Fast Procedure Solving Universal Kepler's Equation

We developed a procedure to solve a modification of the standard form of the universal Kepler’s equation, which is expressed as a nondimensional equation with respect to a nondimensional variable. After reducing the domain of the variable and the argument by using the symmetry and the periodicity of the equation, the method first separates the case where the solution is so small that it is given an inverted series. Second, it separates the cases where the elliptic, parabolic, or hyperbolic standard forms of Kepler’s equation are suitable. Here the separation is done by judging whether detouring these nonuniversal equations will cause a 1-bit loss of information to their nonuniversal solutions or not. Then the nonuniversal equations are solved by the author’s procedures to solve the elliptic Kepler’s equation (Fukushima, 1997a), Barker’s equation (Fukushima, 1998), and the hyperbolic Kepler’s equation (Fukushima, 1997b), respectively. And their nonuniversal solutions are transformed back to the solution of the universal equation. For the rest of the case, we obtain an approximate solution by solving roughly the approximated cubic equation as we did in solving Barker’s equation. Then the correction to the approximate solution is obtained by Halley’s method precisely. There the special function appeared in the universal equation is rewritten into a combination of similar special functions of small arguments, so that they are efficiently evaluated by their Taylor series. Numerical measurements showed that, in the case of Intel Pentium II processor, the new method is 10–25 times as fast as Shepperd’s method (Shepperd, 1985) and 7–13 times as fast as the standard Newton method. This revised version was published online in July 2006 with corrections to the Cover Date.     

Photon path-length distribution function (II)

The determination of the photon path-length distribution function (PLDF) in a semi-infinite plane-parallel homogeneous atmosphere is discussed while the atmosphere scatters radiation according to the 2 × 2 Rayleigh-Cabannes phase matrix. The Piessens-Huysmans method of numerically inverting the Laplace transform which proved to be successful for the non-polarized radiation works in this special case as well. To employ this method we had to define the complex H-matrix and to find a fast method to determine its numerical values. For determining the average path-lengths and the dispersion we set up a system of integral equations the solution of which gave us the H-matrix and its first two derivatives with respect to the albedo of single scattering.The influence of different parameters characterizing the interaction of the polarized radiation with the atmosphere on the PLDF and the average path-length is studied in detail and a sample of average path-lengths is given in Table I.     

Attenuation of small-amplitude oscillations in a prominence–corona model with a transverse magnetic field

Observations show that small-amplitude prominence oscillations are usually damped after a few periods. This phenomenon has been theoretically investigated in terms of non-ideal magnetoacoustic waves, non-adiabatic effects being the best candidates to explain the damping in the case of slow modes. We study the attenuation of non-adiabatic magnetoacoustic waves in a slab prominence embedded in the coronal medium. We assume an equilibrium configuration with a transverse magnetic field to the slab axis and investigate wave damping by thermal conduction and radiative losses. The magnetohydrodynamic equations are considered in their linearised form and terms representing thermal conduction, radiation and heating are included in the energy equation. The differential equations that govern linear slow and fast modes are numerically solved to obtain the complex oscillatory frequency and the corresponding eigenfunctions. We find that coronal thermal conduction and radiative losses from the prominence plasma reveal as the most relevant damping mechanisms. Both mechanisms govern together the attenuation of hybrid modes, whereas prominence radiation is responsible for the damping of internal modes and coronal conduction essentially dominates the attenuation of external modes. In addition, the energy transfer between the prominence and the corona caused by thermal conduction has a noticeable effect on the wave stability, radiative losses from the prominence plasma being of paramount importance for the thermal stability of fast modes. We conclude that slow modes are efficiently damped, with damping times compatible with observations. On the contrary, fast modes are less attenuated by non-adiabatic effects and their damping times are several orders of magnitude larger than those observed. The presence of the corona causes a decrease of the damping times with respect to those of an isolated prominence slab, but its effect is still insufficient to obtain damping times of the order of the period in the case of fast modes.     

Fast computation of Jacobian elliptic functions and incomplete elliptic integrals for constant values of elliptic parameter and elliptic characteristic

In order to accelerate the numerical evaluation of torque-free rotation of triaxial rigid bodies, we present a fast method to compute various kinds of elliptic functions for a series of the elliptic argument when the elliptic parameter and the elliptic characteristic are fixed. The functions we evaluate are the Jacobian elliptic functions and the incomplete elliptic integral of the second and third kinds regarded as a function of that of the first kind. The key technique is the utilization of the Maclaurin series expansion and the addition theorems with respect to the elliptic argument. The new method is around 25 times faster than the method using the incomplete elliptic integral of general kind and around 70 times faster than the method using mathematical libraries given in the latest version of Numerical Recipes.     

