Some special restricted four-body problems—II. From Caledonia to Copenhagen

Special analytical solutions are determined for restricted, coplanar, four-body equal mass problems, including the Caledonian problem, where the masses M_i = M for i = 1,2,3,4. Most of these solutions are shown to reduce to the Lagrange solutions of the Copenhagen problem of three bodies by reducing two of the masses (m_i = m for i = 1,2) in the four-body equal mass problem to zero while maintaining their equality of mass. In so doing, families of special solutions to the four-body problem are shown to exist for any value of the mass ratio μ = m/M.     

MGS electron density profiles: Analysis of the peak magnitudes

We analyze here the behavior of the magnitudes of the F₁ and E peaks of the electron density profiles measured by the Radio Science Subsystem of the Mars Global Surveyor spacecraft, as a function of solar zenith angle χ and solar flux. For each of the 658 days of data in the six occultation seasons in the northern hemisphere, we choose one profile to analyze, which is that for which the F₁ peak is the median value. We assume that the variations of the measured peak densities can be represented as Aa(cosχ) and as Bb(F_10.7), where F_10.7 is the usual solar flux proxy, appropriately shifted to the orbital position of Mars. To minimize the effect of solar activity, we divide the data into 6 F_10.7 bins, fit the data in each bin, and derive the values of the exponent a and the coefficient AF10.7 for each bin. The median values that we derive for the exponent a is 0.46 for the F₁ peak, and 0.395 for the E peak. To minimize the effect of SZA, we divide the data into eight SZA bins, and derive the exponent b and the coefficient Bχ for each SZA bin. We argue that the last three SZA bins should be excluded because the fits were poor, due partly to the small number of data points in each of these bins. If we do so, the median values of b that we derive are 0.27 and 0.40 for the F₁ and E peaks, respectively. Finally we derive a 3-parameter fit to all the data, which expresses the variability of the peak densities as a function of a^′(cosχ) and b^′(F_10.7) simultaneously. The fitted values of the exponents a^′ and b^′ for the F₁ peak are 0.45 and 0.26, respectively; for the E peak, the values are 0.39 and 0.46, respectively. We compare our results to Chapman theory, and to those of other investigators.     

g Dependent particle concentration due to sedimentation

Sedimentation of particles in a fluid has long been used to characterize particle size distribution. Stokes’ law is used to determine an unknown distribution of spherical particle sizes by measuring the time required for the particles to settle a known distance in a fluid of known viscosity and density. In this paper, we study the effects of gravity on sedimentation by examining the resulting particle concentration distributed in an equilibrium profile of concentration C _m,n above the bottom of a container. This is for an experiment on the surface of the Earth and therefore the acceleration of gravity had been corrected for the oblateness of the Earth and its rotation. Next, at the orbital altitude of the spacecraft in orbit around Earth the acceleration due to the central field is corrected for the oblateness of the Earth. Our results show that for experiments taking place in circular or elliptical orbits of various inclinations around the Earth the concentration ratio C _m,n /C _m,ave , the inclination seems to be the most ineffective in affecting the concentration among all the orbital elements. For orbital experiment that use particles of diameter d _p =0.001 μm the concentration ratios for circular and slightly elliptical orbits in the range e=0–0.1 exhibit a 0.009 % difference. The concentration ratio increases with the increase of eccentricity, which increases more for particles of larger diameters. Finally, for particles of the same diameter concentration ratios between Earth and Mars surface experiments are related in the following way .     

The Schwabe and Gleissberg Periods in the Wolf Sunspot Numbers and the Group Sunspot Numbers

Three wavelet functions: the Morlet wavelet, the Paul wavelet, and the DOG wavelet have been respectively performed on both the monthly Wolf sunspot numbers (R_z) from January 1749 to May 2004 and the monthly group sunspot numbers (R_g) from June 1795 to December 1995 to study the evolution of the Gleissberg and Schwabe periods of solar activity. The main results obtained are (1) the two most obvious periods in both the R_z and R_g are the Schwabe and Gleissberg periods. The Schwabe period oscillated during the second half of the eighteenth century and was steady from the 1850s onward. No obvious drifting trend of the Schwabe period exists. (2) The Gleissberg period obviously drifts to longer periods the whole consideration time, and the drifting speed of the Gleissberg period is larger for R_z than for R_g. (3) Although the Schwabe-period values for R_z and R_g are about 10.7 years, the value for R_z seems slightly larger than that for R_g. The Schwabe period of R_z is highly significant after the 1820s, and the Schwabe period of R_g is highly significant over almost the whole consideration time except for about 20 years around the 1800s. The evolution of the Schwabe period for both R_z and R_g in time is similar to each other. (4) The Gleissberg period in R_z and R_g is highly significant during the whole consideration time, but this result is unreliable at the two ends of each of the time series of the data. The evolution of the Gleissberg period in R_z is similar to that in R_g.     

