Generalized energy balance method for the determination of central star parameters and optical thickness of a planetary nebula

A method is proposed allowing a quick self-consistent determination of both the central star parameters (effective temperature, surface gravity, stellar massetc.) and the optical thickness of a planetary nebula (PN). The method is a generalization of the well-known energy balance method. The method has been calibrated and tested using a photoionization model grid computed for this purpose. The internal accuracy of the method is estimated as 0.038dex for the effective temperature of central star and 0.076dex for the surface gravity.The problem of determination of overall energy losses in the nebula required by any kind of energy balance method is considered thoroughly. Approximate expressions are obtained, relating the overall energy losses to the sum of intensities of collisionally excited lines in the optical and ultraviolet spectral ranges and to some other nebular parameters. It is shown that neglecting the energy losses caused by directly unobservable collisional excitation of neutral hydrogen and helium may underestimate the central star temperature by 0.2 or even 0.5dex. Generalized energy balance method is applied to a sample of 41 PN. Central star temperaturesT _GB found by this method show an agreement withHeII Zanstra temperaturesT _z (HeII) whereasT _z (HI) is always less thanT _GB or equal to it within the accuracy of the method. So, we confirm the explanation that the well-known Zanstra discrepancy is caused merely by low optical thickness of many PN in the Lyman continuum of hydrogen. The value ofT _z (HeII) found with modern model atmospheres can be used as good approximation toT _ef for central stars of overwhelming majority of PN whileT _z (HI) is usually close toT _ef for young nebulae only.     

The application of the quickest descent method to the analysis of the Hα line in loops

The quickest descent method and the multi-layer model are proposed to analyse symmetrical and asymmetrical line profiles of loops in this paper. By fitting observed lines, the physical parameters such as Doppler width, Δλ _D , the optical thickness at line centre,τ _λ0, and the line displacement, Δλ ₀, can be directly obtained, Using this method, some symmetrical and asymmetrical line profiles of both the loop prominence on February 18, 1984 and the loop on August 17, 1989 are fitted. It is found that the method mentioned in this paper is characterized by rapid convergence, high precision, and less scatter.     

Numerical solution of MHD free-convection flow in the Stokes problem by the Crank-Nicolson method

A numerical solution of magnetohydrodynamic free-convection flow, in the Stokes' problem, for a porous vertical plate, is obtained, when Prandtl numberP is not equal to one. The magnetic lines of force are assumed to be fixed relative to the plate which is started moving impulsively in its own plane (I.S.P.) or it is uniformly accelerated (U.A.P.). The solution is obtained by the Crank-Nicolson method, which is an implicit second-order method, forP=0.71 (air) andP=7 (water). The obtained results are shown on figures and tables.     

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.     

How the Method of Minimization of Action Avoids Singularities

The method of minimization of action is a powerful technique of proving the existence of particular and interesting solutions of the n-body problem, but it suffers from the possible interference of singularities. The minimization of action is an optimization and, after a short presentation of a few optimization theories, our analysis of interference of singularities will show that:(A) An n-body solution minimizing the action between given boundary conditions has no discontinuity: all n-bodies have a continuous and bounded motion and thus all eventual singularities are collisions;(B) A beautiful extension of Lambert's theorem shows that, for these minimizing solutions, no double collision can occur at an intermediate time;(C) The proof can be extended to triple and to multiple collisions. Thus, the method of minimization of action leads to pure n-body motions without singularity at any intermediate time, even if one or several collisions are imposed at initial and/or final times.This method is suitable for non-infinitesimal masses only. Fortunately, a similar method, with the same general property with respect to the singularities, can be extended to n-body problems including infinitesimal masses.     

A study of collision orbits with the Moon by the method of surface of section

Non-periodic orbits of a natural satellite of the Moon are studied, for the case of the circular three-body problem with the method of surface of section. According to this method, each orbit is represented by a point, in the plane x0\.x, which corresponds to y = 0 and \.y > 0 and a fixed energy. Conclusions are deduced from the shape of this curve for probable collisions of the satellite on the lunar surface. This method of surface of section can be used for the study of orbits which collide with the Moon's surface after a large number of revolutions around the Moon and their study would be difficult to explore with other methods.     

