The KS-transformation in hypercomplex form and the quantization of the negative-energy orbit manifold of the Kepler problem

In a previous note we have shown that the KS-transformation, introduced by Kustaanheimo and Stiefel into Celestial Mechanics for the regularization of the Kepler problem, may be formulated in terms of hypercomplex numbers as the product of a quaternion and its anti-involute, thus representing a particular morphism of the real algebra of quaternions-having for image the physical configuration space of the Kepler problem. In the present note we show, first, that this formulation allows a straight derivation of the Hopf fibering of the sphere S³ (characterized by unit quaternions) having the base space given by the sphere S² (characterized by unit vectors), and secondly that the KS-transformation allows the quantization of the symplectic manifold S² in the sense of Souriau, the associated quantum manifold S³ having a contact structure given by the bilinear relation characteristic of the KS-theory. Furthermore, after presenting a natural extension of the hypercomplex KS-transformation to the full phase space of the Kepler problem, we show that this extension allows the quantization of the manifold of Kepler orbits of fixed negative energy (manifold diffeomorphic to the symplectic product S²×S²). The energy levels satisfy a well known quantum integrality condition and the associated quantum manifold is diffeomorphic to the product manifold S³×S³ quotiented by a suitable equivalence relation.Research supported by the Consiglio Nazionale delle Ricerche of Italy, Gruppo per la Fisica-Matematica.     

Dependence of the correlation of small scale photospheric structures upon resolution

Correlations between continuum intensity, velocity, and equivalent widths of two Mn i lines as observed with two different entrance apertures tend to deteriorate with improvement in spatial resolution. The KPNO multichannel magnetograph was used to make area scans at the center of the disk with entrance apertures 3.5×2.5 arc and 1×1 arc. A coherence analysis shows that this effect is caused by marked differences of fluctuations in temperature and temperature gradients as well as in the velocity structure of photospheric elements of various sizes.     

Frequency spectra of solar image motion

The frequency analysis of image motion (IM) at the solar limb was carried out in the frequency range from 0.5 to 50 Hz using a photoelectric equipment. For a telescopic aperture of 35 cm and a bandwidth of 0.65 Hz a typical frequency spectrum under average observing conditions shows a decrease of amplitude from 2 arcsec at 0.5 Hz to 0.4 arcsec at 5 Hz, 0.03 arsec at 50 Hz (and < 0.01 arcsec at 500 Hz). Visually estimated values of image steadiness seem to be in better agreement with the r.m.s. value of image motion (scattering parameter ) than with the amplitude at a certain frequency (Figures 5a, b). The influence of IM on the quality of photographic pictures or on spectra of solar fine structures is calculated as a function of exposure time. Table II gives the IM scattering parameters (0.01 arcsec to 4 arcsec) calculated for exposure times from 0.001 to 0.5 sec — valid for a time average. The modulation transfer functions (MTF, one-dimensional) derived from the IM scattering parameters are presented in Figure 7 together with the MTF for a diffraction-limited telescope of 35 cm aperture at 6000 Å. Exposure times of less than approximately 0.01 sec (certain within a factor of 2) render the influence of IM negligible compared to the MTF of the objective used for this investigation.Mitteilungen aus dem Fraunhofer Institut Nr. 88.     

Computational advances in gravitational microlensing: A comparison of CPU,GPU, and parallel,large data codes

To assess how future progress in gravitational microlensing computation at high optical depth will rely on both hardware and software solutions, we compare a direct inverse ray-shooting code implemented on a graphics processing unit (GPU) with both a widely-used hierarchical tree code on a single-core CPU, and a recent implementation of a parallel tree code suitable for a CPU-based cluster supercomputer. We examine the accuracy of the tree codes through comparison with a direct code over a much wider range of parameter space than has been feasible before. We demonstrate that all three codes present comparable accuracy, and choice of approach depends on considerations relating to the scale and nature of the microlensing problem under investigation. On current hardware, there is little difference in the processing speed of the single-core CPU tree code and the GPU direct code, however the recent plateau in single-core CPU speeds means the existing tree code is no longer able to take advantage of Moore’s law-like increases in processing speed. Instead, we anticipate a rapid increase in GPU capabilities in the next few years, which is advantageous to the direct code. We suggest that progress in other areas of astrophysical computation may benefit from a transition to GPUs through the use of “brute force” algorithms, rather than attempting to port the current best solution directly to a GPU language – for certain classes of problems, the simple implementation on GPUs may already be no worse than an optimised single-core CPU version.     

