Comparison and analysis on lunar rotation with lunar gravity field models

Understanding the structure of and dynamic processes in the deep interior of planets is crucial for understanding their origin and evolution. An effective way to constrain them is through observation of rotation and subsequent simulation. In this paper, a numerical model of the Moon’s rotation and orbital motion is developed based on previous studies and implemented independently. The Moon is modeled as an anelastic body with a liquid core. The equations of the rotation were nonlinear and the Euler angles are cross coupled. We solve them numerically via the Runge-Kutta-Fehlberg (RKF) and multi-steps Adams-Bashforth-Moulton (ABM) predictor-corrector numerical integration. We have found that adequate accuracy is maintained by taking twelve steps per day using eleventh differences in the integrating polynomial. The lunar orbital and rotational equations are strongly coupled, so we integrated the rotation and motion simultaneously. We refer to other planetary informations from the newest planetary and lunar ephemeris INPOP17a, which is reported had fitted the longest LLR (Lunar Laser Ranging) observation data. Using the model GL660B from GRAIL (Gravity Recovery and Interior Laboratory) mission, we firstly compare our numerical results with the INPOP17a to prove the reasonability of our model. After that we apply the lunar gravity model CEGM02 determined from Chang’E-1 mission and SGM100h from SELENE mission to our model, the difference between results from CEGM02 and GL660B are less than \(-0.20 \sim0.15\) arc-second, and \(-0.25 \sim0.20\) arc-second for GL660B and SGM100h. Compared to SGM100h, the results show that the low degree and order coefficients (less than 6 from this paper) of lunar gravity field were improved in CEGM02 as expected. It is the first time to demonstrate that these models can be applied to lunar rotation model. These results manifest that a development of the gravity field measure will help us to know the rotation motion more precisely.     

Future lunar gravity measurements

Future missions to the Moon should include a detailed high-resolution global gravity survey from a low (15–30 km) polar orbiting spacecraft. The use of gravity gradiometer instruments on board the spacecraft will give higher-resolution data at lower total mission cost that the present Doppler tracking technique. Simulations show that although a three axis gradiometer system is preferred, and can even be used to estimate spacecraft attitude and altitude variation, a properly oriented single rotating gravity gradiometer can be used to resolve closely spaced mascons in both the along-track and cross-track directions.Paper presented at theFuture Lunar Exploration session of the Tenth Lunar and Planetary Science. Conference, Johnson Space Center, Houston, Texas, 19–23 March 1979.     

Strategies for plane change of Earth orbits using lunar gravity and derived trajectories of family G

The dynamics of the circular restricted three-body Earth-Moon-particle problem predicts the existence of the retrograde periodic orbits around the Lagrangian equilibrium point L1. Such orbits belong to the so-called family G (Broucke, Periodic orbits in the restricted three-body problem with Earth-Moon masses, JPL Technical Report 32–1168, 1968) and starting from them it is possible to define a set of trajectories that form round trip links between the Earth and the Moon. These links occur even with more complex dynamical systems as the complete Sun-Earth-Moon-particle problem. One of the most remarkable properties of these trajectories, observed for the four-body problem, is a meaningful inclination gain when they penetrate into the lunar sphere of influence and accomplish a swing-by with the Moon. This way, when one of these trajectories returns to the proximities of the Earth, it will be in a different orbital plane from its initial Earth orbit. In this work, we present studies that show the possibility of using this property mainly to accomplish transfer maneuvers between two Earth orbits with different altitudes and inclinations, with low cost, taking into account the dynamics of the four-body problem and of the swing-by as well. The results show that it is possible to design a set of nominal transfer trajectories that require ΔV _Total less than conventional methods like Hohmann, bi-elliptic and bi-parabolic transfer with plane change.     

