Lunar frozen orbits revisited

Lunar frozen orbits, characterized by constant orbital elements on average, have been previously found using various dynamical models, incorporating the gravitational field of the Moon and the third-body perturbation exerted by the Earth. The resulting mean orbital elements must be converted to osculating elements to initialize the orbiter position and velocity in the lunar frame. Thus far, however, there has not been an explicit transformation from mean to osculating elements, which includes the zonal harmonic \(J_2\), the sectorial harmonic \(C_{22}\), and the Earth third-body effect. In the current paper, we derive the dynamics of a lunar orbiter under the mentioned perturbations, which are shown to be dominant for the evolution of circumlunar orbits, and use von Zeipel’s method to obtain a transformation between mean and osculating elements. Whereas the dynamics of the mean elements do not include \(C_{22}\), and hence does not affect the equilibria leading to frozen orbits, \(C_{22}\) is present in the mean-to-osculating transformation, hence affecting the initialization of the physical circumlunar orbit. Simulations show that by using the newly-derived transformation, frozen orbits exhibit better behavior in terms of long-term stability about the mean values of eccentricity and argument of periapsis, especially for high orbits.     

The equations of relative motion in the orbital reference frame

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

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

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

The Poynting-Robertson effect in the Newtonian potential with a Yukawa correction

We consider a Yukawa-type gravitational potential combined with the Poynting-Robertson effect. Dust particles originating within the asteroid belt and moving on circular and elliptic trajectories are studied and expressions for the time rate of change of their orbital radii and semimajor axes, respectively, are obtained. These expressions are written in terms of basic particle parameters, namely their density and diameter. Then, they are applied to produce expressions for the time required by the dust particles to reach the orbit of Earth. For the Yukawa gravitational potential, dust particles of diameter \(10^{ - 3}\) m in circular orbits require times of the order of \(8.557 \times 10^{6}\) yr and for elliptic orbits of eccentricities \(e =0.1, 0.5\) require times of \(9.396 \times 10^{6}\) and \(2.129 \times 10^{6}\) yr respectively to reach Earth’s orbit. Finally, various cases of the Yukawa potential are studied and the corresponding particle times to reach Earth’s are derived per case along with numerical results for circular and various elliptical orbits.     

A satellite relative motion model including $$J_2$$ and $$J_3$$J3 via Vinti’s intermediary

Vinti’s potential is revisited for analytical propagation of the main satellite problem, this time in the context of relative motion. A particular version of Vinti’s spheroidal method is chosen that is valid for arbitrary elliptical orbits, encapsulating \(J_2\), \(J_3\), and generally a partial \(J_4\) in an orbit propagation theory without recourse to perturbation methods. As a child of Vinti’s solution, the proposed relative motion model inherits these properties. Furthermore, the problem is solved in oblate spheroidal elements, leading to large regions of validity for the linearization approximation. After offering several enhancements to Vinti’s solution, including boosts in accuracy and removal of some singularities, the proposed model is derived and subsequently reformulated so that Vinti’s solution is piecewise differentiable. While the model is valid for the critical inclination and nonsingular in the element space, singularities remain in the linear transformation from Earth-centered inertial coordinates to spheroidal elements when the eccentricity is zero or for nearly equatorial orbits. The new state transition matrix is evaluated against numerical solutions including the \(J_2\) through \(J_5\) terms for a wide range of chief orbits and separation distances. The solution is also compared with side-by-side simulations of the original Gim–Alfriend state transition matrix, which considers the \(J_2\) perturbation. Code for computing the resulting state transition matrix and associated reference frame and coordinate transformations is provided online as supplementary material.     

A vectorial approach to determine frozen orbital conditions

Taking into consideration a probe moving in an elliptical orbit around a celestial body, the possibility of determining conditions which lead to constant values on average of all the orbit elements has been investigated here, considering the influence of the planetary oblateness and the long-term effects deriving from the attraction of several perturbing bodies. To this end, three equations describing the variation of orbit eccentricity, apsidal line and angular momentum unit vector have been first retrieved, starting from a vectorial expression of the Lagrange planetary equations and considering for the third-body perturbation the gravity-gradient approximation, and then exploited to demonstrate the feasibility of achieving the above-mentioned goal. The study has led to the determination of two families of solutions at constant mean orbit elements, both characterised by a co-planarity condition between the eccentricity vector, the angular momentum and a vector resulting from the combination of the orbital poles of the perturbing bodies. As a practical case, the problem of a probe orbiting the Moon has been faced, taking into account the temporal evolution of the perturbing poles of the Sun and Earth, and frozen solutions at argument of pericentre 0\(^{\circ }\) or 180\(^{\circ }\) have been found.     

