A Study on the Precise Design of Direct Transfer Trajectory of Mars Probe

Aiming at the problem of the precise design of the direct transfer trajectory of Mars probe, this paper proposes an algorithm of fast differential correction. It is based on the mathematical model of the difference between the control and target parameters, henceforth the matrix of partial derivatives of the system is solved. This can effectively reduce the number of times of integration in the process of solution. Taking the opportunity of the 2018 Mars probe as an example, the algorithm is verified. The results of emulation show that by using the initial values yielded by the method of patched conical curves, only 6-9 orbit integral iterations are needed to get a standard trajectory. Via the STK (satellite tool kit) technique, the results of computation are compared and justified.     

on the relationship between fast lyapunov indicator and periodic orbits for symplectic mappings

The computation on a relatively short time of a quantity, related to the largest Lyapunov Characteristic Exponent, called Fast Lyapunov Indicator allows to discriminate between ordered and weak chaotic motion and also, under certain conditions, between resonant and non resonant regular orbits. The aim of this paper is to study numerically the relationship between the Fast Lyapunov Indicator values and the order of periodic orbits. Using the two-dimensional standard map as a model problem we have found that the Fast Lyapunov Indicator increases as the logarithm of the order of periodic orbits up to a given order. For higher order the Fast Lyapunov Indicator grows linearly with the order of the periodic orbits. We provide a simple model to explain the relationship that we have found between the values of the Fast Lyapunov Indicator, the order of the periodic orbits and also the minimum number of iterations needed to obtain the Fast Lyapunov Indicator values.     

A fast Stokes inversion technique based on quadratic regression

Stokes inversion calculation is a key process in resolving polarization information on radiation from the Sun and obtaining the associated vector magnetic fields. Even in the cases of simple local thermodynamic equilibrium(LTE) and where the Milne-Eddington approximation is valid, the inversion problem may not be easy to solve. The initial values for the iterations are important in handling the case with multiple minima. In this paper, we develop a fast inversion technique without iterations. The time taken for computation is only 1/100 the time that the iterative algorithm takes. In addition, it can provide available initial values even in cases with lower spectral resolutions. This strategy is useful for a filter-type Stokes spectrograph, such as SDO/HMI and the developed two-dimensional real-time spectrograph(2DS).     

Convergence of Newton's iteration for the expansion of the planetary disturbing function

A method is suggested for choosing the first approximation in Newton's iterations to expand the planetary disturbing function. The method ensures convergence of the process for any planetary orbits. An estimation is given for the number of iterations depending on a given accuracy of calculation.     

A numerical method for determining the temperature structure of planetary atmospheres

A numerical method for calculating the time-average, vertical temperature structure of planetary atmospheres is presented. It is assumed that the atmospheres are in radiative-convective equilibrium, which is a good first approximation to many situations. Numerical tests of the rate of convergence and accuracy of the answer are presented. The method can readily handle molecular sources of opacity. Accurate results can be obtained with a minimum of computer time, because the number of iterations needed (～ 4) is small and the number of pressure levels at which the net flux needs to be evaluated (～ 10) is small. As an application of this procedure, we have calculated some model atmospheres of Jupiter.     

On the construction of model stellar atmospheres

The construction of model atmospheres by means of differential corrections is discussed. Either flux or flux gradients may be minimized, with arbitrary weighting with depth. For the simple atmospheres studied, no convergence problems were encountered even when very poor first approximations were used; and flux constancy in the radiative case was generally attained in three or four iterations.Quantities computed in one iteration may be re-used, not only in subsequent iterations within the model, but also in the construction of other models with differentT _eff org; and so the method is particularly suited for grid computations.     

Data boundary fitting using a generalized least-squares method

In many astronomical problems one often needs to determine the upper and/or lower boundary of a given data set. An automatic and objective approach consists in fitting the data using a generalized least-squares method, where the function to be minimized is defined to handle asymmetrically the data at both sides of the boundary. In order to minimize the cost function, a numerical approach, based on the popular downhill simplex method, is employed. The procedure is valid for any numerically computable function. Simple polynomials provide good boundaries in common situations. For data exhibiting a complex behaviour, the use of adaptive splines gives excellent results. Since the described method is sensitive to extreme data points, the simultaneous introduction of error weighting and the flexibility of allowing some points to fall outside of the fitted frontier, supplies the parameters that help to tune the boundary fitting depending on the nature of the considered problem. Two simple examples are presented, namely the estimation of spectra pseudo-continuum and the segregation of scattered data into ranges. The normalization of the data ranges prior to the fitting computation typically reduces both the numerical errors and the number of iterations required during the iterative minimization procedure.     