A Preliminary Estimate of the Size of the Coming Solar Cycle 24, based on Ohl’s Precursor Method

For many purposes (e.g., satellite drag, operation of power grids on Earth, and satellite communication systems), predictions of the strength of a solar cycle are needed. Predictions are made by using different methods, depending upon the characteristics of sunspot cycles. However, the method most successful seems to be the precursor method by Ohl and his group, in which the geomagnetic activity in the declining phase of a sunspot cycle is found to be well correlated with the sunspot maximum of the next cycle. In the present communication, the method is illustrated by plotting the 12-month running means aa(min ) of the geomagnetic disturbance index aa near sunspot minimum versus the 12-month running means of the sunspot number Rz near sunspot maximum [aa(min ) versus Rz(max )], using data for sunspot cycles 9 – 18 to predict the Rz(max ) of cycle 19, using data for cycles 9 – 19 to predict Rz(max ) of cycle 20, and so on, and finally using data for cycles 9 – 23 to predict Rz(max ) of cycle 24, which is expected to occur in 2011 – 2012. The correlations were good (∼+0.90) and our preliminary predicted Rz(max ) for cycle 24 is 142±24, though this can be regarded as an upper limit, since there are indications that solar minimum may occur as late as March 2008. (Some workers have reported that the aa values before 1957 would have an error of 3 nT; if true, the revised estimate would be 124±26.) This result of the precursor method is compared with several other predictions of cycle 24, which are in a very wide range (50 – 200), so that whatever may be the final observed value, some method or other will be discredited, as happened in the case of cycle 23.     

Statistical Revision of the Moment Method

The moment method is a well known technique, which uses a time series ofthe first 3 moments of a spectral line, to estimate the (discrete) modeparameters and m. The method, contrary to Doppler imaging,also yields other interesting(real-valued) parameters such as the inclination angle i, or v sin i,during its identification procedure.In this paper, we are not only interested in the estimation of thesereal-valued parameters themselves but also inreliable estimates for their uncertainty.We designed a statistical formalism for the moment method based on theso-called generalized estimating equations (GEE). This formalismaims to estimate the uncertainty of the real-valued parameters taking intoaccount that the different moments of a line profile are correlated and –more importantly – that the uncertainty of the observed moments depends onthe pulsation parameters. The latter property of the moment method makesthe least-squares technique a poor choice to estimate the uncertainty ofthe real-valued parameters. We implemented the GEE method and presentan application to a high-resolution spectroscopic dataset of the slowly pulsating B star HD181558.     

Method for the determination of the distribution function for the mean frequency of supernova explosions

A method is suggested for the determination of the distribution function for the mean frequency of supernova explosions, which is similar to the Ambartsumian method for the determination of the distribution function for the mean flare frequency of flare stars. This method has not been applied because of the lack of the required data.Translated fromAstrofizika, Vol. 39, No. 4, pp. 561–566, November, 1996.     

Photo-electric study of the eclipsing binary RU Eridani

B andV observations of the eclipsing binary RU Eridani made on 23 nights during 1974–76 are reported. An improved period of 0^d.63219951 is derived. After rectification ofB andV light curves, new elements are derived using Russel-Merrill method. The system is classified as semidetached.     

Studies in the application of recurrence relations to special perturbation methods

The numerical integration of the differential equations describing dynamical systems has been shown in previous papers of this series to be most effectively accomplished by an explicit Taylor series method.In this paper we show that one explicit Taylor series method, developed earlier in this series and which appears to possess a high degree of versatility, yields considerable gains in efficiency over classical single-step and multi-step methods. (In this context efficiency is a measure of the time taken to carry out a calculation of a specific accuracy).For a given accuracy criterion governing the local truncation error (LTE) it is found that the Taylor series method is generallytwice as fast as the classical multi-step method and up totwenty times faster than the classical single-step method.     

Solution of the transfer equation for interlocked multiplets by probabilistic method

Sobolev's probabilistic method — The method of quantum exit from the medium — has been applied to solve the transfer equation for the case of interlocking without redistribution. The solution contains the function (x) which is same as theH-function involved in the solution given by Busbridge and Stibbs the method of principle of invariance.     