A Proposed lunar orbiting gravity gradiometer experiment

Analysis of the gravity gradiometer developed by R. L. Forward and C. C. Bell at the Hughes Research Laboratories suggest than an accuracy, in the range 0.1 to 0.5 EU can be expected in a lunar orbiter application. This accuracy will allow gradient anomalies associated with mascons to be mapped with 1% accuracy and should reveal a great deal of new information about the lunar gravity field.The proposed experiment calls for putting such a gradiometer into a closely circular polar orbit at an average height of about 30 km above the lunar surface. This orbit allows the entire lunar surface to be covered in fourteen days, the gradiometer to be checked twice per revolution and results in successive passes above the lunar surface being spaced at about the resolution limit of about 30 km set both by the satellite altitude and instrumental integration time. Doppler tracking will be employed and the spacecraft will carry an electromagnetic altimeter. Gradient and altitude data from the far side of the Moon can be stored for replay when communication is re-established.     

Observational effect for the lunar gravity on meteor streams

Meteoroids always posed a great hazard to spacecraft security. A meteoroids stream assembled by a large massive body will further enhance the hazard severalfold. For example, the radiant of Northern Taurids (NTA) will be occulted by the Moon on Nov. 12, 2011. Since the gravitational lensing effect of massive bodies can gather together the orbits of meteoroids, the observable flux of meteoroids will increase. In this paper a set of numerical methods was built to discuss the observational effect of this kind of phenomena. The ZHR of NTA is generally small. But it can be local strong on the Earth by the lunar gravitational assembling. The calculated result suggests that a ten times stronger than normal NTA will appear in the sea area of Tristan da Cunha islands during 00h45m UT to 02h00m UT on Nov. 12, 2011.     

The lunar gravity mission MAGIA: preliminary design and performances

The importance of an accurate model of the Moon gravity field has been assessed for future navigation missions orbiting and/or landing on the Moon, in order to use our natural satellite as an intermediate base for next solar system observations and exploration as well as for lunar resources mapping and exploitation. One of the main scientific goals of MAGIA mission, whose Phase A study has been recently funded by the Italian Space Agency (ASI), is the mapping of lunar gravitational anomalies, and in particular those on the hidden side of the Moon, with an accuracy of 1 mGal RMS at lunar surface in the global solution of the gravitational field up to degree and order 80. MAGIA gravimetric experiment is performed into two phases: the first one, along which the main satellite shall perform remote sensing of the Moon surface, foresees the use of Precise Orbit Determination (POD) data available from ground tracking of the main satellite for the determination of the long wavelength components of gravitational field. Improvement in the accuracy of POD results are expected by the use of ISA, the Italian accelerometer on board the main satellite. Additional gravitational data from recent missions, like Kaguya/Selene, could be used in order to enhance the accuracy of such results. In the second phase the medium/short wavelength components of gravitational field shall be obtained through a low-to-low (GRACE-like) Satellite-to-Satellite Tracking (SST) experiment. POD data shall be acquired during the whole mission duration, while the SST data shall be available after the remote sensing phase, when the sub-satellite shall be released from the main one and both satellites shall be left in a free-fall dynamics in the gravity field of the Moon. SST range-rate data between the two satellites shall be measured through an inter-satellite link with accuracy compliant with current state of art space qualified technology. SST processing and gravitational anomalies retrieval shall benefit from a second ISA accelerometer on the sub-satellite in order to decouple lunar gravitational signal from other accelerations. Experiment performance analysis shows that the stated scientific requirements can be achieved with a low mass and low cost sub-satellite, with a SST gravimetric mission of just few months.     

