On the Dynamics of Weak Stability Boundary Lunar Transfers

Recent studies demonstrate that lunar and solar gravitational assists can offer a good reduction of total variation of velocity Vneeded in lunar transfer trajectories. In particular the spacecraft, crossing regions of unstable equilibrium in the Earth—Moon—Sun system, can be guided by the Sun towards the lunar orbit with the energy needed to be captured ballistically by the Moon. The dynamics of these transfers, called weak stability boundary (WSB) transfers, will be studied here in some detail. The crucial Earth—Moon—Sun configurations allowing such transfers will be defined. The Sun's gravitational effect and lunar gravitational capture will be analyzed in terms of variations of the Jacobi constants in the Earth—Sun and Earth—Moon systems. Many examples will be presented, supporting the understanding of the dynamical mechanism of WSB transfers and analytical formulas will be obtained in the case of quasi ballistic captures.This revised version was published online in October 2005 with corrections to the Cover Date.     

Direct and indirect capture of near-Earth asteroids in the Earth–Moon system

Near-Earth asteroids have attracted attention for both scientific and commercial mission applications. Due to the fact that the Earth–Moon \(\hbox {L}_{1}\) and \(\hbox {L}_{2}\) points are candidates for gateway stations for lunar exploration, and an ideal location for space science, capturing asteroids and inserting them into periodic orbits around these points is of significant interest for the future. In this paper, we define a new type of lunar asteroid capture, termed direct capture. In this capture strategy, the candidate asteroid leaves its heliocentric orbit after an initial impulse, with its dynamics modeled using the Sun–Earth–Moon restricted four-body problem until its insertion, with a second impulse, onto the \(\hbox {L}_{2}\) stable manifold in the Earth–Moon circular restricted three-body problem. A Lambert arc in the Sun-asteroid two-body problem is used as an initial guess and a differential corrector used to generate the transfer trajectory from the asteroid’s initial obit to the stable manifold associated with Earth–Moon \(\hbox {L}_{2}\) point. Results show that the direct asteroid capture strategy needs a shorter flight time compared to an indirect asteroid capture, which couples capture in the Sun–Earth circular restricted three-body problem and subsequent transfer to the Earth–Moon circular restricted three-body problem. Finally, the direct and indirect asteroid capture strategies are also applied to consider capture of asteroids at the triangular libration points in the Earth–Moon system.     

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.     

Density within the moon and implications for lunar composition

Density models for the Moon, including the effects of temperature and pressure, can satisfy the mass and moment of inertia of the Moon and the presence of a low density crust indicated by the seismic refraction results only if the lunar mantle is chemically or mineralogically inhomogeneous. IfC/MR ² exceeds 0.400, the inferred density of the upper mantle must be greater than that of the lower mantle at similar conditions by at least 0.1 g cm^–3 for any of the temperature profiles proposed for the lunar interior. The average mantle density lies between 3.4 and 3.5 g cm^–3, though the density of the upper mantle may be greater. The suggested density inversion is gravitationally unstable, but the implied deviatoric stresses in the mantle need be no larger than those associated with lunar gravity anomalies. UsingC/MR ³=0.400 and the recent seismic evidence suggesting a thin, high density zone beneath the crust and a partially molten core, successful density models can be found for a range of temperature profiles. Temperature distributions as cool as several inferred from the lunar electrical conductivity profile would be excluded. The density and probable seismic velocity for the bulk of the mantle are consistent with a pyroxenite composition and a 100 MgO/(MgO+FeO) molecular ratio of less than 80.Communication presented at the Lunar Science Institute Conference on Geophysical and Geochemical Exploration of the Moon and Planets, January 10–12, 1973.     

