Optimal Initial Conditions Research Articles

Non-modal analysis of transient growth in a liquid annular jet surrounded by gas flow

Transient energy growth is a common mathematical concept in many fluid flow systems, and it has been widely investigated in recent years using non-modal analysis. Non-modal analysis can characterize the short-term energy amplification of perturbations, which is influenced by the Reynolds number, the Weber number, and the initial conditions such as the wavenumber. In gas–liquid coaxial nozzles, annular jets are often produced, and their breakup process is influenced by transient energy growth. However, research in this area has been limited so far. This paper for the first time investigates the transient energy growth of an annular liquid jet in static gas and validates it using a modified annular jet model. In the derivation process, the gas–liquid interfaces inside and outside the annular liquid film are taken into account. It has been found that there exists an optimal initial condition for a certain Reynolds number and a Weber number. The increase in the Reynolds number and ratio of inner and outer radius of the annular jet can maximize the transient growth under a specific initial wavenumber, while the increase in gas/liquid density ratio and the Weber number will minimize the transient growth. It is also found that transient energy growth is caused by the displacement of the free boundary.

Property Enhancement of Recycled Coarse Aggregate and Its Concrete under CO2-Accelerated Curing Treatment.

The poor properties of recycled coarse aggregate (RCA) and recycled coarse aggregate concrete (RCAC) are considered key constraints hindering the reuse of this waste resource in marine engineering. The CO2-based accelerated carbonation method, which utilizes the alkali aggregate properties of RCA to achieve CO2 uptake and sequestration while significantly enhancing its properties, has attracted widespread attention. However, the degree of improvement in the properties of RCA under different initial moisture conditions (IMCs) and aggregate particle sizes (APSs) after CO2-accelerated carbonation remains unclear. Moreover, the quantitative effect of carbonated recycled coarse aggregate (CRCA), which is obtained from RCA samples with the optimal initial moisture conditions, on the improvement of RCAC under optimal accelerated carbonation modification conditions still needs to be studied in depth. For this investigation, a CO2-accelerated carbonation experiment was carried out on RCA samples with different IMCs and APSs, and the variations in the properties of RCA with respect to its IMC and APS were assessed. The degree of accelerated carbonation modification of RCA under different IMCs and APSs was quantified, and the optimal initial moisture conditions for enhancing the properties of the RCA were confirmed. By preparing concrete specimens based on the natural coarse aggregate, RCA, and CRCA with the best initial moisture conditions (considering the same concrete-water proportion), the effect of CRCA on the workability, mechanical properties, and durability of the corresponding concrete specimen was determined. The findings of this study can be used to effectively promote the sustainable development of marine science and engineering in the future and contribute to global dual-carbon goals, which are of great practical significance and scientific value.

Transient amplification due to non-Hermitian interference of dissipative supermodes

Achieving transient amplification of light in coupled waveguides that are characterized by a fully dissipative spectrum, is the context of our study. Physically, material gain is present and spatially distributed in a lossy lattice, and thus makes possible the interesting effect of transient growth. In order to quantify the maximum value of power growth and determine the accompanying optimal initial conditions, we apply pseudospectra and singular value decomposition techniques. Similar to the way we can identify all possible microstates of a paramagnetic crystal for a given value of energy, we can also calculate the maximum transient growth of all different permutations of the gainy channels for given values of gain–loss amplitudes, and thus sort them. The result of such non-Hermitian hierarchy is that we have maximum amplification, when the gain is distributed close to the lattice's edges.

A Hyper-chaotic Medical Image Encryption with Optimized Key Value

This article delves into a medical image encryption/decryption method based on hyper chaotic dynamics and genetic algorithms. The proposed algorithm boasts simplicity in implementation, featuring straightforward operations that render it well-suited for real-time applications while elevating its security measures. Leveraging the sensitivity of chaotic behaviour to initial conditions, a genetic algorithm is employed to select optimal initial conditions for the 5D multi-wing hyper-chaotic system. Initially, a secret key generation method based on the input image is applied, followed by stages of diffusion and encryption utilizing the chaotic system. The secret key undergoes optimization through a genetic algorithm, considering specific parameters within the encrypted image as encryption factors. Subsequently, the encrypted image with the optimized secret key is finalized, serving as the basis for decrypting the cipher image. The proposed method undergoes simulation, testing, and comparison against other image encryption algorithms. Both experimental results and computer simulations affirm the robustness of this cryptographic system, showcasing a significant key space value (2^256), high key sensitivity (Number of Pixels Change Rate: NPCR > 99.55%, Unified Average Changing Intensity: UACI > 33.37%), and its ability to fend off various types of attack.

