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.     

Determining parameters of Moon’s orbital and rotational motion from LLR observations using GRAIL and IERS-recommended models

The aim of this work is to combine the model of orbital and rotational motion of the Moon developed for DE430 with up-to-date astronomical, geodynamical, and geo- and selenophysical models. The parameters of the orbit and physical libration are determined in this work from lunar laser ranging (LLR) observations made at different observatories in 1970–2013. Parameters of other models are taken from solutions that were obtained independently from LLR. A new implementation of the DE430 lunar model, including the liquid core equations, was done within the EPM ephemeris. The postfit residuals of LLR observations make evident that the terrestrial models and solutions recommended by the IERS Conventions are compatible with the lunar theory. That includes: EGM2008 gravitational potential with conventional corrections and variations from solid and ocean tides; displacement of stations due to solid and ocean loading tides; and precession-nutation model. Usage of these models in the solution for LLR observations has allowed us to reduce the number of parameters to be fit. The fixed model of tidal variations of the geopotential has resulted in a lesser value of Moon’s extra eccentricity rate, as compared to the original DE430 model with two fit parameters. A mixed model of lunar gravitational potential was used, with some coefficients determined from LLR observations, and other taken from the GL660b solution obtained from the GRAIL spacecraft mission. Solutions obtain accurate positions for the ranging stations and the five retroreflectors. Station motion is derived for sites with long data spans. Dissipation is detected at the lunar fluid core-solid mantle boundary demonstrating that a fluid core is present. Tidal dissipation is strong at both Earth and Moon. Consequently, the lunar semimajor axis is expanding by 38.20 mm/yr, the tidal acceleration in mean longitude is \(-25.90 {{}^{\prime \prime }}/\mathrm{cy}^2\), and the eccentricity is increasing by \(1.48\times 10^{-11}\) each year.     

On the deviation of the lunar center of mass to the East. Two possible mechanisms based on evolution of the orbit and rounding off the shape of the Moon

From the observations of the gravitational field and the figure of the Moon, it is known that its center of mass (briefly COM) does not coincide with the center of figure (COF), and the line “COF/COM” is not directed to the center of the Earth, but deviates from it to the South–East. Here we study the deviation of the lunar COM to the East from the mean direction to Earth.At first, we consider the optical libration of a satellite with synchronous rotation around the planet for an observer at a point on second (empty) orbit focus. It is found that the main axis of inertia of the satellite has asymmetric nonlinear oscillations with amplitude proportional to the square of the orbit eccentricity. Given this effect, a mechanism of tidal secular evolution of the Moon’s orbit is offered that explains up to \(20\%\) of the known displacement of the lunar COM to the East. It is concluded that from the alternative—evolution of the Moon’s orbit with a decrease or increase in eccentricity—only the scenario of evolution with a monotonous increase in orbit eccentricity agrees with the displacement of lunar COM to the East. The precise calculations available confirm that now the eccentricity of the lunar orbit is actually increasing and therefore in the past it was less than its modern value, \(e = 0.0549\).To fully explain the displacement of the Moon’s COM to the East was deduced a second mechanism, which is based on the reliable effect of tidal changes in the shape of the Moon. For this purpose the differential equation which governs the process of displacement of the Moon’s COM to the East with inevitable rounding off its form in the tidal increase process of the distance between the Earth and the Moon is derived. The second mechanism not only explains the Moon’s COM displacement to the East, but it also predicts that the elongation of the lunar figure in the early epoch was significant and could reach the value \(\varepsilon\approx0.31\). Applying the theory of tidal equilibrium figures, we can estimate how close to the Earth the Moon could have formed.     

Study of the transfer between libration point orbits and lunar orbits in Earth–Moon system

This paper is devoted to the study of the transfer problem from a libration point orbit of the Earth–Moon system to an orbit around the Moon. The transfer procedure analysed has two legs: the first one is an orbit of the unstable manifold of the libration orbit and the second one is a transfer orbit between a certain point on the manifold and the final lunar orbit. There are only two manoeuvres involved in the method and they are applied at the beginning and at the end of the second leg. Although the numerical results given in this paper correspond to transfers between halo orbits around the \(L_1\) point (of several amplitudes) and lunar polar orbits with altitudes varying between 100 and 500 km, the procedure we develop can be applied to any kind of lunar orbits, libration orbits around the \(L_1\) or \(L_2\) points of the Earth–Moon system, or to other similar cases with different values of the mass ratio.     