Orbital mapping of the lunar magnetic field

Magnetometer data obtained during the first four lunations after the deployment of the Apollo 15 subsatellite have been used to construct contour maps of the lunar magnetic field referred to 100 km altitude. These contour maps cover a relatively small band on the lunar surface. Within the region covered there is a marked near side-far side asymmetry. The near-side field is generally weaker and less structured than the far-side field. The strongest intrinsic lunar magnetic field detected is between the craters Van de Graaff and Aitken, centered at 20°S and 172°E. The variation in field strength with altitude for this feature suggests that its scale size is on the order of 80 km. A magnetization contrast between this region and its surroundings of the order of 6 × 10^–5 emu-cm^–3 is obtained assuming a 10-km thick slab. Preliminary Apollo 16 magnetometer data at extremely low altitude (0 to 10 km) show a very structured magnetic field with field strengths up to 56. Large compressions in the magnetic field magnitude, just above the lunar limb regions, are occasionally detected when the Moon is in the solar wind. The occurrence of limb compressions is strongly dependent on the selenographic coordinates of the lunar region on the solar wind terminator beneath the orbit of the sub-satellite. The discovery of remanent magnetization of varying strength over much of the lunar surface and its correlation with limb compression source regions supports the hypothesis that limb compressions are due to the deflection of the solar wind by regions of strong magnetization at the lunar limbs. If this hypothesis is correct, then the map of lunar regions associated with compressions indicates that the northerly equatorial region on the far side is less strongly magnetized than the southerly equatorial region on the far side.Paper dedicated to Professor Harold C. Urey on the occasion of his 80th birthday on 29 April, 1973.     

Study of lunar gravity assist orbits in the restricted four-body problem

In this paper, the lunar gravity assist (LGA) orbits starting from the Earth are investigated in the Sun–Earth–Moon–spacecraft restricted four-body problem (RFBP). First of all, the sphere of influence of the Earth–Moon system (SOIEM) is derived. Numerical calculation displays that inside the SOIEM, the effect of the Sun on the LGA orbits is quite small, but outside the SOIEM, the Sun perturbation can remarkably influence the trend of the LGA orbit. To analyze the effect of the Sun, the RFBP outside the SOIEM is approximately replaced by a planar circular restricted three-body problem, where, in the latter case, the Sun and the Earth–Moon barycenter act as primaries. The stable manifolds associated with the libration point orbit and their Poincaré sections on the SOIEM are applied to investigating the LGA orbit. According to our research, the patched LGA orbits on the Poincaré sections can efficiently distinguish the transit LGA orbits from the non-transit LGA orbits under the RFBP. The former orbits can pass through the region around libration point away from the SOIEM, but the latter orbits will bounce back to the SOIEM. Besides, the stable transit probability is defined and analyzed. According to the variant requirement of the space mission, the results obtained can help us select the LGA orbit and the launch window.     

Photometric properties of lunar terrains derived from Hapke's equation

The photometric parameters of B. W. Hapke's (1986, Icarus 67, 264–280) equation are fit to the lunar disk-integrated visual lightcurve and to disk-resolved data of R. W. Shorthill, J. M. Saari, F. E. Baird, and J. R. LeCompte (1969, Photometric Properties of Selected Lunar Features, NASA Contractor Report CR-1429) for dark, average, and bright lunar terrains. The lunar nearside geometric albedo and phase integral computed from the disk-integrated results are consistent with those of earlier investigators. The single scattering albedos of disk-resolved average and bright lunar terrains are systematically larger than that of lunar mare. Average particles in dark terrain regoliths are more backscattering than those in average and bright lunar terrains. The angular width of the opposition surge is greatest in dark terrains and is found to be best explained by modest differences in regolith particle-size distributions which might accompany the normal regolith maturation process. The total amplitude of the opposition surge for dark terrains is larger than for average and bright terrains. This result appears to be a consequence of the fact that in opaque particles, a larger fraction of singly scattered light at zero phase comes from first-surface reflection. The average subcentimeter macroscopic roughness of dark terrains is significantly lower than that of average and bright terrains. The relative magnitude of this difference is consistent with that obtained from radar measurements at decimeter scales.     

A determination of the parameters of the level surface of lunar gravity

The spectral coefficients of the selenoid have been obtained by inverting the potential series for lunar gravity. The reference value of the lunar level surface has been determined on the base of the mean radius of lunar topography. This enables to evaluate the parameters of the tri-axial reference ellipsoid best fitted to the lunar level surface.     

