Global search for low-thrust transfers to the Moon in the planar circular restricted three-body problem

This paper globally searches for low-thrust transfers to the Moon in the planar, circular, restricted, three-body problem. Propellant-mass optimal trajectories are computed with an indirect method, which implements the necessary conditions of optimality based on the Pontryagin principle. We present techniques to reduce the dimension of the set over which the required initial costates are searched. We obtain a wide range of Pareto solutions in terms of time of flight and mass consumption. Using the Tisserand–Poincaré graph, a number of solutions are shown to exploit high-altitude lunar flybys to reduce fuel consumption.     

Navigating to the Moon along low-energy transfers

This paper presents a navigation strategy to fly to the Moon along a Weak Stability Boundary transfer trajectory. A particular strategy is devised to ensure capture into an uncontrolled relatively stable orbit at the Moon. Both uncertainty in the orbit determination process and in the control of the thrust vector are included in the navigation analysis. The orbit determination process is based on the definition of an optimal filtering technique that is able to meet accuracy requirements at an acceptable computational cost. Three sequential filtering techniques are analysed: an extended Kalman filter, an unscented Kalman filter and a Kalman filter based on high order expansions. The analysis shows that only the unscented Kalman filter meets the accuracy requirements at an acceptable computational cost. This paper demonstrates lunar weak capture for all trajectories within a capture corridor defined by all the trajectories in the neighbourhood of the nominal one, in state space. A minimum Δv strategy is presented to extend the lifetime of the spacecraft around the Moon. The orbit determination and navigation strategies are applied to the case of the European Student Moon Orbiter.     

Optimal low-thrust transfers between libration point orbits

Over the past three decades, ballistic and impulsive trajectories between libration point orbits (LPOs) in the Sun–Earth–Moon system have been investigated to a large extent. It is known that coupling invariant manifolds of LPOs of two different circular restricted three-body problems (i.e., the Sun–Earth and the Earth–Moon systems) can lead to significant mass savings in specific transfers, such as from a low Earth orbit to the Moon’s vicinity. Previous investigations on this issue mainly considered the use of impulsive maneuvers along the trajectory. Here we investigate the dynamical effects of replacing impulsive ΔV’s with low-thrust trajectory arcs to connect LPOs using invariant manifold dynamics. Our investigation shows that the use of low-thrust propulsion in a particular phase of the transfer and the adoption of a more realistic Sun–Earth–Moon four-body model can provide better and more propellant-efficient solution. For this purpose, methods have been developed to compute the invariant tori and their manifolds in this dynamical model.     

Low Energy Transfer to the Moon

In 1991, the Japanese Hiten mission used a low energy transfer with a ballistic capture at the Moon which required less Vthan a standard Hohmann transfer. In this paper, we apply the dynamical systems techniques developed in our earlier work to reproduce systematically a Hiten-like mission. We approximate the Sun–Earth–Moon-spacecraft 4-body system as two 3-body systems. Using the invariant manifold structures of the Lagrange points of the 3-body systems, we are able to construct low energy transfer trajectories from the Earth which execute ballistic capture at the Moon. The techniques used in the design and construction of this trajectory may be applied in many situations.     

Earth,Moon, And Planets

Instructions for Authors

Earth, Moon, And Planets     

An intrinsic volatility scale relevant to the Earth and Moon and the status of water in the Moon

The notion of a dry Moon has recently been challenged by the discovery of high water contents in lunar apatites and in melt inclusions within olivine crystals from two pyroclastic glasses. The highest and most compelling water contents were found in pyroclastic glasses that are not very common on the lunar surface. To obtain more representative constraints on the volatile content of the lunar interior, we measured the Zn content, a moderately volatile element, of mineral and rock fragments in lunar soils collected during Apollo missions. We here confirm that the Moon is significantly more depleted in Zn than the Earth. Combining Zn with existing K and Rb data on similar rocks allows us to anchor a new volatility scale based on the bond energy of nonsiderophile elements in their condensed phases. Extrapolating the volatility curve to H shows that the bulk of the lunar interior must be dry (≤1 ppm). This contrasts with the water content of the mantle sources of pyroclastic glasses, inferred to contain up to approximately 40 ppm water based on H₂O/Ce ratios. These observations are best reconciled if the pyroclastic glasses derive from localized water‐rich heterogeneities in a dominantly dry lunar interior. We argue that, although late addition of 0.015% of a chondritic veneer to the Moon seems required to explain the abundance of platinum group elements (Day et al. 2007), the volatile content of the added material was clearly heterogeneous.     