Local stellar kinematics from RAVE data—VIII. Effects of the Galactic disc perturbations on stellar orbits of red clump stars

We aim to probe the dynamic structure of the extended Solar neighborhood by calculating the radial metallicity gradients from orbit properties, which are obtained for axisymmetric and non-axisymmetric potential models, of red clump (RC) stars selected from the RAdial Velocity Experiment’s Fourth Data Release. Distances are obtained by assuming a single absolute magnitude value in near-infrared, i.e. \(M_{Ks}=-1.54\pm0.04\) mag, for each RC star. Stellar orbit parameters are calculated by using the potential functions: (i) for the MWPotential2014 potential, (ii) for the same potential with perturbation functions of the Galactic bar and transient spiral arms. The stellar age is calculated with a method based on Bayesian statistics. The radial metallicity gradients are evaluated based on the maximum vertical distance (\(z_{max}\)) from the Galactic plane and the planar eccentricity (\(e_{p}\)) of RC stars for both of the potential models. The largest radial metallicity gradient in the \(0< z_{max} \leq0.5\) kpc distance interval is \(-0.065\pm0.005~\mbox{dex}\,\mbox{kpc}^{-1}\) for a subsample with \(e_{p}\leq0.1\), while the lowest value is \(-0.014\pm0.006~\mbox{dex}\,\mbox{kpc}^{-1}\) for the subsample with \(e_{p}\leq0.5\). We find that at \(z_{max}>1\) kpc, the radial metallicity gradients have zero or positive values and they do not depend on \(e_{p}\) subsamples. There is a large radial metallicity gradient for thin disc, but no radial gradient found for thick disc. Moreover, the largest radial metallicity gradients are obtained where the outer Lindblad resonance region is effective. We claim that this apparent change in radial metallicity gradients in the thin disc is a result of orbital perturbation originating from the existing resonance regions.     

Halo orbit transfer trajectory design using invariant manifold in the Sun-Earth system accounting radiation pressure and oblateness

In this paper, we study the invariant manifold and its application in transfer trajectory problem from a low Earth parking orbit to the Sun-Earth \(L_{1}\) and \(L_{2}\)-halo orbits with the inclusion of radiation pressure and oblateness. Invariant manifold of the halo orbit provides a natural entrance to travel the spacecraft in the solar system along some specific paths due to its strong hyperbolic character. In this regard, the halo orbits near both collinear Lagrangian points are computed first. The manifold’s approximation near the nominal halo orbit is computed using the eigenvectors of the monodromy matrix. The obtained local approximation provides globalization of the manifold by applying backward time propagation to the governing equations of motion. The desired transfer trajectory well suited for the transfer is explored by looking at a possible intersection between the Earth’s parking orbit of the spacecraft and the manifold.     

Markarian galaxies in UV and multiwavelength studies

The UV properties of 1152 Markarian galaxies have been investigated based on GALEX data. These objects have been investigated also in other available wavelengths using multi-wavelength data from X-ray to radio. Using our classification for activity types for 779 Markarian galaxies based on SDSS spectroscopy, we have investigated these objects on the GALEX, 2MASS and WISE color-magnitude and color-color diagrams by the location of objects of different activity types and have revealed a number of loci. UV contours overplotted on the optical images revealed additional structures, particularly spiral arms of a number of Markarian galaxies. UV (FUV and NUV) and optical absolute magnitudes and luminosities have been calculated showing graduate transition from AGN to Composites, HIIs and Absorption line galaxies from (average \(M\)) \(-17.56^{m}\) to \(-15.20^{m}\) in FUV, from \(-18.07^{m}\) to \(-15.71^{m}\) in NUV and from AGN to Composites, Absorption line galaxies and HII from \(-21.14^{m}\) to \(-19.42^{m}\) in optical wavelengths and from (average \(L\)) \(7\times10^{9}\) to \(4 \times 10^{8}\) in FUV, from \(1\times 10^{10}\) to \(5\times10^{8}\) in NUV and from AGN to Composites, Absorption line galaxies and HII from \(7\times10^{10}\) to \(1\times10^{10}\) in optical wavelengths.     

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.     

