Robe’s problem: its extension to 2+2 bodies

In the problem of 2+2 bodies in the Robe’s setup, one of the primaries of mass m^*₁m^{*}_{1} is a rigid spherical shell filled with a homogeneous incompressible fluid of density ρ ₁. The second primary is a mass point m ₂ outside the shell. The third and the fourth bodies (of mass m ₃ and m ₄ respectively) are small solid spheres of density ρ ₃ and ρ ₄ respectively inside the shell, with the assumption that the mass and the radius of third and fourth body are infinitesimal. We assume m ₂ is describing a circle around m^*₁m^{*}_{1}. The masses m ₃ and m ₄ mutually attract each other, do not influence the motion of m^*₁m^{*}_{1} and m ₂ but are influenced by them. We also assume masses m ₃ and m ₄ are moving in the plane of motion of mass m ₂. In the paper, the equations of motion, equilibrium solutions, linear stability of m ₃ and m ₄ are analyzed. There are four collinear equilibrium solutions for the given system. The collinear equilibrium solutions are unstable for all values of the mass parameters μ,μ ₃,μ ₄. There exist an infinite number of non collinear equilibrium solutions each for m ₃ and m ₄, lying on circles of radii λ,λ′ respectively (if the densities of m ₃ and m ₄ are different) and the centre at the second primary. These solutions are also unstable for all values of the parameters μ,μ ₃,μ ₄, φ, φ′. Such a model may be useful to study the motion of submarines due to the attraction of earth and moon.     

Bifurcation of central configuration in the 2N+1 body problem

The bifurcation of central configuration in the Newtonian N-body problem for any odd number N ≥ 7 is shown. We study a special case where 2n particles of mass m on the vertices of two different coplanar and concentric regular n-gons (rosette configuration) and an additional particle of mass m₀ at the center are governed by the gravitational law he 2n+1 body problem. This system is of two degrees of freedom and permits only one mass parameter μ =m ₀/m. This parameter μ controls the bifurcation. If n≥ 3, namely any odd N ≥ 7, then the number of central configurations is three when μ ≥ μ _c , and one when μ ≥ μ _c . By combining the results of the preceding studies and our main theorem, explicit examples of bifurcating central configuration are obtained for N ≤ 13, for any odd N ∈ [15,943], and for any N ≥ 945.     

Does the proton-to-electron mass ratio μ = m p /m e vary in the course of cosmological evolution?

The possible cosmological variation of the proton-to-electron mass ratio μ = m _p /m _e was estimated by measuring the H₂ wavelengths in the high-resolution spectrum of the quasar Q 0347-382. Our analysis yielded an estimate for the possible deviation ofμ value in the past, 10 Gyr ago: for the unweighted valueΔ μ / μ = (3.0±2.4)×10^-5; for the weightedvalueΔ μ / μ = (5.02±1.82)×10^-5.Since the significance of the both results does not exceed3σ, further observations are needed to increase the statistical significance. In any case, this result may be considered as the most stringent estimate on an upper limit of a possible variation of μ (95% C.L.):|Δ μ / μ| < 8× 10^-5 .This value serves as an effective tool for selection of models determining a relation between possible cosmological deviations of the fine-structure constant α and the elementary particle masses (m_p, m_e, etc.). This revised version was published online in July 2006 with corrections to the Cover Date.     

Rosette Central Configurations, Degenerate Central Configurations and Bifurcations

In this paper we find a class of new degenerate central configurations and bifurcations in the Newtonian n-body problem. In particular we analyze the Rosette central configurations, namely a coplanar configuration where n particles of mass m₁ lie at the vertices of a regular n-gon, n particles of mass m₂ lie at the vertices of another n-gon concentric with the first, but rotated of an angle π /n, and an additional particle of mass m₀ lies at the center of mass of the system. This system admits two mass parameters μ = m₀/m₁ and ε = m₂/m₁. We show that, as μ varies, if n > 3, there is a degenerate central configuration and a bifurcation for every ε > 0, while if n = 3 there is a bifurcation only for some values of ε.     

