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 \(\frac{1}{n}\).     

$$M_{\bullet } - \sigma $$ relation in spherical systems

To investigate the \(M_\bullet -\sigma \) relation, we consider realistic elliptical galaxy profiles that are taken to follow a single power-law density profile given by \(\rho (r) = \rho _{0}(r/ r_{0})^{-\gamma }\) or the Nuker intensity profile. We calculate the density using Abel’s formula in the latter case by employing the derived stellar potential; in both cases. We derive the distribution function f(E) of the stars in the presence of the supermassive black hole (SMBH) at the center and hence compute the line-of-sight (LoS) velocity dispersion as a function of radius. For the typical range of values for masses of SMBH, we obtain \(M_{\bullet } \propto \sigma ^{p}\) for different profiles. An analytical relation \(p = (2\gamma + 6)/(2 + \gamma )\) is found which is in reasonable agreement with observations (for \(\gamma = 0.75{-}1.4\), \(p = 3.6{-}5.3\)). Assuming that a proportionality relation holds between the black hole mass and bulge mass, \(M_{\bullet } =f M_\mathrm{b}\), and applying this to several galaxies, we find the individual best fit values of p as a function of f; also by minimizing \(\chi ^{2}\), we find the best fit global p and f. For Nuker profiles, we find that \(p = 3.81 \pm 0.004\) and \(f = (1.23 \pm 0.09)\times 10^{-3}\) which are consistent with the observed ranges.     

Predictions on the core mass of Jupiter and of giant planets in general

More than 80 giant planets are known by mass and radius. Their interior structure in terms of core mass, number of layers, and composition however is still poorly known. An overview is presented about the core mass M _core and envelope mass of metals M _Z in Jupiter as predicted by various equations of state. It is argued that the uncertainty about the true H/He EOS in a pressure regime where the gravitational moments J ₂ and J ₄ are most sensitive, i.e. between 0.5 and 4 Mbar, is in part responsible for the broad range \(M_{\mathit{core}}=0{-}18\:M_{\oplus }\), \(M_{Z}=0{-}38\:M_{\oplus }\), and \(M_{\mathit{core}}+M_{Z}=14{-}38\:M_{\oplus }\) currently offered for Jupiter. We then compare the Jupiter models obtained when we only match J ₂ with the range of solutions for the exoplanet \(\mathrm{GJ}\:436\mathrm{b}\), when we match an assumed tidal Love number k ₂ value.     

Secular tidal changes in lunar orbit and Earth rotation

Small tidal forces in the Earth–Moon system cause detectable changes in the orbit. Tidal energy dissipation causes secular rates in the lunar mean motion n, semimajor axis a, and eccentricity e. Terrestrial dissipation causes most of the tidal change in n and a, but lunar dissipation decreases eccentricity rate. Terrestrial tidal dissipation also slows the rotation of the Earth and increases obliquity. A tidal acceleration model is used for integration of the lunar orbit. Analysis of lunar laser ranging (LLR) data provides two or three terrestrial and two lunar dissipation parameters. Additional parameters come from geophysical knowledge of terrestrial tides. When those parameters are converted to secular rates for orbit elements, one obtains dn/dt = \(-25.97\pm 0.05 ''/\)cent\(^{2}\), da/dt = 38.30 ± 0.08 mm/year, and di/dt = ?0.5 ± 0.1 \(\upmu \)as/year. Solving for two terrestrial time delays and an extra de/dt from unspecified causes gives \(\sim \) \(3\times 10^{-12}\)/year for the latter; solving for three LLR tidal time delays without the extra de/dt gives a larger phase lag of the N2 tide so that total de/dt = \((1.50 \pm 0.10)\times 10^{-11}\)/year. For total dn/dt, there is \(\le \)1 % difference between geophysical models of average tidal dissipation in oceans and solid Earth and LLR results, and most of that difference comes from diurnal tides. The geophysical model predicts that tidal deceleration of Earth rotation is \(-1316 ''\)/cent\(^{2}\) or 87.5 s/cent\(^{2}\) for UT1-AT, a 2.395 ms/cent increase in the length of day, and an obliquity rate of 9 \(\upmu \)as/year. For evolution during past times of slow recession, the eccentricity rate can be negative.     