Effect of temperature on the mechanical properties of aluminum polycrystal using molecular dynamics simulation

The initial temperature has a considerable effect on aluminum polycrystals' physical stability and mechanical performance, with the possibility to optimize their mechanical properties for practical applications. Thus, using a molecular dynamics technique, the effect of temperature on the mechanical properties of aluminum polycrystals is studied. Stress-strain curves, ultimate strength, and Young's modulus were all measured at temperatures of 300, 350, 400, and 450 K. The findings from MD simulations show that the initial temperature significantly affects the physical stability and mechanical performance of designed aluminum polycrystals. The aluminum polycrystal experiences a numerical increase in ultimate strength and Young's modulus from 6640 to 74.072 to 7.055 and 79.226 GPa, respectively, when subjected to the optimal initial conditions of 350 K. With further increasing temperature to 450 K, ultimate strength and Young's modulus decrease to 6.461 and 74.413 GPa, respectively. The observed decrease in ultimate strength and Young's modulus of the aluminum polycrystal as the temperature increased from the optimal condition of 350 K–450 K can be attributed to the weakening of interatomic attraction forces at higher temperatures. This reduction in interatomic bonding strength resulted in decreased material stiffness and resistance to deformation, leading to lower ultimate strength and Young's modulus values. This study's novelty lies in its comprehensive assessment of the initial temperature's effects on the mechanical performance of aluminum polycrystals, providing valuable insights for practical applications and advancing beyond previous efforts in the literature.

The fully parabolic multi-species chemotaxis system in

Abstract This article is devoted to the analysis of the parabolic–parabolic chemotaxis system with multi-components over $\mathbb{R}^2$ . The optimal small initial condition on the global existence of solutions for multi-species chemotaxis model in the fully parabolic situation had not been attained as far as the author knows. In this paper, we prove that under the sub-critical mass condition, any solutions to conflict-free system exist globally. Moreover, the global existence of solutions to system with strong self-repelling effect has been discussed even for large initial data. The proof is based on the modified free energy functional and the Moser–Trudinger inequality for system.

Over-30-GeV intense laser phase-locked direct electron acceleration

When an intense laser obliquely irradiates a solid, a pre-pulse will first ionize the solid surface, followed by the main pulse interacting with the plasma and ultimately being reflected by the plasma. Simultaneously, certain electrons within the plasma will be trapped in the accelerating phase of the laser field, subsequently gaining effective acceleration within the field, this phenomenon is known as phase-locked electron acceleration. Given the current intense lasers' electric field intensity nearing the TV/m range, electrons could potentially acquire energy levels on the order of hundreds of GeV or even TeV, provided they stay in the accelerating phase of the laser field long enough. Here, we initially use PIC (Particle-in-Cell) simulations to simulate the interaction process between laser pulses and plasma, thereby obtaining the properties of phase-locked electrons. In order to reduce computational demands, we turn to use a three-dimensional (3D) test particle model to calculate the subsequent interactions of these electrons with the reflected laser field. By this model, we obtain the data of the locked-phase electrons after having interacted with the reflected laser (Fig. (a)). Furthermore, we use this model to calculate the dynamical behavior of electrons under different initial conditions (Fig. (b)). Under the laser intensity of <inline-formula><tex-math id="M1">\begin{document}$ {a}_{0}=350 $\end{document}</tex-math></inline-formula>(<inline-formula><tex-math id="M2">\begin{document}$ {a}_{0} $\end{document}</tex-math></inline-formula> is the normalized laser vector potential), the energy of the electrons directly accelerated by the laser is enhanced to 32 GeV. In contrast, under the same laser intensity, the energy of the electrons accelerated by ponderomotive force is only 0.35 GeV. The research findings indicate that the strong laser with peak power around 10 PW can directly accelerate electrons to approximately 30 GeV. Additionally, this study outlines the optimal initial conditions for injecting electrons into the laser field and the final electron energy within the phase-locked acceleration mechanism, thereby establishing a calibration relationship with the laser field intensity. Given the continual enhancement of laser intensity and the potential application of the laser phase-locked electron acceleration mechanism to positron acceleration, this research holds promise for its implementation in fields such as miniaturized positron-electron colliders and high-energy gamma-ray sources.