CME Velocity and Acceleration Error Estimates Using the Bootstrap Method

The bootstrap method is used to determine errors of basic attributes of coronal mass ejections (CMEs) visually identified in images obtained by the Solar and Heliospheric Observatory (SOHO) mission’s Large Angle and Spectrometric Coronagraph (LASCO) instruments. The basic parameters of CMEs are stored, among others, in a database known as the SOHO/LASCO CME catalog and are widely employed for many research studies. The basic attributes of CMEs (e.g. velocity and acceleration) are obtained from manually generated height-time plots. The subjective nature of manual measurements introduces random errors that are difficult to quantify. In many studies the impact of such measurement errors is overlooked. In this study we present a new possibility to estimate measurements errors in the basic attributes of CMEs. This approach is a computer-intensive method because it requires repeating the original data analysis procedure several times using replicate datasets. This is also commonly called the bootstrap method in the literature. We show that the bootstrap approach can be used to estimate the errors of the basic attributes of CMEs having moderately large numbers of height-time measurements. The velocity errors are in the vast majority small and depend mostly on the number of height-time points measured for a particular event. In the case of acceleration, the errors are significant, and for more than half of all CMEs, they are larger than the acceleration itself.     

The Hypothesis of Additional Acceleration in the Motion of Comet Harrington–Abell from Jupiter

By processing 494 observations of Comet Harrington–Abell, we obtained a unified system of elements that includes its turn around the Sun during which it closely approached Jupiter to a minimum distance of 0.037 AU in 1974. A study of the cometary orbit before and after the approach showed that, probably, at the approach of the comet to Jupiter, apart from the well-known gravitational perturbations, its motion was affected by an additional force. An improvement of the cometary orbit by assuming that an additional acceleration inversely proportional to the square of the distance to Jupiter exists in its motion yielded the following values: (4.57 ± 0.42) × 10^–10 and (–7.20 ± 0.42) × 10^–10 AU day^–2 for the radial and transversal acceleration components, respectively. As a plausible explanation of the changes in the cometary orbit, we additionally considered a model based on the hypothesis of partial disintegration of the cometary nucleus. The parameter that characterizes the instant displacement of the center of inertia along the jovicentric radius vector was estimated to be –1.83 ± 0.75 km. Based on a unified numerical theory of cometary motion, we determined the nongravitational parameters using Marsden's model for two periods: A ₁ = (11.68 ± 1.74) × 10^–10 AU day^–2, A ₂ = (0.53 ± 0.0357) × 10^–10 AU day^–2 for 1975–1999 and A ₁ = (5.92 ± 5.86) × 10^–10 AU day^–2, A ₂ = (0.08 ± 0.028) × 10^–10 AU day^–2 for 1955–1969, under the assumption that the nongravitational acceleration changed at the approach of the comet to Jupiter.     

Interplanetary acceleration of energetic particles at 1 and 5 AU

Energetic particle (0.1 to 100 MeV protons) acceleration is studied by using high resolution interplanetary magnetic field and plasma measurements at 1 AU (HEOS-2) and at 5 AU (Pioneer 10). Energy changes of a particle population are followed by computing test particle trajectories and the energy changes through the particle interaction with the time varying magnetic field. The results show that considerable particle acceleration takes place throughout the interplanetary medium, both in the corotating interaction regions (CIR) (5 AU), and in quiet regions (1 AU). Although shocks may contribute to acceleration we suggest statistical acceleration within the CIRs is sufficient to explain most energetic particle observations (e.g., McDonaldet al., 1975; Barnes and Simpson, 1976).The first and second order statistical acceleration coefficients which include transit time damping and Alfvén resonance interactions, are found to be well represented byD _T 8.5×10^–6 T ^0.5 MeV s^–1 andD _TT 4×10^–6 T ^1.5 MeV² s^–1 at 5 AU.By comparison, Fisk's estimates (1976), based on quasi-linear theory for transit-time damping, gaveD _TT 5×10^–7 T MeV² s^–1 at 1 AU.     

