Analysis of Multicolor Photometric Observations of SS 433

Observations of the rapid photometric variability of SS 433 in the BVR bands, obtained on Mt. Maidanak during 12 nights in 1989-90, are presented. The behavior of the U - B, B - V, and V - R color indices is analyzed on the basis of multicolor photoelectric observations. It is found that, both for rapid variability during a night and for variability on the scale of the orbital period, color-brightness relations exist for U-(U - B) and B-(B - V), analogous to the well-known V-(V - R) relation, with a characteristic property: the color indices decrease with increasing brightness. It is shown that the object's reddening upon a brightness decrease is not due only to eclipses. The behavior of the color characteristics can be explained by the interaction of the relativistic jets with the atmosphere around the accretion disk.     

Motion of rigid bodies in a set of redundant variables

In this article we study the conditions for obtaining canonical transformationsy=f(x) of the phase space, wherey(y ₁,y ₂,...,y _2n ) andx(x ₁,x ₂,...,x _2m ) in such a way that the number of variables is increased. In particular, this study is applied to the rotational motion in functions of the Eulerian parameters (q ₀,q ₁,q ₂,q ₃) and their conjugate momenta (Q ₀,Q ₁,Q ₂,Q ₃) or in functions of complex variables (z ₁,z ₂,z ₃,z ₄) and their conjugate momenta (Z ₁,Z ₂,Z ₃,Z ₄) defined by means of the previous variables. Finally, our article include some properties on the rotational motion of a rigid body moving about a fixed point.     

Photometric Inference of the Corotation Radii of Ten Barred Galaxies

We have used photometric images of ten barred galaxies in the B and I bands to infer the geometrical and dynamical parameters of their bars: Their length, (R _bar), their strengths, (S _b), and their corotation radii, (R _CR).These parameters have been obtained studying azimuthal profiles from the B and I band images of the galaxies. We find that R _CR is in all cases slightly larger than R _bar, with mean values close to 1.2 R _bar. We have also found a dependence of the ratio R _CR/R _bar on S _bar. This revised version was published online in July 2006 with corrections to the Cover Date.     

Extension of the solution of Kepler's equation to high eccentricities

The classic Lagrange's expansion of the solutionE(e, M) of Kepler's equation in powers of eccentricity is extended to highly eccentric orbits, 0.6627 ... <e<1. The solutionE(e, M) is developed in powers of (e–e*), wheree* is a fixed value of the eccentricity. The coefficients of the expansion are given in terms of the derivatives of the Bessel functionsJ _n (ne). The expansion is convergent for values of the eccentricity such that |e–e*|<(e*), where the radius of convergence (e*) is a positive real number, which is calculated numerically.     

Study of faint galaxies in the field of GRB021004

We present an analysis of BV R _c I _c observations of the field sized around 4′ × 4′ centered at the host galaxy of the gamma-ray burst GRB021004 with the 6-m BTA telescope of the Special Astrophysical Observatory of the Russian Academy of Sciences. We measured the magnitudes and constructed the color diagrams for 311 galaxies detected in the field (S/N>3). The differential and integral counts of galaxies up to the limit, corresponding to 28.5 (B), 28.0 (V), 27.0 (R _c ), 26.5 (I _c ) were computed. We compiled the galaxy catalog, consisting of 183 objects, for which the photometric redshifts up to the limiting magnitudes 26.0 (B), 25.5 (V), 25.0 (R _c ), 24.5 (I _c ) were determined using the HyperZ code. We then examined the radial distribution of galaxies based on the z estimates. We have built the curves expected in the case of a uniform distribution of galaxies in space, and obtained the estimates for the size and contrast of the possible super-large-scale structures, which are accessible with the observations of this type.     

Dependence of the aerosol component of optical thickness and the relative concentration of methane on depth in atmospheres of giant planets