A lunar core and the moon's magnetic field

The suggestion that the Moon's magnetic field is due to adiabatic magnetohydrodynamic convection of a molten core has been made by a number of recent authors. Considerations based on petrology, mass and rotational inertia limit the size of this hypothetical core to a few 100 km at the most. A proposal has been made that this core is either molten iron or iron sulfide. Fortunately, we know the properties of both molten iron and iron-sulfide at lunar core pressures. We can find no way of maintaining circulation in a hypothetical lunar core, as circulation is contingent upon a temperature gradient being greater than the adiabatic gradient, or an internal heat source.     

A method to determine regional lunar gravity fields from earth-based satellite tracking data

A method to determine regional gravity fields of the Moon from Earth-based Doppler and range satellite tracking data residuals of a low Moon-orbiting satellite has been developed and thoroughly tested in a controlled simulation environment. A short-arc approach, where one arc consists of the time it takes the satellite to cross the grid of interest on the lunar surface, is used in order to filter out most long-wavelength signal that can still be present in the residuals. Simulation results where the data are contaminated with either typical systematic or stochastic noise show that recovery of the local gravity field down to the level of several mGal is possible. The inclusion of extremely low-altitude data also means that regularisation in the sense of including a priori information in the form of a regularisation matrix is not necessary in order to obtain a good solution at high resolution.     

Effect of 3rd-degree gravity harmonics and Earth perturbations on lunar artificial satellite orbits

In a previous work we studied the effects of (I) the J ₂ and C ₂₂ terms of the lunar potential and (II) the rotation of the primary on the critical inclination orbits of artificial satellites. Here, we show that, when 3rd-degree gravity harmonics are taken into account, the long-term orbital behavior and stability are strongly affected, especially for a non-rotating central body, where chaotic or collision orbits dominate the phase space. In the rotating case these phenomena are strongly weakened and the motion is mostly regular. When the averaged effect of the Earth’s perturbation is added, chaotic regions appear again for some inclination ranges. These are more important for higher values of semi-major axes. We compute the main families of periodic orbits, which are shown to emanate from the ‘frozen eccentricity’ and ‘critical inclination’ solutions of the axisymmetric problem (‘J ₂ + J ₃’). Although the geometrical properties of the orbits are not preserved, we find that the variations in e, I and g can be quite small, so that they can be of practical importance to mission planning.     

Physical librations due to the third and fourth degree harmonics of the lunar gravity potential

The existence of third and fourth harmonics of the lunar gravity potential gives rise to sizable lunar physical librations. Using one recent set of potential estimates, the following effects are noted: the mean sub-Earth point is displaced from the earthward principal moment of inertia axis by 168″; the inclination of the lunar equator to the ecliptic is decreased by 14″.5; and a six year period libration in longitude, with amplitude 13″.1, is induced.     

A determination of the intensity of the ancient lunar magnetic field

Thermal demagnetization of lunar breccia 15498,36 shows that the natural remanent magnetization is a simple thermoremanence carried by metallic iron. Using the classical Thellier-Thellier method the strength of the magnetizing field at the time of sample formation was found to be 2100 ±80 gammas.Paper dedicated to Professor Harold C. Urey on the occasion of his 80th birthday on 29 April 1973.     

A lunar crustal gravity model across the Triesnecker-Hyginus area,Mare Tranquillitatis,and Mare Fecunditatis

