Ballistic capture into distant retrograde orbits around Phobos: an approach to entering orbit around Phobos without a critical maneuver near the moon

This paper discusses an approach for designing missions to Phobos that do not require a critical maneuver in proximity of the moon. A low-energy transfer is designed that utilizes the aspherical mass distribution of Phobos to capture a spacecraft into a distant retrograde orbit (DRO) for the mission duration. The process for generating stable DROs in the Mars–Phobos system is discussed along with the method for generating and surveying a set of ballistic capture trajectories (BCTs) for DROs with altitudes between 0.5 and 14 km above Phobos. Statistical analysis of the BCT set reveals options for designing a mission to the desired DRO. This approach can be used in any three-body system when a significant perturbation is present, such as Phobos’ aspherical co-rotating gravity field.     

Mars interplanetary trajectory design via Lagrangian points

With the increase in complexities of interplanetary missions, the main focus has shifted to reducing the total delta-V for the entire mission and hence increasing the payload capacity of the spacecraft. This paper develops a trajectory to Mars using the Lagrangian points of the Sun-Earth system and the Sun-Mars system. The whole trajectory can be broadly divided into three stages: (1) Trajectory from a near-Earth circular parking orbit to a halo orbit around Sun-Earth Lagrangian point L₂. (2) Trajectory from Sun-Earth L₂ halo orbit to Sun-Mars L₁ halo orbit. (3) Sun-Mars L₁ halo orbit to a circular orbit around Mars. The stable and unstable manifolds of the halo orbits are used for halo orbit insertion. The intermediate transfer arcs are designed using two-body Lambert’s problem. The total delta-V for the whole trajectory is computed and found to be lesser than that for the conventional trajectories. For a 480 km Earth parking orbit, the total delta-V is found to be 4.6203 km/s. Another advantage in the present approach is that delta-V does not depend upon the synodic period of Earth with respect to Mars.     

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.     

Effect of the Earth’s compression on the physical libration of the Moon

The effect of the Earth??s compression on the physical libration of the Moon is studied using a new vector method. The moment of gravitational forces exerted on the Moon by the oblate Earth is derived considering second order harmonics. The terms in the expression for this moment are arranged according to their order of magnitude. The contribution due to a spherically symmetric Earth proves to be greater by a factor of 1.34 × 10⁶ than a typical term allowing for the oblateness. A linearized Euler system of equations to describe the Moon??s rotation with allowance for external gravitational forces is given. A full solution of the differential equation describing the Moon??s libration in longitude is derived. This solution includes both arbitrary and forced oscillation harmonics that we studied earlier (perturbations due to a spherically symmetric Earth and the Sun) and new harmonics due to the Earth??s compression. We posed and solved the problem of spinorbital motion considering the orientation of the Earth??s rotation axis with regard to the axes of inertia of the Moon when it is at a random point in its orbit. The rotation axes of the Earth and the Moon are shown to become coplanar with each other when the orbiting Moon has an ecliptic longitude of L _? = 90° or L _? = 270°. The famous Cassini??s laws describing the motion of the Moon are supplemented by the rule for coplanarity when proper rotations in the Earth-Moon system are taken into account. When we consider the effect of the Earth??s compression on the Moon??s libration in longitude, a harmonic with an amplitude of 0.03?? and period of T ₈ = 9.300 Julian years appears. This amplitude exceeds the most noticeable harmonic due to the Sun by a factor of nearly 2.7. The effect of the Earth??s compression on the variation in spin angular velocity of the Moon proves to be negligible.     

The Soviet ASTRON mission: legacy

2013 marks the 30th anniversary since the launch of Soviet Spacecraft Astron that had been operated for 6 years as the largest ultraviolet telescope during its lifetime. The Astron orbital station was designed for the astrophysical observations. It was launched into orbit by Proton launch system on March 23, 1983. Astron had a 80 cm ultraviolet telescope with mass of 400 kg and a complex of X-ray spectrographs with mass of 300 kg on board as a payload. It’s high apogee orbit (with apogee 200000 km and perigee 2000 km) permitted the influences of the Earth’s umbra and radiation belts to be excluded from the measurements. The main astrophysical results are summarized in this paper.     

An Estimate of the Long-term Tendency of Tidal Evolution of Earth-lunar System

According to the conservation principle of angular momentum, we calculate in this paper the revolution period and the distance between the Earth and the Moon in the equilibrium state of the tidal evolution in the Earth-Moon system. The difference of energy between the current state and the equilibrium state is used to compute the time needed to fulfil the equilibrium state. Then the long-term variations of the Earth-Moon distance and of the Earth rotation rate are further estimated.     