On the possibility of a bimodal solar dynamo

A simple way to couple an interface dynamo model to a fast tachocline model is presented, under the assumption that the dynamo saturation is due to a quadratic process and that the effect of finite shear layer thickness on the dynamo wave frequency is analogous to the effect of finite water depth on surface gravity waves. The model contains one free parameter which is fixed by the requirement that a solution should reproduce the helioseismically determined thickness of the tachocline. In this case it is found that, in addition to this solution, another steady solution exists, characterized by a four times thicker tachocline and 4–5 times weaker magnetic fields. It is tempting to relate the existence of this second solution to the occurrence of grand minima in solar activity. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The time lag of high-energy photons of Cyg X-1 in hard state

The time lag between hard and soft X-rays is an important aspect of the study of high-energy emission. Fourier cross spectrum, often used in calculating the time lag, is unable to measure statistically significant fast light variation above Fourier frequency 30 Hz from the measurements of Cyg X-1. The method of cross correlation function in time domain, developed by Li Ti-pei, can be effectively used in measuring time lags on different time scales. Using Li's method we discuss the properties of the time lag of high-energy photons of Cyg X-1 in hard state at different times. The results show that there is a significant time lap on short times scales (< 0.1 s). We confront different models of hard-state Cygnus X-1 with the observed time lag.     

Modified Newton-Raphson methods for preliminary orbit determination

Preliminary orbit determination is a multipoint boundary value problem which may be solved by the generalized Newton-Raphson iteration. When applied formally the method suffers from extensive computer storage requirements, fairly long execution times and in some cases, insufficient accuracy. In this work we seek to remove these practical difficulties via modification of the computational algorithm in such a way that solution storage is eliminated for the most part and computational speed and tolerance to imprecise integration algorithms is improved. The modified methods are applied to nine typical preliminary orbit determination problems to demonstrate fast convergence and short computation times, even with very poor starting values for the iteration. Excellent precision of the resulting solution is also demonstrated as well as the algorithm's ability to handle circular, elliptic, parabolic and hyperbolic orbits.     

Science, Technology and Detectors for Extremely Large Telescopes

The second decade of the third millennium will hopefully see a new generation of extremely large telescopes. These will have diameters from 30 to 100 meters and use advanced adaptive optics to operate at the diffraction limit in order to detect astronomical objects that are impossible to observe today, such as earth-like planets around nearby stars and the earliest objects in the Universe. Even for small fields of view, the requirements for detectors are daunting, with sizes of several gigapixels, very fast readout times and extremely low readout noise. In this paper I briefly review the science case for ELTs and the requirements they set on telescopes and instruments, and report on the status of the OWL 100 m telescope project and the challenges it poses.     

MHD shock interactions in coronal structures

We consider the magnetohydrodynamic (MHD) interactions of solar coronal fast shock waves of flare and/or nonflare origin with the boundaries of coronal streamers and coronal holes. Boundaries are treated as MHD tangential discontinuities (TD). Different parameters of the observed corona are used in the investigation. The general case of the oblique interaction is studied.It is shown that a solar fast shock wave must be refracted usually as a fast shock wave inside the coronal streamer. For the special case of the velocity shear across TD, a slow shock wave is generated. On the contrary, the shock wave refracted inside the coronal hole is indeed a slow shock wave.The significance of different effects due to the interaction of fast and slow shock waves on the coronal magnetic field is noticed, especially at the time of a coronal mass ejection (CME). It is also shown, that an oblique fast MHD coronal shock wave may trigger an instability at the boundary of a streamer considered as a TD. It might have a relation with the observed process of abrupt disappearance of the streamer's boundary in the solar corona.On leave from the Academy of Sciences, Central Astronomical Observatory Pulkovo, 196140, St. Petersburg, Russia.     

Evaluation of a Wavefront Sensing Concept for a Full Solar Disk Adaptive Optics System

Solar scintillations were investigated as a method of detecting seeing-dependent wavefront curvature variations for solar adaptive optics. The method is applicable to full-disk averaged, adaptive corrections of large-scale near-field seeing distortions. As a test case, seeing measurements were carried out on a single small adaptive mirror, representing one element of what would be a large multielement adaptive array. All observations showed a greater than 5× reduction of seeing distortions as a result of this wavefront correction.     

On the Asynchronous Rotation of Accretors in Interacting Binaries

We show that the gaining component in interacting binaries can rotate faster than its orbital revolution as a consequence of the accretion process. We derive an approximative analytical formula for the Roche lobe radius of asynchronously rotating accretors. We present the case of the semi-detached interacting binary TX UMa, for which we measured directly asynchronous rotation of its accretor. We suggest a method to detect indirectly a fast spinning of accretors in symbiotic binaries based on the the X-ray luminosity of the boundary layer. We demonstrate this possibility for the case of EG And.     