Deep Imitation Learning for Optimal Trajectory Planning and Initial Condition Optimization for an Unstable Dynamic System

In this article, an innovative offline deep imitation learning algorithm for optimal trajectory planning is proposed. While many state‐of‐the‐art works achieved optimal trajectory planning, their systems were stable or quasistable, and their approaches rarely optimized the system's initial conditions (ICs). Here, a new unstable dynamic system task called “internal sliding object stabilization control” is proposed, modeled, and solved by deep imitation learning. Given the system's ICs, the neural networks (NNs) can imitate the iterative linear quadratic regulator (iLQR), generate optimal trajectories, and compute faster. A proportional–integral–derivative (PID) controller is used to track the unstable trajectories. Leveraging on the gradients of NNs, it can optimize the system's ICs, avoid obstacles stepwise, and ensure the worst bounds of NNs for safety. Subsequently, thorough simulations are conducted, including comparing the iLQR and PID controllers in the task, optimizing the system's different ICs by gradient descent, and finding the worst bound of the performance by gradient ascent. Results show that the proposed algorithm achieves considerably improved performance. Finally, experiments are conducted with a real manipulator to compare the proposed structure with the original iLQR. Results indicate that the proposed algorithm resembles the iLQR well. Program code and experiment results are in https://github.com/DanielYamChen/ISOSC.git.

Uncovering personal circadian responses to light through particle swarm optimization

Background and objectivesKronauer's oscillator model of the human central pacemaker is one of the most commonly used approaches to study the human circadian response to light. Two sources of error when applying it to a personal light exposure have been identified: (1) as a populational model, it does not consider inter-individual variability, and (2) the initial conditions needed to integrate the model are usually unknown, and thus subjectively estimated. In this work, we evaluate the ability of particle swarm optimization (PSO) algorithms to simultaneously uncover the optimal initial conditions and individual parameters of a pre-defined Kronauer's oscillator model. MethodsA Canonical PSO, a Dynamic Multi-Swarm PSO and a novel modification of the latter, namely Hierarchical Dynamic Multi-Swarm PSO, are evaluated. Two different target models (under a regular and an irregular schedule) are defined, and the same realistic light profile is fed to them. Based on their output, a fitness function is proposed, which is minimized by the algorithms to find the optimum set of parameters and initial conditions of the model. ResultsWe demonstrate that Dynamic Multi-Swarm and Hierarchical Dynamic Multi-Swarm algorithms can accurately uncover personal circadian parameters under both regular and irregular schedules, but as expected, optimization is easier under a regular schedule. Circadian parameters play the most important role in the optimization process and should be prioritized over initial conditions, although assessment of the impact of misestimating the latter is recommended. The log-log linear relationship between mean absolute error and computational cost shows that the number of particles to use is at the discretion of the user. ConclusionsThe robustness and low errors achieved by the algorithms support their further testing, validation and systematic application to empirical data under a regular or irregular schedule. Uncovering personal circadian parameters can improve the assessment of the circadian status of a person and the applicability of personalized light therapies, as well as help to discover other factors that may lie behind the interindividual variability in the circadian response to light.

Effect of the autochthonous starter culture on physicochemical, microbiological, and sensory characteristics of traditional colombian double-cream cheese: A kinetic approach

Double-cream cheese is a Colombian pasta-filata cheese made from fresh and acidified cow milk. Two kinds of cheese were compared: conventional (QTR) and autochthonous starter culture added (QCP). Double-cream cheeses with yield, moisture, fat, and protein of 10.98%, 48.52 g/100 g, 24.58 g/100 g, and 23.23 g/100 g were obtained, respectively. For both kinds of cheese, no significant differences during the storage period for pH (5.19), acidity (0.3 g/100 g), moisture (46.73 g/100 g), and total solids (53.47 g/100 g) were observed. The values of cohesiveness (0.33), gumminess (18.91 J), hardness (30.03 N), and chewiness (14.75 J) were higher for QCP during the storage time. General quality for QTR and QCP were 9.35 ± 0.87 and 9.25 ± 0.24, respectively. At the end of the storage period, QTR exhibited a higher count of total coliforms (7.20 ± 0.00 log CFU/g), while QCP showed a lower count (3.58 ± 0.98 log CFU/g). The specific growth rates of total coliforms in QTR and QCP were 0.0146 h−1 and 0.0068 h−1, respectively. QCP demonstrates an extended shelf life of 22 days more than QTR under optimal initial conditions (0.00 log CFU/g). The autochthonous starter culture can inhibit coliform growth, enabling the production of cheese with extended shelf life and similar sensory qualities to conventional cheese.

