant point \(L_{3}\) extends the longitudinal coverage much further. The additional vantage points also improve the visibility of the globally influential polar fields. Using a flux-transport model for the solar photospheric field, we model full-disk observations from Earth/\(L_{1}\), \(L_{3}\), \(L_{4}\), and \(L_{5}\) over a solar cycle, construct synoptic maps using a novel weighting scheme adapted for merging magnetogram data from multiple viewpoints, and compute potential-field models for the global coronal field. Each additional viewpoint brings the maps and models into closer agreement with the reference field from the flux-transport simulation, with particular improvement at polar latitudes, the main source of the fast solar wind.     

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.     

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 \(L_5\).     

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.     

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.     

Photometric study of three ultrashort-period contact binaries

We carried out high-precision photometric observations of three eclipsing ultrashort-period contact binaries (USPCBs). Theoretical models were fitted to the light curves by means of the Wilson-Devinney code. The solutions suggest that the three targets have evolved to a contact phase. The photometric results are as follows: (a) 1SWASP?J030749.87?365201.7, \(q=0.439\pm0.003\), \(f=0.0\pm3.6\%\); (b) 1SWASP?J213252.93?441822.6, \(q=0.560\pm0.003\), \(f=14.2\pm1.9\%\); (c) 1SWASP?J200059.78+054408.9, \(q=0.436\pm0.008\), \(f=58.4\pm1.8\%\). The light curves show O’Connell effects, which can be modeled by the assumed cool spots. The cool spots models are strongly supported by the night-to-night variations in the \(I\)-band light curves of 1SWASP?J030749.87?365201.7. For a comparative study, we collected the whole set of 28 well-studied USPCBs with \(P < 0.24\) day. Thus, we found that most of them (17 of 28) are in shallow contact (i.e. fill-out factors \(f<20\%\)). Only four USPCBs have deep fill-out factors (i.e. \(f>50\%\)). Generally, contact binaries with deep fill-out factors are going to merge, but it is believed that USPCBs have just evolved to a contact phase. Hence, the deep USPCB 1SWASP?J200059.78+054408.9 seems to be a contradiction, making it very interesting. Particularly, 1SWASP?J030749.87?365201.7 is a zero contact binary in thermal equilibrium, implying that it should be a turn-off sample as predicted by the thermal relaxation oscillation (TRO) theory.     

Intermittency of the Solar Magnetic Field and Solar Magnetic Activity Cycle

Small-scale solar magnetic fields demonstrate features of fractal intermittent behavior, which requires quantification. For this purpose we investigate how the observational estimate of the solar magnetic flux density \(B\) depends on resolution \(D\) in order to obtain the scaling \(\ln B_{D} = - k \ln D +a\) in a reasonably wide range. The quantity \(k\) demonstrates cyclic variations typical of a solar activity cycle. In addition, \(k\) depends on the magnetic flux density, i.e. the ratio of the magnetic flux to the area over which the flux is calculated, at a given instant. The quantity \(a\) demonstrates some cyclic variation, but it is much weaker than in the case of \(k\). The scaling obtained generalizes previous scalings found for the particular cycle phases. The scaling is typical of fractal structures. In our opinion, the results obtained trace small-scale action in the solar convective zone and its coexistence with the conventional large-scale solar dynamo based on differential rotation and mirror-asymmetric convection.     

Kelvin–Helmholtz Instability in a Cool Solar Jet in the Framework of Hall Magnetohydrodynamics: A Case Study

We investigate the conditions under which the magnetohydrodynamic (MHD) modes in a cylindrical magnetic flux tube moving along its axis become unstable against the Kelvin–Helmholtz (KH) instability. We use the dispersion relations of MHD modes obtained from the linearized Hall MHD equations for cool (zero beta) plasma by assuming real wave numbers and complex angular wave frequencies/complex wave phase velocities. The dispersion equations are solved numerically at fixed input parameters and varying values of the ratio \(l_{\mathrm{Hall}}/a\), where \(l_{\mathrm{Hall}} = c/\omega_{\mathrm{pi}}\) (\(c\) being the speed of light, and \(\omega_{\mathrm{pi}}\) the ion plasma frequency) and \(a\) is the flux tube radius. It is shown that the stability of the MHD modes depends upon four parameters: the density contrast between the flux tube and its environment, the ratio of external and internal magnetic fields, the ratio \(l_{\mathrm{Hall}}/a\), and the value of the Alfvén Mach number defined as the ratio of the tube axial velocity to Alfvén speed inside the flux tube. It is found that at high density contrasts, for small values of \(l_{\mathrm{Hall}}/a\), the kink (\(m = 1\)) mode can become unstable against KH instability at some critical Alfvén Mach number (or equivalently at critical flow speed), but a threshold \(l_{\mathrm{Hall}}/a\) can suppress the onset of the KH instability. At small density contrasts, however, the magnitude of \(l_{\mathrm{Hall}}/a\) does not affect noticeably the condition for instability occurrence – even though it can reduce the critical Alfvén Mach number. It is established that the sausage mode (\(m = 0\)) is not subject to the KH instability.     