Interstellar Extinction and the Intrinsic Colors of Classical Cepheids in the Galaxy, the LMC, and the SMC

New methods are applied to samples of classical cepheids in the galaxy, the Large Magellanic Cloud, and the Small Magellanic Cloud to determine the interstellar extinction law for the classical cepheids, R _B:R _V:R _I:R _J:R _H:R _K= 4.190:3.190:1.884:0.851:0.501:0.303, the color excesses for classical cepheids in the galaxy, E(B-V)=-0.382-0.168logP+0.766(V-I), and the color excesses for classical cepheids in the LMC and SMC, E(B-V)=-0.374-0.166logP+0.766(V-I). The dependence of the intrinsic color (B-V)₀ on the metallicity of classical cepheids is discussed. The intrinsic color (V-I)₀ is found to be absolutely independent of the metallicity of classical cepheids. A high precision formula is obtained for calculating the intrinsic colors of classical cepheids in the galaxy: (<B>-<V>)₀=0.365(±0.011)+0.328(±0.012)logP.     

Population I pulsating stars

The radiiR and surface gravitiesg of Population I pulsating stars (89 Delta Scuti-variables and 155 classical cepheids) have been investigated. Semi-empirical period-radius (P-R) and period-gravity (P-g) relations are obtained for Delta Scuti-stars (for the four lowest modes of radial pulsations) and for classical cepheids. For Delta Scuti-stars, the uncertainties of radius and gravity estimations calculated from theP-R andP-g relations for different modes, are evaluated. There is a good agreement both between theP-R relations and between theP-g relations for Delta Scuti-stars and for classical cepheids, but a gap exists between the two types of variables. From models of Delta Scuti-stars, theoreticalP-R andP-g relations for the four lowest modes of radial pulsations are obtained, in a good agreement with the corresponding semi-empirical relations. There is an excellent agreement between the theoretical and semi-empirical period ratios of radial pulsations as derived from theP-R andP-g relations for Delta Scuti-stars. It is not necessary to take into account the colours (in addition to the periods), in order to estimate the radii and gravities of the variables under study.     

The dependence of solar flare energetics on flare volumes

Solar X-ray flare images from Skylab and data from full Sun detectors were used in a statistical analysis to determine the relationship between flare volumes and flare energetics. Data from the rise phases of 45 flares were used in the analysis. For each event the diameter D, length L, and volume V of the flare loops were determined and then compared to the thermal energy, rate of increase of thermal energy, and rise time of the soft X-ray flux. The latter three quantities were all found to be positively correlated with D, L, and V. However, the thermal energy per unit volume and rate of increase of thermal energy per unit volume decrease with increasing volume. No correlation was found between emission measure Y and volume V, indicating that the electron density tends to be smaller for larger flare volumes. We find a larger dynamic range for V than for Y, hence knowledge of V is more critical than that of Y for calculating the thermal energy of the X-ray emitting structure, which is proportional to Y ^0.5 V ^0.5. Using certain assumptions, the results were compared to several flare models. The classical neutral sheet model, the sheared loop model of Spicer and even models using the magnetic field in a passive role for the energy release were all found to be consistent with the results.     

Influence of short-duration matter temperature perturbations on the cosmological hydrogen recombination spectrum

A perturbation in the ratio of the matter temperature to the radiation temperature in the form of a Gaussian with amplitude A and width σ (in units of the redshift z) centered at some redshift z _c is considered, with some “standard” temperature ratio obtained from a simultaneous solution of the cosmological recombination kinetics and energy equations being taken as the initial (unperturbed) one. Comparatively small (A = ± 0.01), fast (σ = 17) perturbations are shown to give rise to distinct narrow absorption (for A > 0) or emission (for A < 0) quasi-lines in each of the subordinate continua. The positions of these quasi-lines correlate with the position of the perturbation center, while their intensities are very sensitive to the perturbation amplitude. At the same time, the manifestation of the perturbation is much less clear in hydrogen lines (subordinate ones and the Ly-α line) and two-photon emission. As a result, the full perturbed spectrum is characterized by the presence of the narrow quasi-lines mentioned above and by a general decrease (for A > 0) or increase (for A < 0) in intensity with increasing wavelength.     

