Element sets for high-order Poincaré mapping of perturbed Keplerian motion

The propagation and Poincaré mapping of perturbed Keplerian motion is a key topic in Celestial Mechanics and Astrodynamics, e.g., to study the stability of orbits or design bounded relative trajectories. The high-order transfer map (HOTM) method enables efficient mapping of perturbed Keplerian orbits using the high-order Taylor expansion of a Poincaré or stroboscopic map. The HOTM is only accurate close to the expansion point and therefore the number of revolutions for which the map is accurate tends to be limited. The proper selection of coordinates is of key importance for improving the performance of the HOTM method. In this paper, we investigate the use of different element sets for expressing the high-order map in order to find the coordinates that perform best in terms of accuracy. A new set of elements is introduced that enables extremely accurate mapping of the state, even for high eccentricities and higher-order zonal perturbations. Finally, the high-order map is shown to be very useful for the determination and study of fixed points and center manifolds of Poincaré maps.     

Dealing with uncertainties in angles-only initial orbit determination

A method to deal with uncertainties in initial orbit determination (IOD) is presented. This is based on the use of Taylor differential algebra (DA) to nonlinearly map uncertainties from the observation space to the state space. When a minimum set of observations is available, DA is used to expand the solution of the IOD problem in Taylor series with respect to measurement errors. When more observations are available, high order inversion tools are exploited to obtain full state pseudo-observations at a common epoch. The mean and covariance of these pseudo-observations are nonlinearly computed by evaluating the expectation of high order Taylor polynomials. Finally, a linear scheme is employed to update the current knowledge of the orbit. Angles-only observations are considered and simplified Keplerian dynamics adopted to ease the explanation. Three test cases of orbit determination of artificial satellites in different orbital regimes are presented to discuss the feature and performances of the proposed methodology.     

Asteroid close encounters characterization using differential algebra: the case of Apophis

A method for the nonlinear propagation of uncertainties in Celestial Mechanics based on differential algebra is presented. The arbitrary order Taylor expansion of the flow of ordinary differential equations with respect to the initial condition delivered by differential algebra is exploited to implement an accurate and computationally efficient Monte Carlo algorithm, in which thousands of pointwise integrations are substituted by polynomial evaluations. The algorithm is applied to study the close encounter of asteroid Apophis with our planet in 2029. To this aim, we first compute the high order Taylor expansion of Apophis’ close encounter distance from the Earth by means of map inversion and composition; then we run the proposed Monte Carlo algorithm to perform the statistical analysis.     

数值历表对动力系统参数的偏导数

本文推导了天体运动方程的数值解对积分初始条件、天体质量等动力系统参数的偏导数所满足的微分方程和初始条件。     

Optimal two-impulse rendezvous using constrained multiple-revolution Lambert solutions

A solution to the fixed-time minimum-fuel two-impulse rendezvous problem for the general non-coplanar elliptical orbits is provided. The optimal transfer orbit is obtained using the constrained multiple-revolution Lambert solution. Constraints consist of lower bound for perigee altitude and upper bound for apogee altitude. The optimal time-free two-impulse transfer problem between two fixed endpoints implies finding the roots of an eighth order polynomial, which is done using a numerical iterative technique. The set of feasible solutions is determined by using the constraints conditions to solve for the short-path and long-path orbits semimajor axis ranges. Then, by comparing the optimal time-free solution with the feasible solutions, the optimal semimajor axis for the two fixed-endpoints transfer is identified. Based on the proposed solution procedure for the optimal two fixed-endpoints transfer, a contour of the minimum cost for different initial and final coasting parameters is obtained. Finally, a numerical optimization algorithm (e.g., evolutionary algorithm) can be used to solve this global minimization problem. A numerical example is provided to show how to apply the proposed technique.     