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 Connection of Solar Wind Parameters with Radio and UV Emission from Coronal Holes

This paper presents the results of a comparison between observations of coronal holes in UV (SOHO EIT) and radio emission (17, 5.7 GHz, 327 and 150.9 MHz, from NoRH, SSRT and Nançay radioheliographs), and solar wind parameters, from ACE spacecraft data over the period 12 March?–?31 May 2007. The increase in the solar wind velocity up to ～?600 km?s^?1 was found to correlate with a decrease in the UV flux in the central parts of the solar disk. A connection between the parameters of the radio emission from three different layers of the solar atmosphere and the solar wind velocity near the Earth’s orbit was discovered. Such a connection is suggestive of a common mechanism of solar wind acceleration from chromospheric heights to the upper corona.     

Generation of a secular system in the analytical theory for the motion of the moon

In order to generate an analytical theory of the motion of the Moon by considering planetary perturbations, a procedure of general planetary theory (GPT) is used. In this case, the Moon is considered as an addition planet to the eight principal planets. Therefore, according to the GPT procedure, the theory of the Moon’s orbital motion can be presented in the form of series with respect to the evolution of eccentric and oblique variables with quasi-periodic coefficients, which are the functions of mean longitudes for principal planets and the Moon. The relationship between evolution variables and the time is determined by a trigonometric solution for the independent secular system that describes the secular motion of a perigee and the Moon node by considering secular planetary inequalities. Principal planetary coordinates required for generating the theory of the motion of the Moon includes only Keplerian terms, the intermediate orbit, and the linear theory with respect to eccentricities and inclinations in the first order relative to the masses. All analytical calculations are performed by means of the specialized echeloned Poisson Series Processor EPSP.     

Numerical models for the study of motion of lunar satellites

The present study deals with numerical modeling of the elliptic restricted three-body problem as well as of the perturbed elliptic restricted three-body (Earth-Moon-Satellite) problem by a fourth body (Sun). Two numerical algorithms are established and investigated. The first is based on the method of the series solution of the differential equations and the second is based on a 5th-order Runge-Kutta method. The applications concern the solution of the equations and integrals of motion of the circular and elliptical restricted three-body problem as well as the search for periodic orbits of the natural satellites of the Moon in the Earth-Moon system in both cases in which the Moon describes circular or elliptical orbit around the Earth before the perturbations induced by the Sun. After the introduction of the perturbations in the Earth-Moon-Satellite system the motions of the Moon and the Satellite are studied with the same initial conditions which give periodic orbits for the unperturbed elliptic problem.     

Magnetospheric protons and electrons encountered by the moon in the plasma sheet

During its passage through the geomagnetic tail, the Moon encounters the plasma sheet. Properties of plasma sheet electrons and protons, first detected at lunar distances by Explorer 35, are described. The electrons have a rapidly fluctuating non-Maxwellian energy distribution with a mean energy of several hundred electron volts and density ç 0.2 cm^?3. The protons, of energy ç 1 keV, were usually detected above the instrument background when flowing towards the Earth at ç 200 km s^?1. Implications for migration of grains on the lunar surface are also pointed out and it is suggested that strong terrestrial polar winds in the early history of the Earth-Moon system may have caused some erosion of the Earth-facing side of the Moon, and that gravitational shielding of interplanetary rock flux by the Earth may also be an explanation of the relative smoothness of the front side.     

Utilisation of the lunar field for interplanetary travel by the Oslo system

This paper presents preliminary results of orbital investigations by a data processing machine aimed at the utilisation of the lunar gravitational field in interplanetary travel. The lunar field is utilised in two successive steps. In the first step a spacecraft in an Earth-bound orbit is deviated into an orbit similar to but separated from the Earth's orbit around the Sun. In a successive approach between the spacecraft and the Earth-Moon system the combined fields of the Earth and the Moon are utilised as a means to convey to the spacecraft a main portion of the momentum required in order to carry it to the proximity of Venus.A substantial portion of the fuel required for space travel can be saved by this kind of procedure.The CD 3300 machine time needed for this investigation was supplied by the computing facility of the University of Oslo at Blindern.     

The effect of the Earth’s oblateness on the Moon’s physical libration in latitude