Accurate masses and radii of normal stars

Summary Binary stars are the main source of fundamental data on stellar masses and radii (M, R). Considerable progress has been made in recent years in the quality and quantity of such data, and stellar masses and radii of high accuracy have led to a number of qualitatively new and interesting results on the properties and evolution of normal stars. This paper reviews the current status of fundamentalM andR determinations which (i) have errors 2%, the limit for non-trivial results in many applications, and (ii) can be presumed valid for single stars. These two conditions limit the discussion to data fromdetached, doublelined eclipsing binary systems.After a brief discussion (Sect. 2) of the main tests for accuracy and consistency which must be met for observational data to be included in the sample, data for 45 binary systems (90 single stars) are presented in Sect. 3 (Table 1 and Figs. 2–5). Spectral types are O8-M1 on the main sequence, with only two stars clearly in the red-giant region. From the review by Popper (1980), data for only 6 systems survive unchanged in the present list, while improved data are given for 18 systems; 21 systems are new additions. Broadband colours, effective temperatures, and luminosities are also given, but are scale-dependent and considerably less reliably determined thanM andR.The observed ranges inM andR for a given colour far exceed the observational errors, primarily due to evolutionary effects within the main sequence. For this reason, single-parameter relations used to predictM andR for single stars are limited to an accuracy of some ±15% inM and ±50% inR, basically independent of the number and accuracy of the data used to establish the relations. Two-parameter calibrations are discussed (Sect. 4) which can eventually reduce these errors to & 5% in bothM andR. At this level, abundance effects become significant and presumably account for the residual scatter.Comparison of the data with stellar evolution models is the topic of Sect. 5. Characteristic features of the data which are crucial in such work are emphasized, rather than attempts to prove the validity of any particular set of models. Already fromM andR alone, some significant constraints can be derived (Fig. 4). When bothM, R, andT _e are known, the initial helium abundanceY can be estimated if the metal-abundance parameter Z is assumed or determined. Studies in which binaries with accurate values ofM, R, and Z are fit by models calculated for the precise observed masses, and withY and mixing length constrained to solar values, provide the most stringent tests of the models. Probing further model refinements such as convective overshooting requires full use of the potential of the data. For example, models may yield general main-sequence limits which are consistent with the observations, but still be unable to fit any single system to the precision of the data. Conditions for critical, informative tests are discussed. Tidal effects in binaries are briefly discussed in Sect. 6. As tidal forces are extremely sensitive to the dimensions and internal structure of the stars, the present sample is well suited for such studies. Recent success in matching computed and observed apsidal-motion parameters for early-type binaries is mentioned. Finally, main priorities for future work are outlined.     

The extinction curve in the visible and the value of RV: Addendum to AN 333, 160

This paper corrects and completes a previous study of the shape of the extinction curve in the visible and the value of R_V. A continuous visible/infrared extinction law proportional to 1/λ^p with p close to 1 (± 0.4) is indistinguishable from a perfectly linear law (p = 1) in the visible within observational precision, but the shape of the curve in the infrared can be substantially modified. Values of p slightly larger than 1 would account for the increase of extinction (compared to the p = 1 law) reported for λ > 1 μ m and deeply affect the value of R_V. In the absence of gray extinction R_V must be 4.04 if p = 1. It becomes 3.14 for p = 1.25, 3.00 for p = 1.30, and 2.76 for p = 1.40. Values of p near 1.3 are also attributed to extinction by atmospheric aerosols, which indicates that both phenomena may be governed by similar particle size distributions. A power extinction law may harmonize visible and infrared data into a single, continuous, and universal interstellar extinction law (© 2012 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Strong Chaos in N-body problem and Microcanonical Thermodynamics of Collisionless Self Gravitating Systems