Based on the data on a spectral dependence of the geometric albedo of giant planet discs, we obtained depth variations in the optical thickness τ _a of the aerosol component and relative concentration γ of methane (Uranium, Neptune) lnτ _a = −0.720 + 1.507Δlnp (for −2.2085 ≤ lnp ≤ −1.0018), lnτ _a = +1.224 + 1.160Δlnp (for −1.0018 ≤ lnp ≤ −0.0595), lnτ _a = +2.318 + 0.192Δlnp (for −0.0595 ≤ lnp), γ = 0.0027 for Jupiter; lnτ _a = −0.846 + 1.598Δlnp (for −3.3619 ≤ lnp ≤ −2.0575), lnτ _a = +1.238 + 1.342Δlnp (for −2.0575 ≤ lnp ≤ −1.2074), lnτ _a = +2.379 + 0.722 (for −1.2074 ≤ lnp ≤ −0.6501), lnτ _a = +2.781 + 0.326Δlnp (for 0.6501 ≤ lnp), γ = 0.0027 for Saturn; lnτ _a = −2.694 + 0.087Δlnp (for +0.3685 ≤ lnp ≤ +1.2314), lnτ _a = −2.619 + 7.341Δlnp (for +1.2314 ≤ lnp ≤ +1.7556), lnτ _a = +1.229 + 0.956Δlnp (for +1.7556 ≤ lnp) for Uranium; lnτ _a = −1.861 + 1.248Δlnp (for +0.3204 ≤ lnp ≤ +0.9051), lnτ _a = −1.131 + 0.347Δlnp (for +0.9051 ≤ lnp) for Neptune; depth-averaged relative methane concentration lnγ = −9.982 + 2.676Δlnp(0.3584 ≤ lnp ≤ 1.5445); ln γ = −9.738 + 2.561Δlnp(0.3237 ≤ lnp ≤ 1.6156) and γ = 0.00382(lnp ≥ 1.6156); 0.00554(lnp ≥ 1.6156) for Uranium and Neptune, respectively (p is in bar).     

On the influence of electronic concentration fluctuations of the dispersion measure and rotation measure

The question of the influence of electronic concentration fluctuations along the line of sight on the dispersion measure (DM) and rotation measure (RM) is under consideration. it is shown that the measuring ofDM andRM enables us to determine the number of electronsN _e =n _e ds and the quantityM _e =n _e H cos ds, respectively, along the line of sight in accordance with the general assertion. Under the conditions considered, the conclusion (Lerche, 1970a, b) that the connection ofDM andRM withN _e andM _e greatly depends on the electronic concentration fluctuations, is found to be incorrect.     

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.     

The existence of families of periodic orbits in theN-body problem

It is proved that monoparametric families of periodic orbits of theN-body problem in the plane, for fixed values of all masses, exist in a rotating frame of reference whosex axis contains always two of the bodiesP ₁ andP ₂. A periodic motion of theN-body problem is obtained as a continuation ofN–2 symmetric periodic orbits of the circular restricted three-body problem whose periods are in integer dependence, by increasing the masses of the originallyN–2 massless bodiesP ₃, ...,P _k. The analytic continuation, for sufficiently small values of theN–2 bodiesP ₃ ...P _k and finite values for the masses ofP ₁ andP ₂ has been proved by the continuation method and the solution itself has been found explicitly to a linear approximation in the small masses. Also, the possible application of the above periodic orbits to the study of the Solar system and of stellar systems is mentioned.     

Effect of perturbation on the location of libration point in the robe restricted problem of three bodies

The effect of small perturbation in the Coriolis and centrifugal forces on the location of libration point in the ‘Robe (1977) restricted problem of three bodies’ has been studied. In this problem one body,m ₁, is a rigid spherical shell filled with an homogeneous incompressible fluid of densityϱ ₁. The second one,m ₂, is a mass point outside the shell andm ₃ is a small solid sphere of densityϱ ₃ supposed to be moving inside the shell subject to the attraction ofm ₂ and buoyancy force due to fluidϱ ₁. Here we assumem ₃ to be an infinitesimal mass and the orbit of the massm ₂ to be circular, and we also suppose the densitiesϱ ₁, andϱ ₃ to be equal. Then there exists an equilibrium point (−μ + (ɛ′μ)/(1 + 2μ), 0, 0).     

Dependence of the U-B and B-V colors of quasars and Seyfert galaxies on their redshifts

We show that the strong emission lines(Mg II, C III, C IV, and Ly and the absorption lineO VI (1035 A) of quasars, when passed through the maximum transmission ofU, B, V, andR light filters, leads to a continuous, and sometimes periodic variation in the dependence ofU-B onB-V. The correlation between the(U-B) and(B-V) colors in quasars with z<1.5 is expressed by the relation(U-B)=–0.92+1.15(B-V). We also give the dependence ofU-B onB-V for Seyfert galaxies of typeS1. We determine the mean variation of the(U-B, B-V) colors due to the strongest emission lines (Mg II, C IV, Ly). It amounts to magnitude0 _. ^m 15.Translated fromAstrofizika, Vol. 38, No. 2, 1995.     