Multilevel Line Transfer with the Implicit Integral Method

Once the need for an iterative procedure in order to solve the problem of the formation of spectral lines in the case of a model atom with many energy levels, the sequel is to seek for the most effective form of such an iterative scheme. It is an almost universal is assumed within all the iterative methods for the solution of those radiative transfer problems, in which the transfer equations are coupled to the state of the matter, to take as the input of each step of iterations the values of the opacity coefficients obtained as a result of the previous one. This is, for instance, the procedure used to correct the temperature in the computation of stellar atmosphere models, or that to build the -operator (either the exact or the approximated one) within the Accelerated Lambda Iteration methods. Yet, if we assume, in order to solve the multilevel line transfer problem, that at each step of iterations the line opacities are known, we can express via the statistical equilibrium equations the populations of the energy levels - and consequently the source functions of the relevant spectral lines - as a linear function of the full set of the corresponding mean intensities of the radiation field. Once such linear forms for the source functions, we are able to solve without the need of any further approximation the radiative transfer equations for are obtained lines, now linearly coupled through the above linear forms of the statistical equilibrium equations. This is achieved by means of the Implicit Integral Method that we already presented in a series of previous papers.     

The accuracy of light curve interpretation programs revisited

It is stressed that poor convergence criteria are often used in light-curve synthesis programs. A large number of iterations (several tens) is needed to arrive at a definite conclusion about convergence. The result is usually that of some parameters little or no knowledge is gained and that the formal error estimates of all parameters are unrealistically small.     

Determination of the mass-ratio distribution,II: Double-lined spectroscopic binary stars

The application of the Richardson-Lucy iterative technique for the rectification of measurement errors, to the mass-ratio distribution of the double-lined spectroscopic binary stars (SBII) in theSixth Catalogue of the Orbital Elements of Spectroscopic Binary Stars, is evaluated for the SBII systems in the Eighth version of the Catalogue.The estimates of the real distribution, produced by the Richardson-Lucy method after successive iterations, do not converge in a trivial way. The stop criterion, suggested in the original paper on the method, is used in the application of the method to both real and simulated distributions. It is shown that, in the application to the mass-ratio distribution of the SBII systems, the iterations should stop considerably earlier than was previously assumed.The enhanced peak at mass-ratios near unity, that was found to exist in the real mass-ratio distribution, based on the earlier application of the iterative technique to the SBII systems, is likely to be due to an overshoot effect that occurs when the iterations are carried too far. When the stop criterion is used in the application of the method to the SBII systems in the Eighth version of the Catalogue, the resulting estimate of the real mass-ratio distribution does not shown evidence for anenhanced number of binary stars with mass ratios near unity.     

Density models for the upper atmosphere

Modeling the effects of atmospheric drag is one of the more important problems associated with the determination of the orbit of a near-earth satellite. Errors in the drag model can lead to significant errors in the determination and prediction of the satellite motion. The uncertainty in the drag acceleration can be attributed to three separate effects: (a) errors in the atmospheric density model, (b) errors in the ballistic coefficient, and (c) errors in the satellite relative velocity. In a number of contemporary satellite missions, the requirements for performing the orbit determination and predictions in near real-time has placed an emphasis on density model computation time as well as the model accuracy. In this investigation, a comparison is made of three contemporary atmospheric density models which are candidates for meeting the current orbit computation requirements. The models considered are the analytic Jacchia-Roberts model, the modified Harris-Priester model, and the USSR Cosmos satellite derived density model. The computational characteristics of each of the models are compared and a modification to the modified Harris-Priester model is proposed which improves its ability to represent the diurnal variation in the atmospheric density.This investigation was supported by the NASA Goddard Spaceflight Center under contract NAS5-20946 and Contract NSG 5154.     

Bandlimited implicit Runge–Kutta integration for Astrodynamics

We describe a new method for numerical integration, dubbed bandlimited collocation implicit Runge–Kutta (BLC-IRK), and compare its efficiency in propagating orbits to existing techniques commonly used in Astrodynamics. The BLC-IRK scheme uses generalized Gaussian quadratures for bandlimited functions. This new method allows us to use significantly fewer force function evaluations than explicit Runge–Kutta schemes. In particular, we use a low-fidelity force model for most of the iterations, thus minimizing the number of high-fidelity force model evaluations. We also investigate the dense output capability of the new scheme, quantifying its accuracy for Earth orbits. We demonstrate that this numerical integration technique is faster than explicit methods of Dormand and Prince 5(4) and 8(7), Runge–Kutta–Fehlberg 7(8), and approaches the efficiency of the 8th-order Gauss–Jackson multistep method. We anticipate a significant acceleration of the scheme in a multiprocessor environment.     