LOS Bouguer gravity anomalies have been calculated from a low altitude LOS free air Doppler gravity profile across northern Mare Fecunditatis, southern Mare Tranquillitatis and the Aridaeus Rille. The Hyginus-Triesnecker area has been included in model calculations, though here only free air anomalies are present. A crustal density model has been fitted to the Bouguer anomalies and to the free air anomalies in the case of the Hyginus-Triesnecker area.On a regional scale northern Fecunditatis has Bouguer anomalies up to 80 mgal and lithostatic stresses of 29 bar and thus is nearly in isostatic equilibrium. Tranquillitatis can be divided into three regions of different crustal structure: (1) northern Tranquillitatis with only minor free air gravity anomalies is more or less in isostatic balance, (2) the southeastern region with Bouguer anomalies to –100 mgal and lithostatic stresses of –73 bar has a considerable mass deficit, (3) the southwestern basin is dominated by the local structure Lamont with a Bouguer maximum of 200 mgal and extremely high lithostatic stresses of 285 bar.The Bouguer minimum of –180 mgal of the Aridaeus area has been modelled by two alternative models: (i) a crustal thickening of 33 km and associated lithostatic stresses of –164 bar, and (ii) a crustal thickening of 20 km plus a low density intrusion. The free air maximum of the Hyginus-Triesnecker area has been fitted by a mantle plug connected with stresses of 116 bar.As the old irregular maria could not sustain large mascon stresses, it has been concluded that the local high stresses of Lamont, Aridaeus, and Hyginus-Triesnecker have been evolved after the impacts of the circular maria. Intrusional activities in these areas could have proceeded to fault zones generated by the large impacts.Contribution No. 211, Institut für Geophysik der Universität Kiel, F.R.G.     

Lunar gravity derived from long-period satellite motion — A proposed method

A new method has been devised to determine the spherical harmonic coefficients of the lunar gravity field. This method consists of a two-step data reduction and estimation process. In the first step, a weighted least-squares empirical orbit determination scheme is applied to Doppler tracking data from lunar orbits to estimate longpperiod Kepler elements and rates. Each of the Kepler elements is represented by an independent function of time. The long-period perturbing effects of the Earth, Sun, and solar radiation are explicitly modeled in this scheme. Kepler element variations estimated by this empirical processor are then ascribed to the non-central lunar gravitation features. Doppler data are reduced in this manner for as many orbits as are available. In the second step, the Kepler element rates are used as input to a second least-squares processor that estimates lunar gravity coefficients using the long-period Lagrange perturbation equations.Pseudo Doppler data have been generated simulating two different lunar orbits. This analysis included the perturbing effects of the L1 lunar gravity field, the Earth, the Sun, and solar radiation pressure. Orbit determinations were performed on these data and long-period orbital elements obtained. The Kepler element rates from these solutions were used to recover L1 lunar gravity coefficients. Overall results of this controlled experiment show that lunar gravity coefficients can be accurately determined and that the method is dynamically consistent with long-period perturbation theory.     

Venus gravity field: Pioneer Venus orbiter navigation results

The gravity field of Venus has been modeled by a spherical harmonic expansion of the potential to degree and order seven. The estimates of these coefficients were obtained by combining information from 43 short arcs (4 hr) of line-of-sight Doppler data centered at periapsis. The data arcs were distributed in longitude and time over more than two circulations of Venus by the Pioneer Venus Orbiter subperiapsis point which was confined to the band of latitudes from 14°N to 17°N. Convergence of the solution has been assured by iterating upon the initial estimate. All estimates were performed with zero a priori information on the gravity coefficients. Since the altitude of periapsis for most of the orbits was within the sensible Venusian atmosphere, drag effects on the estimated harmonics have been removed using an exponential atmosphere density model. Estimates of the mass parameter (GM) of Venus using this dataset are also evaluated.     

Scalar field cosmology in $$ gravity via Noether symmetry

This paper investigates the existence of Noether symmetries of isotropic universe model in \(f(R,T)\) gravity admitting minimal coupling of matter and scalar fields. The scalar field incorporates two dark energy models such as quintessence and phantom models. We determine symmetry generators and corresponding conserved quantities for two particular \(f(R,T)\) models. We also evaluate exact solutions and investigate their physical behavior via different cosmological parameters. For the first model, the graphical behavior of these parameters indicate consistency with recent observations representing accelerated expansion of the universe. For the second model, these parameters identify a transition form accelerated to decelerated expansion of the universe. The potential function is found to be constant for the first model while it becomes \(V(\phi )\approx \phi ^{2}\) for the second model. We conclude that the Noether symmetry generators and corresponding conserved quantities appear in all cases.     

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.