A new kind of 3-body problem

A new kind of restricted 3-body problem is considered. One body,m ₁, is a rigid spherical shell filled with an homogeneous incompressible fluid of density ₁. The second one,m ₂, is a mass point outside the shell andm ₃ a small solid sphere of density ₃ supposed movinginside the shell and subjected to the attraction ofm ₂ and the buoyancy force due to the fluid ₁. There exists a solution withm ₃ at the center of the shell whilem ₂ describes a Keplerian orbit around it. The linear stability of this configuration is studied assuming the mass ofm ₃ to beinfinitesimal. Explicitly two cases are considered. In the first case, the orbit ofm ₂ aroundm ₁ is circular. In the second case, this orbit is elliptic but the shell is empty (i.e. no fluid inside it) or the densities ₁ and ₃ are equal. In each case, the domain of stability is investigated for the whole range of the parameters characterizing the problem.     

Collinear Equilibrium Solutions of Four-body Problem

We discuss the equilibrium solutions of four different types of collinear four-body problems having two pairs of equal masses. Two of these four-body models are symmetric about the center-of-mass while the other two are non-symmetric. We define two mass ratios as μ ₁ = m ₁/M _T and μ ₂ = m ₂/M _T, where m ₁ and m ₂ are the two unequal masses and M _T is the total mass of the system. We discuss the existence of continuous family of equilibrium solutions for all the four types of four-body problems.     

Distribution of stars perpendicular to the plane of the galaxy

We present here rigorous analytical solutions for the Boltzmann-Poisson equation concerning the distribution of stars above the galactic plane. The number density of stars is considered to follow a behaviour n(m,0) ∼H(m - m₀)m^−x, wherem is the mass of a star andx an arbitrary exponent greater than 2 and also the velocity dispersion of the stars is assumed to behave as < v²(m)> ∼ m^−θ the exponent θ being arbitrary and positive. It is shown that an analytic expression can be found for the gravitational field K_z, in terms of confluent hypergeometric functions, the limiting trends being K_z∼z for z →0, while K_z → constant for z → infinity. We also study the behaviour of < |z(m)|²>,i.e. the dispersion of the distance from the galactic disc for the stars of massm. It is seen that the quantity < |z(m)|²>∼ m^t-θ, for m→ t, while it departs significantly from this harmonic oscillator behaviour for stars of lighter masses. It is suggested that observation of < |z(m)|²> can be used as a probe to findx and hence obtain information about the mass spectrum.     

WR 140 (=V1687 Cyg): Infrared photometry, 2001–2010

We present the results of our infrared observations of WR 140 (=V1687 Cyg) in 2001–2010. Analysis of the observations has shown that the J brightness at maximum increased near the periastron by about 0 ^m .3; the M brightness increased by ∼2 ^m in less than 50 days. The minimum J brightness and the minimum L and M brightnesses were observed 550–600 and 1300–1400 days after the maximum, respectively. The JHKLM brightness minimum was observed in the range of orbital phases 0.7–0.9. The parameters of the primary O5 component of the binary have been estimated to be the following: R(O5) ≈ 24.7R _⊙, L(O5) ≈ 8 × 10⁵ L _⊙, and M _bol(O5) ≈ −10 ^m . At the infrared brightness minimum, T _g ∼ 820–880 K, R _g ≈ 2.6 × 10⁵ R _⊙, the optical depth of the shell at 3.5 μm is ∼5.3 × 10⁻⁶, and its mass is ≈1.4 × 10⁻⁸ M _⊙. At the maximum, the corresponding parameters are ∼1300 K, 8.6 × 10⁴ R _⊙, ∼2 × 10⁻⁴, and ∼6 × 10⁻⁸ M _⊙; the mean rate of dust inflow (condensation) into the dust structure is ∼3.3 × 10⁻⁸ M _⊙ yr⁻¹. The mean escape velocity of the shell from the heating source is ∼10³ km s⁻¹ and the mean dispersal rate of the shell is ∼1.1 × 10⁻⁸ M _⊙ yr⁻¹.     