Fourier analysis of the light curves of eclipsing variables,XXII

The theoretical values of the momentsA _2m for any type of eclipses, expressed in terms of the elementsL ₁,a andc ₀, have been derived in the simple forms of rapidly convergent expansions to the series of Chebyshev polynomials, Jacobi polynomials and KopalJ-integrals (Kopal, 1977c) and hold good for any real (not necessarily integral) value ofm0.The aim of the present paper has been to establish explicit expressions for the Jacobian and its fast enough computation in the light changes of close eclipsing systems, arising from the partial derivative of different pairs ofg-functions (Kopal and Demircan, 1978, Paper XIV) with respect toa andc ₀ ² , for any type of eclipses (be these occultations or transit, partial, total or annular) and for any arbitrary degreel of the adopted law of limb-darkening. The functional behaviour of this Jacobian would determine the reasonable light curve in connection with geometrical determinacy of the parametersa andc ₀. In the expansion of Jacobian, the terms consist of two polynomials which satisfy certain three-term recursion relations having the eclipse parametersa andc ₀, as their arguments.Closed form expressions forf-functions, as well as of the Jacobian (e.g.,m=1, 2, 3), obtaining in the case of total eclipses, are given for a comparative discussion with the theoretical values of Jacobian derived from partial derivative of different pairs ofg-functions.The numerical magnitude of Jacobian would determine the best combination of the momentsA _2m in the different pairs ofg-functions and definite results would follow in the subsequent paper of this series (Edalati, 1978c, Paper XXIV).     

Sensitivity of solar eigenfrequencies to the age of the sun

The increasing central concentration of the Sun with age modifies the acoustic eigenfrequencies. In particular, the frequency separation d _l =3(2l+3)^–1v _n,l –v _n–1,l+2 for modes with l + 1/2 n decreases as nuclear reactions augment the molecular-weight gradient in the energy-generating core. If, for example, the Sun were older than is generally believed, one might therefore expect d _l to be smaller than current theoretical predictions. On the other hand, to ensure that the luminosity is consistent with observations, the presumed initial hydrogen abundance would need to be enhanced, thereby reducing the resultant molecular-weight gradient. Thus there is some degree of cancellation of the two major factors that determine d _l .Various authors have either reported directly on the sensitivity of d _l , or have provided the information from which it can be calculated. We have added our own computations. There is broad agreement amongst the results: d _l diminishes with the presumed age of the Sun at the rate of about 1 Hz per Gy for l = 0; the magnitude of the rate appears to decline with increasing l.     

Charged analogues of Schwarzschild interior solution in terms of pressure

Recently, Bijalwan (Astrophys. Space Sci., doi:, 2011a) discussed charged fluid spheres with pressure while Bijalwan and Gupta (Astrophys. Space Sci. 317, 251–260, 2008) suggested using a monotonically decreasing function f to generate all possible physically viable charged analogues of Schwarzschild interior solutions analytically. They discussed some previously known and new solutions for Schwarzschild parameter u( = \fracGMc²a ) ￡ 0.142u( =\frac{GM}{c^{2}a} ) \le 0.142, a being radius of star. In this paper we investigate wide range of u by generating a class of solutions that are well behaved and suitable for modeling Neutron star charge matter. We have exploited the range u≤0.142 by considering pressure p=p(ω) and f = ( f₀(1 - \fracR²(1 - w)a²) +f_a\fracR²(1 - w)a² )f = ( f_{0}(1 - \frac{R^{2}(1 - \omega )}{a^{2}}) +f_{a}\frac{R^{2}(1 - \omega )}{a^{2}} ), where w = 1 -\fracr²R²\omega = 1 -\frac{r^{2}}{R^{2}} to explore new class of solutions. Hence, class of charged analogues of Schwarzschild interior is found for barotropic equation of state relating the radial pressure to the energy density. The analytical models thus found are well behaved with surface red shift z _s ≤0.181, central red shift z _c ≤0.282, mass to radius ratio M/a≤0.149, total charge to total mass ratio e/M≤0.807 and satisfy Andreasson’s (Commun. Math. Phys. 288, 715–730, 2009) stability condition. Red-shift, velocity of sound and p/c ² ρ are monotonically decreasing towards the surface while adiabatic index is monotonically increasing. The maximum mass found to be 1.512 M _Θ with linear dimension 14.964 km. Class of charged analogues of Schwarzschild interior discussed in this paper doesn’t have neutral counter part. These solutions completely describe interior of a stable Neutron star charge matter since at centre the charge distribution is zero, e/M≤0.807 and a typical neutral Neutron star has mass between 1.35 and about 2.1 solar mass, with a corresponding radius of about 12 km (Kiziltan et al., [astro-ph.GA], 2010).     