Some effects of elasticity on lunar rotation

A general Hamiltonian for a rotating Moon in the field of the Earth is expanded in terms of parameters orienting the spin angular momentum relative to the pricipal axes of the Moon and relative to coordinate axes fixed in the orbital plane. The effects of elastic distortion are included as modifications of the moment of inertia tensor, where the magnitude of the distortion is parameterized by the Love numberk ₂. The principal periodic terms in the longitude of a point on the Moon due to variations of the tide caused by the Earth are shown to have amplitudes between 3.9 × 10^–3 and 1.6 × 10^–2 with a period of an anomalistic month, 3.0 × 10^–4 and 1.2 × 10^–3 with a period of one-half an anomalistic month and 2.4 × 10^–4 and 9.6 × 10^–4 with a period of one-half of a nodical month. The extremes in the amplitudes correspond to rigidities of 8 × 10¹¹ cgs and 2 × 10¹¹ cgs, respectively, the former rigidity being comparable to that of the Earth. Only the largest amplitude given above is comparable to that detectable by the projected precision of the laser ranging to the lunar retrorereflectors, and this amplitude corresponds to an improbably low rigidity for the Moon. A detailed derivation of the free wobble of the lunar spin axis about the axis of maximum moment of inertia is given, where it is shown that elasticity can alter the period of the free wobble of 75.3 yr by only 3 × 10^–4 to 10^–3 of this period. Also, the effect of elasticity on the period of free libration is completely negligible by many orders of magnitude. If the Moon's rigidity is close to that of the Earth there is no effect of elasticity on the rotation which can be measured with the laser ranging and, therefore, no elastic properties of the Moon can be determined from variations in the rotation.Currently on leave from the Dept. of Physics, University of California, Santa, Barbara, California.Communication presented at the conference on Lunar Dynamics and Observational Coordinate Systems held January 15–17, 1973 at the Lunar Science Institute, Houston, Tex., U.S.A.     

Mean field escapers in non-equilibrium systems

We present some results of our study on non-equilibriumN-body systems that undergo initial collapse. This collapse is followed by global pulsations of the system. During these pulsations, the mean gravitational field fluctuates violently. Some particles pick up enough energy to be ejected from the system into the halo or even fly off to infinity. We discuss the region in phase space from which these mean field escapers originate and their energy frequency distribution for the illustrative case of a uniform spherical initial state. The pulsations lead to the production of shells in the halo.Paper presented at the 11th European Regional Astronomical Meetings of the IAU on New Windows to the Universe, held 3–8 July, 1989, Tenerife, Canary Islands, Spain.     

Constructing ballistic capture orbits in the real Solar System model

A method to design ballistic capture orbits in the real Solar System model is presented, so extending previous works using the planar restricted three-body problem. In this generalization a number of issues arise, which are treated in the present work. These involve reformulating the notion of stability in three-dimensions, managing a multi-dimensional space of initial conditions, and implementing a restricted \(n\) -body model with accurate planetary ephemerides. Initial conditions are categorized into four subsets according to the orbits they generate in forward and backward time. These are labelled weakly stable, unstable, crash, and acrobatic, and their manipulation allows us to derive orbits with prescribed behavior. A post-capture stability index is formulated to extract the ideal orbits, which are those of practical interest. Study cases analyze ballistic capture about Mercury, Europa, and the Earth. These simulations show the effectiveness of the developed method in finding solutions matching mission requirements.     

Earth–Mars transfers with ballistic escape and low-thrust capture

In this paper novel Earth–Mars transfers are presented. These transfers exploit the natural dynamics of n-body models as well as the high specific impulse typical of low-thrust systems. The Moon-perturbed version of the Sun–Earth problem is introduced to design ballistic escape orbits performing lunar gravity assists. The ballistic capture is designed in the Sun–Mars system where special attainable sets are defined and used to handle the low-thrust control. The complete trajectory is optimized in the full n-body problem which takes into account planets’ orbital inclinations and eccentricities. Accurate, efficient solutions with reasonable flight times are presented and compared with known results.     

N-body simulations of non-gaussian distributions: Topological analysis and evolution

WithN-body simulations, we study the evolution of the large-scale structure of the Universe, assuming non-Gaussian initial conditions. We also discuss the sensitivity of different statistical tests to the change of the initial conditions.Paper presented at the 11th European Regional Astronomical Meetings of the IAU on New Windows to the Universe, held 3–8 July, 1989, Tenerife, Canary Islands, Spain.     

A note on the determination of ozone concentrations from radio-meteor duration distributions.

