Reduction to proper elements for the entire solar system

The solution of Brouwer and van Woerkom (1950), for reduction to prope, elements of asteroidal osculating elements at epoch, is extended to the range of semimajor axes, 0.1<a<55 AU. The locations and strengths of secular resonances are noted.     

Planetary elements for 10 000 000 years

In 1950 Brouwer and van Woerkom published a secular theory of the variations of the planetary elements in analytical form. In the present paper we provide a graphical representation of this theory in the form of element plots for a time span of ten million years.     

Fourier techniques for an analysis of eclipsing binary light curves. V

An integral transform called the momentsA _2m of the light curves has been introduced by Kopal (1975) and utilized in the subsequent papers for an analysis of the light curves of eclipsing variables. The aim of the present paper is to generalize this integral transform by two distinct ways: (i) by introducing an exponential factor, and (ii) a Jacobi polynomial as multiplicative factor into the integrand of the transformA _2m. Observational values of these general transforms are likewise obtainable. They have been expressed in terms of eclipse elementsr _1,2,i andL ₁. These expressions can be used to solve the eclipse elements in terms of observed quantities. Free parameters in the expressions increase the flexibility in applications and may be utilized to improve the determinacy of the elements.     

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.     

Fourier analysis of the light curves of eclipsing variables,VIII

The main aim of this paper will be to develop explicit form of the moments of the light curvesA _2m(r ₁,r ₂,i) required for the solution for the geometrical elementsr _1,2 andi of eclipsing systems exhibiting annular eclipses (Sections 2 and 3), as well as partial eclipses (Section 4).In the concluding Section 5 we shall demonstrate that — regardless of the type of eclipse and distribution of brightness on the apparent disc of the eclipsed star, or indeed of the shape of the eclipsing as well as eclipsed components — the momentsA _2m satisfy certain simple functional equations — a fact which relates them to other classes of functions previously studied in applied mathematics.     

Fourier analysis of the light curves of eclipsing variables,XVI

The practical procedures for the solutions of the elements of any eclipsing system in the frequency-domain have been described in a previous paper of this series (Kopal and Demircan, 1978, Paper XIV). The fundamental quantities from which we depart in quest of our solution are twog-functions defining by the momentsA _2m (see Equations (2.13)–(2.16) in Paper XIV, or Equations (3.2)–(3.6) in Paper XV: Demircan, 1978b). If we establish the observational values for these functions, they constitute two independent relations between the unknown parametersa andc _o, and can be numerically solved for them with the aid of the general expressions for the respective moments. However, the determinacy of these parameters depends on not only the accuracy of observations but also the employedg-functions. For better understanding of the geometrical determinacy of the eclipse parametersa andc _o, different combinations of the momentsA _2m have been worked out asg-functions. For the index 2m, the values between 0 and 6 were applied. It has been noted that the behaviour of these functions vary but very little with applied different combinations of the moments. A choice of the most convenient moments to obtain a good determinacy for the eclipse elements were discussed. In this connection, (i) them-dependence of the moments, and the errors in their observational values have been considered, (ii) different practical procedures for the solution of eclipse elements were introduced, and (iii) different type of moments were tested.     

Fourier analysis of the light curves of eclipsing variables,XXIV

The aim of the present paper will be to evaluate numerically Jacobian and other functions which have been discussed in more detail in a previous paper of this series (Edalati, 1978b, Paper XXII), and also choose the most convenient moments to obtain a good determination for the unknown eclipse parametersa andc ₀. More than 12 different pairs ofg-functions for real values ofm have been investigated numerically and diagrammatically. The behaviour ofg-functions depends but very little on different combination of the moments, and related diagrams are approximately the same asg ₂ andg ₄ (Kopal and Demircan, 1978, Paper XIV).The behaviour of the vanishing Jacobian, arising from different pairs ofg-functions for real values ofm1 has been shown diagrammatically in terms ofa andc ₀. Accordingly, we obtain the optimum combination of the moments (i.e.,A ₆,A ₇,A ₈ andA ₉) ing-functionsg ₇ andg ₈. It has been noted that the behaviour of theg-functions which depend on the combinations of the higher order moments (i.e.,m5) have been ruled out, because the proportional error of the momentsA _2m increases with increasing values of realm. The automated method has been tested successfully on the light curve of RT Per (Mancusoet al., 1977; Edalati, 1978a). Finally, a comparison is given of the elements of RT Per arising from two different pairs ofg-functions, i.e.,g ₂,g ₄ (Edalati, 1978a) andg ₇,g ₈ for the light curves analysis.     

