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1.
This paper is devoted to the study of the transfer problem from a libration point orbit of the Earth–Moon system to an orbit around the Moon. The transfer procedure analysed has two legs: the first one is an orbit of the unstable manifold of the libration orbit and the second one is a transfer orbit between a certain point on the manifold and the final lunar orbit. There are only two manoeuvres involved in the method and they are applied at the beginning and at the end of the second leg. Although the numerical results given in this paper correspond to transfers between halo orbits around the \(L_1\) point (of several amplitudes) and lunar polar orbits with altitudes varying between 100 and 500 km, the procedure we develop can be applied to any kind of lunar orbits, libration orbits around the \(L_1\) or \(L_2\) points of the Earth–Moon system, or to other similar cases with different values of the mass ratio.  相似文献   

2.
Lunar frozen orbits, characterized by constant orbital elements on average, have been previously found using various dynamical models, incorporating the gravitational field of the Moon and the third-body perturbation exerted by the Earth. The resulting mean orbital elements must be converted to osculating elements to initialize the orbiter position and velocity in the lunar frame. Thus far, however, there has not been an explicit transformation from mean to osculating elements, which includes the zonal harmonic \(J_2\), the sectorial harmonic \(C_{22}\), and the Earth third-body effect. In the current paper, we derive the dynamics of a lunar orbiter under the mentioned perturbations, which are shown to be dominant for the evolution of circumlunar orbits, and use von Zeipel’s method to obtain a transformation between mean and osculating elements. Whereas the dynamics of the mean elements do not include \(C_{22}\), and hence does not affect the equilibria leading to frozen orbits, \(C_{22}\) is present in the mean-to-osculating transformation, hence affecting the initialization of the physical circumlunar orbit. Simulations show that by using the newly-derived transformation, frozen orbits exhibit better behavior in terms of long-term stability about the mean values of eccentricity and argument of periapsis, especially for high orbits.  相似文献   

3.
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.  相似文献   

4.
We consider a Yukawa-type gravitational potential combined with the Poynting-Robertson effect. Dust particles originating within the asteroid belt and moving on circular and elliptic trajectories are studied and expressions for the time rate of change of their orbital radii and semimajor axes, respectively, are obtained. These expressions are written in terms of basic particle parameters, namely their density and diameter. Then, they are applied to produce expressions for the time required by the dust particles to reach the orbit of Earth. For the Yukawa gravitational potential, dust particles of diameter \(10^{ - 3}\) m in circular orbits require times of the order of \(8.557 \times 10^{6}\) yr and for elliptic orbits of eccentricities \(e =0.1, 0.5\) require times of \(9.396 \times 10^{6}\) and \(2.129 \times 10^{6}\) yr respectively to reach Earth’s orbit. Finally, various cases of the Yukawa potential are studied and the corresponding particle times to reach Earth’s are derived per case along with numerical results for circular and various elliptical orbits.  相似文献   

5.
Small tidal forces in the Earth–Moon system cause detectable changes in the orbit. Tidal energy dissipation causes secular rates in the lunar mean motion n, semimajor axis a, and eccentricity e. Terrestrial dissipation causes most of the tidal change in n and a, but lunar dissipation decreases eccentricity rate. Terrestrial tidal dissipation also slows the rotation of the Earth and increases obliquity. A tidal acceleration model is used for integration of the lunar orbit. Analysis of lunar laser ranging (LLR) data provides two or three terrestrial and two lunar dissipation parameters. Additional parameters come from geophysical knowledge of terrestrial tides. When those parameters are converted to secular rates for orbit elements, one obtains dn/dt = \(-25.97\pm 0.05 ''/\)cent\(^{2}\), da/dt = 38.30 ± 0.08 mm/year, and di/dt = ?0.5 ± 0.1 \(\upmu \)as/year. Solving for two terrestrial time delays and an extra de/dt from unspecified causes gives \(\sim \) \(3\times 10^{-12}\)/year for the latter; solving for three LLR tidal time delays without the extra de/dt gives a larger phase lag of the N2 tide so that total de/dt = \((1.50 \pm 0.10)\times 10^{-11}\)/year. For total dn/dt, there is \(\le \)1 % difference between geophysical models of average tidal dissipation in oceans and solid Earth and LLR results, and most of that difference comes from diurnal tides. The geophysical model predicts that tidal deceleration of Earth rotation is \(-1316 ''\)/cent\(^{2}\) or 87.5 s/cent\(^{2}\) for UT1-AT, a 2.395 ms/cent increase in the length of day, and an obliquity rate of 9 \(\upmu \)as/year. For evolution during past times of slow recession, the eccentricity rate can be negative.  相似文献   