On some exceptional cases in the integrability of the three-body problem

We consider the Newtonian planar three-body problem with positive masses m ₁, m ₂, m ₃. We prove that it does not have an additional first integral meromorphic in the complex neighborhood of the parabolic Lagrangian orbit besides three exceptional cases ∑m _i m _j /(∑m _k )² = 1/3, 2³/3³, 2/3² where the linearized equations are shown to be partially integrable. This result completes the non-integrability analysis of the three-body problem started in papers [Tsygvintsev, A.: Journal für die reine und angewandte Mathematik N 537, 127–149 (2001a); Celest. Mech. Dyn. Astron. 86(3), 237–247 (2003)] and based on the Morales–Ramis–Ziglin approach.     

UBV observations of the eclipsing binary FW Monocerotis

Photoelectric observations of FW Monocerotis inU,B, andV filters have been discussed. The depths of the primary minimum inU BV colours are 2 _. ^m 63, 2 _. ^m 17 and 1 _. ^m 76, respectively. No variation in the period has been found and its revised value is 3 _. ^d 8735868.     

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.     

Spectroscopy and photometry of the protoplanetary nebula candidate stHα62 = IRAS 07171+1823

We present the results of spectroscopic and photometric observations for the B star StHα62 with an IR excess, a post-AGB candidate identified with the IR source IRAS 07171+1823. High-resolution spectroscopy has allowed the λ4330–7340 Å spectrum of the star to be identified: it contains absorption lines of an early B star and emission lines of a gaseous shell. The residual line intensities have been measured. The heliocentric radial velocities measured from absorption lines of the star and emission lines of the shell are 〈V _r 〉 = +45 ± 1 and +52 ± 1 km s^?1, respectively. The line-of-sight velocities of gas-dust clouds determined from the interstellar Na I lines are 12 and 33 km s^?1. The He I λ5876 Å line exhibits a P Cyg profile, which is indicative of an ongoing mass loss by the star. The expansion velocity of the outer shell estimated from forbidden lines is 12–13 km s^?1. Quantitative classification gives the spectral type B0.51 for the star. The parameters of the gaseous shell have been determined: N _e = 3.1 × 10³ cm^?3 and T _e ～ 21 000 K. Over 4 years of its observations, the star showed rapid irregular light variations with the amplitudes ΔV = We present the results of spectroscopic and photometric observations for the B star StHα62 with an IR excess, a post-AGB candidate identified with the IR source IRAS 07171+1823. High-resolution spectroscopy has allowed the λ4330–7340 ? spectrum of the star to be identified: it contains absorption lines of an early B star and emission lines of a gaseous shell. The residual line intensities have been measured. The heliocentric radial velocities measured from absorption lines of the star and emission lines of the shell are 〈V _r 〉 = +45 ± 1 and +52 ± 1 km s⁻¹, respectively. The line-of-sight velocities of gas-dust clouds determined from the interstellar Na I lines are 12 and 33 km s⁻¹. The He I λ5876 ? line exhibits a P Cyg profile, which is indicative of an ongoing mass loss by the star. The expansion velocity of the outer shell estimated from forbidden lines is 12–13 km s⁻¹. Quantitative classification gives the spectral type B0.51 for the star. The parameters of the gaseous shell have been determined: N _e = 3.1 × 10³ cm⁻³ and T _e ∼ 21 000 K. Over 4 years of its observations, the star showed rapid irregular light variations with the amplitudes ΔV = , ΔB = , and ΔU = and no color-magnitude correlation. We estimate the total extinction for the star from our photometric observations as A _v = . Near-IR observations have revealed dust radiation with a temperature of ∼1300 K. We estimate the distance to StHα62 to be r = 5.2 ± 1.2 kpc by assuming that the star is a low-mass (M = 0.55 ± 0.05 M _⊙) protoplanetary nebula. Original Russian Text ? V.P. Arkhipova, V.G. Klochkova, E.L. Chentsov, V.F. Esipov, N.P. Ikonnikova, G.V. Komissarova, 2006, published in Pis’ma v Astronomicheskiĭ Zhurnal, 2006, Vol. 32, No. 10, pp. 737–747.     