Accumulation of Fourier component phases during object observation through a turbulent atmosphere. III

We investigate whether it is possible to overcome the limitations on the efficient application of Fourier-component phase accumulation method. We suggest the use of an instrument with a composite aperture for observation. It is shown that, in this case, the limitations are shifted as many times as the subaperture’s size is less than the typical size of atmospheric inhomogeneities. The interferograms obtained by means of a multibeam interpherometer are processed with the help of the total phase accumulation method using a computer model and the results are presented in the paper.     

A Method Solving Kepler’s Equation for Hyperbolic Case

We developed a method to solve Keplers equation for the hyperboliccase. The solution interval is separated into three regions; F 1, F 1, F 1. For the region F is large, we transformed the variablefrom F to exp F and applied the Newton method to the transformed equation.For the region F is small, we expanded the equation with respect to F andapplied the Newton method to the approximated equations. For the middleregion, we adopted a discretization of the Newton method and used the Taylorseries expansion for the evaluation of transcendental functions (Fukushima,1997). Numerical measurements showed that, in the case of Intel Pentiumprocessor, the new method is more than 3.7 times as fast as the combinationof a fourth order correction formula (Fukushima, 1997) and Roysstarting procedure (Roy, 1988) and others.     

A Fast Current-Field Iteration Method for Calculating Nonlinear Force-Free Fields

Existing methods for calculating nonlinear force-free magnetic fields are slow, and are likely to be inadequate for reconstructing coronal magnetic fields based on high-resolution vector magnetic field data from a new generation of spectro-polarimetric instruments. In this paper a new implementation of the current-field iteration method is presented, which is simple, fast, and accurate. The time taken by the method scales as N ⁴, for a three-dimensional grid with N ³ points. The method solves the field-updating part of the iteration by exploiting a three-dimensional Fast Fourier Transform solution of Ampere’s law with a current density field constructed to satisfy the required boundary conditions, and uses field line tracing to solve the current-updating part of the iteration. The method is demonstrated in application to a known nonlinear force-free field and to a bipolar test case.     

Fast galactic dynamos and interstellar magnetic bubbles

Large ( > 100 pc) interstellar magnetic bubbles are necessary in the cosmic-ray-driven fast galactic dynamo, as pioneered by Parker in 1992. In a first part, a look is made at the available data on nearby (< 1000 pc) large interstellar magnetic bubbles. Here the magnetic field strengthB in a large shell of densityn around an OB association is found to be a few times greater than that outside in the general interstellar medium, varying typically likeB ~n, as expected for a shocked medium. In a second part, some tests are made of the predictions about interstellar magnetic bubbles made by the theory of a cosmic-ray driven fast galactic dynamo. The bubble tests generally support the idea of a cosmic-ray-driven fast galactic dynamo for the Milky Way.     

Automatic removal of false image stars in disk-resolved images of the Cassini Imaging Science Subsystem

Taking a large number of images, the Cassini Imaging Science Subsystem(ISS) has been routinely used in astrometry. In ISS images, disk-resolved objects often lead to false detection of stars that disturb the camera pointing correction. The aim of this study was to develop an automated processing method to remove the false image stars in disk-resolved objects in ISS images. The method included the following steps:extracting edges, segmenting boundary arcs, fitting circles and excluding false image stars. The proposed method was tested using 200 ISS images. Preliminary experimental results show that it can remove the false image stars in more than 95% of ISS images with disk-resolved objects in a fully automatic manner,i.e., outperforming the traditional circle detection based on Circular Hough Transform(CHT) by 17%. In addition, its speed is more than twice as fast as that of the CHT method. It is also more robust(no manual parameter tuning is needed) when compared with CHT. The proposed method was also applied to a set of ISS images of Rhea to eliminate the mismatch in pointing correction in automatic procedure. Experiment results showed that the precision of final astrometry results can be improve by roughly 2 times that of automatic procedure without the method. It proved that the proposed method is helpful in the astrometry of ISS images in a fully automatic manner.     

Removing cosmic-ray hits from CCD images in real-time mode by means of an artificial neural network