Dust color temperature distribution of two FIR cavities at IRIS and AKARI maps

By systematically searching the region of far infrared loops, we found a number of huge cavity-like dust structures at \(60\,\mu \hbox {m}\) and \(100\,\mu \hbox {m}\) IRIS maps. By checking these with AKARI maps (\(90\,\mu \hbox {m}\) and \(140\,\mu \hbox {m}\)), two new cavity-like structures (sizes \(\sim \) \( 2.7\,\hbox {pc} \times 0.8\,\hbox {pc}\) and \(\sim \) \( 1.8\,\hbox {pc} \times 1\,\hbox {pc}\)) located at R.A. (\(\hbox {J}2000)=14^{h}41^{m}23^{s}\) and Dec. \((\hbox {J}2000)=-64^{\circ }04^{\prime }17^{{\prime }{\prime }}\) and R.A. \((\hbox {J}2000)=05^{h}05^{m}35^{s}\) and Dec. \((\hbox {J}2000)=-\,69^{\circ }35^{\prime } 25^{{\prime }{\prime }}\) were selected for the study. The difference in the average dust color temperatures calculated using IRIS and AKARI maps of the cavity candidates were found to be \(3.2\pm 0.9\,\hbox {K}\) and \(4.1\pm 1.2\,\hbox {K}\), respectively. Interestingly, the longer wavelength AKARI map gives larger values of dust color temperature than that of the shorter wavelength IRIS maps. Possible explanation of the results will be presented.     

Halo orbit of regularized circular restricted three-body problem with radiation pressure and oblateness

In this paper, computation of the halo orbit for the KS-regularized photogravitational circular restricted three-body problem is carried out. This work extends the idea of Srivastava et al. (Astrophys. Space Sci. 362: 49, 2017) which only concentrated on the (i) regularization of the 3D-governing equations of motion, and (ii) validation of the modeling for small out-of-plane amplitude (\(A_z =110000\) km) assuming the third-order analytical approximation as an initial guess with and without differential correction. This motivated us to compute the halo orbits for the large out-of-plane amplitudes and to study their stability analysis for the regularized motion. The stability indices are described as a function of out-of-plane amplitude, mass reduction factor and oblateness coefficient. Three different Sun–planet systems: the Sun–Earth, Sun–Mars and the Sun–Jupiter are chosen in this study. Stable halo orbits do not exist around the \(L_{1}\) point, however, around the \(L_{2}\) point stable halo orbits are found for the considered systems.     

Motion of the fast exoplanets

It is shown that a number of superfast, with periods \(< 2\) d, exoplanets revolve around parent stars with periods, near-commensurate with \(P_{E}\) and/or \(2 P_{E} / \pi\), where the exoplanet resonance timescale \(P_{E}=9603(85)\) s agrees fairly well with the period \(P_{0}= 9600.606(12)\) s of the so-called “cosmic oscillation” (the probability that the two timescales would coincide by chance is near \(3 \times10^{-4}\); the \(P_{0}\) period was discovered first in the Sun, and later on—in other objects of Cosmos). True nature of the exoplanet \(P_{0}\) resonance is unknown.     

Higher harmonics electrostatic ion cyclotron parallel flow velocity shear instability with inhomogeneous DC electric field in the magnetosphere of Saturn

In present paper higher harmonic electrostatic ion-cyclotron (EIC) parallel flow velocity shear instability in presence of perpendicular inhomogeneous DC electric field with the ambient magnetic field has been studied, in different regions of the magnetosphere of Saturn. Dimensionless growth rate variation of EIC waves has been observed with respect to \(k_{ \bot } \rho _{i}\) for various plasma parameters. Effect of velocity shear scale length (\(A_{i}\)), temperature anisotropy (\(T_{ \bot } /T_{\|}\)), magnetic field (\(B\)), electric field (\(E\)), inhomogeneity (\(P/a\)), angle of propagation (\(\theta \)), ratio of electron to ion temperature (\(T_{e}/T_{i}\)) and density gradient (\(\varepsilon _{n}\rho _{i}\)) on the growth of EIC waves in the inner magnetosphere of Saturn has been studied and analyzed. The mathematical formulation for dispersion relation and growth rate has been done by using the method of characteristic solution and kinetic approach. This theoretical analysis has been done taking the data from the Cassini in the inner magnetosphere of Saturn in the extended region where ion cyclotron waves have been observed. The change in the growth of these waves due to the presence of Enceladus has been analyzed.     

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 \(\frac{1}{n}\).     

