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.     

A tentative derivation of the main cosmological parameters

Based on the assumption that some apparent properties of the observable universe are accurate at a reasonable level of approximation, a tentative is made to independently derive the values of the baryon density parameter, the Hubble constant, the cosmic microwave background temperature and the helium mass fraction. The obtained values are in excellent agreement with those given by the most recent observational data.     

A new absolute magnitude calibration with 2MASS for cataclysmic variables

Using reliable trigonometric measurements, we find that the absolute magnitude of cataclysmic variables depends on the orbital period and de-reddened (J − H)₀ and (H − K _s)₀ colours of 2MASS (Two Micron All Sky Survey) photometric system. The calibration equation covers the ranges 0.032^d < P_orb ⩽ 0.454^d, −0.08 < (J − H)₀ ⩽ 1.54, −0.03 < (H − K_s)₀ ⩽ 0.56 and 2.0 < M_J < 11.7; It is based on trigonometric parallaxes with relative errors of (σ_π/π) ⩽ 0.4. By using the period-luminosity-colours (PLCs) relation, we estimated the distances of cataclysmic variables with orbital periods and 2MASS observations and compared them with distances found from other methods. We suggest that the PLCs relation can be a useful statistical tool to estimate the distances of cataclysmic variables.     

A classical derivation of inverse Compton radiation

A classical method is used to derive formulas describing inverse Compton radiation in vacuo. It is shown that several expressions, including the power radiated and the rate at which photons are scattered, can be calculated exactly for the case of isotropically distributed unpolarized waves scattered by particles with any energy.     

Concerning the derivation of the KS-transformation

The Kustaanheimo-Stiefel (KS) transformation is shown to follow naturally from the general solution of the two-body motion if half-arguments are introduced. Application to collision orbits and to the exact triangular solutions of Lagrange (vide E. Stiefel and G. Scheifele: 1971,Linear and Regular Celestial Mechanics, Springer, Berlin-Heidelberg-New York, p. 23–35).Notations x Position vector (x, y, z) - r=|x| Distance from the origin - 1/2h Energy constant or Kepler motion - c Angular momentum vector of Kepler motion - t physical time ()^·=d/dt () - new independent variable ()=d/d () Note by editor: This is the well-known Three-dimensional regularization, published in 1965 by P. Kustaanheimo and E. Stiefel, Perturbation Theory of Kepler Motion Based on Spinor Regularization,J. reine angewandte Mathematik 218, 204. The present article was written during Professor Volk's stay at the Zurich Technische Hochschule in 1972, when he also celebrated his 80th birthday.     

Another derivation of the titius-bode law

Using merely mathematical means the author points out that in case of planets the Titius-Bode Law actully states, that if considering the distance between the Sun and the Earth as a unit, the mean-distance of any planet being farther than Mercury can be obtained if 0.4 is substracted from the double of the mean distance of the preceding planet. But in reality the fact is that for planets not the immediate preceding one mean-distance but the immediate preceding two mean-distances must be known in determining a mean-distance. The conformity between the measured values and those calculated by the Law and – in case of Neptune and Pluto – the discrepancy between them, stem from this fact. If in case of planets we consider not one, but two preceding mean-distances, a ”︁correction” will occur in the formula obtained which eliminates the considerable discrepancy. In case of satellites it is enough to know the immediate preceding one mean-distance which means, that the application of the Titius-Bode Law proves successful.     

Fundamental parameters and new variables of the galactic open cluster NGC 7128

CCD photometry in Johnson UBV and Strömgren uvby systems and medium-resolution spectroscopy of the galactic open cluster NGC 7128 are presented. Spectral types of the brightest 12 stars in the cluster field were determined based on equivalent widths of the H α and the He  i 6678-Å line. The spectroscopic observations also revealed two obvious and one probable Be-type stars showing H α emission. The analysis of the photometric diagrams gave a colour excess of E ( B − V )=1.03±0.06 mag, a distance modulus DM =13.0±0.2 mag and an age above 10 Myr. Time-resolved photometric observations obtained on one night resulted in the detection of short time-scale light variations of seven new and three already known variable stars in the cluster field.     