Behaviour of physical parameters in extended gravity with hyperbolic function

In this paper, we have constructed the cosmological model of the universe in f(R, T) theory of gravity in a Bianchi type \(\mathrm{VI}_h\) universe for the functional f(R, T) in the form \(f(R,T)=\mu R+\mu T\), where R and T are respectively Ricci scalar and trace of energy momentum tensor and \(\mu \) is a constant. We have made use of the hyperbolic scale factor to find the physical parameters and metric potentials defined in the space-time. The physical parameters are constrained from different representative values to build up a realistic cosmological model aligned with the observational behaviour. The state finder diagnostic pair is found to be in the acceptable range. The energy conditions of the model are also studied.     

On a family of well behaved perfect fluid balls as astrophysical objects in general relativity

A family of well behaved perfect fluid balls has been derived starting with the metric potential g ₄₄=B(1+Cr ²) ⁿ for all positive integral values of n. For n≥4, the members of this family are seen to satisfy the various physical conditions e.g. c ² ρ≥p≥0,dp/dr<0,dρ/dr<0, along with the velocity of sound \((\sqrt{dp/c^{2}d\rho} )< 1\) and the adiabatic index ((p+c ² ρ)/p)(dp/(c ² dρ))>1. Also the pressure, energy density, velocity of sound and ratio of pressure and energy density are of monotonically decreasing towards the pressure free interface (r=a). The fluid balls join smoothly with the Schwarzschild exterior model at r=a. The well behaved perfect fluid balls so obtained are utilised to construct the superdense star models with their surface density 2×10¹⁴  gm/cm³. We have found that the maximum mass of the fluid balls corresponding to various values of n are decreasing with the increasing values of n. Over all maximum mass for the whole family turns out to be 4.1848M _Θ and the corresponding radius as 19.4144 km while the red shift at the centre and red shift at surface as Z ₀=1.6459 and Z _a =0.6538 respectively this all happens for n=4. It is interesting to note that for higher values of n viz n≥170, the physical data start merging with that of Kuchowicz superdense star models and hence the family of fluid models tends to the Kuchowicz fluid models as n→∞. Consequently the maximum mass of the family of solution can not be less than 1.6096 M _Θ which is the maximum mass occupied by the Kuchowicz superdense ball. Hence each member of the family for n≥4 provides the astrophysical objects like White dwarfs, Quark star, typical neutron star.     

Inverse problem of central configurations and singular curve in the collinear 4-body problem

In this paper, we consider the inverse problem of central configurations of n-body problem. For a given \({q=(q_1, q_2, \ldots, q_n)\in ({\bf R}^d)^n}\), let S(q) be the admissible set of masses denoted \({ S(q)=\{ m=(m_1,m_2, \ldots, m_n)| m_i \in {\bf R}^+, q}\) is a central configuration for m}. For a given \({m\in S(q)}\), let S _m (q) be the permutational admissible set about m = (m ₁, m ₂, . . . , m _n ) denoted

$S_m(q)=\{m^\prime | m^\prime\in S(q),m^\prime \not=m \, {\rm and} \, m^\prime\,{\rm is\, a\, permutation\, of }\, m \}.$

The main discovery in this paper is the existence of a singular curve \({\bar{\Gamma}_{31}}\) on which S _m (q) is a nonempty set for some m in the collinear four-body problem. \({\bar{\Gamma}_{31}}\) is explicitly constructed by a polynomial in two variables. We proved:

1. (1)
   If \({m\in S(q)}\), then either ^# S _m (q) = 0 or ^# S _m (q) = 1.
    
2. (2)
   ^#S _m (q) = 1 only in the following cases:

   1. (i)
      If s = t, then S _m (q) = {(m ₄, m ₃, m ₂, m ₁)}.
       
   2. (ii)
      If \({(s,t)\in \bar{\Gamma}_{31}\setminus \{(\bar{s},\bar{s})\}}\), then either S _m (q) = {(m ₂, m ₄, m ₁, m ₃)} or S _m (q) = {(m ₃, m ₁, m ₄, m ₂)}.
       
    

     

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.     