Periodic orbits of solar sail equipped with reflectance control device in Earth–Moon system

In this paper, families of Lyapunov and halo orbits are presented with a solar sail equipped with a reflectance control device in the Earth–Moon system. System dynamical model is established considering solar sail acceleration, and four solar sail steering laws and two initial Sun-sail configurations are introduced. The initial natural periodic orbits with suitable periods are firstly identified. Subsequently, families of solar sail Lyapunov and halo orbits around the \(L_{1}\) and \(L_{2}\) points are designed with fixed solar sail characteristic acceleration and varying reflectivity rate and pitching angle by the combination of the modified differential correction method and continuation approach. The linear stabilities of solar sail periodic orbits are investigated, and a nonlinear sliding model controller is designed for station keeping. In addition, orbit transfer between the same family of solar sail orbits is investigated preliminarily to showcase reflectance control device solar sail maneuver capability.     

Deprojected Trajectory of Blobs in the Inner Corona

We have carried out a statistical analysis of the kinematical behavior of small white-light transients (blobs) as tracers of the slow solar wind. The characterization of these faint white-light structures gives us insight on the origin and acceleration of the slow solar wind. The vantage observing points provided by the SECCHI and LASCO instruments on board the STEREO and SOHO spacecraft, respectively, allow us to reconstruct the 3D trajectories of these blob-like features and hence calculate their deprojected kinematical parameters. We have studied 44 blobs revealed in LASCO C2/C3 and SECCHI COR2 data from 2007 to 2008, a period within the solar minimum between Solar Cycles 23 and 24. We found that the blobs propagate along approximately constant position angles with accelerations from 1.40 to \(15.34~\mbox{m}\,\mbox{s}^{-2}\) between 3.42 \(R_{\astrosun }\) and 14.80 \(R_{\astrosun }\), their radial sizes ranging between 0.57 \(R_{\astrosun }\) and 1.69 \(R_{\astrosun }\). We also studied the global corona magnetic field morphology for a subset of blobs using a potential field source surface model for cases where blob detachments persist for two to five days. The study of localized blob releases indicates that these plasma structures start their transit at a distance of \(\sim\,{3.40}~R_{\astrosun }\) and their origin is connected either with the boundaries of weak coronal holes or with streamers at equatorial latitudes.     

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.     

An interacting new holographic dark energy in the framework of fractal cosmology

In this paper, we study an interacting holographic dark energy model in the framework of fractal cosmology. The features of fractal cosmology could pass ultraviolet divergencies and also make a better understanding of the universe in different dimensions. We discuss a fractal FRW universe filled with the dark energy and cold dark matter interacting with each other. It is observed that the Hubble parameter embraces the recent observational range while the deceleration parameter demonstrates an accelerating universe and a behavior similar to \(\Lambda \mbox{CDM}\). Plotting the equation of state shows that it lies in phantom region for interaction mode. We use \(\mathit{Om}\)-diagnostic tool and it shows a phantom behavior of dark energy which is a condition of avoiding the formation of black holes. Finally we execute the StateFinder diagnostic pair and all the trajectories for interacting and non-interacting state of the model meet the fixed point \(\Lambda \mbox{CDM}\) at the start of the evolution. A behavior similar to Chaplygin gas also can be observed in statefinder plane. We find that new holographic dark energy model (NHDE) in fractal cosmology expressed the consistent behavior with recent observational data and can be considered as a model to avoid the formation of black holes in comparison with the main model of NHDE in the simple FRW universe. It has also been observed that for the interaction term varying with matter density, the model generates asymptotic de-Sitter solution. However, if the interaction term varies with energy density, then the model shows Big-Rip singularity. Using our modified CAMB code, we observed that the interacting model suppresses the CMB spectrum at low multipoles \(l<50\) and enhances the acoustic peaks. Based on the observational data sets used in this paper and using Metropolis-Hastings method of MCMC numerical calculation, it seems that the best value with \(1\sigma \) and \(2\sigma \) confidence interval are \(\Omega _{m0}=0.278^{+0.008~+0.010} _{-0.007~-0.009}\), \(H_{0}=69.9^{+0.95~+1.57}_{-0.95~-1.57}\), \(r_{c}=0.08^{+0.02~+0.027}_{-0.002~-0.0027}\), \(\beta =0.496^{+0.005~+0.009} _{-0.005~-0.009}\), \(c= 0.691^{+0.024~+0.039}_{-0.025~-0.037}\) and \(b^{2}=0.035\) according to which we find that the proposed model in the presence of interaction is compatible with the recent observational data.