High yield of heat-resistant spores of Bacillus megaterium in bioreactors

Microbial inoculants have been proposed as a sustainable and cost-effective alternative to conventional agricultural practices. However, the effectiveness of these bio-based products can be negatively impacted by reduced cell viability during product development, storage, and commercialization. To address these limitations, it is critical to identify and optimize the cultivation conditions that can lead to the production of highly resistant spores for use as inoculants. In this study, optimal cultivation conditions for obtaining Bacillus megaterium B119 spores in bioreactors were reached, with increases in the sporulation efficiency and heat resistance. The best nutrient medium and optimal temperature and initial pH conditions were obtained on shake flask experiments and subsequently validated in two bench-scale bioreactor models (stirred and aerated tank and the bubble column). The optimized temperature and initial pH were found to be 37 °C and 5.6. The highest concentration of heat-resistant Bacillus spores of 1.31·1012 CFU/mL was obtained in the bubble column bioreactor with a sporulation efficiency of 100.0%, a heat resistance of 78.4%, and an energetic efficiency on spores production of 4.83·1010 CFU/J. The production of highly resistant spores can significantly improve the survival rate of microorganisms during inoculant production and application, thereby enhancing their effectiveness in the field.

DETERMINING OPTIMAL SIMULATION SETTINGS FOR THE CENTRIFUGAL PUMP PARTS OPTIMIZATION PROCESS

This paper explores the optimal simulation settings required by the optimization process of centrifugal canned motor pump parts using the CH 6,3/32-2,2-2 pump as an example. The optimization of centrifugal pump parts is particularly important in order to increase pump characteristics: pump efficiency, net positive suction head, as well as other optimization criteria. The optimal conditions to obtain a geometric model and optimal mesh properties, initial and boundary conditions, solver settings were investigated. The mathematical models used in the simulation were also presented. In conclusion, there have been revealed optimal simulation settings.

Optimal initial conditions for science orbits around Ganymede

An important aspect concerning the dynamics of a probe around a natural satellite is represented by the probe lifetime. In fact, the science orbits, usually characterised by low altitude and high inclination, are subjected to an important increase in eccentricity, due to the mother planet’s gravitational attraction, which can lead the probe to collide with the surface of the natural satellite. In the present work, the issue is addressed at the Jupiter’s moon Ganymede, which represents one of most important targets in the exploration of the Solar System. Thus, taking advantage from theoretical aspects and analytical relationships concerning the evolution of the mean orbit elements of the probe, a numerical investigation aimed at determining the initial conditions of the probe orbit which allow the longest lifetime has been carried out.

Effects of phase boundary and shear on diffusive instability

We study the diffusive instability subject to a solid–liquid interface and a Kolmogorov flow using modal, non-modal analyses and energy analysis. It is found that the phase boundary has different effects on the threshold of diffusive convection for weak and strong salinity stratification. In the context of oceanography where the salinity Rayleigh number RS is very high, the ice–water interface has negligible influence on the onset of diffusive convection. In the presence of shear, the diffusive convection for RS < 106 tends to be inhibited and with the increase of shear intensity, the oscillatory, steady convective and Kelvin–Helmholtz instabilities will be successively dominant. For RS > 106, the shear has a destabilizing effect on the diffusive convection, due to the generation of thermohaline-sheared instability found by Radko (J. Fluid Mech., vol. 805, 2016, pp. 147–170). Non-modal analysis indicates that for realistic parameters of high-latitude oceans, with the transition of the ultimate energy source of instability from the density gradient to background current, the thermohaline-shear instability is expected to transition from oscillatory to steady instability. The initial transient amplification due to double diffusion has also been studied, which is due to the generation of overstable instability at the initial phase. For RS < 106, the optimal initial condition to achieve the maximum transient growth favours longitudinal rolls. For thermohaline-shear instability, however, it favours transverse rolls and specifically, in oscillatory thermohaline-shear instability, the transient amplification can be enhanced by the shear by one order of magnitude, thus having important influence on the stability of the system.