An effective Microcanonical Thermodynamics of self gravitating systems(SGS) is proposed, analyzing the well known obstacles thought to prevent the formulation of a rigorous Statistical Mechanics (SM), as those due to the formal unboundedness of available phase space and to the unscreened, long range, nature of the interaction. The latter feature entails the well known inequivalence of statistical ensembles, puts clearly into question the meaning, for these systems, of the Thermodynamic Limit, and rules out the use of canonical and grand-canonical ensembles. As to the first obstacle, we argue nevertheless that a hierarchy of timescales exist such that, at any finite time, the volume of the effectively available region of phase space is indeed finite, and that the dynamics satisfies a strong chaos criterion, leading to a fast, increasingly uniform, spreading of orbits over an effectively invariant subset of the constant (N,V,E) surface; thus leading to the definition of a secularly evolving, generalized microcanonical ensemble, which allows to define an (almost extensive) effective entropy and to derive self-consistent definitions for other thermodynamic variables, giving thus an orthode for SGS. Moreover, a Second Law-like criterion allows to single out the hierarchy of secular equilibria describing, for any finite time, the macroscopic behaviour of SGS. This revised version was published online in July 2006 with corrections to the Cover Date.     

Stationary Configurations for Co-orbital Satellites with Small Arbitrary Masses

We derive general results on the existence of stationary configurations for N co-orbital satellites with small but otherwise arbitrary masses m _i , revolving on circular and planar orbits around a massive primary. The existence of stationary configurations depends on the parity of N. If N is odd, then for any arbitrary angular separation between the satellites, there always exists a set of masses (positive or negative) which achieves stationarity. However, physically acceptable solutions (m _i > 0 for all i) restrict this existence to sub-domains of angular separations. If N is even, then for given angular separations of the satellites, there is in general no set of masses which achieves stationarity. The case N=3 is treated completely for small arbitrary satellite masses, giving all the possible solutions and their stability, to within our approximations.     

New times of the minima and light elements of SW Lacertae

SW Lacertae is a short-period variable star of the W UMa-type. A total of 261 photoelectric observations for eachU, B, andV filter were obtained in 1986 while 522 photoelectric observations for eachB andV filter were obtained in 1987. All these observations were transformed to theU, B, andV colours of theUBV standard system. Nine light curves for primary and secondary eclipses were obtained, their times of minima were determined and a new linear ephemeris was obtained. The period changes of the system were also discussed.     

The significance of observed rotational magnetic hysteresis in lunar samples

Rotational Magnetic hysteresis (W _R ) curves for lunar soils 10084, 12070, 14259, and rock 14053, have been published. There is no adequate explanation to date for the observed largeW _R at high fields. Lunar rock magnetism researchers consider fine particle iron to be the primary source of stable magnetic remanence in lunar samples. Iron has cubic anisotropy with added shape anisotropy for extreme particle shapes. The observed high fieldW _R must have its source in uniaxial or unidirectional anisotropy. This implies the existence of minerals with uniaxial anisotropy or exchange coupled spin states. Therefore, the source of this observed high fieldW _R must be identified and understood before serious paleointensity studies are made. It is probable that the exchange coupled spin states and/or the source of uniaxial anisotropy responsible for the high fieldW _R might be influenced by the lunar surface diurnal temperature cycling. The possible sources of high fieldW _R in lunar samples are presented and considered.     

Inhomogeneities in the spatial distribution of gamma-ray bursts

The distribution of pairwise distances f(l) for different dependences r(z) of the metric distance is used to reveal inhomogeneities in the spatial distribution of 201 long (T ₉₀>2^s) gamma-ray bursts with measured redshifts z. For a fractal set with dimensionality D, this function behaves asymptotically as f(l) ∼ l ^D−1 for small l. Signs of fractal behavior with dimensionality D = 2.2–2.5 show up in all the models considered for the spatial distribution of the gamma-ray bursts. Several spatially distinct groups of gamma-ray bursts are identified. The group with equatorial coordinates ranging from 23^h56^m to 0^h49^m and δ from +19° to +23° with redshifts of 0.81–0.94 is examined separately.     