6.
Vinti’s potential is revisited for analytical propagation of the main satellite problem, this time in the context of relative motion. A particular version of Vinti’s spheroidal method is chosen that is valid for arbitrary elliptical orbits, encapsulating \(J_2\), \(J_3\), and generally a partial \(J_4\) in an orbit propagation theory without recourse to perturbation methods. As a child of Vinti’s solution, the proposed relative motion model inherits these properties. Furthermore, the problem is solved in oblate spheroidal elements, leading to large regions of validity for the linearization approximation. After offering several enhancements to Vinti’s solution, including boosts in accuracy and removal of some singularities, the proposed model is derived and subsequently reformulated so that Vinti’s solution is piecewise differentiable. While the model is valid for the critical inclination and nonsingular in the element space, singularities remain in the linear transformation from Earth-centered inertial coordinates to spheroidal elements when the eccentricity is zero or for nearly equatorial orbits. The new state transition matrix is evaluated against numerical solutions including the \(J_2\) through \(J_5\) terms for a wide range of chief orbits and separation distances. The solution is also compared with side-by-side simulations of the original Gim–Alfriend state transition matrix, which considers the \(J_2\) perturbation. Code for computing the resulting state transition matrix and associated reference frame and coordinate transformations is provided online as supplementary material.  相似文献   

7.
In this note a study of the convergence properties of some starters \( E_0 = E_0(e,M)\) in the eccentricity–mean anomaly variables for solving the elliptic Kepler’s equation (KE) by Newton’s method is presented. By using a Wang Xinghua’s theorem (Xinghua in Math Comput 68(225):169–186, 1999) on best possible error bounds in the solution of nonlinear equations by Newton’s method, we obtain for each starter \( E_0(e,M)\) a set of values \( (e,M) \in [0, 1) \times [0, \pi ]\) that lead to the q-convergence in the sense that Newton’s sequence \( (E_n)_{n \ge 0}\) generated from \( E_0 = E_0(e,M)\) is well defined, converges to the exact solution \(E^* = E^*(e,M)\) of KE and further \( \vert E_n - E^* \vert \le q^{2^n -1}\; \vert E_0 - E^* \vert \) holds for all \( n \ge 0\). This study completes in some sense the results derived by Avendaño et al. (Celest Mech Dyn Astron 119:27–44, 2014) by using Smale’s \(\alpha \)-test with \(q=1/2\). Also since in KE the convergence rate of Newton’s method tends to zero as \( e \rightarrow 0\), we show that the error estimates given in the Wang Xinghua’s theorem for KE can also be used to determine sets of q-convergence with \( q = e^k \; \widetilde{q} \) for all \( e \in [0,1)\) and a fixed \( \widetilde{q} \le 1\). Some remarks on the use of this theorem to derive a priori estimates of the error \( \vert E_n - E^* \vert \) after n Kepler’s iterations are given. Finally, a posteriori bounds of this error that can be used to a dynamical estimation of the error are also obtained.  相似文献   