Contractivity-preserving explicit multistep Hermite–Obrechkoff series differential equation solvers

New optimal, contractivity-preserving (CP), explicit, d-derivative, k-step Hermite–Obrechkoff series methods of order p up to \(p=20\), denoted by CP HO(d, k, p), with nonnegative coefficients are constructed. These methods are used to solve nonstiff first-order initial value problems \(y'=f(t,y)\), \(y(t_0)=y_0\). The upper bound \(p_u\) of order p of HO(d, k, p) can reach, approximately, as high as 2.4 times the number of derivatives d. The stability regions of HO(d, k, p) have generally a good shape and grow with decreasing \(p-d\). We, first, note that three selected CP HO methods: 4-derivative 7-step HO of order 13, denoted by HO(4, 7, 13), 5-derivative 6-step HO of order 13, denoted by HO(5, 6, 13), and 9-derivative 2-step HO of order 13, denoted by CMDAHO(13) compare favorably with Adams–Cowell of order 13, denoted by AC(13), in solving standard N-body problems over an interval of 1000 periods on the basis of the relative error of energy as a function of the CPU time. Next, the three HO methods compare positively with AC(13) in solving standard N-body problems on the basis of the growth of relative positional error and relative energy error over 10, 000 periods of integration. Finally, these three methods compare also well with P-stable methods of Cash and Franco et al. on some quasi periodic, second-order linear and nonlinear problems. The coefficients of selected HO methods are listed in the appendix.     

A photometric study of faint galaxies in the field of GRB 000926

We present our B, V, R_c, and I_c observations of a \(3'.6 \times 3'\) field centered on the host galaxy of GRB 000926 (α_2000.0=17^h04^m11^s, \(\delta _{2000.0} = + 51^ \circ 47'9\mathop .\limits^{''} 8\)). The observations were carried out on the 6-m Special Astrophysical Observatory telescope using the SCORPIO instrument. The catalog of galaxies detected in this field includes 264 objects for which the signal-to-noise ratio is larger than 5 in each photometric band. The following limiting magnitudes in the catalog correspond to this limitation: 26.6 (B), 25.7 (V), 25.8 (R), and 24.5 (I). The differential galaxy counts are in good agreement with previously published CCD observations of deep fields. We estimated the photometric redshifts for all of the cataloged objects and studied the color variations of the galaxies with z. For luminous spiral galaxies with M(B)z～1.     

Ratio of the Core to the Extended Emissions in the Comoving Frame for Blazars

In a two-component jet model, the emissions are the sum of the core and extended emissions: \(S^{\mathrm{ob}}=S_{\mathrm{core}}^{\mathrm{ob}}+S_{\mathrm{ext}}^{\mathrm{ob}}\), with the core emissions, \(S_{\mathrm{core}}^{\mathrm{ob}}= f S_{\mathrm{ext}}^{\mathrm{ob}}\delta ^{q}\) being a function of the Doppler factor \(\delta \), the extended emission \(S_{\mathrm{ext}}^{\mathrm{ob}}\), the jet type dependent factor q, and the ratio of the core to the extended emissions in the comoving frame, f. The f is an unobservable but important parameter. Following our previous work, we collect 65 blazars with available Doppler factor \(\delta \), superluminal velocity \(\beta _{\mathrm{app}}\), and core-dominance parameter, R, and calculated the ratio, f, and performed statistical analyses. We found that the ratio, f, in BL Lacs is on average larger than that in FSRQs. We suggest that the difference of the ratio f between FSRQs and BL Lacs is one of the possible reasons that cause the difference of other observed properties between them. We also find some significant correlations between \(\log f\) and other parameters, including intrinsic (de-beamed) peak frequency, \(\log \nu _{\mathrm{p}}^{\mathrm{in}}\), intrinsic polarization, \(\log P^{\mathrm{in}}\), and core-dominance parameter, \(\log R\), for the whole sample. In addition, we show that the ratio, f, can be estimated by R.     