Hajduk, et al (1992) have drawn attention to a possible source of error in our earlier analysis of visual and radar observations of shower meteors from which we calculated the ozone concentration at meteor heights. We have considered their main criticism that the features of our data which we used to calculate the ozone concentration may have been due to the constraint that our meteors had to be observed by both methods. We have shown that observations collected from radar-only systems show very similar characteristics and that these can be analysed without recourse to visual data to produce very similar results to those from our original analysis. Presented at theInternational Conference on Meteoroids 23 – 31 August 1994 in Bratislava.Submitted to Earth, Moon and Planets     

A combination of SPH andN-body 2 for gas dynamics in star clusters

When stars are being formed in a region of a molecular cloud, the presence of the gas of this cloud will influence the dynamics of these newly formed stars. We discuss an implementation of SPH or Smooth Particle Hydrodynamics within the framework ofN-body2, which will permit the study of this kind of mixed systems. It is our intention to include various gas removal processes, and to study their influence on the behaviour of stars and gas.Paper presented at the 11th European Regional Astronomical Meetings of the IAU on New Windows to the Universe, held 3–8 July, 1989, Tenerife, Canary Islands, Spain.     

The comparative spectra of cosmic-ray protons and helium nuclei

We have re-examined and extended the measurements of the primary cosmic ray proton and helium nuclei intensities in the range from a few MeV nuc^–1 to 100 GeV nuc^–1 using a considerable body of recently published data. The differential spectra obtained from this data are determined as a function of both energy and rigidity. The exponents of the energy spectra of both protons and helium nuclei are found to be different at the same energy/nucleon and to increase with increasing energy between 1 and 100 GeV nuc^–1 reaching a value=–2.70 at higher energies and in addition, theP/He ratio changes from a value 5 at 1 GeV nuc^–1 and below to a value 30 at 100 GeV nuc^–1. On a rigidity representation the spectral exponent for each species is nearly identical and remains virtually constant above several GV at a value of –2.70, and in addition, theP/He ratio is also a constant 7 above 3 GeV. The changingP/He ratio and spectral exponent on an energy representation occur at energies well above those at which interplanetary modulation effects or interstellar ionization energy loss effects can significantly affect the spectra. In effect by comparing energy spectra and rigidity spectra in the intermediate energy range above the point where solar modulation effects and interstellar energy loss effects are important, but in the range where there are significant differences between energy and rigidity spectra, we deduce that the cosmic ray source spectra are effectively rigidity spectra. This fact has important implications regarding the mechanism of acceleration of this radiation and also with regard to the form of the assumed galactic spectrum at low energies. The relationship between the proton and helium spectra derived here and the heavier nuclei spectral differences recently reported in the literature is also examined.If rigidity spectra are adopted for protons and helium nuclei, then the source abundance ratio of these two components is determined to be 7:1. Some cosmological implications of this ratio are discussed.     

The Earth-figure perturbations in the Lunar theory

We compute the perturbations on the motion of the Moon due to the shape of the Earth. The zonal terms inJ ₂,J ₃, andJ ₄ are considered. The accuracy is estimated at 3×10^–5 and the results compared with previous theories.     

Hypothesis of variable g and the secular accelerations of the Sun and the Moon

It is known that the observed secular accelerations of the Sun and Moon are not consistent with the tidal interactions of the Earth with the Sun and Moon. Following Dicke, the hypothesis of variable constant of gravity is adopted and expressions for the accelerations are derived. It is shown that if the theoretical ratio of the acceleration is equated the observed one, a unique value for —/G can be calculated. Adopting the accelerations obtained by Fotheringham, Newton, Muller and Stephenson, and Stephenson, it is found that — /G ranges from 1.4 × 10^–11 to 3.3 × 10^–11 yr^–1. This estimate is consistent with the one based upon the comparison of the lunar accelerations measured with respect to atomic and ephemis times.     

Irregular extrasolar systems and dynamical instability

The astrometric search for extrasolar planetary systems can regard, at least in the first stage, only systems with high mass ratio (10^–2) of the largest planets to the central star. The formation of such systems is probably a common occurrence in the galaxy, as indicated by various theoretical arguments; these systems could have evolved in a regular way (near-circular and coplanar orbits, Titius-Bode law, etc.) until the most massive accumulating protoplanet exceeded the critical value of the mass ratio corresponding to dynamical instability. This phenomenon has been extensively investigated in recent years, showing that a three- orN-body system becomes unstable (for relative separations similar to the planetary ones) for >10^–2. Therefore, it seems likely that observations of systems within this range of mass ratio will show the irregular end-products of processes related to the instability (close encounters or ejections), which affected drastically the orbital configuration in the last phase of the planet formation.Paper presented at the European Workshop on Planetary Sciences, organised by the Laboratorio di Astrofisica Spaziale di Frascati, and held between April 23–27, 1979, at the Accademia Nazionale del Lincei in Rome, Italy.     