8.
The aim of this work is to combine the model of orbital and rotational motion of the Moon developed for DE430 with up-to-date astronomical, geodynamical, and geo- and selenophysical models. The parameters of the orbit and physical libration are determined in this work from lunar laser ranging (LLR) observations made at different observatories in 1970–2013. Parameters of other models are taken from solutions that were obtained independently from LLR. A new implementation of the DE430 lunar model, including the liquid core equations, was done within the EPM ephemeris. The postfit residuals of LLR observations make evident that the terrestrial models and solutions recommended by the IERS Conventions are compatible with the lunar theory. That includes: EGM2008 gravitational potential with conventional corrections and variations from solid and ocean tides; displacement of stations due to solid and ocean loading tides; and precession-nutation model. Usage of these models in the solution for LLR observations has allowed us to reduce the number of parameters to be fit. The fixed model of tidal variations of the geopotential has resulted in a lesser value of Moon’s extra eccentricity rate, as compared to the original DE430 model with two fit parameters. A mixed model of lunar gravitational potential was used, with some coefficients determined from LLR observations, and other taken from the GL660b solution obtained from the GRAIL spacecraft mission. Solutions obtain accurate positions for the ranging stations and the five retroreflectors. Station motion is derived for sites with long data spans. Dissipation is detected at the lunar fluid core-solid mantle boundary demonstrating that a fluid core is present. Tidal dissipation is strong at both Earth and Moon. Consequently, the lunar semimajor axis is expanding by 38.20 mm/yr, the tidal acceleration in mean longitude is \(-25.90 {{}^{\prime \prime }}/\mathrm{cy}^2\), and the eccentricity is increasing by \(1.48\times 10^{-11}\) each year.  相似文献   

9.
We review the origin and evolution of the atmospheres of Earth, Venus and Mars from the time when their accreting bodies were released from the protoplanetary disk a few million years after the origin of the Sun. If the accreting planetary cores reached masses \(\ge 0.5 M_\mathrm{Earth}\) before the gas in the disk disappeared, primordial atmospheres consisting mainly of H\(_2\) form around the young planetary body, contrary to late-stage planet formation, where terrestrial planets accrete material after the nebula phase of the disk. The differences between these two scenarios are explored by investigating non-radiogenic atmospheric noble gas isotope anomalies observed on the three terrestrial planets. The role of the young Sun’s more efficient EUV radiation and of the plasma environment into the escape of early atmospheres is also addressed. We discuss the catastrophic outgassing of volatiles and the formation and cooling of steam atmospheres after the solidification of magma oceans and we describe the geochemical evidence for additional delivery of volatile-rich chondritic materials during the main stages of terrestrial planet formation. The evolution scenario of early Earth is then compared with the atmospheric evolution of planets where no active plate tectonics emerged like on Venus and Mars. We look at the diversity between early Earth, Venus and Mars, which is found to be related to their differing geochemical, geodynamical and geophysical conditions, including plate tectonics, crust and mantle oxidation processes and their involvement in degassing processes of secondary \(\hbox {N}_2\) atmospheres. The buildup of atmospheric \(\hbox {N}_2\), \(\hbox {O}_2\), and the role of greenhouse gases such as \(\hbox {CO}_2\) and \(\hbox {CH}_4\) to counter the Faint Young Sun Paradox (FYSP), when the earliest life forms on Earth originated until the Great Oxidation Event \(\approx \) 2.3 Gyr ago, are addressed. This review concludes with a discussion on the implications of understanding Earth’s geophysical and related atmospheric evolution in relation to the discovery of potential habitable terrestrial exoplanets.  相似文献   