The precise positioning of lunar farside lander using a four-way lander-orbiter relay tracking mode

China is planning to land a spacecraft on the farside of the Moon, a premiere, by 2018. In essence, the traditional tracking modes, based on direct visibility, cannot operate for the lunar farside lander tracking, and therefore a relay satellite, visible at the same time by both the lander and the Earth, will be required, operating in the so-called four-way mode (Earth-relay satellite-lander-relay satellite-Earth). In this paper, we firstly give the mathematical formulation of the four-way relay tracking mode and of its partial derivatives with respect to the relevant parameters, implemented in our POD software WUDOGS (Wuhan University Deep-space Orbit determination and Gravity recovery System). In a second step, in simulation mode, we apply this relay mode to determining lander coordinates, which are absolutely needed for a sample return mission, or to add constraints on rotation models of the Moon. The results show that with Doppler measurements at a 0.1 mm/s error level, the positioning of the farside lander could be done at centimeters level (1-\(\delta\)) in the case of a circumlunar relay satellite; and at a 5 meters level (1-\(\delta\)) in the case of a Lagrange point (L2) Halo relay satellite.     

Preferred design and error analysis for the future dedicated deep-space Mars-SST satellite gravity mission

Because the precise measurement of the Martian gravitational field plays a significant role in the future Mars exploration program, the future dedicated Mars satellite-to-satellite tracking (Mars-SST) gravity mission in China is investigated in detail for producing the next generation of the Mars gravity field model with high accuracy. Firstly, a new semi-numerical synthetical error model of the cumulative Martian geoid height influenced by the major error sources of the space-borne instruments is precisely established and efficiently verified. Secondly, the deep space network in combination with the satellite-to-satellite tracking in the low-low (DSN-SST-LL) mode is a preferred design owing to the high precision determination of the gravity maps, the low technical complexity of the satellite system and the successful experiences with the Earth’s Gravity Recovery and Climate Experiment (GRACE) projects and the lunar Gravity Recovery and Interior Laboratory (GRAIL) program. Finally, the future twin Mars-SST satellites plan to adopt the optimal matching accuracy indices of the satellite-equipped sensors (e.g., \(10^{-7}\) m/s in the inter-satellite range-rate from the interferometric laser ranging system (ILRS), 35 m in the orbital position tracked by the DSN and \(3\times 10^{-11}\) m/s² in the non-conservative force from the drag-free control system (DFCS)) and the preferred orbital parameters (e.g., the orbital altitude of \(100\pm 50\) km and the inter-satellite range of \(50\pm 10\) km).     

Existence of static wormhole solutions in $$ gravity

This work investigates some feasible regions for the existence of traversable wormhole geometries in \(f(R,G)\) gravity, where \(R\) and \(G\) represent the Ricci scalar and the Gauss-Bonnet invariant respectively. Three different matter contents anisotropic fluid, isotropic fluid and barotropic fluid have been considered for the analysis. Moreover, we split \(f(R,G)\) gravity model into Strobinsky like \(f(R)\) model and a power law \(f(G)\) model to explore wormhole geometries. We select red-shift and shape functions which are suitable for the existence of wormhole solutions for the chosen \(f(R,G)\) gravity model. It has been analyzed with the graphical evolution that the null energy and weak energy conditions for the effective energy-momentum tensor are usually violated for the ordinary matter content. However, some small feasible regions for the existence of wormhole solutions have been found where the energy conditions are not violated. The overall analysis confirms the existence of the wormhole geometries in \(f(R,G)\) gravity under some reasonable circumstances.     

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.     

FLRW cosmological models with quark and strange quark matters in $$ gravity

In this paper, we have studied the magnetized quark matter (QM) and strange quark matter (SQM) distributions in the presence of \(f(R,T)\) gravity in the background of Friedmann-Lemaître-Robertson-Walker (FLRW) metric. To get exact solutions of modified field equations we have used \(f(R,T ) = R + 2 f(T)\) model given by Harko et al. with two different parametrization of geometrical parameters i.e. the parametrization of the deceleration parameter \(q \), and the scale factor \(a \) in hybrid expansion form. Also, we have obtained Einstein Static Universe (ESU) solutions for QM and SQM distributions in \(f(R,T)\) gravity and General Relativity (GR). All models in \(f(R,T)\) gravity and GR for FRW and ESU Universes with QM also SQM distributions, we get zero magnetic field. These results agree with the solutions of Akta? and Aygün in \(f(R,T)\) gravity. However, we have also discussed the physical consequences of our obtained models.     

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.     

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.     

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 \(A\).     