Basin-ring spacing on the Moon,Mercury, and Mars

Radial spacing between concentric rings of impact basins that lack central peaks is statistically similar and nonrandom on the Moon, Mercury, and Mars, both inside and outside the main ring. One spacing interval, (2.0 ± 0.3)^0.5D, or an integer multiple of it, dominates most basin rings. Three analytical approaches yield similar results from 296 remapped or newly mapped rings of 67 multi-ringed basins: least-squares of rank-grouped rings, least-squares of rank and ring diameter for each basin, and averaged ratios of adjacent rings. Analysis of 106 rings of 53 two-ring basins by the first and third methods yields an integer multiple (2 ×) of 2.0^0.5D. There are two exceptions: (1) Rings adjacent to the main ring of multi-ring basins are consistently spaced at a slightly, but significantly, larger interval, (2.1 ± 0.3)^0.5D; (2) The 88 rings of 44 protobasins (large peak-plus-inner-ring craters) are spaced at an entirely different interval (3.3 ± 0.6)^0.5D.The statistically constant and target-invariant spacing of so many rings suggests that this characteristic may constrain formational models of impact basins on the terrestrial planets. The key elements of such a constraint include: (1) ring positions may not have been located by the same process(es) that formed ring topography; (2) ring location and emplacement of ring topography need not be coeval; (3) ring location, but not necessarily the mode of ring emplacement, reflects one process that operated at the time of impact; and (4) the process yields similarly-disposed topographic features that are spatially discrete at 2^0.5D intervals, or some multiple, rather than continuous. These four elements suggest that some type of wave mechanism dominates the location, but not necessarily the formation, of basin rings. The waves may be standing, rather than travelling. The ring topography itself may be emplaced at impact by this and/or other mechanisms and may reflect additional, including post-impact, influences.     

On Choosing a Rational Flight Trajectory to the Moon

The algorithm for choosing a trajectory of spacecraft flight to the Moon is discussed. The characteristic velocity values needed for correcting the flight trajectory and a braking maneuver are estimated using the Monte Carlo method. The profile of insertion and flight to a near-circular polar orbit with an altitude of ~100 km of an artificial lunar satellite (ALS) is given. The case of two corrections applied during the flight and braking phases is considered. The flight to an ALS orbit is modeled in the geocentric geoequatorial nonrotating coordinate system with the influence of perturbations from the Earth, the Sun, and the Moon factored in. The characteristic correction costs corresponding to corrections performed at different time points are examined. Insertion phase errors, the errors of performing the needed corrections, and the errors of determining the flight trajectory parameters are taken into account.     

Optimal low-thrust trajectories to asteroids through an algorithm based on differential dynamic programming

In this paper an optimisation algorithm based on Differential dynamic programming is applied to the design of rendezvous and fly-by trajectories to near Earth objects. Differential dynamic programming is a successive approximation technique that computes a feedback control law in correspondence of a fixed number of decision times. In this way the high dimensional problem characteristic of low-thrust optimisation is reduced into a series of small dimensional problems. The proposed method exploits the stage-wise approach to incorporate an adaptive refinement of the discretisation mesh within the optimisation process. A particular interpolation technique was used to preserve the feedback nature of the control law, thus improving robustness against some approximation errors introduced during the adaptation process. The algorithm implements global variations of the control law, which ensure a further increase in robustness. The results presented show how the proposed approach is capable of fully exploiting the multi-body dynamics of the problem; in fact, in one of the study cases, a fly-by of the Earth is scheduled, which was not included in the first guess solution.     

Stresses in the Moon

It is argued that a reliable theory of the stress history of the Moon should take into account several factors; and that direct observations of the Moon's surface can throw much light on this subject.     

Perturbations due to the shape of the Moon in lunar theory

We compute the perturbations on the motion of the Moon due to its shape. The accuracy is estimated at 1.10^–5 in longitude and latitude and 5 parts in 10¹¹ in distance.     