10.
We aim to probe the dynamic structure of the extended Solar neighborhood by calculating the radial metallicity gradients from orbit properties, which are obtained for axisymmetric and non-axisymmetric potential models, of red clump (RC) stars selected from the RAdial Velocity Experiment’s Fourth Data Release. Distances are obtained by assuming a single absolute magnitude value in near-infrared, i.e. \(M_{Ks}=-1.54\pm0.04\) mag, for each RC star. Stellar orbit parameters are calculated by using the potential functions: (i) for the MWPotential2014 potential, (ii) for the same potential with perturbation functions of the Galactic bar and transient spiral arms. The stellar age is calculated with a method based on Bayesian statistics. The radial metallicity gradients are evaluated based on the maximum vertical distance (\(z_{max}\)) from the Galactic plane and the planar eccentricity (\(e_{p}\)) of RC stars for both of the potential models. The largest radial metallicity gradient in the \(0< z_{max} \leq0.5\) kpc distance interval is \(-0.065\pm0.005~\mbox{dex}\,\mbox{kpc}^{-1}\) for a subsample with \(e_{p}\leq0.1\), while the lowest value is \(-0.014\pm0.006~\mbox{dex}\,\mbox{kpc}^{-1}\) for the subsample with \(e_{p}\leq0.5\). We find that at \(z_{max}>1\) kpc, the radial metallicity gradients have zero or positive values and they do not depend on \(e_{p}\) subsamples. There is a large radial metallicity gradient for thin disc, but no radial gradient found for thick disc. Moreover, the largest radial metallicity gradients are obtained where the outer Lindblad resonance region is effective. We claim that this apparent change in radial metallicity gradients in the thin disc is a result of orbital perturbation originating from the existing resonance regions.  相似文献   

11.
In this paper, we deal with a Hill’s equation, depending on two parameters \(e\in [0,1)\) and \(\varLambda >0\), that has applications to some problems in Celestial Mechanics of the Sitnikov type. Due to the nonlinearity of the eccentricity parameter e and the coexistence problem, the stability diagram in the \((e,\varLambda )\)-plane presents unusual resonance tongues emerging from points \((0,(n/2)^2),\ n=1,2,\ldots \) The tongues bounded by curves of eigenvalues corresponding to \(2\pi \)-periodic solutions collapse into a single curve of coexistence (for which there exist two independent \(2\pi \)-periodic eigenfunctions), whereas the remaining tongues have no pockets and are very thin. Unlike most of the literature related to resonance tongues and Sitnikov-type problems, the study of the tongues is made from a global point of view in the whole range of \(e\in [0,1)\). Indeed, an interesting behavior of the tongues is found: almost all of them concentrate in a small \(\varLambda \)-interval [1, 9 / 8] as \(e\rightarrow 1^-\). We apply the stability diagram of our equation to determine the regions for which the equilibrium of a Sitnikov \((N+1)\)-body problem is stable in the sense of Lyapunov and the regions having symmetric periodic solutions with a given number of zeros. We also study the Lyapunov stability of the equilibrium in the center of mass of a curved Sitnikov problem.  相似文献   

12.
We will show that the period T of a closed orbit of the planar circular restricted three body problem (viewed on rotating coordinates) depends on the region it encloses. Roughly speaking, we show that, \(2 T=k\pi +\int _\Omega g\) where k is an integer, \(\Omega \) is the region enclosed by the periodic orbit and \(g:{\mathbb {R}}^2\rightarrow {\mathbb {R}}\) is a function that only depends on the constant C known as the Jacobian constant; it does not depend on \(\Omega \). This theorem has a Keplerian flavor in the sense that it relates the period with the space “swept” by the orbit. As an application we prove that there is a neighborhood around \(L_4\) such that every periodic solution contained in this neighborhood must move clockwise. The same result holds true for \(L_5\).  相似文献   