Analysis of a Failed Eclipse Plasma Ejection Using EUV Observations

The photometry of eclipse white-light (W-L) images showing a moving blob is interpreted for the first time together with observations from space with the PRoject for On Board Autonomy (PROBA-2) mission (ESA). An off-limb event seen with great details in W-L was analyzed with the SWAP imager (Sun Watcher using Active pixel system detector and image Processing) working in the EUV near 174 Å. It is an elongated plasma blob structure of 25 Mm diameter moving above the east limb with coronal loops under. Summed and co-aligned SWAP images are evaluated using a 20-h sequence, in addition to the 11 July, 2010 eclipse W-L images taken from several sites. The Atmospheric Imaging Assembly (AIA) instrument on board the Solar Dynamics Observatory (SDO) recorded the event suggesting a magnetic reconnection near a high neutral point; accordingly, we also call it a magnetic plasmoid. The measured proper motion of the blob shows a velocity up to \(12~\mbox{km}\,\mbox{s}^{-1}\). Electron densities of the isolated condensation (cloud or blob or plasmoid) are photometrically evaluated. The typical value is \(10^{8}~\mbox{cm}^{-3}\) at \(r=1.7~\mathrm{R}_{\odot}\), superposed on a background corona of \(10^{7}~\mbox{cm}^{-3}\) density. The mass of the cloud near its maximum brightness is found to be \(1.6\times10^{13}\) g, which is typically \(0.6\times10^{-4}\) of the overall mass of the corona. From the extrapolated magnetic field the cloud evolves inside a rather broad open region but decelerates, after reaching its maximum brightness. The influence of such small events for supplying material to the ubiquitous slow wind is noticed. A precise evaluation of the EUV photometric data, after accurately removing the stray light, suggests an interpretation of the weak 174 Å radiation of the cloud as due to resonance scattering in the Fe IX/X lines.     

Correlations Between CME Parameters and Sunspot Activity

Smoothed monthly mean coronal mass ejection (CME) parameters (speed, acceleration, central position angle, angular width, mass, and kinetic energy) for Cycle 23 are cross-analyzed, showing that there is a high correlation between most of them. The CME acceleration (a) is highly correlated with the reciprocal of its mass (M), with a correlation coefficient r=0.899. The force (Ma) to drive a CME is found to be well anti-correlated with the sunspot number (R _z), r=?0.750. The relationships between CME parameters and R _z can be well described by an integral response model with a decay time scale of about 11 months. The correlation coefficients of CME parameters with the reconstructed series based on this model (\(\overline{r}_{\mathrm{f1}}=0.886\)) are higher than the linear correlation coefficients of the parameters with R _z (\(\overline{r}_{\mathrm{0}}=0.830\)). If a double decay integral response model is used (with two decay time scales of about 6 and 60 months), the correlations between CME parameters and R _z improve (\(\overline{r}_{\mathrm{f2}}=0.906\)). The time delays between CME parameters with respect to R _z are also well predicted by this model (19/22=86%); the average time delays are 19 months for the reconstructed and 22 months for the original time series. The model implies that CMEs are related to the accumulation of solar magnetic energy. These relationships can help in understanding the mechanisms at work during the solar cycle.     

Kinematics of B-F Stars as a Function of Their Dereddened Color from Gaia and PCRV Data

Parallaxes with an accuracy better than 10% and proper motions from the Gaia DR1 TGAS catalogue, radial velocities from the Pulkovo Compilation of Radial Velocities (PCRV), accurate Tycho-2 photometry, theoretical PARSEC, MIST, YaPSI, BaSTI isochrones, and the most accurate reddening and interstellar extinction estimates have been used to analyze the kinematics of 9543 thin-disk B-F stars as a function of their dereddened color. The stars under consideration are located on the Hertzsprung–Russell diagram relative to the isochrones with an accuracy of a few hundredths of a magnitude, i.e., at the level of uncertainty in the parallax, photometry, reddening, extinction, and the isochrones themselves. This has allowed us to choose the most plausible reddening and extinction estimates and to conclude that the reddening and extinction were significantly underestimated in some kinematic studies of other authors. Owing to the higher accuracy of TGAS parallaxes than that of Hipparcos ones, the median accuracy of the velocity components U, V, W in this study has improved to 1.7 km s^?1, although outside the range ?0.1 ^m < (B _T ? V _T )₀ < 0.5 ^m the kinematic characteristics are noticeably biased due to the incompleteness of the sample. We have confirmed the variations in the mean velocity of stars relative to the Sun and the stellar velocity dispersion as a function of their dereddened color known from the Hipparcos data. Given the age estimates for the stars under consideration from the TRILEGAL model and the Geneva–Copenhagen survey, these variations may be considered as variations as a function of the stellar age. A comparison of our results with the results of other studies of the stellar kinematics near the Sun has shown that selection and reddening underestimation explain almost completely the discrepancies between the results. The dispersions and mean velocities from the results of reliable studies fit into a ±2 km s^?1 corridor, while the ratios σ _V /σ _U and σ _W /σ _U fit into ±0.05. Based on all reliable studies in the range ?0.1 ^m < (B _T ? V _T )₀ < 0.5^m, i.e., for an age from 0.23 to 2.4 Gyr, we have found: W_⊙ = 7.15 km s^?1, \({\sigma _U} = 16.0{e^{1.29({B_T} - {V_T})o}}\), \({\sigma _V} = 10.9{e^{1.11({B_T} - {V_T})o}}\), \({\sigma _W} = 6.8{e^{1.46({B_T} - {V_T})o}}\), the stellar velocity dispersions in km s^?1 are proportional to the age in Gyr raised to the power β _U = 0.33, β _V = 0.285, and β _W = 0.37.     