Fourier techniques for an analysis of eclipsing binary light curves,VII

A Laplace and a Hankel transform of the light curves of eclipsing binary star systems have been expressed in terms of the eclipse elements (L ₁,r _1,2 andi) of the spherical basic model. These general expressions can be used for the solution of the eclipse elements of the systems in terms of the observed quantities.     

Green's theorem and Green's functions for the steady-state cosmic-ray equation of transport

Green's Theorem is developed for the spherically-symmetric steady-state cosmic-ray equation of transport in interplanetary space. By means of it the momentum distribution functionF _o(r,p), (r=heliocentric distance,p=momentum) can be determined in a regionr _arr_bwhen a source is specified throughout the region and the momentum spectrum is specified on the boundaries atr _a andr _b . Evaluation requires a knowledge of the Green's function which corresponds to the solution for monoenergetic particles released at heliocentric radiusr _o , Examples of Green's functions are given for the caser _a =0,r _b = and derived for the cases of finiter _a andr _b . The diffusion coefficient is assumed of the form = _o(p)r ^b . The treatment systematizes the development of all analytic solutions for steady-state solar and galactic cosmic-ray propagation and previous solutions form a subset of the present solutions.     

A study of the contrast of sunspots from photometric images

The thermal contrast , and the umbra-penumbraA _u/A _p, were calculated for 63 sunspots of various sizes and morphologies. Contrary to the assumptions of the PSI model, andA _u/A _p were found to be quite variable. The values of ranged from 0.1807 to 0.4266;A _u/A _p ranged from 0.0089 to 0.4899. The values of andA _u/A _p correlated well (r = 0.6018;p<0.005) and the regression for andA _u/A _p was obtained: = (0.220 ± 0.016) + (0.340 ± O.06)A _u/A _p. The values of andA _u/A _p were then compared with complexity ratings, magnetic field strength, time, and . The quantities andA _u/A _p were found to be independent of the complexity, magnetic field strength, and time factors. The correlation between andA _u/A _p lead to the proposed division of into an umbral thermal contrast _u, and a penumbral thermal contrast _p. These values were calculated from the photometric data: _u = 0.57 ± 0.01 and _p = 0.26 ± 0.006.     

Transition probabilities and dissociation energy of astrophysical molecule GeH

The Franck-Condon factors andr-centroids, which are very closely related to vibrational transition probabilities, have been evaluated by the more reliable numerical integration procedure for the bands of A ² Δ - X² π_r system of astrophysical molecule GeH, using a suitable potential. The dissociation energy for the electronic ground state of astrophysical molecule GeH has been estimated precisely as D ⁰ ₀ = 2.69 ± 0.05 eV by fitting the empirical potential function to the experimental potential energy curve using correlation coefficient. This revised version was published online in July 2006 with corrections to the Cover Date.     

Photometry of the contact binary system V471 Cas

UBV photoelectric observations of the W UMa-type binary V471 Cas were made from September to November 1984. Its colour indices were found to be(B-V)=0 _. ^m 771±0 _. ^m 02 and(U-B)=0 _. ^m 196±0 _. ^m 02. TheU, B, andV light curves of V471 Cas show some photometric fluctuations. We found that its orbital period is not 0.335998 days which was given by GCVS (1986), but 0.405356 days.Photometric orbital elements of V471 Cas were found using the Wilson-Devinney method. V471 Cas is a contact system, in which the overcontact factor is 0.19, its mass ratio of two components is 0.5947, and orbital minclination is in 83.29 degrees.     

Fourier techniques for an analysis of eclipsing binary light curves: I

The Fourier techniques developed so far for an analysis of eclipsing binary light curves have been re-discussed. The Fourier coefficients for the analysis have been derived in a simple form of series expansions, in terms of eclipse elements, valid for any type of eclipse (regardless of whetherr ₁r ₂).These coefficients may be utilized to solve the eclipse elements in terms of the observed characteristics of the light curves. A general relation between the observed quantitiesl and , and the eclipse elementsr _1,2,i andL ₁ has also been given in the form of series expansions which can be used for the synthesis of the light curves.     

Determinacy of the eclipse elements from a part of the light curves of eclipsing binaries

A part of the light curves of two detached (CD Tau and V909 Cyg) and one semi-detached (Algol) binaries have been analysed with the emphasis on the determinacy of the parameters of different systems from a part of their eclipse curves. A model light curve for whichr ₁=0.050,r ₂=0.150,i=90°, andL ₁=0.800 were chosen, has also been considered for a further clarification of the determinacy problem of the eclipse parameters. Some insight into the determinacy is gained by varying one parameter and adjusting the others for the best fit to the light curves in the considered eclipse phases. It was realised that the determinacy problem of the unknown parameters is stored in the determinacy of the type of the eclipse minima. The attempts of the solutions with a false choice of initial elements result in finding (i) no minimum of ² to give any acceptable solution, (ii) rapid departure from the false start and convergence on the correct solution, or (iii) a false solution mostly with a true inclinationi, but wrong sense of the ratio of the radii.     