Identification of new cataclysmic variables in the 1RXS and USNO-B1.0 catalogs

As a result of applying the original optical variability search method on publicly available data, we have found eight new cataclysmic variables and two possible Optically Violent Variable quasars among the previously unidentified X-ray sources in the ROSAT catalog. We describe the search method and present the characteristics of the newly identified variable stars. The obtained results demonstrate the large potential of the concept of Virtual Observatory for identifying new objects of astrophysical interest.     

On the dynamical derivation of the Titius-Bode law

The Principle of Least Action Interaction, developed by the dynamical astronomer Michael W. Ovenden, is tested using a new algorithm based on the ergodic hypothesis that the time mean of the disturbing function is equal to the space mean. This algorithm is an improvement over the one that Ovenden (1972) used in testing his principle, i.e. it can be applied to systems having more than three satellites without violating the conservation law of angular momentum and these satellites may have significant inclinations. This algorithm treats the problem of finding the configuration of least action interaction as a Lagrange multiplier problem. Renormalization group techniques and existing non-gradient optimization algorithms are incorporated into this new algorithm to reduce some of the numerical complexities.This algorithm is tested on the planets and asteroids in our solar system and on the satellite systems of Jupiter, Saturn, and Uranus. In most cases the results show that the current distances of the satellites from their primary is very close to the minimum interaction-action configuration for that system. The possibility of a planet lying beyond Pluto is investigated using this algorithm.Finally, some of my results are compared with those of Ovenden (1972) for our solar system. The results indicate that the interaction-action potential is lower using this new algorithm than the potential obtained from Ovenden's. Also, greater skepticism is raised concerning the one-time existence of a planet of 90 earth masses lying between Mars and Jupiter.     

An analysis of the derivation of Friedmann's Equation

In the introductive section a deeper specification is made of the physical meaning of the terms in the Friedmann Equation (FE). After that, assuming validity of Mach's Principle, it is deduced (Section 2) that the FE for the radiation-dominated phase of the evolution of the Universe (which is considered to be closed) is valid even for the matter-dominated era. On the other hand, such a form of the FE is shown to be irrelevant when used for deducing how the frequency of the 3K-photons depends on the radius of the Universe. The solution of the dilemma turns out to have its roots in the way in which the relativistic gravitational potentials are composed, together with the fact that the cosmic potential of the closed Universe is equal to –1. Consequently, the equation one uses for the matter-dominated era is shown to be proper for both the main phases of the cosmological evolution (Section 3). In Section 4, the deduction of the critical density-value is revised, and the new result shortly discussed. In Section 5 it is concluded that there is only one possible value for the maximum radius of our Universe, specific for the initial condition due to the FE. Nevertheless, it is not excluded that other smaller universes can exist. In the final section a modified soabubble analogy with the first law of thermodynamics is used to describe the dynamics of the cosmological expansion, and, in this context, the importance of Mach's Principle is pointed out.     

A derivation of the luminosity function of the Kuiper belt from a broken power-law size distribution

We have derived a model of the Kuiper belt luminosity function exhibited by a broken power-law size distribution. This model allows direct comparison of the observed luminosity function to the underlying size distribution. We discuss the importance of the radial distribution model in determining the break diameter. We determine a best-fit break-diameter of the Kuiper belt size-distribution of 30<Db<90 km via a maximum-likelihood fit of our model to the observed luminosity function. We also confirm that the observed luminosity function for m(R)∼21-28 is consistent with a broken power-law size distribution, and exhibits a break at .     