The Pleiades apex and its kinematical structure

A study of cluster characteristics and internal kinematical structure of the middle-aged Pleiades open star cluster is presented. The individual star apexes and various cluster kinematical parameters including the velocity ellipsoid parameters are determined using both Hipparcos and Gaia data. Modern astrometric parameters were taken from the Gaia Data Release 1 (DR1) in combination with the Radial Velocity Experiment Fifth Data Release (DR5). The necessary set of parameters including parallaxes, proper motions and radial velocities are used for \(n=17\) stars from Gaia DR1+RAVE DR5 and for \(n=19\) stars from the Hipparcos catalog using SIMBAD data base. Single stars are used to improve accuracy by eliminating orbital movements. RAVE DR5 measurements were taken only for the stars with the radial velocity errors not exceeding \(2~\mbox{km}/\mbox{s}\). For the Pleiades stars taken from Gaia, we found mean heliocentric distance as \(136.8 \pm 6.4\) pc, and the apex position is calculated as: \(A_{CP}=92^{\circ }.52\pm 1^{\circ }.72\), \(D_{CP}=-42^{\circ }.28\pm 2^{\circ }.56\) by the convergent point method and \(A_{0}=95^{\circ }.59\pm 2^{\circ }.30\) and \(D_{0}=-50^{\circ }.90\pm 2^{\circ }.04\) using AD-diagram method (\(n=17\) in both cases). The results are compared with those obtained historically before the Gaia mission era.     

Relativistic satellite orbits: central body with higher zonal harmonics

Satellite orbits around a central body with arbitrary zonal harmonics are considered in a relativistic framework. Our starting point is the relativistic Celestial Mechanics based upon the first post-Newtonian approximation to Einstein’s theory of gravity as it has been formulated by Damour et al. (Phys Rev D 43:3273–3307, 1991; 45:1017–1044, 1992; 47:3124–3135, 1993; 49:618–635, 1994). Since effects of order \((\mathrm{GM}/c^2R) \times J_k\) with \(k \ge 2\) for the Earth are very small (of order \( 7 \times 10^{-10}\,\times \,J_k\)) we consider an axially symmetric body with arbitrary zonal harmonics and a static external gravitational field. In such a field the explicit \(J_k/c^2\)-terms (direct terms) in the equations of motion for the coordinate acceleration of a satellite are treated first with first-order perturbation theory. The derived perturbation theoretical results of first order have been checked by purely numerical integrations of the equations of motion. Additional terms of the same order result from the interaction of the Newtonian \(J_k\)-terms with the post-Newtonian Schwarzschild terms (relativistic terms related to the mass of the central body). These ‘mixed terms’ are treated by means of second-order perturbation theory based on the Lie-series method (Hori–Deprit method). Here we concentrate on the secular drifts of the ascending node \(<\!{\dot{\Omega }}\!>\) and argument of the pericenter \(<\!{\dot{\omega }}\!>\). Finally orders of magnitude are given and discussed.     

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.     

New Hamiltonian expansions adapted to the Trojan problem

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

New holographic dark energy model with constant bulk viscosity in modified $$ gravity theory

The aim of this paper is to study new holographic dark energy (HDE) model in modified \(f(R,T)\) gravity theory within the framework of a flat Friedmann-Robertson-Walker model with bulk viscous matter content. It is thought that the negative pressure caused by the bulk viscosity can play the role of dark energy component, and drive the accelerating expansion of the universe. This is the motive of this paper to observe such phenomena with bulk viscosity. In the specific model \(f(R,T)=R+\lambda T\), where \(R\) is the Ricci scalar, \(T\) the trace of the energy-momentum tensor and \(\lambda \) is a constant, we find the solution for non-viscous and viscous new HDE models. We analyze new HDE model with constant bulk viscosity, \(\zeta =\zeta _{0}= \text{const.}\) to explain the present accelerated expansion of the universe. We classify all possible scenarios (deceleration, acceleration and their transition) with possible positive and negative ranges of \(\lambda \) over the constraint on \(\zeta _{0}\) to analyze the evolution of the universe. We obtain the solutions of scale factor and deceleration parameter, and discuss the evolution of the universe. We observe the future finite-time singularities of type I and III at a finite time under certain constraints on \(\lambda \). We also investigate the statefinder and \(\mathit{Om}\) diagnostics of the viscous new HDE model to discriminate with other existing dark energy models. In late time the viscous new HDE model approaches to \(\varLambda \mathit{CDM}\) model. We also discuss the thermodynamics and entropy of the model and find that it satisfies the second law of thermodynamics.     

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.     

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.     

Gravitational lunar capture based on bicircular model in restricted four body problem

Gravitational capture is a useful phenomenon in the design of the low energy transfer (LET) orbit for a space mission. In this paper, gravitational lunar capture based on the Sun–Earth–Moon bicircular model (BCM) in the restricted four body problem is studied. By the mechanical analysis in the space near the Moon, we first propose a new parameter \(k\) , the corrected ratio of the radial force, to investigate the influence of the radial force on the capture eccentricity in the BCM. Then, a parametric analysis is performed to detect the influences on the corrected ratio \(k\) . Considering the restriction of time-of-flight and corrected ratio, we investigate, respectively, the minimum capture eccentricity and the corrected minimum capture eccentricity. Via numerical analysis, we discover two special regions on the sphere of capture, in which the capture point possesses the global minimum capture eccentricity and corrected capture eccentricity. They denote the optimal capture regions in terms of minimizing the fuel consumption of the maneuver. According to the results obtained, some suggestions on the design of the LET orbit are given.