Time transformations and Cowell's method

The precise numerical integration of Cowell's equations of satellite motion is frequently performed with an independent variables defined by an equation of the form dt=cr ⁿ ds, wheret represents time,r the radial distance from the center of attraction,c is a constant, andn is a parameter. This has been primarily motivated by the uniformizing effects of such a transformation resulting in desirable analytic stepsize control for elliptical orbits. This report discusses the proper choice of the parametern defining the independent variables for various types of orbits and perturbation models, and develops a criterion for its selection.     

Stabilized Kepler motion connected with analytic step adaptation

In the publication Baumgarte and Stiefel (1974a) a stabilization of the Keplerian motion was offered by making use of a manipulation of the Hamiltonian. By this stabilization technique the given HamiltonianH(p _i,q _i) is replaced by a new HamiltonianH ^*, which leads to Lyapunov-stable differential equations of motion.Whereas, in the quoted publication, the physical timet was used as the independent variable we now develop a generalization which allows to combine the stabilization with the introduction of a new independent variables. Such a fictitious times is popular for achieving an analytic step-size adaptation (Baumgarte and Stiefel, 1974c). Perturbations of Kepler motion are discussed.     

A new time element for a general time transformation

The paper introduces a new time element to be used with a general time transformation for satellite equations of motion. The purpose of this time element is to reduce the growth of the numerical errors with respect to the time integration. It is characteristic for the new time element , that the relation between the timet and the element does not depend on the independent variables.     

The uniform,regular differential equations of the KS transformed perturbed two-body problem

The Newtonian differential equations of motion for the two-body problem can be transformed into four, linear, harmonic oscillator equations by simultaneously applying the regularizing time transformation dt/ds=r and the Kustaanheimo-Stiefel (KS) coordinate transformation. The time transformation changes the independent variable from time to a new variables, and the KS transformation transforms the position and velocity vectors from Cartesian space into a four-dimensional space. This paper presents the derivation of uniform, regular equations for the perturbed twobody problem in the four-dimensional space. The variation of parameters technique is used to develop expressions for the derivatives of ten elements (which are constants in the unperturbed motion) for the general case that includes both perturbations which can arise from a potential and perturbations which cannot be derived from a potential. These element differential equations are slightly modified by introducing two additional elements for the time to further improve long term stability of numerical integration.Originally presented at the AAS/AIAA Astrodynamics Specialists Conference, Vail, Colorado, July 1973     

Effect of noise on sunspot magnetic field parameters

Using a perturbated (noised) dipole model of a sunspot magnetic field structure we simulated the influence of background noise or apparent noise (unresolved small-scale magnetic field structure) on sunspot magnetic field parameters. We evaluated mean values of the vertical and horizontal electric current densities |j|_∥ and |j|_⊥, respectively, of the force-free parameter α and of the Lorentz force |F|. For comparison we estimated |j_⊥| and |F| of a standard sunspot magnetic field model (return-flux model, OSHEROVICH 1982). Furthermore, we compared our results with those from observations resulting in estimated values of |j|_∥ for quiet sunspots. Our investigation led to the following results: the estimated values of 〈|F|〉 show clearly that due to the noise the axisymmetric magnetic dipole model is clustered into several subsystems of fluxbundles. The latter are connected with a system of electric current densities of the order of |j|_∥ ∼ 10⁻³ Am⁻² and |j|_⊥ = 10⁻¹ Am⁻², i.e., this system is a noise-generated nonaxisymmetric magnetohydrostatic model.     