A theory of the equilibrium figure and gravitational field of the Galilean satellite Io: The second approximation

We construct a theory of the equilibrium figure and gravitational field of the Galilean satellite Io to within terms of the second order in the small parameter α. We show that to describe all effects of the second approximation, the equation for the figure of the satellite must contain not only the components of the second spherical function, but also the components of the third and fourth spherical functions. The contribution of the third spherical function is determined by the Love number of the third order h₃, whose model value is 1.6582. Measurements of the third-order gravitational moments could reveal the extent to which the hydrostatic equilibrium conditions are satisfied for Io. These conditions are J₃=C₃₂=0 and C₃₁/C₃₃=?6. We have calculated the corrections of the second order of smallness to the gravitational moments J₂ and C₂₂. We conclude that when modeling the internal structure of Io, it is better to use the observed value of k₂ than the moment of inertia derived from k₂. The corrections to the lengths of the semiaxes of the equilibrium figure of Io are all positive and equal to ～64.5, ～26, and ～14 m for the a, b, and c axes, respectively. Our theory allows the parameters of the figure and the fourth-order gravitational moments that differ from zero to be calculated. For the homogeneous model, their values are:\(s_4 = \frac{{885}}{{224}}\alpha ^2 ,s_{42} = - \frac{{75}}{{224}}\alpha ^2 ,s_{44} = \frac{{15}}{{896}}\alpha ^2 ,J_4 = - \frac{{885}}{{224}}\alpha ^2 ,C_{42} = \frac{{75}}{{224}}\alpha ^2 ,C_{44} = \frac{{15}}{{896}}\alpha ^2 \).     

On the exactness of estimates for irregularly structured bodies of the general term of Laplace series

The main form of the representation of a gravitational potential V for a celestial body T in outer space is the Laplace series in solid spherical harmonics \((R/r)^{n+1}Y_n(\theta ,\lambda )\) with R being the radius of the enveloping T sphere. The surface harmonic \(Y_n\) satisfies the inequality

$$\begin{aligned} \langle Y_n\rangle < Cn^{-\sigma }. \end{aligned}$$

The angular brackets mark the maximum of a function’s modulus over a unit sphere. For bodies with an irregular structure \(\sigma = 5/2\), and this value cannot be increased generally. However, a class of irregular bodies (smooth bodies with peaked mountains) has been found recently in which \(\sigma = 3\). In this paper, we will prove the exactness of this estimate, showing that a body belonging to the above class does exist and

$$\begin{aligned} 0<\varlimsup n^3\langle Y_n\rangle <\infty \end{aligned}$$

for it.

     

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.     