Theoretical and experimental study on FBG bending sensor for quantitative monitoring of bolt looseness

A quantitative monitoring method for bolt looseness using an elastic substrate fiber Bragg grating (FBG) bending sensor is presented in this paper, which is achieved by measuring the wavelength shift of the FBG sensor response to the bending curvature change induced by bolt loose angle variation. The theoretical sensing principle is established based on the deformation model of an inextensible elastic beam under prescribed end rotations. The influence parameters on the FBG bending sensor are investigated numerically under the bolt loose angle variation from 0° to 15°, and the position on the sensor with maximum curvature difference varying linearly to bolt loose angle variation is obtained and selected to arrange the FBG. Based on the theoretical and numerical analysis, a general solution of the wavelength shifts of the FBG bending sensor versus bolt loose angle under an optimal initial installation condition (i.e., d/l=0.9, and s/l = 0.85) is established for quantitative monitoring bolt looseness. The feasibility and performance of the proposed method are experimentally verified and the maximum error between numerical and experimental results is less than 6.19%. The results indicate that the proposed method can quantitatively monitor bolt loose angle with high resolution (about 0.005°), also prove that the theoretical derivation in this paper and the proposed design criteria can theoretically guide the monitoring of bolt looseness using FBG bending sensor.

A Statistical Hypothesis Testing Strategy for Adaptively Blending Particle Filters and Ensemble Kalman Filters for Data Assimilation

Abstract Particle filters avoid parametric estimates for Bayesian posterior densities, which alleviates Gaussian assumptions in nonlinear regimes. These methods, however, are more sensitive to sampling errors than Gaussian-based techniques such as ensemble Kalman filters. A recent study by the authors introduced an iterative strategy for particle filters that match posterior moments—where iterations improve the filter’s ability to draw samples from non-Gaussian posterior densities. The iterations follow from a factorization of particle weights, providing a natural framework for combining particle filters with alternative filters to mitigate the impact of sampling errors. The current study introduces a novel approach to forming an adaptive hybrid data assimilation methodology, exploiting the theoretical strengths of nonparametric and parametric filters. At each data assimilation cycle, the iterative particle filter performs a sequence of updates while the prior sample distribution is non-Gaussian, then an ensemble Kalman filter provides the final adjustment when Gaussian distributions for marginal quantities are detected. The method employs the Shapiro–Wilk test to determine when to make the transition between filter algorithms, which has outstanding power for detecting departures from normality. Experiments using low-dimensional models demonstrate that the approach has a significant value, especially for nonhomogeneous observation networks and unknown model process errors. Moreover, hybrid factors are extended to consider marginals of more than one collocated variables using a test for multivariate normality. Findings from this study motivate the use of the proposed method for geophysical problems characterized by diverse observation networks and various dynamic instabilities, such as numerical weather prediction models. Significance Statement Data assimilation statistically processes observation errors and model forecast errors to provide optimal initial conditions for the forecast, playing a critical role in numerical weather forecasting. The ensemble Kalman filter, which has been widely adopted and developed in many operational centers, assumes Gaussianity of the prior distribution and solves a linear system of equations, leading to bias in strong nonlinear regimes. On the other hand, particle filters avoid many of those assumptions but are sensitive to sampling errors and are computationally expensive. We propose an adaptive hybrid strategy that combines their advantages and minimizes the disadvantages of the two methods. The hybrid particle filter–ensemble Kalman filter is achieved with the Shapiro–Wilk test to detect the Gaussianity of the ensemble members and determine the timing of the transition between these filter updates. Demonstrations in this study show that the proposed method is advantageous when observations are heterogeneous and when the model has an unknown bias. Furthermore, by extending the statistical hypothesis test to the test for multivariate normality, we consider marginals of more than one collocated variable. These results encourage further testing for real geophysical problems characterized by various dynamic instabilities, such as real numerical weather prediction models.