Orbits of colliding galaxies

A simple, semi-analytic method is developed for obtaining the orbits of galaxies undergoing fast collisions in which the galaxies are represented by Plummer models. The results are found to agree fairly well with those of N-body simulations.A simple formula for obtaining the angle of deflection is deduced. The maximum angle of deflection is 180° forV _p/V _esc(p)=1.00, about 36° forV _p/V _esc(p)=1.50, and about 18° forV _p/V _esc(p)=2.00, whereV _p is the velocity at closest approachp, andV _esc(p) is the parabolic velocity of escape atp. The angle of deflection of a pair of colliding elliptical galaxies without halos is about twice that for a pair of galaxies with halos for the same relative velocity at infinite separation.     

Stability of motion in the Sitnikov 3-body problem

We study the stability of motion in the 3-body Sitnikov problem, with the two equal mass primaries (m ₁ = m ₂ = 0.5) rotating in the x, y plane and vary the mass of the third particle, 0 ≤ m ₃ < 10⁻³, placed initially on the z-axis. We begin by finding for the restricted problem (with m ₃ = 0) an apparently infinite sequence of stability intervals on the z-axis, whose width grows and tends to a fixed non-zero value, as we move away from z = 0. We then estimate the extent of “islands” of bounded motion in x, y, z space about these intervals and show that it also increases as |z| grows. Turning to the so-called extended Sitnikov problem, where the third particle moves only along the z-axis, we find that, as m ₃ increases, the domain of allowed motion grows significantly and chaotic regions in phase space appear through a series of saddle-node bifurcations. Finally, we concentrate on the general 3-body problem and demonstrate that, for very small masses, m ₃ ≈ 10⁻⁶, the “islands” of bounded motion about the z-axis stability intervals are larger than the ones for m ₃ = 0. Furthermore, as m ₃ increases, it is the regions of bounded motion closest to z = 0 that disappear first, while the ones further away “disperse” at larger m ₃ values, thus providing further evidence of an increasing stability of the motion away from the plane of the two primaries, as observed in the m ₃ = 0 case.     

Integrability of the Yang-Mills Hamiltonian system

This paper considers the integrability of generalized Yang-Mills system with the HamiltonianH _a (p, q)=1/2(p ₁ ² +p ₂ ² +a ₁ q ₁ ² +a ₂ q ₂ ² )+1/4q ₁ ⁴ +1/4a ₃ q ₂ ⁴ + 1/2a ₄ q ₁ ² q ₂ ² . We prove that the system is integrable for the cases: (A)a ₁=a ₂,a ₃=a ₄=1; (b)a ₁=a ₂,a ₃=1,a ₄=3; (C)a ₁=a ₂/4,a ₃=16,a ₄=6. Our main result is the presentation of these integrals. Only for cases A and B does the Yang-Mills Hamiltonian possess the Painlevé property. Therefore the Painlevé test does not take account of the integrability for the case C.     

A comparison of theoretical Sv emission line strengths with extreme ultraviolet observations of a sunspot

Electron impact excitation rates for transitions in the S v ion, calculated with theR-matrix code, are used to derive the electron temperature sensitive emission line ratiosR ₁ =I(854.8 Å)/I(786.9 Å),R ₂ =I(852.2 Å)/I(786.9 Å),R ₃ =I(849.2 Å)/I(786.9 Å), andR ₄ =I(1199.1 Å)/I(786.9 Å), which are found to be significantly different from previous estimates. A comparison of the present results with observational data for a sunspot obtained with the Harvard S-055 spectrometer on boardSkylab reveals generally good agreement between theory and experiment, except in the case ofR ₁, which is probably due to blending in the 854.8 Å feature. The possible effects of Lyman continuum absorption on the observed line ratios is briefly discussed.     

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.