Exact analytic solution for the rotation of a rigid body having spherical ellipsoid of inertia and subjected to a constant torque

The exact analytic solution is introduced for the rotational motion of a rigid body having three equal principal moments of inertia and subjected to an external torque vector which is constant for an observer fixed with the body, and to arbitrary initial angular velocity. In the paper a parametrization of the rotation by three complex numbers is used. In particular, the rows of the rotation matrix are seen as elements of the unit sphere and projected, by stereographic projection, onto points on the complex plane. In this representation, the kinematic differential equation reduces to an equation of Riccati type, which is solved through appropriate choices of substitutions, thereby yielding an analytic solution in terms of confluent hypergeometric functions. The rotation matrix is recovered from the three complex rotation variables by inverse stereographic map. The results of a numerical experiment confirming the exactness of the analytic solution are reported. The newly found analytic solution is valid for any motion time length and rotation amplitude. The present paper adds a further element to the small set of special cases for which an exact solution of the rotational motion of a rigid body exists.     

A Hamiltonian approach to the planar optimization of mid-course corrections

Lawden’s primer vector theory gives a set of necessary conditions that characterize the optimality of a transfer orbit, defined accordingly to the possibility of adding mid-course corrections. In this paper a novel approach is proposed where, through a polar coordinates transformation, the primer vector components decouple. Furthermore, the case when transfer, departure and arrival orbits are coplanar is analyzed using a Hamiltonian approach. This procedure leads to approximate analytic solutions for the in-plane components of the primer vector. Moreover, the solution for the circular transfer case is proven to be the Hill’s solution. The novel procedure reduces the mathematical and computational complexity of the original case study. It is shown that the primer vector is independent of the semi-major axis of the transfer orbit. The case with a fixed transfer trajectory and variable initial and final thrust impulses is studied. The acquired related optimality maps are presented and analyzed and they express the likelihood of a set of trajectories to be optimal. Furthermore, it is presented which kind of requirements have to be fulfilled by a set of departure and arrival orbits to have the same profile of primer vector.     

Near distance approximation in astrodynamical applications of Lambert’s theorem

The smallness parameter of the approximation method is defined in terms of the non-dimensional initial distance between target and chaser satellite. In the case of a circular target orbit, compact analytical expressions are obtained for the interception travel time up to third order. For eccentric target orbits, an explicit result is worked out to first order, and the tools are prepared for numerical evaluation of higher order contributions. The possible transfer orbits are examined within Lambert’s theorem. For an eventual rendezvous it is assumed that the directions of the angular momenta of the two orbits enclose an acute angle. This assumption, together with the property that the travel time should vanish with vanishing initial distance, leads to a condition on the admissible initial positions of the chaser satellite. The condition is worked out explicitly in the general case of an eccentric target orbit and a non-coplanar transfer orbit. The condition is local. However, since during a rendezvous maneuver, the chaser eventually passes through the local space, the condition propagates to non-local initial distances. As to quantitative accuracy, the third order approximation reproduces the elements of Mars, in the historical problem treated by Gauss, to seven decimals accuracy, and in the case of the International Space Station, the method predicts an encounter error of about 12 m for an initial distance of 70 km.     

Polynomial approximation of Poincaré maps for hamiltonian systems

Different methods are proposed and tested for transforming a nonlinear differential system, and more particularly a hamiltonian one, into a map without having to integrate the whole orbit as in the well known Poincaré map technique. We construct piecewise polynomial maps by coarse-graining the phase surface of section into parallelograms using values of the Poincaré maps at the vertices to define a polynomial approximation within each cell. The numerical experiments are in good agreement with the standard map taken as a model problem. The agreement is better when the number of vertices and the order of the polynomial fit increase. The synthetic mapping obtained is not symplectic even if at vertices there is an exact interpolation. We introduce a second new method based on a global fitting . The polynomials are obtained using at once all the vertices and fitting by least square polynomes but in such a way that the symplectic character is not lost.     

A Simplified Kinetic Element Formulation for the Rotation of a Perturbed Mass-Asymmetric Rigid Body

Euler's equations, describing the rotation of an arbitrarily torqued mass asymmetric rigid body, are scaled using linear transformations that lead to a simplified set of first order ordinary differential equations without the explicit appearance of the principal moments of inertia. These scaled differential equations provide trivial access to an analytical solution and two constants of integration for the case of torque-free motion. Two additional representations for the third constant of integration are chosen to complete two new kinetic element sets that describe an osculating solution using the variation of parameters. The elements' physical representations are amplitudes and either angular displacement or initial time constant in the torque-free solution. These new kinetic elements lead to a considerably simplified variation of parameters solution to Euler's equations. The resulting variational equations are quite compact. To investigate error propagation behaviour of these new variational formulations in computer simulations, they are compared to the unmodified equations without kinematic coupling but under the influence of simulated gravity-gradient torques.     