UBV photometry at geographic latitude 69°22′ North

UBV measurements of the light of the night sky in the auroral zone during three seasons are presented. The mean brightness of the night sky in theV band is found to be equal one 18^m1 star (arc sec)^–2, with considerable variations. The observed meanB-V andU-B indicies are +0 _. ^m 7 and –1 _. ^m 6, respectively.Light curves of variable stars during strong auroral activities are also shown. On the basis of measurements we briefly discuss the possibility of accurate stellar photometry in the auroral region.     

Types I and II supernovae and the neutrino mechanism of thermonuclear explosion of degenerate carbon-oxygen stellar cores

The present work studies the hydrodynamic process of thermonuclear explosion of hydrostatic equilibrium, degenerate carbon-oxygen cores withM _C=1.40M _⊙ with different values of central densityϱ _c within the interval 2 × 10⁹ <ϱ _c < 3 × 10¹⁰ g cm⁻³. The initial temperature distribution has been determined by the preceding thermal stage of explosion. The calculations successively include the kinetics of thermonuclear burning, the kinetics of β-processes, and neutrino energy losses. By considering the neutrino mechanism of heating and carbon ignition we obtained in our numerical hydrodynamic calculations two characteristic versions of the development of an explosion: (a) at 2 × 10⁹ <ϱ _c < 9 × 10⁹ g cm⁻³ there is disruption of the whole star with either complete or partial burning of the carbon and a 10⁵⁰–10⁵¹ erg kinetic energy; and (b) at 9 × 10⁹ <ϱ _c < 3 × 10¹⁰ g cm⁻³ the stellar core collapses into a neutron star with partial outburst of the outer envelope with a smaller kinetic energy of 10⁴⁹–10⁵⁰ erg. The paper proposes and details a hypothesis (the scenario of supernovae and the formation of neutron stars) on the first version of explosion, corresponding to SNII, and on the second, supplemented by some mechanism of slow energy release into the envelope expelled from the newly formed neutron star, corresponding to SNI. On the basis of the proposed hypothesis a satisfactory agreement with the observed masses and energies of the supernovae envelope, their light curves and spectra, as well as with the data on their chemical composition has been obtained. For this agreement we must assume that type I pre-supernovae are almost bare compact carbon-oxygen stellar cores, and that type II presupernovae are red supergiants. It is most probable that the evolution of type I pre-supernovae occurs in close binaries while the evolution of type II pre-supernovae seems to be very similar to the evolution of a single star.     

Spectroscopic Orbit of the Eclipsing Binary AI Draconis

New radial velocity measurements of the Algol-type eclipsing binary AI Dra, based on Reticon observations, are presented. The velocity measures themselves are based on fitting theoretical profiles, generated by a physical model of the binary, to the observed cross-correlation function (ccf). Such profiles match this function very well, much better in fact that Gaussian profiles which are generally used. Measuring the ccf's with Gaussian profiles yields following results: m_p sin³ i=2.55± 0.05m_⊙, m_s sin³ i = 1.14 ± 0.03m_⊙, (a_p + a_s) sin i=7.34 ±0.05R_⊙, and m_p/m_s =2.23± 0.05. Where as measuring the ccf's with theoretical profiles yields a mass ratio of 2.33 and following results: m_p sin³ i=2.84± 0.05m_⊙, m_s sin³ i=1.22 ± 0.03m_⊙, (a_p +a_s) sin i=7.56± 0.05R_⊙. The system comprising a semi-detached configuration. From the solution of a previously published light curved and combining it's results with the spectroscopic orbit, one can lead to the following physical parameters: m_p =2.99m_⊙, m_s =1.28m_⊙, > T_p < =9600 K, > T_s < =5400 K, > R_p < =2.35R_⊙, > R_s < =2.12R_⊙. The system comprising an AO primary and a secondary of G2 spectral type. This revised version was published online in July 2006 with corrections to the Cover Date.     