Interstellar extinction in the direction of the Merope dark cloud in pleiades

The dependence of interstellar extinction on distance in the direction of the dark cloud south of Merope is determined using photoelectric photometry of 93 stars in the Vilnius photometric system. The cloud front edge is detected at 120–130 pc from the Sun and the distance of the Pleiades cluster is found to be 127 pc. Mean extinctionA _V in the Merope cloud is of the order of 1.0 mag. There is no evidence of extinction at distances exceeding the Merope cloud distance. Variable extinction method yieldsR=A _V/E_B-V=3.6, while the maximum polarization wavelength method gives the value 3.4. Some Pleiades stars are suspected to be unresolved binaries.     

Photometric reductions and the lightcurve of (51) Nemausa

The lightcurve of (51) Nemausa is determined by 1779 photoelectric observations in B and V and reduction by a rigorous least-squares method is discussed. For the lost (1026) Ingrid is obtained B(1,0) = 14^m.7.     

Cosmological models with the variable gravitational and cosmological constants

Cosmology with the gravitational and cosmological constants generalized as coupling scalars in Einsteins theory is considered. A general method of solving the field equations is given. Exact solution for Zeldovich fluid satisfying G=G ₀(R/R ₀) ⁿ is given.     

The use of the Goldberg-Unno method for the investigation of small-scale photosphere motions

The Goldberg-Unno method is analysed. Accounting for the instrumental profile correction reduces the derived microturbulent velocities only slightly. A similar effect may be caused by an unresolved macroturbulence. The method of accounting for the damping effect is considered. The correction for the influence of the damping effect does not change substantially the general trend of the variation of _t with ₀. The microturbulent velocity _t is reduced appreciably.An attempt to analyse the microturbulent velocities by the Goldberg-Unno method under deviation from LTE is made.The main conclusion is that the Goldberg-Unno method, especially in its modified form, is valid and useful.     

Infinitesimal orbits around lagrange points in the elliptic,restricted three-body problem

This study presents a method of obtaining asymptotic approximations for motions near a Lagrange point in the planar, elliptic, restricted three-body problem by using a von Zeipel-type method. The calculations are carried out for a second-order escape solution in the proximity of the equilateral Lagrange point, L ₄, where the primaries' orbital eccentricity is taken as the small parameter .     

Estimation of the dissociation energy of CO+ from spectroscopic data

The potential energy curves for theX ² Σ⁺ andB ² Σ⁺ states of CO⁺ have been constructed by the Rydberg-Klein-Rees (RKR) method as modified by van der Sliceet al. The dissociation energy is estimated to be 7.70±0.19 eV by the method of curve fitting using the five parameter Hulburt-Hirschfelder’s function. The estimated value is in good agreement with the value (7.839 eV) given by Misraet al. Carefull observation of the results reveals that accurateD ₀ value for CO⁺ is 8.33 eV     

Distribution of free electrons in the galactic plane

A method is proposed for constructing a map of the electron density distribution in the galactic plane. Data on the dispersion measures DM of more than 1500 pulsars and their distances, found by an independent method, are used. The independent distances of the pulsars are estimated using an empirical relation of the form L=γP ^α Ṗ ^β W ^δ between the radio luminosity L of the pulsars and their periods P, the rate of change Ṗ. of their periods, and the half width W of their pulses. A map of the electron density distribution in the galactic plane within a ±400 pc layer is provided. __________ Translated from Astrofizika, Vol. 49, No. 2, pp. 277–287 (May 2006).     

A typical example of the application of the `Multi-Cloud Model'method to the asymmetric profiles processing

Multi-Cloud Model(MCM) is an effective method for dealing with asymmetric profiles of spectral line formed by active objects on the sun. A typical example is given for the application of the MCM method in this paper. The H _β spectral data of a large post-flare loop (PFL) system which occurred on the solar western limb are obtained with MW-SSHG at the Yunnan Observatory and analysed based on the MCM method. The line-of-sight velocity fields observed in different times are derived and discussed. This revised version was published online in July 2006 with corrections to the Cover Date.