Relativistic satellite orbits: central body with higher zonal harmonics

Satellite orbits around a central body with arbitrary zonal harmonics are considered in a relativistic framework. Our starting point is the relativistic Celestial Mechanics based upon the first post-Newtonian approximation to Einstein’s theory of gravity as it has been formulated by Damour et al. (Phys Rev D 43:3273–3307, 1991; 45:1017–1044, 1992; 47:3124–3135, 1993; 49:618–635, 1994). Since effects of order \((\mathrm{GM}/c^2R) \times J_k\) with \(k \ge 2\) for the Earth are very small (of order \( 7 \times 10^{-10}\,\times \,J_k\)) we consider an axially symmetric body with arbitrary zonal harmonics and a static external gravitational field. In such a field the explicit \(J_k/c^2\)-terms (direct terms) in the equations of motion for the coordinate acceleration of a satellite are treated first with first-order perturbation theory. The derived perturbation theoretical results of first order have been checked by purely numerical integrations of the equations of motion. Additional terms of the same order result from the interaction of the Newtonian \(J_k\)-terms with the post-Newtonian Schwarzschild terms (relativistic terms related to the mass of the central body). These ‘mixed terms’ are treated by means of second-order perturbation theory based on the Lie-series method (Hori–Deprit method). Here we concentrate on the secular drifts of the ascending node \(<\!{\dot{\Omega }}\!>\) and argument of the pericenter \(<\!{\dot{\omega }}\!>\). Finally orders of magnitude are given and discussed.     

Periodic orbits of solar sail equipped with reflectance control device in Earth–Moon system

In this paper, families of Lyapunov and halo orbits are presented with a solar sail equipped with a reflectance control device in the Earth–Moon system. System dynamical model is established considering solar sail acceleration, and four solar sail steering laws and two initial Sun-sail configurations are introduced. The initial natural periodic orbits with suitable periods are firstly identified. Subsequently, families of solar sail Lyapunov and halo orbits around the \(L_{1}\) and \(L_{2}\) points are designed with fixed solar sail characteristic acceleration and varying reflectivity rate and pitching angle by the combination of the modified differential correction method and continuation approach. The linear stabilities of solar sail periodic orbits are investigated, and a nonlinear sliding model controller is designed for station keeping. In addition, orbit transfer between the same family of solar sail orbits is investigated preliminarily to showcase reflectance control device solar sail maneuver capability.     

Solar Wind and Chromospheric Network

A physical model of the transition region, including upflow of the plasma in magnetic field funnels that are open to the overlying corona, is presented. A numerical study of the effects of Alfvén waves on the heating and acceleration of the nascent solar wind originating in the chromospheric network is carried out within the framework of a two-fluid model for the plasma. It is shown that waves with reasonable amplitudes can, through their pressure gradient together with the thermal pressure gradient, cause a substantial initial acceleration of the wind (on scales of a few Mm) to locally supersonic flows in the rapidly expanding magnetic field trunks of the transition region network. The concurrent proton heating is due to the energy supplied by cyclotron damping of the high-frequency Alfvén waves, which are assumed to be created through small-scale magnetic activity. The wave energy flux of the model is given as a condition at the upper chromosphere boundary, located above the thin layer where the first ionization of hydrogen takes place.Among the new numerical results are the following: Alfvén waves with an assumed f ^-1 power spectrum in the frequency range from 1 to ⁴ Hz, and with an integrated mean amplitude ranging between 25 and 75 km s⁴, can produce very fast acceleration and also heating through wave dissipation. This can heat the lower corona to a temperature of 5× 10⁵ K at a height of h=12,000 km, starting from 5× 10⁴ K at h=3000 km. The resulting thermal and wave pressure gradients can accelerate the wind to speeds of up to 150 km s^-1 at h=12,000 km, starting from 20 km s^-1 at h=3000 km in a rapidly diverging flux tube. Thus the nascent solar wind becomes supersonic at heights well below the classical Parker-Type sonic point. This is a consequence of the fact that any large wave-energy flux, if it is to be conducted through the expanding funnel to the corona, implies the building-up of an associated wave-pressure gradient. Because of the diverging field geometry, this might lead to a strong initial acceleration of the flow. There is a multiplicity of solutions, depending mainly on the coronal pressure. Here we discuss two new (as compared with a static transition region model) possibilities, namely that either the flow remains supersonic or slows down abruptly by shock formation, which then yields substantial coronal heating up to the canonical 10⁶ K for the proton temperature.     