On bridges B(pL, qS) around triangular libration points

When μ is smaller than Routh’s critical value μ ₁ = 0.03852 . . . , two planar Lyapunov families around triangular libration points exist, with the names of long and short period families. There are periodic families which we call bridges connecting these two Lyapunov families. With μ increasing from 0 to 1, how these bridges evolve was studied. The interval (0,1) was divided into six subintervals (0, μ ₅), (μ ₅, μ ₄), (μ ₄, μ ₃), (μ ₃, μ ₂), (μ ₂, μ ₁), (μ ₁, 1), and in each subinterval the families B(pL, qS) were studied, along with the families B(qS, qS′). Especially in the interval (μ ₂, μ ₁), the conclusion that the bridges B(qS, qS′) do not exist was obtained. Connections between the short period family and the bridges B(kS, (k + 1)S) were also studied. With these studies, the structure of the web of periodic families around triangular libration points was enriched.     

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.     

VV Orionis – improved elements

TheUBV light curves obtained by Duerbeck (1975) andHa (wide) and Ha (narrow) light curves obtained by Chambliss & Davan (1987) of the detached eclipsing binary VV Orionis (VV Ori) were analysed using the Wilson-Devinney method fixing the two parametersT _h (25,000 K) and q(0.4172), resulting in the following absolute elements:A = 13.605 ± 0.03 LR_⊙,R _h = 5.03 ±0.03R _⊙, R_c = 2.43 ±0.02R _⊙,M _bol,h = -5.18 ± 0.11,M _bol,c = -1.54 ± 0.06,m _h =10.81 + 0.42m _⊙ andm _c = 4.51 ± 0.41m _⊙. The de-reddened colours obtained from applying the reddening corrections ofE(B-V) = 0^m.05 andE(U-B) = O ^m .04, and the derived temperatures of the components, gave spectral types ofB 1.5V for the primary and 54-5V with anUV excess of 0 ^m ·3 for the secondary component. A comparison of the logL and logT _e of the components with the observed ZAMS shows the primary component to be a little above and the secondary component to be a little below/or on the ZAMS. A comparison of the properties of the components of VV Ori and a few other detached systems with the normal stars in the logL, logR and logT _e versus logm planes, indicated a need for either a readjustment of the scales of the above parameters or modifications in the theoretical models. From the position of the components on the evolutionary tracks of Pop I composition computed by Schaller et al. (1992) it is noticed that while the primary component of W Ori had slightly evolved along the main-sequence, its secondary is still unevolved. The age of VV Ori is found to be 10 ± 1 million years and it is at a distance of 368 ± 10 pc.     

Motion close to the Hopf Bifurcation Of the vertical family of periodic orbits of L4

The paper deals with different kinds of invariant motions (periodic orbits, 2D and 3D invariant tori and invariant manifolds of periodic orbits) in order to analyze the Hamiltonian direct Hopf bifurcation that takes place close to the Lyapunov vertical family of periodic orbits of the triangular equilibrium point L₄ in the 3D restricted three-body problem (RTBP) for the mass parameter, μ greater than (and close to) μ_R (Routh’s mass parameter). Consequences of such bifurcation, concerning the confinement of the motion close to the hyperbolic orbits and the 3D nearby tori are also described.