Etude variationnelle des decalages spectraux

In a static gravitational field the paths of light are curved, as noticed by H. Weyl. This property can bea priori stated for aV ₃ Riemannian manifold: through any two points ofV ₃ it is possible to draw two families of curves, the straight lines of Euclidean geometry and the photon trajectoriesz. We can perform a fibration of the Galilean space-time in an original way, by taking thez-trajectories of the photons as the base, the isochronic surfaces as fibres, and ‘the equal length time on az trajectory to reach a given point’ as the equivalence relation. The straight lines of Euclidean geometry can then carry the classical mechanics timet, and thez trajectories can carry the optics time t. These times are related by dt=F(x,t) dt. If we class the Universe as a pseudo-Riemannian manifold of normal hyperbolic typeC ^∞, the time t determined above can be taken as the time coordinate inV ₄. Under these conditions we have \(d\overline s ^2 \) =F ² \(d\overline s ^2 \) , where \(d\overline s ^2 \) is the metric of the Riemannian manifold, conforming to the metric ds ² and allowing t as the cosmic time. We can then use the results previously achieved by the author (Peton, 1979) and write: 1 +Z _G =F(A _s,t _s,)/F(Ao_s,t _o) wherez _G denotes the shift of the spectral lines due to the metric. In the case of relative motion betweenO andS, we have $${\text{1 + z' = (1 + }}z_{\text{G}} {\text{)(1 + }}\beta _{\text{r}} {\text{)(1 }} - {\text{ }}\beta ^2 {\text{)}}^{ - 1/2} $$ The Doppler-Fizeau effect therefore appears as a result of the application of the Fermat principle.     

Cosmic behavior, statefinder diagnostic and w?w′ analysis for?interacting new agegraphic dark energy model in non-flat universe

We investigate the interacting NADE model in non-flat universe. The effects of spatial curvature Ω _k , interaction coefficient α and the main parameter of NADE, n, on EoS parameter w _d and deceleration parameter q are studied. We obtain a minimum value for n in both early and present time, in order to that our DE model crosses the phantom divide. Also in a closed universe, changing the sign of q is strongly dependent on α. It has been shown that the quantities w _d and q have a different treatment for various spatial curvature. At last, we calculate the statefinder diagnostic and w−w ^′ analysis in non flat universe. In non flat universe, the statefinder trajectories are discriminated by both n and α.     

Identifying the quasi-regular and stochastic components of solar cyclicity and their properties

We have averaged over every Carrington semi-rotation (C.s.-r.), the daily Wolf numbers (RW), total areas of sunspot groups (SA), the 10.7-cm radio flux (F _10.7), and the modulus of the mean magnetic field (|SMMF|). The fractal method of scaling the variance of time series was used to separate the regular and stochastic components. The manifestation of chaotic and stochastic properties of these components was investigated by testing with the methods of chaotic dynamics, as well as with two new methods: (1) close return maps; and (2) multivariate scaling analysis. Results: (1) by separating time series of global indices of solar activity, it is possible to identify the quasi-regular (the quasi-regularity is caused not by the absolute smoothness of the function) component on time scales longer than two years, and the irregular component on time scales shorter than two years; (2) the regular component has the properties of a nonlinear quasi-periodic oscillator; (3) the irregular component is a random one and has the properties of chromatic noise; and (4) by investigating the nonlinear connection of the solar activity indices under consideration it was found that such a connection is strong between F _10.7and RW. A nonlinear correlation between the attractors RW–|SMMF| and F _10.7–|SMMF| was also revealed.     

Theoretical intensities of Fexiv in the solar EUV spectrum

Equilibrium population of Fexiv levels in coronal conditions was calculated including configurations 3s ²3p, 3s3p ², 3s ²3d, 3p ³, 3s3p3d, 3s ²4s, 3s ²4p, 3s ²4d, 3s ²4f. Relative populations of selected levels are given in Table VII. Figure 1 shows the dependence of relative intensities of the strongest lines on electron density. Certain line ratios can be used for the determination of N _e .E.g., at T=2 × 10⁶ K and with a dilution factor 0.4, the intensity ratio of λ211.3 and λ219.0 changes by a factor of 65 if N _e increases from 10⁷ to 10¹¹ (Table VIII). Cascades from the 3s3p3d and 3p ³ configurations are important in the population of some levels of 3s3p ² (Table VI). A possibility of identification of additional lines in the solar spectrum is indicated. NAS-NRC Resident Research Associate.     