Measurement of coronal properties of Seyfert galaxies from NuSTAR’s hard X-ray spectrum

Precise measurement of the coronal properties of Active Galactic Nuclei (AGN) requires the availability of high signal-to-noise ratio data covering a wide range of X-ray energies. The Nuclear Spectroscopic Telescope Array (NuSTAR) which is highly sensitive to earlier missions in its operational energy range of 3–79 keV, allows us to arrive at precise estimates of the coronal parameters such as cut-off energy (\(E_\mathrm{cut}\)), coronal temperature (\(\textit{kT}_e\)) and geometry of the corona at least for sources that have \(E_\mathrm{cut}\) within the energy range of NuSTAR. In this paper, we present our preliminary results on the spectral analysis of two Seyfert galaxies namely 3C 120 and NGC 4151 using NuSTAR observations in the 3–79 keV band. We investigated the continuum and coronal parameters, the photon index \(\Gamma \), \(E_\mathrm{cut}\) and \(\textit{kT}_{e}\). By fitting the X-ray spectrum of 3C 120 and NGC 4151 with a simple phenomenological model, we found that both the sources showed a clear cut-off in their spectrum.     

A study of the geomagnetic indices asymmetry based on the interplanetary magnetic field polarities

Data of geomagnetic indices (aa, Kp, Ap, and Dst) recorded near 1 AU over the period 1967–2016, have been studied based on the asymmetry between the interplanetary magnetic field (IMF) directions above and below of the heliospheric current sheet (HCS). Our results led to the following conclusions: (i) Throughout the considered period, 31 random years (62%) showed apparent asymmetries between Toward (\(\mathbf{T}\)) and Away (\(\mathbf{A}\)) polarity days and 19 years (38%) exhibited nearly a symmetrical behavior. The days of \(\mathbf{A}\) polarity predominated over the \(\mathbf{T}\) polarity days by 4.3% during the positive magnetic polarity epoch (1991–1999). While the days of \(\mathbf{T}\) polarity exceeded the days of \(\mathbf{A}\) polarity by 5.8% during the negative magnetic polarity epoch (2001–2012). (ii) Considerable yearly North–South (N–S) asymmetries of geomagnetic indices observed throughout the considered period. (iii) The largest toward dominant peaks for \(aa\) and \(Ap\) indices occurred in 1995 near to minimum of solar activity. Moreover, the most substantial away dominant peaks for \(aa\) and \(Ap\) indices occurred in 2003 (during the descending phase of the solar cycle 23) and in 1991 (near the maximum of solar activity cycle) respectively. (iv) The N–S asymmetry of \(Kp\) index indicated a most significant away dominant peak occurred in 2003. (v) Four of the away dominant peaks of Dst index occurred at the maxima of solar activity in the years 1980, 1990, 2000, and 2013. The largest toward dominant peak occurred in 1991 (at the reversal of IMF polarity). (vi) The geomagnetic indices (aa, Ap, and \(Kp\)) all have northern dominance during positive magnetic polarity epoch (1971–1979), while the asymmetries shifts to the southern solar hemisphere during negative magnetic polarity epoch (2001–2012).     

The fundamental plane and other scaling relations for galaxy groups and clusters

In this paper we study the relations between the main characteristics of groups and clusters of galaxies using the archival data of the SDSS and 2MASX catalogs. We have developed and implemented a new method of determining the size of galaxy systems and their effective radius which contains half of the galaxies and not half the luminosity, since the luminosity of the brightest galaxy in a group can account for over 50% of the total luminosity of the group. The derived parameters (log L_K, logR_e, and log σ₂₀₀) for 94 systems of galaxies (0.0038 < z < 0.09) determine the Fundamental Plane (FP), which, with a scatter of 0.15, is similar in form to the FP of galaxy clusters obtained by Schaeffer et al. (1993) and D’Onofrio et al. (2013) with other methods and for different bands. We show that the FP in the near-infrared region (NIR) for 94 galaxy systems has the form of L_K ∝ \(R_e^{0.70 \pm {{0.13}_\sigma }1.34 \pm 0.13}\), whereas in x-rays it has the form of—L_X ∝ \(R_e^{1.15 \pm {{0.39}_\sigma }2.56 \pm 0.40}\). The form of the FP for groups and clusters is consistent with the FP for early-type galaxies determined in the same way. The form of the FP for galaxy systems deviates from the shape that one would expect from virial predictions. Adding the mass-to-light ratio as a fourth independent parameter has little effect on this deviation, but decreases the scatter of the FP for a sample of rich galaxy clusters by 12%.