CCD <Emphasis Type="Italic">UBV</Emphasis> photometric study of five open clusters—Dolidze 36, NGC 6728, NGC 6800, NGC 7209, and Platais 1

In this study, we present CCD UBV photometry of poorly studied open star clusters, Dolidze 36, NGC 6728, NGC 6800, NGC 7209, and Platais 1, located in the first and second Galactic quadrants. Observations were obtained with T100, the 1-m telescope of the TÜB?TAK National Observatory. Using photometric data, we determined several astrophysical parameters such as reddening, distance, metallicity and ages and from them, initial mass functions, integrated magnitudes and colours. We took into account the proper motions of the observed stars to calculate the membership probabilities. The colour excesses and metallicities were determined independently using two-colour diagrams. After obtaining the colour excesses of the clusters Dolidze 36, NGC 6728, NGC 6800, NGC 7209, and Platais 1 as \(0.19\pm0.06\), \(0.15\pm0.05\), \(0.32\pm0.05\), \(0.12\pm 0.04\), and \(0.43\pm0.06\) mag, respectively, the metallicities are found to be \(0.00\pm0.09\), \(0.02\pm0.11\), \(0.03\pm0.07\), \(0.01\pm0.08\), and \(0.01\pm0.08\) dex, respectively. Furthermore, using these parameters, distance moduli and age of the clusters were also calculated from colour-magnitude diagrams simultaneously using PARSEC theoretical models. The distances to the clusters Dolidze 36, NGC 6728, NGC 6800, NGC 7209, and Platais 1 are \(1050\pm90\), \(1610\pm190\), \(1210\pm150\), \(1060\pm90\), and \(1710\pm250\) pc, respectively, while corresponding ages are \(400\pm100\), \(750\pm150\), \(400\pm100\), \(600\pm100\), and \(175\pm50\) Myr, respectively. Our results are compatible with those found in previous studies. The mass function of each cluster is derived. The slopes of the mass functions of the open clusters range from 1.31 to 1.58, which are in agreement with Salpeter’s initial mass function. We also found integrated absolute magnitudes varying from ?4.08 to ?3.40 for the clusters.     

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.     

Corotating Shock Waves and the Solar-wind Source of Energetic Ion bundances: Power Laws in $$/Q$Q$

We find that element abundances in energetic ions accelerated by shock waves formed at corotating interaction regions (CIRs) mirror the abundances of the solar wind modified by a decreasing power-law dependence on the mass-to-charge ratio \(A\)/\(Q\) of the ions. This behavior is similar in character to the well-known power-law dependence on \(A\)/\(Q\) of abundances in large gradual solar energetic particles (SEP). The CIR ions reflect the pattern of \(A\)/\(Q\), with \(Q\) values of the source plasma temperature or freezing-in temperature of 1.0?–?1.2 MK typical of the fast solar wind in this case. Thus the relative ion abundances in CIRs are of the form \((A\mbox{/}Q)^{a}\), where \(a\) is nearly always negative and evidently decreases with distance from the shocks, which usually begin beyond 1 AU. For one unusual historic CIR event where \(a \approx 0\), the reverse shock wave of the CIR seems to occur at 1 AU, and these abundances of the energetic ions become a direct proxy for the abundances of the fast solar wind.     

Stellar parameters of the two binary systems: HIP 14075 and HIP 14230

We present the stellar parameters of the individual components of the two old close binary systems HIP 14075 and HIP 14230 using synthetic photometric analysis. These parameters are accurately calculated based on the best match between the synthetic photometric results within three different photometric systems with the observed photometry of the entire system. From the synthetic photometry, we derive the masses and radii of HIP 14075 as \({\mathcal {M}}^A=0.99\pm 0.19 \mathcal {M_\odot }\), \(R_{A}=0.877\pm 0.08 R_\odot \) for the primary and \({\mathcal {M}}^B=0.96\pm 0.15 \mathcal {M_\odot }\), \(R_{B}=0.821\pm 0.07 R_\odot \) for the secondary, and of HIP 14230 as \({\mathcal {M}}^A=1.18\pm 0.22 \mathcal {M_\odot }\), \(R_{A}=1.234\pm 0.05 R_\odot \) for the primary and \({\mathcal {M}}^B=0.84\pm 0.12 \mathcal {M_\odot }\) , \(R_{B}=0.820\pm 0.05 R_\odot \) for the secondary. Both systems depend on Gaia parallaxes. Based on the positions of the components of the two systems on a theoretical Hertzsprung–Russell diagram, we find that the age of HIP 14075 is \(11.5\pm 2.0\) Gyr and of HIP 14230 is \(3.5\pm 1.5\) Gyr. Our analysis reveals that both systems are old close binary systems (\(\approx > 4\) Gyr). Finally, the positions of the components of both the systems on the stellar evolutionary tracks and isochrones are discussed.     

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.     

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.     

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.     

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.     

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.     

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.     

Nonlinear kinetic Alfven waves in space plasmas with generalized ($$) distribution

In this paper, we have investigated linear and nonlinear propagation of kinetic Alfven waves in which the electrons have been assumed to follow generalized (\(r,q\)) distribution. We have shown that (\(r,q\)) distribution gives us most of the distributions observed in space plasmas. We have varied the flatness parameter \(r\) and the tail parameter \(q\) to explore the linear and nonlinear propagation characteristics of kinetic Alfven waves. We have also discussed the limiting cases. It has been shown that our results agree well with Fast and Freja observations of the nonlinear kinetic Alfven waves. An important feature of our study is the formation of rarefactive solitary structures. It has been shown that this result cannot be obtained with Maxwellian distribution and that it agrees well with the observations of Fast and Freja satellites.