Principe cosmologique,principe variationnel et theorie des groupes

The Friedmann universes are built on the cosmological principle only. The Robertson-Walker metric is common to all the theories based on a homogeneous, isotropic and irrotational universe. In the present work we examine ways of constructing a metric conformal with that of Robertson and Walker, by means of a variational principle which takes into account the cosmological principle as stated by Weinberg (1972), and based on the existence of orbits generated by a one-parameter group of diffeomorphisms of physical space. The application of the cosmological principle to variational methods allows the determination of first integrals which can characterize the physical properties of the Universe. To this end, we show that the Lagrangian of the Universe, considered as a mechanical system, can be chosen from the germs of functions, and that the form variations δq ⁱ are tangent vectors of the group orbits in a Riemannian manifold. Thus the variation of the action vanishes automatically. There appears a first integral of the Euler equations, which is δq ⁱ (?L/?q1 ⁱ ) = C ^te , and also the condition ?L/?t=0, which means the uniformity of time in a Lagrangian conservative system, and which is a direct application of the cosmological principle. These conditions allow the effective determination of a form invariant Lagrangian in the case of isometries. These conditions can be generalized to the case in which the group trajectories are a partition of physical space. Thus, it is possible to define a time from the group trajectories inV ₃: a second of the group time is a lengthm measured along any orbit θ _p of the group. Any pointp of the manifold can then be considered as the starting point of a bundle of orbits, along which the tangent vectors δq ⁱ could be calculated. From this group time, we can build a metric ds ² conformal to the initial ds ² and for which the orbits, which are geodesic, are orthogonal to the transitivity surfaces of the group in the manifold. This implies new statements of the cosmological principle:

1. At any point of space-time it is possible to construct a metric ds ² from the trajectories generated by a one-parameter group of diffeomorphisms ofV ₄.
2. Any two points of space-time can always be joined by means of trajectories of group.

The variational implications of these two principles are the appearance of spectral line shifts such as 1+z=F(p, t _p)/F(q, t_q), wherep andq are arbitrary points of the manifold, andF the transformation function which allows passage from one metric to another. The identification of group trajectories with physical trajectories depends on these two principles. The photon trajectories inV ₃ is an example of this identification. The trajectories of charged particles inV ₄ are another. Principle (b) stated an entropy condition; its application allows a new expression of action variation, this one leading to a general formulation of the shift of spectral lines by a variational method. If we choose the parabolic Friedmann universe as a realistic model, it is the expansion itself which is the generator of the diffeomorphisms allowing the establishment of a group structure in the manifold. The photons are carried away by expansion and do not resist it. The massive particles moderate this expansion locally, and their trajectories inV ₃ are the result of the reaction. In this scheme there is no theoretical difference between the treatment of particles of vanishing proper mass and massive particles. The Robertson-Walker metric fork=0 corresponds to a picture of the Universe which can be drawn by study of the movement of photons in physical space. Only the study of particles can allow the generalization of this scheme and, from this, make a real Universe which is not just a reflection of the physical properties of the photons alone.     

The radiation of energy by a current in a plasma

The derivation of the differential power emitted in any given direction by a current J in a linear, homogeneous and non-absorbing plasma is reviewed in detail. The conventional derivation is shown to give the poweremitted; a formalism for the powerreceived is established by evaluating the Poynting vector in terms of the far field. It is pointed out that the two power expressions differ because the same energy dE is emitted in a time dt _e but received over a different time dt _r . Moreover, a careful scrutiny of both the formalism for the power emission and for the power reception exposes implicit assumptions which do not hold if the plasma is anisotropic. The necessary steps for establishing a valid formalism for anisotropic media are briefly sketched.     

A Quantitative Study on Magnetic Configuration for Active Regions

As compared with the Mount Wilson Magnetic Classification (MWMC), effective distance (d _E) is a useful parameter because it gives a quantitative measure of magnetic configuration in active regions. We have analyzed magnetograms of 24 active regions of different types with MWMC. We have studied the evolution of magnetic fields of five active regions using d _E, total flux (F _t) and tilt angle (Tilt) quantitatively. Furthermore, 43 flare-associated and 25 CME-associated active regions have been studied to investigate and quantify the statistical correlation between flares/CMEs and the three parameters. The main results are as follows: (1) There is a basic agreement between d _E and MWMC. (2) The evolution of magnetic fields can be described in three aspects quantitatively and accurately by the three parameters, in particular by d _E on the analysis of δ-type active regions. (3) The high correlation between d _E and flares/CMEs means that d _E could be a promising measure to predict the flare-CME activity of active regions.     