The Moon’s physical libration in latitude generated by gravitational forces caused by the Earth’s oblateness has been examined by a vector analytical method. Libration oscillations are described by a close set of five linear inhomogeneous differential equations, the dispersion equation has five roots, one of which is zero. A complete solution is obtained. It is revealed that the Earth’s oblateness: a) has little effect on the instantaneous axis of Moon’s rotation, but causes an oscillatory rotation of the body of the Moon with an amplitude of 0.072″ and pulsation period of 16.88 Julian years; b) causes small nutations of poles of the orbit and of the ecliptic along tight spirals, which occupy a disk with a cut in a center and with radius of 0.072″. Perturbations caused by the spherical Earth generate: a) physical librations in latitude with an amplitude of 34.275″; b) nutational motion for centers of small spiral nutations of orbit (ecliptic) pole over ellipses with semi-major axes of 113.850″ (85.158″) and the first pole rotates round the second one along a circle with radius of 28.691″; c) nutation of the Moon’s celestial pole over an ellipse with a semi-major axis of 45.04″ and with an axes ratio of about 0.004 with a period of T = 27.212 days. The principal ellipse’s axis is directed tangentially with respect to the precession circumference, along which the celestial pole moves nonuniformly nearly in one dimension. In contrast to the accepted concept, the latitude does not change while the Moon’s poles of rotation move. The dynamical reason for the inclination of the Moon’s mean equator with respect to the ecliptic is oblateness of the body of the Moon.     

Molecular dynamics estimates for the thermodynamic properties of the Fe–S liquid cores of the Moon,Io, Europa,and Ganymede

A molecular dynamics (MD) simulation is performed for the physical and chemical properties of solid and liquid Fe–S solutions using the embedded atom model (EAM) potential as applied to the internal structure of the Moon, Io, Europa, and Ganymede under the assumption that the satellites' cores can be described by a two-component iron–sulfur system. Calculated results are presented for the thermodynamic parameters including the caloric, thermal, and elastic properties (specific heat, thermal expansion, Grüneisen parameter, density, compression module, velocity of sound, and adiabatic gradient) of the Fe–S solutions at sulfur concentrations of 0–18 at %, temperatures of up to 2500 K, and pressures of up to 14 GPa. The velocity of sound, which increases as pressure rises, is weakly dependent on sulfur concentration and temperature. For the Moon’s outer Fe–S core (~5 GPa/2000 K), which contains 6–16 at % (3.5–10 wt %) sulfur, the density and the velocity of sound are estimated at 6.3–7.0 g/cm³ and 4000 ± 50 m/s, respectively. The MD calculations are compared with the interpretation of the Apollo observations (Weber et al., 2011) to show a good consistency of the velocity of P-waves in the Moon’s liquid core whereas the thermodynamic density of the Fe–S core is not consistent with the seismic models with ρ = 5.1–5.2 g/cm³ (Garcia et al., 2011; Weber et al., 2011). The revision the density values for the core leads to the revision of its size and mass. At sulfur concentrations of 3.5–10 wt %, the density of the Fe–S melt is 20–30% higher that the seismic density of the core. Therefore, the most likely radius of the Moon’s outer core must be less than 330 km (Weber et al., 2011) because, provided that the constraint on the Moon’s mass and moment of inertia is satisfied, an increase in the density of the core must lead to a reduction of its radius. For Jupiter’s Galilean moons Io, Europa, and Ganymede, constraints are obtained on the size, density, and sound velocity of the Fe–S liquid cores. The geophysical and geochemical characteristics of the internal structure of the Moon and Jupiter’s moons are compared. The calculations of the adiabatic gradient at the P–T conditions for the Fe–S cores of the Moon, Io, Europa, and Ganymede suggest the top-down crystallization of the core (Fe-snow scenario).     

Lunar cratering asymmetries with high lunar orbital obliquity and inclination of the Moon

Accurate estimation of cratering asymmetry on the Moon is crucial for understanding Moon evolution history. Early studies of cratering asymmetry have omitted the contributions of high lunar obliquity and inclination. Here, we include lunar obliquity and inclination as new controlling variables to derive the cratering rate spatial variation as a function of longitude and latitude. With examining the influence of lunar obliquity and inclination on the asteroids population encountered by the Moon, we then have derived general formulas of the cratering rate spatial variation based on the crater scaling law. Our formulas with addition of lunar obliquity and inclination can reproduce the lunar cratering rate asymmetry at the current Earth-Moon distance and predict the apex/ant-apex ratio and the pole/equator ratio of this lunar cratering rate to be 1.36 and 0.87, respectively. The apex/ant-apex ratio is decreasing as the obliquity and inclination increasing. Combining with the evolution of lunar obliquity and inclination, our model shows that the apex/ant-apex ratio does not monotonically decrease with Earth-Moon distance and hence the influences of obliquity and inclination are not negligible on evolution of apex/ant-apex ratio. This model is generalizable to other planets and moons, especially for different spin-orbit resonances.     

Analysis of Stability of Orbits of Artificial Lunar Satellites and Configuring of a Lunar Satellite Navigation System

The analysis of the Moon artificial satellite orbits stability and satellite system configuring are important issues of lunar orbital navigational system development. The article analyses the influence of different combinations of perturbations on Moon artificial satellite’s obits evolution. The method of Moon artificial satellite’s orbital evolution analysis is offered; general stability regions of Moon artificial satellite’s orbits are defined and the quality characteristics of the selected orbital groups of the satellite system are evaluated.     