The origin of the Moon and the single-impact hypothesis,II

This is the second paper devoted to the numerical study of planetary collisions as a possible scenario for forming the Moon. We present a series of nine simulations of a collision between the protoearth and an impactor of various sizes. The mass ratio between the protoearth and the impactor ranged from 0.1 to 0.25. We were able to model both planets with iron cores, having modified our smoothed particle hydrodynamics code to allow the inclusion of up to 10 different material types. Two different relative velocities at infinity for the impactor were considered: ν_∞ = 0 km/sec and ν_∞ = 10 km/sec. We show that for a low-velocity collision and an impactor in the mass range 6.5 × 10²⁶ ≤ M_impactor ≤ 8.2 × 10²⁶ g, more than a lunar mass of iron-poor material is thrown into orbit. For an impactor with a mass within this range, the ejected mass that goes into orbit is for the most part divided comparably into material orbiting inside the Roche limit and into material orbiting outside the Roche limit. This material is either spread out in the form of a disk, or, for a relatively narrow range of masses toward the lower end of the range, clumped into an object of about lunar mass beyond the Roche limit. For impactors more massive than about 8.2 × 10²⁶ g we found that there is too little mass thrown into orbit. For very small mass impactors well over a lunar mass is placed in orbit, but a large amount of it is iron. In the high-velocity range we did not find a possible mass range for the impactor that would lead to the formation of an iron-poor disk massive enough to form the Moon.     

From interstellar gas to the Earth‐Moon system

Abstract— This paper reports the current status of my smoothed particle hydrodynamic (SPH) simulations of the formation of the Moon. Since the Moon has recently been found to have been formed approximately 50 Ma after the solar nebula itself was formed, I have placed the lunar formation problem in the entire context of the formation and early evolution of the solar nebula. This set of processes remains controversial, and I have outlined what I believe to be the essential physical processes involved. These start with the formation of short‐lived (now extinct) radioactive nuclides in a massive supernova. Then follows the probable role of the supernova ejecta in triggering the collapse of a core in a molecular cloud to form the solar nebula, and the injection of the radioactivities into the collapsing cloud core. Most of the solar nebula dissipates to form the Sun, and what remains becomes relatively quiescent. Gas drag acting on interstellar grains and the dustballs formed from them, due both to vertical descent to midplane and inward spiralling in midplane, quickly causes growth of the solid materials to form planetesimals. When these bodies reach the kilometer size range and beyond, gravitational forces dominate the accumulation process. The accumulation of the Earth requires of the order of 10⁸ years. About half‐way through that process the giant impact occurs with the next largest accumulating body near the protoearth. I have been simulating the giant impact using SPH with 100 000 particles. The simulations of three of these runs are depicted in detail with a series of color images. It is shown that conventional accumulation simulations that assume Keplerian orbits and that merge bodies upon collision are misleading because they cannot take account of tidal stripping nor of loss and gain of particles during the accumulation. In addition, the large rotational flattening of the protoearth renders the orbital motions nonkeplerian. The simulations that are shown in detail have been followed for just over a week of real time, and in that time the largest accumulating clump has reached about half or more of the mass of the Moon and additional clumps have accumulated into bodies in the range of 1 to 20% of a lunar mass. It is important to note that although these runs have given very promising results, the parameter space that could plausibly be associated with the giant impact is not yet adequately explored.     

Stress history of the Moon

The character of the lunar surface indicates that surface faulting has not been an important mechanism for the build-up of the lunar surface. If the radioactive content of the Moon is of the same order as that of chondritic meteorites, then the absence of major surface faults can be explained in a number of ways. A near-surface concentration of radioactivity will provide an equality of heat production and surface heat flow necessary for the maintenance of a constant lunar radius. Alternatively, the radioactivity could be deeply buried, with the radius still remaining constant over the past 2,000,000,000 years. Heat transported by mechanisms other than radiation and thermal conduction will also tend to keep the radius of the Moon at a constant value.

Even though the radius of the Moon remains constant, there is a major build-up of strain energy throughout the Moon. The rate is such that, on the average, something on the order of 10²⁴–10²⁵ ergs of distortional energy should be released per year throughout the Moon, provided the radioactivity is uniformly distributed. A near-surface concentration of the radioactivity might decrease this rate of energy release but certainly by no more than an order of magnitude. Under all circumstances it would appear that a Moon of chondritic composition would have strong Scismic activity.     

Topographic mapping of the Moon

Contour maps of the Mooon have been compiled by photogrammetric methods that use stereoscopic combinations of all available metric photographs from the Apollo 15, 16, and 17 missions. The maps utilize the same format as the existing NASA shaded-relief Lunar Planning Charts (LOC-1, -2, -3, and -4), which have a scale of 1:2 750 000. The map contour interval is 500m. A control net derived from Apollo photographs by Doyle and others was used for the compilation. Contour lines and elevations are referred to the new topographic datum of the Moon, which is defined in terms of spherical harmonics from the lunar gravity field. Compilation of all four LOC charts was completed on analytical plotters from 566 stereo models of Apollo metric photographs that cover approximately 20% of the Moon. This is the first step toward compiling a global topographic map of the Moon at a scale of 1:5 000 000.