Scaling laws in decaying helical hydromagnetic turbulence

We study the evolution of growth and decay laws for the magnetic field coherence length ξ, energy E_M and magnetic helicity H in freely decaying 3D MHD turbulence. We show that with certain assumptions, self‐similarity of the magnetic power spectrum alone implies that ξ ∼ t^1/2. This in turn implies that magnetic helicity decays as H ∼ t^–2s, where s = (ξ_diff/ξ_H)², in terms of ξ_diff, the diffusion length scale, and ξ_H, a length scale defined from the helicity power spectrum. The relative magnetic helicity remains constant, implying that the magnetic energy decays as E_M ∼ t^–1/2–2s. The parameter s is inversely proportional to the magnetic Reynolds number Re_M, which is constant in the self‐similar regime. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A short derivation of the sperling-Burdet equations

A short derivation is given of the regularized equations of motion for the perturbed two-body problem. This method is then applied to the slightly modified time transformation dt/ds=r/ω.     

Theorie de Weyl et principe de moindre action: Application a l'etude des mouvements

Sommaire Les lois du mouvement dans une variété riemannienneV ₄ peuvent être déduites d'un principe de moindre action. Nous établissons dans cet article l'équivalence des relations ds=0 et dL=-L _k d^k, où ds ²=L ² est une métrique riemannienne et d _k /dt une fonction homogène de degré 1 des variables dx ⁱ/dt qui définit un espace de Weyl. Ce théorème permet de ramener une théorie de jaugen à un principe de moindre action. Il peut être utilisé dans la théorie de la double métrique de Dirac, obtenue en choissant la théorie des grands nombres comme condition de jauge. Une fibration de l'espace physiqueV ₃ basée sur le théorème de Huyghens permet de déduire les propriétés dynamiques des particules des propriétés des photons dansV ₃, et constitue en ce sens une unification des propriétés dynamiques des particules.

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The laws of motion in a RiemannianV ₄ manifold can be deduced from the principle of least action. We state in this work the equivalence between the equations ds=0 and dL=-L _k d^k, where ds ² =L ² is the Riemannian metric and d _k /dt the homogeneous functions of first degree of the dx ⁱ/dt which define a Weylian space. This theorem can then reduce a gauge theory to a principle of least action. It can be used in the double metric theory of Dirac, obtained by means of the Large Number Hypothesis as a gauging condition. A fibration of the physical spaceV ₃ based on Huyghens' theorem allows the deduction of the dynamical properties of particles by means of the properties of photons inV ₃, and constitutes in this way an unification of the dynamical properties of particles.

     

Dirac quasinormal modes of a black hole with quintessence-like matter and a deficit solid angle

Using the third-order WKB approximation, we evaluate the quasinormal frequencies of massless Dirac field perturbation around a black hole with quintessence-like matter and a deficit solid angle. We discuss carefully the properties of quasinormal frequencies with the change of quintessential state parameter w _q , the deficit solid angle parameter ε, the energy density of quintessence-like matter ρ ₀, and the total angular momentum number |k| and the overtone number n, respectively.     