Origin and evolution of the earth-moon system

The origin and evolution of the Earth-Moon system is studied by comparing it to the satellite systems of other planets. The normal structure of a system of secondary bodies orbiting around a central body depends essentially on the mass of the central body. The Earth with a mass intermediate between Uranus and Mars should have a normal satellite system that consists of about half a dozen satellites each with a mass of a fraction of a percent of the lunar mass. Hence, the Moon is not likely to have been generated in the environment of the Earth by a normal accretion process as is claimed by some authors.Capture of satellites is quite a common process as shown by the fact that there are six satellites in the solar system which, because they are retrograde, must have been captured. There is little doubt that the Moon is also a captured satellite, but its capture orbit and tidal evolution are still incompletely understood.The Earth and the Moon are likely to have been formed from planetesimals accreting in particle swarms in Kepler orbits (jet streams). This process leads to the formation of a cool lunar interior with an outer layer accreted at increasingly higher temperatures. The primeval Earth should similarly have formed with a cool inner core surrounded in this case by a very strongly heated outer core and with a mantle accreted slowly and with a low average temperature but with intense transient heating at each individual impact site.     

Lunar paleotides and the origin of the Earth-Moon system

A new method for determining the early history of the Earth-Moon system is described. Called the study of lunar paleotides, it describes a method for explaining features of the remnant lunar gravity field, and the generation of the lunar mascons. A method for the determination of Earth-Moon distances compared with the radiometric ages of the maria is developed. It is shown that the Moon underwent strong anomalous gravitational tidal forces, for a durationt<10⁶yr, prior to the formation of the mascon surfaces. As these tidal forces had not been present at the time of the formation of the Moon, this shows that the Moon could not have been formed in orbit about the Earth.There are tides in the affairs of men which, taken at the flood, lead on to fortune... William Shakespeare 1564–1616     

Magnetization of celestial bodies with special application to the primeval Earth and Moon

The magnetic fields of celestial bodies are usually supposed to be due to a ‘hydromagnetic dynamo’. This term refers to a number of rather speculative processes which are supposed to take place in the liquid core of a celestial body. In this paper we shall follow another approach which is more closely connected with hydromagnetic processes well-known from the laboratory, and hence basically less speculative. The paper should be regarded as part of a general program to connect cosmical phenomena with phenomena studied in the laboratory. As has been demonstrated by laboratory experiments, a poloidal magnetic field may be increased by the transfer of energy from a toroidal magnetic field through kink instability of the current system. This mechanism can be applied to the fluid core of a celestial body. Any differential rotation will produce a toroidal field from an existing poloidal field, and the kink instability will feed toroidal energy back to the poloidal field, and hence amplify it. In the Earth-Moon system the tidal braking of the Earth's mantle acts to produce a differential angular velocity between core and mantle. The braking will be transferred to the core by hydromagnetic forces which at the same time give rise to a strong magnetic field. The strength of the field will be determined by the rate of tidal braking. It is suggested that the magnetization of lunar rocks from the period ?4 to ?3 Gyears derives from the Earth's magnetic field. As the interior of the Moon immediately after accretion probably was too cool to be melted, the Moon could not produce a magnetic field by hydromagnetic effects in its core. The observed lunar magnetization could be produced by such an amplified Earth field even if the Moon never came closer than 10 or 20 Earth's radii. This hypothesis might be checked by magnetic measurements on the Earth during the same period.     

The 27-Day Cosmic Ray Intensity Variations During Solar Minimum 23/24

We have studied the 27-day variations and their harmonics in Galactic cosmic ray (GCR) intensity, solar wind velocity, and interplanetary magnetic field (IMF) components during the recent prolonged solar minimum 23/24. The time evolution of the quasi-periodicity in these parameters connected with the Sun’s rotation reveals that the synodic period of these variations is ≈?26?–?27 days and is stable. This means that the changes in the solar wind speed and the IMF are related to the Sun’s near-equatorial regions in considering the differential rotation of the Sun. However, the solar wind parameters observed near the Earth’s orbit provide only the conditions in the limited local vicinity of the equatorial region in the heliosphere (within ±?7^° in latitude). We also demonstrate that the observed period of the GCR intensity connected with the Sun’s rotation increased up to ≈?33?–?36 days in 2009. This means that the process that drives the 27-day GCR intensity variations takes place not only in the limited local surroundings of the equatorial region but in the global 3-D space of the heliosphere, covering also higher latitude regions. A relatively long period (≈?34 days) found for 2009 in the GCR intensity gives possible evidence of the onset of cycle 24 due to active regions at higher latitudes and rotating slowly because of the Sun’s differential rotation. We also discuss the effect of differential rotation on the theoretical model of the 27-day GCR intensity variations.