On the spectra of type-III solar radio bursts observed at low frequencies

The spectra of strong bursts observed at low frequencies by OGO-5 during 1968–1970 are presented. They usually exhibit an intense main peak between 100 kHz and 1 MHz, and sometimes a less intense secondary peak between 1 and 3.5 MHz. Main peaks of 10^–12 Wm^–2 Hz^–1 or more were obtained in very strong events, but because of antenna calibration problems those could be one or two orders of magnitude too high. Recently published work supports the finding that type III bursts at low frequencies can be at least four orders of magnitude more intense than at ground-based frequencies of observation. It is found that the energy received at the Earth increases with decreasing frequency approximately as f ^–n, where 3 n 4.     

Energy Spectra of low-Flux Protons in the Inner Heliosphere under Quiet Solar Conditions

The energy spectra of protons at energies in the range of about 1–100 MeV are investigated during time periods of low solar activity using data sets from near Earth spacecraft. These populations pose a tough experimental and theoretical problem that remains unsolved up to now. We attempt to provide a consistent definition of low-flux quiet-time periods relevant to low solar activity as well as quasi-stationary periods useful at higher levels of solar activity. Using statistical methods, the possible instrumental contribution to the lowest observed proton fluxes for various detectors is estimated. We suggest and prove that there exists a low-flux population of charged particles in the energy range of about 1–10 MeV, which is present in the inner heliosphere even during the quietest conditions at lowest solar activity. The dynamics of the variations of proton spectra over the solar cycle is investigated. A series of low-flux periods is examined in detail and energy spectra of protons are approximated in the form of J(E)=AE ^–+CE. By determining the best fitting parameters to the energy spectra correlations are made among them as well as with monthly sunspot numbers characterizing solar activity. It has been demonstrated that the value of the energy minimum of proton spectrum E _min that `divides' the two populations – `solar/heliospheric' and `galactic' – is shifted towards higher values with increasing solar activity. Protons have been argued to be predominantly of solar origin up to several MeV near the solar cycle minimum and up to 20–30 MeV at maximum. The slope of the lower spectrum branch (parameter ) slightly decreases with increasing solar activity. The minimum fluxes observed during the last 3 minima of solar activity are compared; the lowest fluxes were those during the 1985–1987 period.     

A hypothesis to explain the virial discrepancy in clusters of galaxies

On the basis of numerical experiments onn-body binding energies we tentatively consider the following hypothesis: If the distance between two galaxies forming a binary system isa _g, and a cluster of galaxies that is substructured in a hierarchical fashion onall scales froma _g upwards has a total massM, then the total gravitational binding energy of the cluster is _TH = –GM ²/2a _g . As an explanation for missing masses up to order 100 we test this hypothesis in three different ways, finding remarkable agreement with observation, with no need for physical missing mass.     

Sunspot umbral oscillations in Mgii k

Time series observations of the profile of the Mgii k line 2795.52 have been obtained in five sunspots with the Ultraviolet Spectrometer and Polarimeter (UVSP) on the Solar Maximum Mission. The three sunspots with umbrae larger than the 3 × 3 pixel size show significant oscillations in integrated line intensity and line centroid, with frequencies in the range 5.29–7.55 mHz (periods of 132–190 s).The frequencies of significant peaks in average umbral power spectra agree well with the frequencies of the three lowest-frequency transmission peaks predicted by a model of resonant transmission of acoustic waves. If radiative delays are unimportant, and the line centroid can be interpreted straightforwardly as a Doppler shift, the measured velocity-intensity phase differences indicate the superposition of upward- and downward-propagating waves in the umbral chromosphere; this is further evidence for the resonant transmission model.A single, quiet Sun time series of k core profiles yields power spectra and a phase difference consistent with the existence of a chromospheric p-mode.The SMM data used in this work were available only because of the repair of the SMM spacecraft by the crew of Challenger on mission 41-C. The pilot for that mission and the commander of Challenger's last mission was Francis R. Scobee.This work is dedicated to his memory.