Radiative Transfer in Inhomogeneous Atmospheres. II

This paper is a continuation of a study of radiative transfer in one-dimensional inhomogeneous atmospheres. Two of the most important characteristics of multiple scattering in these media are calculated: the photon escape probability and the average number of scattering events. The latter is determined separately for photons leaving the medium and for photons that have undergone thermalization in the medium. The problem of finding the radiation field in an inhomogeneous atmosphere containing energy sources is also examined. It is assumed that the power of these sources, as well as the scattering coefficient, can vary arbitrarily with depth. It is shown that knowledge of the reflection and transmission coefficients of the atmosphere makes it possible to reduce all these problems to solving some first order linear differential equations with specified initial conditions. A series of new analytic results are obtained. Numerical calculations are done for two types of atmosphere with different depth dependences for the scattering coefficient. These are interpreted physically.     

Analytical design methods for transfer trajectories between the Earth and the Lunar Orbital Station

Lunar Orbital Station (LOS) is proposed as support of manned lunar exploration missions. A fast-converging iteration method for determining the initial conditions of two-impulse transfer trajectories between the Earth and the LOS is proposed based on the patched conic approach. In the Earth phase, near Earth state is connected with the state at the lunar sphere of influence (LSOI) based on the relationship between the initial and terminal orbital state. Then, an analytical algorithm is proposed to find the state vector at LSOI, such to satisfy the LOS orbital constraint. An iterative process is finally adopted to generate favorable initial solutions that satisfy the constraint near the Earth and at the perilune. The algorithm convergence is investigated, and two types of transfer trajectories are found for both Earth-LOS and LOS-Earth transfer. Based on the algorithm, orbital transfer windows, velocity impulse and time of flight are analyzed in the typical years 2025 and 2034. At last, the initial solution is corrected with a high fidelity model based on the active-set method, which shows the precision of this algorithm. The novel procedure for the transfer trajectories design and the analytic result can be used as a basis for rapid mission evaluation and design for future manned lunar missions based on the LOS.     

Map-projection-independent crater size-frequency determination in GIS environments—New software tool for ArcGIS

Statistical analysis of crater size-frequency distributions (CSFDs) of impact craters on planetary surfaces is a well-established method to derive absolute ages on the basis of remotely-sensed image data. Although modelling approaches and the derivation of absolute ages from a given CSFD have been described and discussed in considerable depth since the late 1960s, there is no standardised methodology or guideline for the measurement of impact-crater diameters and area sizes that are both needed to determine absolute ages correctly. Distortions of distances (i.e., diameters) and areas within different map projections are considerable error sources during crater and area measurements.In order to address this problem and to minimize such errors, a software extension for Environmental Systems Research Institute's (ESRI's) ArcMap (ArcGIS) has been developed measuring CSFDs on planetary surfaces independently of image and data frame map projections, which can also be theoretically transferred to every Geographic Information System (GIS) capable of working with different map projections.Using this new approach each digitized impact crater is internally projected to a stereographic map projection with the crater's central-point set as the projection center. In this projection, the circle is defined without any distortion of its shape (i.e., conformality). Using a sinusoidal map projection with a center longitude set to the crater's central-point, the diameter of the impact crater is measured along this central meridian which is true-scale and does not show any distortion. The crater is re-projected to the map projection of the current data frame and stored as vector geometry with attributes. Output from this workflow comprises correct impact-crater diameters and area sizes in sinusoidal map projections and can be used for further processing, i.e. absolute age determinations (e.g., using the software CraterStats). The ArcMap toolbar CraterTools developed in this context significantly helps to improve and simplify the crater size-frequency (CSF) measurement process. For GIS-based measurements, we strongly recommend our procedure as the standard method for determining CSFDs on planetary surfaces to minimize map distortion effects for further analysis.     