Satellite onboard orbit propagation using Deprit’s radial intermediary

Short-term satellite onboard orbit propagation is required when GPS position measurements are unavailable due to an obstruction or a malfunction. In this paper, it is shown that natural intermediary orbits of the main problem provide a useful alternative for the implementation of short-term onboard orbit propagators instead of direct numerical integration. Among these intermediaries, Deprit’s radial intermediary (DRI), obtained by the elimination of the parallax transformation, shows clear merits in terms of computational efficiency and accuracy. Indeed, this proposed analytical solution is free from elliptic integrals, as opposed to other intermediaries, thus speeding the evaluation of corresponding expressions. The only remaining equation to be solved by iterations is the Kepler equation, which in most of cases does not impact the total computation time. A comprehensive performance evaluation using Monte-Carlo simulations is performed for various orbital inclinations, showing that the analytical solution based on DRI outperforms a Dormand–Prince fixed-step Runge–Kutta integrator as the inclination grows.     

Procedures for solving Kepler's equation

We review starting formulae and iteration processes for the solution of Kepler's equation, and give details of two complete procedures. The first has been in use for a number of years, but the second is entirely new. The new procedure operates with an iterative process that always gives fourth-order convergence and is taken to only two iterations. The error in the resulting solution then never exceeds 7×10^–15 rad.     

On universal elements,and conversion procedures to and from position and velocity

An element set is advocated that is familiar (in traditional terms), and yet applicable to every type of conic-section orbit without loss of accuracy. It is not free of singularity, but this is not a serious deficiency. Conversion procedures, to and from position and velocity, are outlined, with Fortran-77 listings appended. Tests have indicated that the errors in the pair of procedures are minimal, accuracy being limited only by computer precision and the (fixed) number of iterations used in the Kepler-equation solutions.     

Recurrence time in the homoclinic tangle

We find the minimum recurrence time for the lobes of an unstable periodic orbit (i.e. the number of iterations required in order that an image of a given lobe intersects itself). This time is much shorter than the usual recurrence time derived by applying Poincaré's theorem. As the energy of the system increases the minimum recurrence time decreases. The minimum recurrence time can be found also when the energy exceeds the escape energy, in which case the usual Poincaré recurrence time is not defined.     

Effects of partial frequency redistribution on the level population densities in a resonance line

We have obtained a simultaneous solution of the statistical equilibrium equation for a non-LTE two-level atom and the radiative transfer equation in the comoving frames by employing the angle-averaged partial frequency redistribution.R _i with isotropic scattering. In the first iteration we have set the population density of the upper level equal to zero and allow it to be populated in the subsequent iterations. The solution converges within two to four iterations. The process of iteration is terminated when the ratios of population densities in two successive iterations at each radial point, attain an accuracy of 1%. The effects of partial frequency redistribution is to increase the population density of the upper level. Radial gas motions do not seem to have significant effects, although in highly extend geometries, velocity gradients change the population densities considerably.     

An analytical theory for tesseral gravitational harmonics

An analytical method has been developed for the treatment of tesseral harmonic perturbations. The procedure is an iterative Lie transformation technique which avoids the typical eccentricity expansions as well as the numerical singularities normally associated with resonance conditions. At each iteration, terms of the perturbing potential become multiplied by the ratio of the satellite's orbital period to the earth's rotational period. Following a suitable number of iterations, the potential is deemed to be sufficiently small that it may be ignored, with the tesseral effects captured in the transformation. This revised version was published online in July 2006 with corrections to the Cover Date.     

Optimization of Low-Thrust Trajectories Using an Indirect Shooting Method without Guesses of Initial Costates

According to the optimal control theory, the optimal control problem of the low-thrust tra jectory can be converted into a solution of nonlinear two- point boundary-value problem (TPBVP). To solve the TPBVP, the repeated random guesses for the initial costate variables and iterative computations are needed. In order to enhance the convergence of the iterations, we select an appropriate performance index, and then linearize the equations of the TPBVP around a Keplerian orbit. For multi-revolution transfers, instead of the multi- revolution Lambert tra jectory, multiple segmented Keplerian arcs are used to ensure the effectiveness of the linearization. The method is totally automatic with multiple iterations. With this method, we can get the results within 3 ∼ 5 iterations, and the random guess of the initial costates is unnecessary. Finally by the iterative optimization of the performance index, a better control strategy approaching to the bang-bang control is obtained.     

Computation of the atmosphere-less light intensity curve during a total solar eclipse by using Lunar Reconnaissance Orbiter topography data and the DE430 astronomical ephemeris

Observations of the sky irradiation intensity in the visible wavelengths during a solar eclipse permit to model the Sun diameter,a key number to constrain the internal structure of our star.In this paper,we present an algorithm that takes advantage of the precise Moon topography from Lunar Reconnaissance Orbiter to compute,with a high resolution in time,the geometrical part(i.e.top-of-atmosphere,and for a given wavelength)of the sky irradiation at any given location on the Earth during these events.The algorithm is also able to model the Baily’s beads.We give as an application the theoretical computation of the light curve corresponding to the solar eclipse observed at Lakeland(Queensland,North Australia)on 2012 November 13.The application to real data,with the introduction of atmospheric and instrumental passbands,will be considered in a forthcoming paper.