Topological directed amplification

A phenomenon of topological directed amplification of certain initial perturbations is revealed theoretically to emerge in a class of asymptotically stable skin-effect lattices described by nonnormal Toeplitz operators $H_g$ with positive ``numerical ordinate" $\omega(H_g)>0$. Nonnormal temporal evolution, even in the presence of global dissipation, is shown to manifest a counterintuitive transient phase of edge-state amplification -- a behavior, drastically different from the asymptote, that spectral analysis of $H_g$ fails to directly reveal. A consistent description of the effect is provided by the general tool of ``pseudospectrum", and a quantitative estimation of the maximum power amplification is provided by the {\it Kreiss constant}. A recipe to determine an optimal initial condition that will attain maximum amplification power is given by singular value decomposition of the propagator $e^{-i H_g t}$. It is further predicted that the interplay between nonnormality and nonlinearity in a skin-effect laser array can facilitate narrow-emission spectra with scalable stable-output power.

Searching for Singularities in Navier–Stokes Flows Based on the Ladyzhenskaya–Prodi–Serrin Conditions

In this investigation we perform a systematic computational search for potential singularities in 3D Navier-Stokes flows based on the Ladyzhenskaya-Prodi-Serrin conditions. They assert that if the quantity $\int_0^T \| \mathbf{u}(t) \|_{L^q(\Omega)}^p \, dt$, where $2/p+3/q \le 1$, $q > 3$, is bounded, then the solution $\mathbf{u}(t)$ of the Navier-Stokes system is smooth on the interval $[0,T]$. In other words, if a singularity should occur at some time $t \in [0,T]$, then this quantity must be unbounded. We have probed this condition by studying a family of variational PDE optimization problems where initial conditions $\mathbf{u}_0$ are sought to maximize $\int_0^T \| \mathbf{u}(t) \|_{L^4(\Omega)}^8 \, dt$ for different $T$ subject to suitable constraints. These problems are solved numerically using a large-scale adjoint-based gradient approach. Even in the flows corresponding to the optimal initial conditions determined in this way no evidence has been found for singularity formation, which would be manifested by unbounded growth of $\| \mathbf{u}(t) \|_{L^4(\Omega)}$. However, the maximum enstrophy attained in these extreme flows scales in proportion to $\mathcal{E}_0^{3/2}$, the same as found by Kang et al. (2020) when maximizing the finite-time growth of enstrophy. In addition, we also consider sharpness of an a priori estimate on the time evolution of $\| \mathbf{u}(t) \|_{L^4(\Omega)}$ by solving another PDE optimization problem and demonstrate that the upper bound in this estimate could be improved.

Forecasting Solar Flares by Data Assimilation in Sandpile Models

The prediction of solar flares is still a significant challenge in space weather research, with no techniques currently capable of producing reliable forecasts performing significantly above climatology. In this paper, we present a flare forecasting technique using data assimilation coupled with computationally inexpensive cellular automata called sandpile models. Our data assimilation algorithm uses the simulated annealing method to find an optimal initial condition that reproduces well an energy-release time series. We present and empirically analyze the predictive capabilities of three sandpile models, namely the Lu and Hamilton model (LH) and two deterministically-driven models (D). Despite their stochastic elements, we show that deterministically-driven models display temporal correlations between simulated events, a needed condition for data assimilation. We present our new data assimilation algorithm and demonstrate its success in assimilating synthetic observations produced by the avalanche models themselves. We then apply our method to GOES X-Ray time series for 11 active regions having generated multiple X-class flares in the course of their lifetime. We demonstrate that for such large flares, our data assimilation scheme substantially increases the success of ``All-Clear'' forecasts, as compared to model climatology.

Probe lifetime around natural satellites with obliquity

The dynamics of a probe orbiting a moon can be significantly influenced by the non-coincidence between the moon’s equatorial and orbital planes. Thus, we performed a general analysis about the effects of the angle (obliquity) between the above-mentioned planes and of the angle (nodal phasing) between the nodal lines of the mother planet’s apparent orbit and the probe orbit on the lifetime of the probe. The lifetime, strictly correlated to the variations in eccentricity of the probe orbit, was evaluated starting from low values of the semi-major axis, moderate eccentricity, and high inclination to offer high ground spatial resolution and extend latitudinal coverage of the natural satellite. This investigation, carried out through numerical simulations, may be useful for identifying the optimal initial conditions of the probe’s orbit elements, leading to an important increase in the probe lifetime in missions devoted to the exploration of natural satellites.