Mixed convection over a horizontal plate with vectored mass transfer in a transverse magnetic field

An exact similarity solution is presented for developing mixed convection flows of electrically conducting fluids over a semi-infinite horizontal plate with vectored mass transfer at the wall which are subjected to an applied transverse magnetic field. This solution is given for the case of a wall temperature that is inversely proportional to the square root of the distance from the leading edge. By application of appropriate coordinate transformations, the governing momentum and energy boundary-layer equations are expressed as a set of coupled ordinary differential equations that depend on a magnetic parameter, the buoyancy parameter, and the Prandtl number. The shear stress, the total heat transfer, and the displacement thickness are calculated for different values of both buoyancy and magnetic parameters.     

A numerical method for differential equations of second order

A new algorithm of order five is presented for the solution of the initial value problem where the system of ordinary differential equations is of second order and does not contain the first derivative.     

基于改进VNet太阳暗条检测方法

太阳暗条作为太阳大气磁场的示踪,对研究太阳磁场有极其重要的意义。针对现有的暗条检测方法存在检测精度不高,弱小暗条错检、漏检等问题,提出一种基于改进VNet网络的太阳暗条检测方法。首先,使用大熊湖天文台Hα全日面图像并结合磁图制作了太阳暗条数据集;其次,在VNet网络下采样部分采用Inception模块融合不同尺度特征图的特征,同时加入注意力机制增强特征图中暗条部分的语义信息;最后在上采样部分引入深度监督模块,更多地保留太阳暗条的细节特征。为验证算法性能,采用191幅Hα全日面图像数据集,其中包含暗条共3372条。算法在测试数据集上平均准确率达到0.9883,F1值达到0.8385。实验结果证明,该方法可以有效识别Hα全日面图中的暗条。     

Differential rotation of photospheric magnetic fields associated with coronal holes

An interesting aspect of solar rotation is the fact that coronal holes seem to exhibit little or no differential rotation. We set out to investigate the question of whether or not the photospheric magnetic fields underlying coronal holes also exhibit reduced differential rotation. In order to accomplish this we measured the daily positions of filaments and plages surrounding a large coronal hole that lasted for several disk passages. The resulting differential rotation curve was considerably flatter than the standard curve for long-lived filaments and was in remarkably good agreement with the curve found for the overlying coronal hole itself.     

Radiative Transfer in Inhomogeneous Atmospheres. III

The approach proposed in the previous parts of this series of papers is used to solve the radiative transfer problem in scattering and absorbing multicomponent atmospheres. Linear recurrence relations are obtained for both the reflectance and transmittance of these kinds of atmospheres, as well as for the emerging intensities when the atmosphere contains energy sources. Spectral line formation in a one-dimensional inhomogeneous atmosphere is examined as an illustration of the possibility of generalizing our approach to the matrix case. It is shown that, in this case as well, the question reduces to solving an initial value problem for linear differential equations. Some numerical calculations are presented.     

Elliptical multi-sun-synchronous orbits for Mars exploration

The multi-sun-synchronous orbits allow cycles of observation of the same area in which solar illumination repetitively changes according to the value of the orbit elements and returns to the initial condition after a temporal interval multiple of the repetition of observation. This paper generalizes the concept of multi-sun-synchronous orbits, whose classical sun-synchronous orbits represent particular solutions, taking into consideration the elliptical case. The feasibility of using this typology of orbits, referred to as elliptical periodic multi-sun-synchronous orbits, has been investigated for the exploration of Mars and particular solutions have been selected. Such solutions considerably reduce the manoeuvre of velocity variation at the end of the interplanetary transfer with respect to the case of a target circular orbit around Mars. They are based on the use of quasi-critical inclinations in order to minimize the apsidal line motion and thus reduce orbit maintenance costs. Moreover, in the case of high eccentricities, the argument of pericentre may be set in order to obtain, around the apocentre, a condition of quasi-synchronism with the planet (the footprint of the probe on the surface presents a small shift with respect to a fixed point on the Martian surface). The low altitude of pericentre allows observation of the planet at a higher spatial resolution, while the orbit arc around the apocentre may be used to observe Mars with a wide spatial coverage in quasi-stationary conditions. This latter characteristic is useful for analysing atmospheric and meteorological phenomena and it allows for most of the orbital period a link between a rover on the surface of Mars and a probe orbiting around the planet.