Upper limit on the steady emission of the 2.223 MeV neutron capture γ-ray line from the Sun

Data accumulated by the Solar Maximum Mission Gamma Ray Spectrometer (GRS) have been searched for evidence of the 2.223 MeV neutron capture line from the Sun, outside the times of -ray-emitting solar flares. Background-corrected spectra accumulated over 3-day intervals between 1980 and 1989 show no evidence of the line. Upper limits are reported separately for periods of high and low solar activity.A conservative 3 upper limit of 5.7 × 10^–5 (cm² s)^–1 is placed on the steady flux in the 2.223 MeV line during inactive periods, which is nearly two orders of magnitude lower than previously published results. After correction for limb darkening of the line emission from off-center positions, this upper limit becomes 7.1 × 10^–5 (cm²s)^–1. Our 3 upper limit on the steady flux in the line during periods of high solar activity is 6.9 × 10^–5 (cm² s)^–1, or 8.6 × 10^–5 (cm² s)^–1 after correction for limb darkening. Our results imply that the quiescent solar corona cannot be heated by ions accelerated above 1 MeV in microflares (or a continuous acceleration process), so long as the ion energy spectrum is similar to that measured in large flares. We also use our results to derive the rate of tritium production at the solar surface; our upper limit of 9 nuclei (cm² s)^–1 is about a factor of 9 below the upper limit from searches for ³H in the solar wind. We place upper limits of the order 10³³ on the number of energetic (> 30 MeV) protons which can be stored in active regions prior to being released in solar flares, which imply that the strongest observed flares cannot be produced by such a mechanism.     

A multi-arc approach for chaotic orbit determination problems

Chaotic dynamical systems are characterized by the existence of a predictability horizon, connected to the notion of Lyapunov time, beyond which predictions of the state of the system are meaningless. In order to study the main features of orbit determination in the presence of chaos, Spoto and Milani (Celest Mech Dyn Astron 124:295–309, 2016) applied the classical least-squares fit and differential correction algorithm to determine a chaotic orbit and a dynamical parameter of a simple discrete system—Chirikov standard map (cf. Chirikov in Phys Rep 52:263, 1979)—with observations distributed beyond the predictability horizon. They found a time limit beyond which numerical calculations are affected by numerical instability: the computability horizon. In this article, we aim at pushing forward such inherent obstacle to numerical calculations in chaotic orbit determination by applying the classical and the constrained multi-arc method (cf. Alessi et al. in Mon Not R Astron Soc 423:2270–2278, 2012) to the same dynamical system. These strategies entail the determination of an orbit when observations are grouped in separate observed arcs. For each arc, a set of initial conditions is determined and, in the case of the constrained multi-arc method, all subsequent arcs are constrained to belong to the same trajectory. We show that the use of these techniques in place of the standard least-squares method has significant advantages: Not only can we perform accurate numerical calculations well beyond the computability horizon, in particular, the constrained multi-arc strategy improves considerably the determination of the dynamical parameter.     