A study of the close binary star TU Monocerotis

New photometric elements,i=89°.5,r _a =0.24,r _b =0.25 andL _a =0.82(Y), 0.88(B), 0.94(U), are deduced for the eclipsing system TU Mon, using the incomplete Fourier method for the analysis of its light curve. They are based upon three-colour photoelectric observations obtained in 1966–68 with the 36-inch reflector of the Okayama Astrophysical Observatory. From discussion combined with the spectroscopic data byDeutsch (1945) and byPopper (1967), it seems fair to conclude that TU Mon is an ordinary semi-detached close binary system consisting of a detached brighter B5V star and an A5 subgiant in contact at its Roche limit.     

Analysis of light variations of the eclipsing binary system XY ceti with Am spectra

The photoelectric light curves of XY Cet of Srivastava and Padalia (1975) and of Morrison and Morrison (1968), from which considerably different sets of photometric elements have previously been deduced, were re-analyzed in detail. The elements derived in the present study from Srivastava and Padalia's (1975)UBV observations arer _g=0.157±0.006 (p.e.),r _s=0.142±0.007,i=88°.4±0°.5,L _g=0.600±0.017, 0.602±0.015 and 0.584±0.018 forU, B andV, respectively. It is also found from their light curves that the system has an eccentric orbit withe cos =0.0026±0.0005. The elements deduced from theV observations of Morrison and Morrison (1968) are found to be in satisfactory agreement with those presented above.The existence of theAm characteristics of both components, which was recognized by earlier spectroscopic observations, are supported by theB-V andU-B color indices and [m ₁] value of the system. Both components are also found to have appreciably evolved, as expected for metallicline stars.     

Spherical variable-energy blast wave in self-gravitating gas

Modified similarity method has been used to study the propagation of spherical-variable energy blast waves through a self-gravitating gas. For an energy inputE =E ₀t^4/3, whereE is the energy released up to timet andE ₀ is a functional constant, the similarity solutions correct up to third approximation have been obtained. It is found that the effects of self-gravitational forces are of third order. An increase in the parameterA ² (characterising the gravitational field) increases the shock velocity.     

Fourier techniques for an analysis of eclipsing binary light curves: II

Some properties of the quantitiesB _2m (Smith, 1977) inherent in the frequency-domain approach have been deduced, and a general expression for them in terms of the eclipse elementsr _1,2,i andL ₁ of the basic model has been presented (Section 2).An expansion for the loss of light (1–l) into a Fourier sine series alone have been introduced, and its coefficientsb _m presented (Section 3) in terms of the same eclipse elements. A method of increasing the rate of convergence of this series has been given in Section 4. The methods for obtaining the elements of eclipsing binaries by making use of all these quantities in the frequency-domain can likewise be generalized to cover the photometric effects of gravitational and radiative interaction between the components.     

On the possibility of using metal unblanketed model atmospheres for analysis of population II stars

A comparison between model atmosphere grids with and without metal blanketing has been performed in the low-temperature range (T _eff=6000 K to 10000 K) and for different metal contents (logA=0, –1, –2;A=scaled solar abundance). ForA10^–1 andT _eff7000 K, the Johnson colour indicesU-B andB-V, together with the Strömgrenu-b, b-y and the bolometric correction (BC) are little affected by metal blanketing ( mag<0 _. ^m 04). On the other hand, the trend of the physical quantities and them ₁ andc ₁ colour indices reflect the inadequateness of the models even withA=10^–2 andT _eff=9500 K, where hydrogen line blanketing is expected to dominate. This fact discourages once and for all the use of metal unblanketed atmospheres other than for comparison of colour indices or the calculation of bolometric corrections for Population II A-F spectral types.     

The solution of radiation transfer problem for a slab with space-dependent albedo and anisotropic scattering

The transport of thermal radiation has been considered within a finite slab which absorb and scatter anisotropically. The problem involves the space-dependent single-scattering albedow(x). Two approximations are taken forw(x). In the first it is represented in exponential form asw(x)=w ₀ exp(–x/s), wherew ₀ ands are given constants andx is the optical variable. The second approximation assumes the formw(x) = _r=0 ^R d _r ^* p _r (x/a), whered _r ^* are known expansion coefficients anda is the half optical thickness of the slab. Analytic expressions for the forward, backward radiation intensities and fluxes are given in each approximation. The solution of the linear transport equation is performed on the basis of integral Fourier transforms.