13.
Taking into consideration a probe moving in an elliptical orbit around a celestial body, the possibility of determining conditions which lead to constant values on average of all the orbit elements has been investigated here, considering the influence of the planetary oblateness and the long-term effects deriving from the attraction of several perturbing bodies. To this end, three equations describing the variation of orbit eccentricity, apsidal line and angular momentum unit vector have been first retrieved, starting from a vectorial expression of the Lagrange planetary equations and considering for the third-body perturbation the gravity-gradient approximation, and then exploited to demonstrate the feasibility of achieving the above-mentioned goal. The study has led to the determination of two families of solutions at constant mean orbit elements, both characterised by a co-planarity condition between the eccentricity vector, the angular momentum and a vector resulting from the combination of the orbital poles of the perturbing bodies. As a practical case, the problem of a probe orbiting the Moon has been faced, taking into account the temporal evolution of the perturbing poles of the Sun and Earth, and frozen solutions at argument of pericentre 0\(^{\circ }\) or 180\(^{\circ }\) have been found.  相似文献   

14.
Tidal torque drives the rotational and orbital evolution of planet–satellite and star–exoplanet systems. This paper presents one analytical tidal theory for a viscoelastic multi-layered body with an arbitrary number of homogeneous layers. Starting with the static equilibrium figure, modified to include tide and differential rotation, and using the Newtonian creep approach, we find the dynamical equilibrium figure of the deformed body, which allows us to calculate the tidal potential and the forces acting on the tide generating body, as well as the rotation and orbital elements variations. In the particular case of the two-layer model, we study the tidal synchronization when the gravitational coupling and the friction in the interface between the layers is added. For high relaxation factors (low viscosity), the stationary solution of each layer is synchronous with the orbital mean motion (n) when the orbit is circular, but the rotational frequencies increase if the orbital eccentricity increases. This behavior is characteristic in the classical Darwinian theories and in the homogeneous case of the creep tide theory. For low relaxation factors (high viscosity), as in planetary satellites, if friction remains low, each layer can be trapped in different spin-orbit resonances with frequencies \(n/2,n,3n/2,2n,\ldots \). When the friction increases, attractors with differential rotations are destroyed, surviving only commensurabilities in which core and shell have the same velocity of rotation. We apply the theory to Titan. The main results are: (i) the rotational constraint does not allow us to confirm or reject the existence of a subsurface ocean in Titan; and (ii) the crust-atmosphere exchange of angular momentum can be neglected. Using the rotation estimate based on Cassini’s observation (Meriggiola et al. in Icarus 275:183–192, 2016), we limit the possible value of the shell relaxation factor, when a deep subsurface ocean is assumed, to \(\gamma _s\lesssim 10^{-9}\,\hbox {s}^{-1}\), which corresponds to a shell’s viscosity \(\eta _s\gtrsim 10^{18}\,\hbox {Pa}\,\hbox {s}\), depending on the ocean’s thickness and viscosity values. In the case in which a subsurface ocean does not exist, the maximum shell relaxation factor is one order of magnitude smaller and the corresponding minimum shell’s viscosity is one order higher.  相似文献   

15.
China is planning to land a spacecraft on the farside of the Moon, a premiere, by 2018. In essence, the traditional tracking modes, based on direct visibility, cannot operate for the lunar farside lander tracking, and therefore a relay satellite, visible at the same time by both the lander and the Earth, will be required, operating in the so-called four-way mode (Earth-relay satellite-lander-relay satellite-Earth). In this paper, we firstly give the mathematical formulation of the four-way relay tracking mode and of its partial derivatives with respect to the relevant parameters, implemented in our POD software WUDOGS (Wuhan University Deep-space Orbit determination and Gravity recovery System). In a second step, in simulation mode, we apply this relay mode to determining lander coordinates, which are absolutely needed for a sample return mission, or to add constraints on rotation models of the Moon. The results show that with Doppler measurements at a 0.1 mm/s error level, the positioning of the farside lander could be done at centimeters level (1-\(\delta\)) in the case of a circumlunar relay satellite; and at a 5 meters level (1-\(\delta\)) in the case of a Lagrange point (L2) Halo relay satellite.  相似文献   