Fast numerics for the spin orbit equation with realistic tidal dissipation and constant eccentricity

We present an algorithm for the rapid numerical integration of a time-periodic ODE with a small dissipation term that is \(C^1\) in the velocity. Such an ODE arises as a model of spin–orbit coupling in a star/planet system, and the motivation for devising a fast algorithm for its solution comes from the desire to estimate probability of capture in various solutions, via Monte Carlo simulation: the integration times are very long, since we are interested in phenomena occurring on timescales of the order of \(10^6\)–\(10^7\) years. The proposed algorithm is based on the high-order Euler method which was described in Bartuccelli et al. (Celest Mech Dyn Astron 121(3):233–260, 2015), and it requires computer algebra to set up the code for its implementation. The payoff is an overall increase in speed by a factor of about 7.5 compared to standard numerical methods. Means for accelerating the purely numerical computation are also discussed.     

Research on Wide-field Imaging Techniques for Low-frequency Radio Arraytwo

Wide-field imaging of low-frequency radio telescopes is subject to a number of difficult problems. One particularly pernicious problem is the non-coplanar baseline effect. It will lead to distortion of the final image when the phase of $w$-direction called $w$-term is ignored. The image degradation effects are amplified for the telescopes with a wide field of view. This paper summarizes and analyzes several $w$-term correction methods and their technical principles. Their advantages and disadvantages are analyzed after comparing their computational cost and computational complexity. We conduct simulations with two of these methods, i.e., faceting and $w$-projection, based on the configuration of the first-phase Square Kilometre Array (SKA) low-frequency array. The resulted images are also compared with the result of the two-dimensional Fourier transform method. The results show that the image quality and correctness derived from both faceting and $w$-projection are better than the two-dimensional Fourier transform method in wide-field imaging. The effects of the number of facets and the $w$-direction step length on the image quality and running time are evaluated. The results indicate that the number of facets and the $w$-direction step length must be reasonable. Finally, we analyze the effect of data size on the running times of the faceting and $w$-projection algorithms. The results show that the faceting and $w$-projection algorithms need to be optimized before the huge amount of data processing. The research of the present paper initiates the analysis of wide-field imaging techniques and their applications in the existing and future low-frequency arrays, and will foster their applications in even broader fields.     

Study of the transfer between libration point orbits and lunar orbits in Earth–Moon system

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.     

Comparisons between stellar models and reliability of the theoretical models

The high quality of the asteroseismic data provided by space missions such as CoRoT (Michel et al. in The CoRoT Mission, ESA Spec. Publ. vol. 1306, p. 39, 2006) or expected from new operating missions such as Kepler (Christensen-Dalsgaard et al. in Commun. Asteroseismol. 150:350, 2007) requires the capacity of stellar evolution codes to provide accurate models whose numerical precision is better than the expected observational errors (i.e. below 0.1 μHz on the frequencies in the case of CoRoT). We present a review of some thorough comparisons of stellar models produced by different evolution codes, involved in the CoRoT/ESTA activities (Monteiro in Evolution and Seismic Tools for Stellar Astrophysics, 2009). We examine the numerical aspects of the computations as well as the effects of different implementations of the same physics on the global quantities, physical structure and oscillations properties of the stellar models. We also discuss a few aspects of the input physics.     

Bidimensional analysis of solar active regions and flares

We describe a method of solar bidimensional spectroscopy exploiting the performances of a Universal Birefringent Filter (UBF) like that of the Sacramento Peak Observatory, which enable an estimate of the profile of some chromospheric lines with moderate spectral resolution ( \((\lambda /\Delta \lambda = 2.5 x 10^4 )\) ). The numerical inversion technique of Backus and Gilbert has been used to retrieve the estimated line profiles; the capabilities of the proposed method is fully analyzed with some numerical tests and examples. Correction procedures for errors in the positions of the UBF passband, random fluctuations of the exposure times and non-uniform brightness distribution on the filtergrams are also presented. The whole method has been tested on the recovery of quiet atmosphere line profiles and the results derived for the Na D₂ line show that the proposed method is completely suitable for many investigations in solar physics.