A Hopf variables view on the libration points dynamics

This study analyzes a recently discovered class of exterior transfers to the Moon. These transfers terminate in retrograde ballistic capture orbits, i.e., orbits with negative Keplerian energy and angular momentum with respect to the Moon. Yet, their Jacobi constant is relatively low, for which no forbidden regions exist, and the trajectories do not appear to mimic the dynamics of the invariant manifolds of the Lagrange points. This paper shows that these orbits shadow instead lunar collision orbits. We investigate the dynamics of singular, lunar collision orbits in the Earth–Moon planar circular restricted three-body problem, and reveal their rich phase space structure in the medium-energy regime, where invariant manifolds of the Lagrange point orbits break up. We show that lunar retrograde ballistic capture trajectories lie inside the tube structure of collision orbits. We also develop a method to compute medium-energy transfers by patching together orbits inside the collision tube and those whose apogees are located in the appropriate quadrant in the Sun–Earth system. The method yields the novel family of transfers as well as those ending in direct capture orbits, under particular energetic and geometrical conditions.     

A derivation of the rate of spin-down of the Earth using ancient Chinese records of central eclipses

We describe a method of studying the variation in the rotation of the Earth in historical times using ancient records of central solar eclipses. From the records of central eclipses covering a period of some one thousand four hundred years (7th century B.C. to 7th century A.D.), given in the Chinese annals, we selected 88 entries and calculated the rotation variation parameter ΔT. Our result shows that the Earth is spinning down at a relative rate of − (1.88 ± 0.10) × 10^−10/yr.     

Machine derivation of the tesseral perturbation of artificial satellites

A non-conservative Lie transformation is used to establish the theory of tesseral perturbation including the cross terms from the zonal harmonic J₂ with the tesseral harmonics. The formulae for the perturbations are derived with a computer. The storage of the Poisson series is effected through a one-to-one correspondence between the multi-dimensional index of a term in the series and the store address of the coefficient of that term. Rules for storing some typical series in celestial mechanics in computers are also given.     

A note on harmonic variables for the Brink-Schwarz covariant superparticle

We clarify the geometric meaning of auxiliary variables that were introduced for covariant quantization of the Brink-Schwarz superparticle (and the Green-Schwarz heterotic superstring) by KALLOSH and RAHMANOV (1988).     

Two new systems with enhanced UV line flux ratios as indication of the evolution of cataclysmic variables

We present ultraviolet spectra of 62 cataclysmic variables observed with Hubble Space Telescope Space Telescope Imaging Spectrograph & Faint Object Spectrograph (HST STIS FOS) to diagnose cataclysmic variables with enhanced NV/C IV line flux ratios. We concentrated on calculating the line flux ratios of the above mentioned sample. We found that a number of cataclysmic variables reveal enhanced NV/C IV line flux ratios in agreement with the previous studies (de Martino and Gänsicke, 2009, Gänsicke et al., 2003). Also we found that two new systems (U SCO, V1974 CYG) reveal enhanced NV/C IV line flux ratios. Such anomalous line flux ratios confirmed that these CVs went through a phase of thermal timescale mass transfer (TTSMT) and now accrete CNO processed material from a companion stripped of its external layers (Gänsicke et al., 2003, Gänsicke, 2004, de Martino and Gänsicke, 2009).     

A simple derivation of microwave solar brightness temperatures and polarizations from thermal regions

Using the full magnetoionic theory in the microwave region along a ray-path through the solar chromosphere and corona we have obtained a convenient expression, and from it a simple diagram which permits the direct computation of the brightness temperature and circular polarization from a thermal source, for any combination of the physical parameters which enter in their computation.     

Observation of variables in ophiuchus-scorpius region. New rapid variables and nebular variables

Preliminary results of the search for flare stars and nebular variables in Ophiuchus and Scorpius are reported. From April to August 1975, 26 single and multiple exposure plates centering at α = 16^h25^m and δ = −24°30′ and using 103a0 Eastman Kodak plates without filters were obtained with the 40 cm double astrograph (ƒ;7.5) of Purple Mountain Observatory and the 40 cm double astrograph (ƒ5) of Peking Observatory. From this photographic material, 1 new flare star and 10 new possible nebular variables in which some are rapid variables have been identified. Table 1 gives the data for the 11 variables found.The method of reduction of the photographic plates is presented and the light curves of new variables as well as their identification charts are given.