Regularization of Motion Equations with L-Transformation and Numerical Integration of the Regular Equations

The sets of L-matrices of the second, fourth and eighth orders are constructed axiomatically. The defining relations are taken from the regularization of motion equations for Keplerian problem. In particular, the Levi-Civita matrix and KS-matrix are L-matrices of second and fourth order, respectively. A theorem on the ranks of L-transformations of different orders is proved. The notion of L-similarity transformation is introduced, certain sets of L-matrices are constructed, and their classification is given. An application of fourth order L-matrices for N-body problem regularization is given. A method of correction for regular coordinates in the Runge–Kutta–Fehlberg integration method for regular motion equations of a perturbed two-body problem is suggested. Comparison is given for the results of numerical integration in the problem of defining the orbit of a satellite, with and without the above correction method. The comparison is carried out with respect to the number of calls to the subroutine evaluating the perturbational accelerations vector. The results of integration using the correction turn out to be in a favorable position.     

Improved values for the solar micro- and macro-turbulent filter functions

The contributions of any arbitrary photospheric velocity field to (macroturbulent) line displacement, and to (microturbulent) line broadening can be expressed by the macro- and micro-turbulent filters f _M (k) and f _t(_k), where _k is the wavenumber of the energy spectrum in which the line-of-sight component of the velocity field can be decomposed. As a correction to a previous computation of f _M and f _t we give in this Note improved values for the filter functions for weak lines in LTE. An example of the way to use the filter functions is given.     

Fourier analysis of the light curves of eclipsing variables,XV

A new general expression for the theoretical momentsA _2m of the light curves of eclipsing systems has been presented in the form of infinite series expansion. In this expansion, the terms have been given as the product of two different polynomials which satisfy certain three-term recursion formulae, and the coefficients diminish rapidly with increasing number of terms. Thus, the numerical values of the theoretical momentsA _2m can be generated recursively up to four significant figures for any given set of eclipse elements. This can be utilized to solve the eclipse elements in two ways: (i) with an indirect method (for the procedures see Paper XIV, Kopal and Demircan, 1978), (ii) with a direct method as minimization to the observational momentsA _2m (area fitting). The procedures given in Paper XIV for obtaining the elements of any eclipsing system consisting of spherical stars have been automated by making use of the new expression for the momentsA _2m of the light curves. The theoretical functionsf ₀,f ₂,f ₄,f ₆,g ₂ andg ₄ which are the functions ofa andc ₀, have been used to solve the eclipse elements from the observed photometric data. The closed-form expressions for the functionsf ₂,f ₄ andf ₆ have also been derived (Section 3) in terms of Kopal'sI-integrals.The automated methods for obtaining the eclipse elements from one minimum alone have been tested on the light curves of YZ (21) Cassiopeiae under the spherical model assumptions. The results of these applications will be given in Section 5 which follows a brief introduction to the procedure we followed.     

The observational evidence for mass distribution in the meteoritic complex

The integrated mass indexS for solid particles in the solar system is defined by the equationN = (const)m ^–S, whereN is the number of particles counted down to a lower limit of massm. Independent values ofS found from various observational programs are reviewed. These lie generally in the range from 0.4 to 1.4 for the average background of particles, and include data from lunar craters, satellite impacts, meteors, meteorites and asteroids. The trend ofS with mass is reasonably well established for masses less than one gram, but there are many gaps in our knowledge concerning the objects of greater mass.Paper dedicated to Prof. Harold C. Urey on the occasion of his 80th birthday on 29 April 1973.     

Nonuniform Target Ionization and Fitting Thick Target Electron Injection Spectra to RHESSI Data

Past analyses of flare hard X-ray (HXR) spectra have largely ignored the effect of nonuniform ionization along the electron paths in the thick-target model, though it is very significant for well-resolved spectra. The inverse problem (photon spectrum to electron injection spectrum F ₀(E ₀)) is disturbingly non-unique. However, we show that it is relatively simple to allow for the effect in forward fitting of parametric models of F ₀(E ₀)) and provide an expression to evaluate it for the usual single power-law form of F ₀(E ₀)).The expression involves the column depth N _* of the transition region in the flare loop as one of the parameters so data fitting can enable derivation of N _* (and its evaporative evolution) as part of the fitting procedure. The fit to RHESSI data on four flares for a single power law F ₀(E ₀)) is much improved when ionization structure is included compared to when the usual fully ionized approximation is used. This removes the need, in these events at least, to invoke broken power laws, or other forms, of the acceleration spectrum F ₀(E ₀)) to explain the observed photon spectrum