16.
We consider an elliptic restricted four-body system including three primaries and a massless particle. The orbits of the primaries are elliptic, and the massless particle moves under the mutual gravitational attraction. From the dynamic equations, a quasi-integral is obtained, which is similar to the Jacobi integral in the circular restricted three-body problem (CRTBP). The energy constant \(C\) determines the topology of zero velocity surfaces, which bifurcate at the equilibrium point. We define the concept of Hill stability in this problem, and a criterion for stability is deduced. If the actual energy constant \(C_{\mathrm{ac}}\ ( {>} 0 ) \) is bigger than or equal to the critical energy constant \(C_{\mathrm{cr}}\), the particle will be Hill stable. The critical energy constant is determined by the mass and orbits of the primaries. The criterion provides a way to capture an asteroid into the Earth–Moon system.  相似文献   

17.
The term “jumping” Trojan was introduced by Tsiganis et al. (Astron Astrophys 354:1091–1100, 2000) in their studies of long-term dynamics exhibited by the asteroid (1868) Thersites, which had been observed to jump from librations around \(L_4\) to librations around \(L_5\). Another example of a “jumping” Trojan was found by Connors et al. (Nature 475:481–483, 2011): librations of the asteroid 2010 TK7 around the Earth’s libration point \(L_4\) preceded by its librations around \(L_5\). We explore the dynamics of “jumping” Trojans under the scope of the restricted planar elliptical three-body problem. Via double numerical averaging we construct evolutionary equations, which allow analyzing transitions between different regimes of orbital motion.  相似文献   

18.
In this note, by using Smale’s \(\alpha \)-theorem on the convergence of Newton’s method, the \(\alpha \)-sets of convergence of some starters of solving the elliptic Kepler’s equation are derived. For each starter we compute the exact \(\alpha \)-set in the eccentricity-main anomaly \((e,M)\in [0,1)\times [0,\pi ]\), showing that these sets are larger than those derived by Avendaño et al. (Celest Mech Dyn Astron 119:27–44, 2014). Further, new convergence tests based on the Newton–Kantorowitch theorem are given comparing with the derived from Smale’s \(\alpha \)-test.  相似文献   

19.
In this paper we address an \(n+1\)-body gravitational problem governed by the Newton’s laws, where n primary bodies orbit on a plane \(\varPi \) and an additional massless particle moves on the perpendicular line to \(\varPi \) passing through the center of mass of the primary bodies. We find a condition for the described configuration to be possible. In the case when the primaries are in a rigid motion, we classify all the motions of the massless particle. We study the situation when the massless particle has a periodic motion with the same minimal period as the primary bodies. We show that this fact is related to the existence of a certain pyramidal central configuration.  相似文献   

20.
Because the precise measurement of the Martian gravitational field plays a significant role in the future Mars exploration program, the future dedicated Mars satellite-to-satellite tracking (Mars-SST) gravity mission in China is investigated in detail for producing the next generation of the Mars gravity field model with high accuracy. Firstly, a new semi-numerical synthetical error model of the cumulative Martian geoid height influenced by the major error sources of the space-borne instruments is precisely established and efficiently verified. Secondly, the deep space network in combination with the satellite-to-satellite tracking in the low-low (DSN-SST-LL) mode is a preferred design owing to the high precision determination of the gravity maps, the low technical complexity of the satellite system and the successful experiences with the Earth’s Gravity Recovery and Climate Experiment (GRACE) projects and the lunar Gravity Recovery and Interior Laboratory (GRAIL) program. Finally, the future twin Mars-SST satellites plan to adopt the optimal matching accuracy indices of the satellite-equipped sensors (e.g., \(10^{-7}\) m/s in the inter-satellite range-rate from the interferometric laser ranging system (ILRS), 35 m in the orbital position tracked by the DSN and \(3\times 10^{-11}\) m/s2 in the non-conservative force from the drag-free control system (DFCS)) and the preferred orbital parameters (e.g., the orbital altitude of \(100\pm 50\) km and the inter-satellite range of \(50\pm 10\) km).  相似文献   

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