Investigation of the Neupert Effect in the Various Intervals of Solar Flares

The Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI) gives us a chance to investigate the theoretical Neupert effect using the correlation between the thermal-energy derivative and the nonthermal energy, or the thermal energy and the integral nonthermal energy. Based on this concept, we analyze four M-class RHESSI flares on 13 November 2003, 4 November 2004, 3 and 25 August 2005. According to the evolution of the temperature [T], emission measure [EM], and thermal energy [E _th], each event is divided into three phases during the nonthermal-energy input [ \frac dE_nthdt\frac {\mathrm{d}E_{\mathrm{nth}}}{\mathrm{d}t} in the units of erg s⁻¹]. Phase 1 is identified as the interval before the temperature maximum, while after the thermal-energy maximum is phase 3, between them is phase 2. We find that these four flares show the Neupert effect in phase 1, but not in phase 3. The Neupert effect still works well in the second phase, although the cooling becomes slightly important. We define the parameter μ in the relation of \fracdE_thdt=m\fracdE_nth(t)dt\frac{\mathrm {d}E_{\mathrm{th}}}{\mathrm{d}t}=\mu\frac{\mathrm{d}E_{\mathrm {nth}}(t)}{\mathrm{d}t} or E_th(t₀)=mò₀^t₀\fracdE_nth(t)dt dtE_{\mathrm{th}}(t_{0})=\mu\int_{0}^{t_{0}}\frac{\mathrm{d}E_{\mathrm{nth}}(t)}{\mathrm{d}t}\,\mathrm{d}t when the cooling is ignored in phase 1. Considering the uncertainties in estimating the energy from the observations, it is not possible to precisely determine the fraction of the known energy in the nonthermal electrons transformed into the thermal energy of the hottest plasma observed by RHESSI. After a rough estimate of the flare volume and the assumption of the filling factor, we investigate the parameter μ in these four events. Its value ranges from 0.02 to 0.20, indicating that a small fraction (2% – 20%) of the nonthermal energy can be efficiently transformed into thermal energy, which is traced by the soft X-ray emission, and the bulk of the energy is lost possibly due to cooling.     

Evidence for a disaggregation of the universe

Combining the kinematical definitions of the two dimensionless parameters, the deceleration q(x) and the Hubble t ₀ H(x), we get a differential equation (where x=t/t ₀ is the age of the universe relative to its present value t ₀). First integration gives the function H(x). The present values of the Hubble parameter H(1) [approximately t ₀ H(1)≈1], and the deceleration parameter [approximately q(1)≈−0.5], determine the function H(x). A second integration gives the cosmological scale factor a(x). Differentiation of a(x) gives the speed of expansion of the universe. The evolution of the universe that results from our approach is: an initial extremely fast exponential expansion (inflation), followed by an almost linear expansion (first decelerated, and later accelerated). For the future, at approximately t≈3t ₀ there is a final exponential expansion, a second inflation that produces a disaggregation of the universe to infinity. We find the necessary and sufficient conditions for this disaggregation to occur. The precise value of the final age is given only with one parameter: the present value of the deceleration parameter [q(1)≈−0.5]. This emerging picture of the history of the universe represents an important challenge, an opportunity for the immediate research on the Universe. These conclusions have been elaborated without the use of any particular cosmological model of the universe.     

Identification of forbidden coronal lines of Fe x and Ni xii

The transitions 3s ²3p ⁴3d ⁴ D, ⁴ F–3s3p ⁵3d ⁴ F have been studied in the Cl i isoelectronic sequence from Ca iv to Fe x. The determination of the 3s ²3p ⁴3d quartet intervals in Fe x has led to the identification of eleven coronal lines in the region from 1463 Å to 5539 Å as forbidden transitions within this configuration. By extrapolation, an additional coronal line is identified with a similar transition in Ni xii.     

Impact crater formed on sintered snow surface simulating porous icy bodies

To improve the scaling parameter controlling the impact crater formation in the strength regime, we conducted impact experiments on sintered snow targets with the dynamic strength continuously changed from 20 to 200 kPa, and the largest crater size formed on small icy satellites was considered by using the revised scaling parameter. Ice and snow projectiles were impacted on a snow surface with 36% porosity at an impact velocity from 31 m s⁻¹ to 150 m s⁻¹. The snow target was sintered at the temperature from −5 °C to −18 °C, and the snow dynamic strength was changed with the sintering duration at each temperature. We found that the mass ejected from the crater normalized by the projectile mass, π_V, was related to the ratio of the dynamic strength to the impact pressure, , as follows: , where the impact pressure was indicated by P = ρ_tC_0tv_i/2 with the target density of ρ_t, when the impact velocity, v_i, was much smaller than the bulk sound velocity C_0t (typically 1.8 km s⁻¹ in our targets). The ratio of the largest crater diameter to the diameter of the target body, d_max/D, was estimated by calculating the crater diameter at the impact condition for catastrophic disruption and then compared to the observed d_max/D of jovian and saturnian small satellites, in order to discuss the formation condition of these large d_max/D in the strength regime.