A feed-forward artificial neural network has been implemented to the problem of removing cosmic-ray hits (CRH) from CCD images. The results of a number of tests demonstrate the effectiveness of this method especially for undersampled stellar profiles. The problem of optimal and low price preparing of training data, which could enable real-time or at least fast post-processing filtering out of CRH is discussed. The training and test ensembles were composed of a number of synthetic stellar profiles involving different S/N ratios and CRH images taken from real data. Certain aspects of the network’s architecture and its training efficiency for different modes of the back-propagation procedure as well as for the pre-process normalization of data have been examined. It is shown that for training set composed of stellar images and CRH at a ratio of 1:2 recognition can reach 99% in the case of stars and 96% for CRH. To determine the extent to which the cognition power of a network trained using an ensemble of circular symmetric stellar profiles of a given radius can be generalised the test data included stellar profiles of different radii, as well as elongated profiles. The goal was to mimic temporal changes in seeing as well as such problems as image defocusing, the lack of isoplanatism and improper sideral tracking of a telescope. The experiments provided us with the conclusion that for S/N > 10 excellent classification property is maintained in cases where the change in the radius of a circular profile is up to 30%, as well as for elongated profiles where the longest dimension is almost double that of the shortest one. Moreover, the generalization capability has been investigated for test images of synthetic pairs of overlapping stars with different distances between components. Almost 99% recognition efficiency was achieved even if the separation was nearly three times the radius of the stellar profile, a case when two stars could be analyzed by appropriate software as separate objects. The example of removal of CRH from real CCD images is presented to give an idea of how an algorithm based on a neural network can work in practice. The result of such an experiment appears fully consistent with the conclusions drawn from the tests made on synthetic data.     

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.     

Spatial Structure and Spectra of X-Ray Sources During the 0.8 – 4.5 GHz Reverse Drift Bursts Observations

In the years 2002 – 2005, 38 groups of the reverse drift bursts (RDBs) were observed in the 0.8 – 4.5 GHz frequency range by the Ondřejov radiospectrograph. In 21 cases, which were observed at the times of the RHESSI observations, spatial structure, positional changes, and spectra of X-ray sources during RDB observations are studied in detail. First, based on the frequency drift and the spatial structure of the associated X-ray source, the events are classified as: (a) fast drifting RDBs with a compact X-ray source, (b) fast drifting RDBs with a multiple X-ray source (FM), and slowly drifting RDBs. Then, the spectra of X-ray sources at the times of RDBs are analyzed. It is found that most fast drifting RDBs (16 of 17 cases) are associated with the spectra having a distinct power-law (non-thermal) component. In contrast, the X-ray spectra associated with the slowly drifting RDBs are predominantly purely thermal (in three out of four cases; in the 26 July 2004, case the X-ray spectrum is thermal and high temperature, with non-thermal component). Two special cases of RDBs observed during the 28 October 2003, and 23 July 2004, flares are added for comparison. The most frequent events are those with fast drifting RDBs, a compact short-lasting X-ray sources, and a power-law X-ray spectrum. The individual reverse drift bursts (∼1 s duration) do not show a clear temporal association with individual peaks of hard X-ray bursts. During slowly drifting RDBs the shape of the associated X-ray source changed or expanded. Among them the most interesting one was observed in 26 July 2004, when the very slowly drifting RDBs (+40 MHz s⁻¹) were associated with an X-ray loop-like source continuously elongating in the southwest direction. In the most cases the model of RDBs with electron beams is compatible with the observations, but in flares on 26 July 2004, and 28 October 2003, the RDBs are probably generated by some other type of an agent; we propose here a thermal conduction front.     

Prediction Test for the Two Extremely Strong Solar Storms in October 2003

In late October and early November 2003, a series of space weather hazard events erupted in solar-terrestrial space. Aiming at two intense storm (shock) events on 28 and 29 October, this paper presents a Two-Step method, which combines synoptic analysis of space weather–`observing’ and quantitative prediction – ‘palpating’, and uses it to test predictions. In the first step, ‘observing’, on the basis of observations of the source surface magnetic field, interplanetary scintillation (IPS) and ACE spacecraft, we find that the propagation of the shock waves is asymmetric and northward relative to the normal direction of their solar sources due to the large-scale configuration of the coronal magnetic fields, and the Earth is located near the direction of the fastest speed and greatest energy of the shocks. Being two fast ejection shock events, the fast explosion of extremely high temperature and strong magnetic field, and background solar wind velocity as high as 600 and 1000 km s⁻¹, are also helpful to their rapid propagation. According to the synoptic analysis, the shock travel times can be estimated as 21 and 20 h, which are close to the observational results of 19.97 and 19.63 h, respectively. In the second step, ‘palpating’, we adopt a new membership function of the fast shock events for the ISF method. The predicted results here show that for the onset time of the geomagnetic disturbance, the relative errors between the observational and the predicted results are 1.8 and 6.7%, which are consistent with the estimated results of the first step; and for the magnetic disturbance magnitude, the relative errors between the observational and the predicted results are 4.1 and 3.1%, respectively. Furthermore, the comparison among the predicted results of our Two-Step method with those of five other prevailing methods shows that the Two-Step method is advantageous in predicting such strong shock event. It can predict not only shock arrival time, but also the magnitude of magnetic disturbance. The results of the present paper tell us that understanding the physical features of shock propagation thoroughly is of great importance in improving the prediction efficiency.